Solution of the eta-4He problem with quasi-particle formalism

The Alt-Grassberger-Sandhas equations for the five-body eta-4N problem are solved for the case of the driving eta N and NN potentials limited to s-waves. The quasi-particle (Schmidt) method is employed to convert the equations into the effective two-body form. Numerical results are presented for the eta-4He scattering length.


I. INTRODUCTION
During the last years considerable attention has been paid to interaction of η-mesons with four nucleons [1][2][3][4][5][6][7][8][9].Analysis of different data is mainly focused on the search for η 4 He bound states.According to the available experimental results, the rise of the dd → η 4 He experimental cross section at E η → 0 seems to be not as steep as in the pd → η 3 He reaction.As discussed, e.g., in Ref. [3] the most natural interpretation of this fact is that due to additional attraction caused by one extra nucleon the pole in the η 4 He scattering matrix is shifted into the region of negative values of Re E η and turns out to be farther from the physical region than the η 3 He pole.It is therefore concluded that formation of the bound η 4 He state is highly probable.
In view of general complexity of the five-body η − 4N problem there are still no rigorous few-body calculations of this system.At the same time, a systematic practical way of handling the n-body interaction is provided by the quasi-particle formalism in which the kernels of integral equations are represented by series of separable terms.This method becomes especially efficient if the driving two-particle potentials are governed by the nearly lying resonances or bound (virtual) states, like in the N N and ηN case.Then reasonable accuracy may be achieved with only few separable terms retained in the series.In particular, the quasi-particle formalism is shown to be very well suited for practical calculation of ηN N [10][11][12] as well as η − 3N [13] scattering (in Ref. [14] another method based on the hypospherical function expansion has been developed).
In this letter we apply the quasi-particle method to study the five-body system η − 4N .As a formal basis we use the Alt-Grassberger-Sandhas n-body equations derived in Ref. [15].For the sake of simplicity we neglect influence of the spin and isospin on the interaction between nucleons, treating them as spinless indistinguishable particles.Furthermore, since only the threshold η 4 He energies are considered, we restrict all interactions to s-waves only.

II. FORMALISM
As is well known, separable expansion of the kernels allows one to reduce the n-body integral equations to the (n − 1)-body equations, where two of n particles in each state are effectively treated as a composite particle (quasiparticle).Therefore, the essence of the method is to approximate the (n − 1)-particle interaction obtained in the separable-potential model again by the separable ansatz.In this respect, to simplify presentation of the formalism, we start directly from successive application of the quasi-particle technique to 2-, 3-, and 4-body subamplitudes, occurring when the five-body system is divided into groups of mutually interacting particles.
In what follows, we use the concept of partitions as introduced, e.g., in Ref. [16].Different partitions (as well as the quasi-particles related to these partitions) are further denoted by the symbols α, β, . .., whereas the Latin letters a, b, . . .are used for numbering the terms in the separable expansions of the subamplitudes.The notation α n refers to the partition obtained by dividing the η − 4N system into n groups.Writing α n+1 ⊂ α n means that the partition α n+1 is obtained from α n via further division of the quasi-particle α n into two groups of particles.
The basic ingredient of the formalism is a separable expansion of the quasi-particle amplitudes Then the integral equations for the amplitudes X αn−1 αn,βn are transformed exactly into the quasi-two-body equations which in the operator form read or more explicitly with µ γn being the reduced mass associated with the partition γ n .The effective potentials Z αn−1 αn,βn are determined as matrix elements of the 'resolvent' ∆ γn+1 between the form factors appearing in the expansion (1) The structure of Eq. ( 4) is conveniently illustrated in the form of diagrams.In Fig. 1 we show as an example one of the effective potentials Z α2 α3,β3 , connecting two configurations of the type (ηN N ) + N .Since the nucleons are identical, the condition α n = β n in Eq. ( 4) means that the nucleon lines, entering the quasi-particles α n and β n and not included into the quasi-particle γ n+1 should be different.To calculate the form factors u αn(a) γn+1(k) and the propagators ∆ γn+1 kl we employed the energy dependent pole expansion (EDPE) method of Ref. [17].

A. Four-body partitions
Considering nucleons as indistinguishable particles we have only two different types of four-particle partitions: The partitions 1 and 2 and the related two-particle subsystems N N and ηN will further be labeled by the index α 4 = 1, 2. In the present calculation, the N N and ηN s-wave interactions were approximated by simplest rank-one separable potentials.For N N we employed The corresponding t-matrix has the usual form with the N N propagator where M N is the nucleon mass.The form factors were chosen in the Yamaguchi form Since we treat nucleons as spinless particles, the strength λ N N was taken as an average of the singlet and the triplet strength The singlet and the triplet scattering lengths, a 0 and a 1 , as well as the cut-off momentum β were taken directly from the analysis [18] of the low-energy np scattering It is well known that the Yamaguchi N N potential overestimates attraction at high momenta and yields significant overbinding already in the 3 He case (see Table I).Therefore we also adopted the spin-independent N N potential with exponential form factors which yields the same binding energy E N N of two nucleons.The form factors (12) with parameters listed in Table I give for the three-and four-nucleon binding energies, E 3N and E 4N , the values which are rather close to those of the 3 He and 4 He nuclei.At the same time, with the Gauss form factors we have a visibly larger value of the N N effective range r 0 (see Table I).
The ηN s-wave interaction was reduced to excitation of the resonance N (1535)1/2 − only.To include pions we used a conventional coupled channel formalism, where the resulting separable t-matrix has the matrix form with The propagator with W = z + M N + M η , where M η is the η mass, is determined by the N (1535)1/2 − self-energies Σ η (W ) and Σ π (W ).The two-pion channel was included via the ππN decay width Γ ππ parametrized in the form The parameters g η , β η , g π , β π , M 0 , and γ ππ were chosen in such a way that the scattering amplitude f ηN corresponding to our t-matrix t ηη (13) is close to that obtained in the coupled-channel analyses in the energy region from 20 MeV above the ηN threshold to 100 MeV below the threshold.Here we took the results of two works [19] and [20] predicting rather different values of Re f ηN (see Fig. 2).FIG.2: The S11 partial wave of the ηN scattering amplitude calculated with Sets I and II of the parameters listed in Table II.Notations: solid curve: real part, dashed curve: imaginary part.Crosses and squares represent the results of the coupled channel analysis of Refs.[19] and [20], respectively.

B. Three-body partitions
We have four different three-body partitions which in the following are numerated by the index α 3 = 1, . . ., 4. In the latter two cases there are two pairs of interacting particles propagating independently.The effective potentials Z α3 α4,β4 determined by Eq. ( 4) for n = 4 are matrix elements of the free resolvent G 0 between the form factors g α4 (α 4 = 1, 2) The functions g α4 (q) are given by Eqs.(9) (or (12)) and ( 14) with g 2 (q) ≡ g η (q).Here we omit the superfluous indices a, b, since our separable ansatz for N N and ηN amplitudes contains in both cases only one term (see Eqs. ( 7) and ( 13)).
FIG. 3: Effective quasi-two-body equations for the (ηN ) − (N N N ) amplitudes X 2 α 3 ,β 3 .Notation of the lines as in Fig. 1.The lower and the upper indices in u α 3 α 4 refer to the numbers of the four-and three-body partitions listed in Eqs. ( 5) and ( 16), respectively.The numerical coefficients appear due to symmetrization of the nucleon states.
The calculation of the N N N N (α 2 = 1) and ηN N N (α 2 = 4) amplitudes with separable N N potentials may be found, e.g., in Refs.[21] and [13], and we refer the reader to these works.The effective (3 + 2) amplitudes (α 2 = 2, 3) describe propagation of two groups of mutually interacting particles.The corresponding integral equations are schematically presented in Figs. 3 and 4.
After the separable expansions (1) for n = 2 are calculated we build the effective potentials Z α2a,β2b (4) as The corresponding system of the five-body η −4N equations is diagrammatically presented in Fig. 5.After this system is solved, the η 4 He scattering amplitude can be calculated as Here N is the normalization constant of the 4 He wave function, µ is the η− 4 He reduced mass, and the momentum p is fixed by the on-mass-shell condition where E 4N > 0 is the four-nucleon binding energy given in Table I.
FIG. 5: Graphical representation of the effective quasi-two-body equations for η − 4N scattering.Notations as in Fig. 1.The lower and the upper indices in u α 2 α 3 refer to the three-and two-body partitions, as given in Eqs. ( 16) and (18).The numerical factors arise from the identity of the nucleons.
In Table III we present the value of the η 4 He scattering length calculated with different number N α2 of terms retained in the separable expansion (1) of the amplitudes X α2 α3,β3 .As one can see, satisfactory accuracy is achieved with N 1 = N 2 = 6, N 3 = N 4 = 8.In principle, already with first four terms in each expansion the resulting scattering length is within less than 2% of the correct value.Thus, also in the five-body case η − 4N the quasi-particle approach based on the EDPE method of Ref. [17] is very suitable for practical applications.The minimum number of separable terms N α2 only slightly exceeds that for the four-body kernels, where convergence is achieved already with first four-six terms in each subamplitude.

III. DISCUSSION AND CONCLUSION
As our main result we present the η 4 He scattering length a η 4 He = f η 4 He (0).It is given in Table IV for two versions of the N N potential.For comparison purposes also the η 3 He scattering length calculated with the same sets of the N N and ηN − πN parameters is presented.It is remarkable, that despite the larger number of nucleons in 4 He the predicted value of a η 4 He is smaller than a η 3 He .Direct calculation shows that the main reason of this somewhat unexpected result is rather rapid decrease of the ηN scattering amplitude in the subthreshold region (see Fig. 2).Because of essentially stronger binding of 4 He in comparison to 3 He, in the former case the effective in-medium ηN interaction acts at lower internal ηN energies, thus leading to general reduction of the attractive ηN forces (this question was addressed in detail in Refs.[22][23][24]).This effective weakening may qualitatively explain why the peculiar slope in the η spectrum at low energies seen in the data for dd → η 4 He [7] and pd → η 3 He [25,26] becomes less steep, when we turn from η 3 He to η 4 He.Summarizing, η 4 He interaction is calculated for the first time correctly dealing with the few-body aspects of the problem.Applying separable representation firstly to the (3 + 1) and (2 + 2) and then to the (4 + 1) and (3 + 2) kernels we have solved the five-body Alt-Grassberger-Sandhas equations reducing them to a coupled set of quasi-two-body equations having Lippmann-Schwinger structure.
The predicted value of Re a η 4 He is positive and turns out to be smaller than Re a η 3 He .This finding should be attributed to effective weakening of the in-medium ηN interaction.According to our calculation, increase of the attractive forces due to an extra nucleon in 4 He is overwhelmed by stronger suppression of the subthreshold ηN interaction in a more dense nucleus.The resulting attraction in the η − 4N system is too weak and does not support existence of the η 4 He bound state, at least with the ηN parameters, used in the present calculation.This might be the key reason why no signal of η 4 He bound state formation is still revealed, e.g., in the dd → 3 He nπ 0 and dd → 3 He pπ − reactions [27,28].
Finally, we note that although our results obviously suffer from oversimplified treatment of the N N potential, they demonstrate applicability of the quasi-particle formalism to the five-body η 4 He problem.The EDPE method provides rather rapid convergence of the separable expansion, so that transition from η − 3N to the η − 4N case is performed without drastic increase of numerical complexity.At the same time, more refined treatment requires inclusion of the nucleon spin as well as more sophisticated nucleon-nucleon potential instead of our simple rank-one ansatz.

TABLE I :
The N N potential parameters.ENN , E3N , and E4N are the two-, three-, and four-nucleon binding energies calculated with our model.

TABLE III :
The scattering length a η 4 He as a function of Nα 2 (α2 = 1, . . ., 4), the number of separable terms retained in the separable expansion (1) for the (4+1) and (3+2) subamplitudes X α 2 .The calculation is performed with the Gauss N N potential and Set I of the ηN − πN parameters.

TABLE IV :
The η 3 He and η 4 He scattering lengths predicted by our calculation.The first and the second rows for each version of the N N potential list the values obtained with Set I and Set II of the ηN − πN parameters, respectively.