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

TL;DR

This paper solves the eta-4He scattering problem using a quasi-particle formalism to simplify complex five-body equations, providing numerical results for scattering length.

Contribution

It introduces a quasi-particle approach to reduce five-body eta-4N equations to an effective two-body form, enabling practical calculations.

Findings

Calculated eta-4He scattering length

Demonstrated effectiveness of quasi-particle formalism

Provided numerical benchmarks for eta-4N interactions

Abstract

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.

Tables

Table 1. Table 1: The N ​ N 𝑁 𝑁 NN potential parameters. E N ​ N subscript 𝐸 𝑁 𝑁 E_{NN} , E 3 ​ N subscript 𝐸 3 𝑁 E_{3N} , and E 4 ​ N subscript 𝐸 4 𝑁 E_{4N} are the two-, three-, and four-nucleon binding energies calculated with our model.

Type	${\lambda }_{NN}$	$\beta$	$E_{NN}$	$r_0$	$E_{3N}$	$E_{4N}$
	fm2	fm-1	MeV	fm	MeV	MeV
Yamaguchi	4.17	1.45	0.428	1.89	12.6	54.8
Gauss	6.51	1.24	0.428	2.33	8.05	30.3

Table 2. Table 2: The η ​ N − π ​ N 𝜂 𝑁 𝜋 𝑁 \eta N-\pi N parameters.

Set [Ref.]	$g_{\eta }$	${\beta }_{\eta }$	$g_{\pi }$	${\beta }_{\pi }$	$M_0$	${\gamma }_{\pi \pi }$
		MeV		MeV	MeV	MeV
I Wycech	1.91	636	0.651	850	1577	4.0
II KSW	1.23	636	1.28	350	1527	1.0

Table 3. Table 3: The scattering length a η 4 ​ He subscript 𝑎 superscript 𝜂 4 He a_{\eta^{4}\mathrm{He}} as a function of N α 2 subscript 𝑁 subscript 𝛼 2 N_{\alpha_{2}} ( α 2 = 1 , … , 4 subscript 𝛼 2 1 … 4 \alpha_{2}=1,\ldots,4 ), the number of separable terms retained in the separable expansion ( 1 ) for the (4+1) and (3+2) subamplitudes X α 2 superscript 𝑋 subscript 𝛼 2 X^{\alpha_{2}} . The calculation is performed with the Gauss N ​ N 𝑁 𝑁 NN potential and Set I of the η ​ N − π ​ N 𝜂 𝑁 𝜋 𝑁 \eta N-\pi N parameters.

$N_1$	$N_2$	$N_3$	$N_4$	$a_{{\eta }^4\mathrm{He}}$ [fm]
2	2	2	2	$5.56+0.96i$
4	4	4	4	$4.88+1.23i$
4	4	6	6	$4.83+1.23i$
6	6	8	8	$4.79+1.22i$
10	10	12	12	$4.79+1.22i$
20	20	20	20	$4.80+1.22i$

Table 4. Table 4: The η 3 superscript 𝜂 3 \eta^{3} He and η 4 superscript 𝜂 4 \eta^{4} He scattering lengths predicted by our calculation. The first and the second rows for each version of the N ​ N 𝑁 𝑁 NN potential list the values obtained with Set I and Set II of the η ​ N − π ​ N 𝜂 𝑁 𝜋 𝑁 \eta N-\pi N parameters, respectively.

$NN$	$\eta N-\pi N$	$a_{{\eta }^3\mathrm{He}}$ [fm]	$a_{{\eta }^4\mathrm{He}}$ [fm]
Yamaguchi	I	$6.5+3.6i$	$2.2+0.3i$
	II	$1.1+0.5i$	$0.5+0.1i$
Gauss	I	$6.7+4.0i$	$4.8+1.2i$
	II	$1.3+0.7i$	$1.0+0.3i$

Equations

$$X_{{\alpha }_{n+1}a,{\beta }_{n+1}b}^{{\alpha }_n}(z)=\sum _{k,l=1}^{N_{{\alpha }_n}}|u_{{\alpha }_{n+1}(a)}^{{\alpha }_n(k)}⟩{\Delta }_{kl}^{{\alpha }_n}(z)⟨u_{{\beta }_{n+1}(b)}^{{\alpha }_n(l)}|,$$

$${\alpha }_{n+1},{\beta }_{n+1}\subset {\alpha }_n.$$

$$X_{{\alpha }_{n}a,{\beta }_{n}b}^{{\alpha }_{n-1}}=Z_{{\alpha }_{n}a,{\beta }_{n}b}^{{\alpha }_{n-1}}+\sum _{{\gamma }_n}\sum _{k,l=1}^{N_{{\gamma }_n}}{Z}_{{\alpha }_{n}a,{\gamma }_{n}k}^{{\alpha }_{n-1}}{\Delta }_{kl}^{{\gamma }_n}{X}_{{\gamma }_{n}l,{\beta }_{n}b}^{{\alpha }_{n-1}},$$

$${\alpha }_n,{\beta }_n,{\gamma }_n\subset {\alpha }_{n-1},$$

$$X_{{\alpha }_{n}a,{\beta }_{n}b}^{{\alpha }_{n-1}}(z;p,p^{\prime })$$

$$Z_{{\alpha }_{n}a,{\beta }_{n}b}^{{\alpha }_{n-1}}=\sum _{{\gamma }_{n+1}}\sum _{k,l}⟨u_{{\gamma }_{n+1}(k)}^{{\alpha }_n(a)}|{\Delta }_{kl}^{{\gamma }_{n+1}}|u_{{\gamma }_{n+1}(l)}^{{\beta }_n(b)}⟩,$$

$${\gamma }_{n+1}\subset {\alpha }_n,{\gamma }_{n+1}\subset {\beta }_n,{\alpha }_n\ne {\beta }_n.$$

$$1:(NN)+N+N+\eta ,2:(\eta N)+N+N+N.$$

$$v_1(z)=-|g_1⟩⟨g_1|.$$

$$t_1(z)=|g_1⟩{\tau }_1(z)⟨g_1|$$

$$\tau_{1}(z)=-\bigg{[}1+\frac{1}{2\pi^{2}}\int_{0}^{\infty}\frac{g_{1}(q)^{2}}{z-q^{2}/M_{N}}\,q^{2}dq\bigg{]}^{-1}\,,$$

$$g_1(q)=\frac{\sqrt{{\lambda }_{NN}}}{1+(q/\beta {)}^2}.$$

$${\lambda }_{NN}=\frac{1}{2}({\lambda }_{NN}^{(0)}+{\lambda }_{NN}^{(1)}),{\lambda }_{NN}^{(s)}=\frac{8\pi a_s}{M_N(a_s\beta -2)}.$$

$$a_0=23.690\text{mbox}fm,a_1=-5.378\text{mbox}fm,\beta =1.4488\text{mbox}{fm}^{-1}.$$

$$g_1(q)=\sqrt{{\lambda }_{NN}}{e}^{-q^2/{\beta }^2},$$

$$t_{\mu \nu }(z)=\frac{1}{W-M_0}|g_{\mu }⟩{\tau }_2(z)⟨g_{\nu }|,\mu ,\nu \in \{\pi ,\eta \}$$

$$g_{\mu }(q)=\frac{g_{\mu }}{1+(q/{\beta }_{\mu }{)}^2}.$$

$${\tau }_2(z)=\frac{1}{W-M_0-{\Sigma }_{\eta }(W)-{\Sigma }_{\pi }(W)+\frac{i}{2}{\Gamma }_{\pi \pi }(W)}$$

$${\Gamma }_{\pi \pi }(W)={\gamma }_{\pi \pi }\frac{W-M_N-2M_{\pi }}{M_{\pi }}.$$

$$\begin{array}[]{llll}1:&(NNN)+N+\eta\,,&2:&(\eta NN)+N+N\,,\\ 3:&(\eta N)+(NN)+N\,,\phantom{xxx}&4:&(NN)+(NN)+\eta\end{array}$$

$$Z_{{\alpha }_4,{\beta }_4}^{{\alpha }_3}=⟨g_{{\alpha }_4}|G_0|g_{{\beta }_4}⟩.$$

$$\begin{array}[]{llll}1:&\eta+(NNNN)\,,&2:&(\eta N)+(NNN)\,,\\ 3:&(\eta NN)+(NN)\,,\phantom{xxxxx}&4:&(\eta NNN)+N\end{array}$$

$$Z_{{\alpha }_{3}a,{\beta }_{3}b}^{{\alpha }_2}=\sum _{{\gamma }_4=1,2}⟨u_{{\gamma }_4}^{{\alpha }_3(a)}|{\tau }_{{\gamma }_4}|u_{{\gamma }_4}^{{\beta }_3(b)}⟩.$$

$$Z_{{\alpha }_{2}a,{\beta }_{2}b}=\sum _{{\gamma }_3=1}^4\sum _{k,l=1}^{N_{{\gamma }_3}}⟨u_{{\gamma }_3(k)}^{{\alpha }_2(a)}|{\Delta }_{kl}^{{\gamma }_3}|u_{{\gamma }_3(l)}^{{\beta }_2(b)}⟩.$$

$$f_{{\eta }^4\mathrm{He}}(p)=-N^2\frac{\mu }{2\pi }{X}_{11,11}(z;p,p).$$

$$p=\sqrt{2\mu (z+E_{4N})},$$

Full text

I Introduction

During the last years considerable attention has been paid to interaction of $\eta$-mesons with four nucleons KruscheWilkin ; Machner ; Willis ; Wronska ; Budzanowski ; Moskal ; Adlarson1 ; Kelkar ; ScRiska . Analysis of different data is mainly focused on the search for $\eta^{4}$He bound states. According to the available experimental results, the rise of the $dd\to\eta^{4}$He experimental cross section at $E_{\eta}\to 0$ seems to be not as steep as in the $pd\to\eta^{3}$He reaction. As discussed, e.g., in Ref. Willis the most natural interpretation of this fact is that due to additional attraction caused by one extra nucleon the pole in the $\eta^{4}$He scattering matrix is shifted into the region of negative values of $Re\,E_{\eta}$ and turns out to be farther from the physical region than the $\eta^{3}$He pole. It is therefore concluded that formation of the bound $\eta^{4}$He state is highly probable.

In view of general complexity of the five-body $\eta-4N$ 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 $NN$ and $\eta N$ 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 $\eta NN$ Shev ; Pena ; FiAr2N as well as $\eta-3N$ FiAr3N scattering (in Ref. Barnea 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 $\eta-4N$. As a formal basis we use the Alt-Grassberger-Sandhas $n$-body equations derived in Ref. GS . 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 $\eta^{4}$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 (quasi-particle). 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. Yakubovsky . Different partitions (as well as the quasi-particles related to these partitions) are further denoted by the symbols $\alpha,\beta,\ldots$, whereas the Latin letters $a,b,\ldots$ are used for numbering the terms in the separable expansions of the subamplitudes. The notation $\alpha_{n}$ refers to the partition obtained by dividing the $\eta-4N$ system into $n$ groups. Writing $\alpha_{n+1}\subset\alpha_{n}$ means that the partition $\alpha_{n+1}$ is obtained from $\alpha_{n}$ via further division of the quasi-particle $\alpha_{n}$ into two groups of particles.

The basic ingredient of the formalism is a separable expansion of the quasi-particle amplitudes

[TABLE]

Then the integral equations for the amplitudes $X^{\alpha_{n-1}}_{\alpha_{n},\beta_{n}}$ are transformed exactly into the quasi-two-body equations which in the operator form read

[TABLE]

or more explicitly

[TABLE]

with $\mu_{\gamma_{n}}$ being the reduced mass associated with the partition $\gamma_{n}$. The effective potentials $Z^{\alpha_{n-1}}_{\alpha_{n},\beta_{n}}$ are determined as matrix elements of the ’resolvent’ $\Delta^{\gamma_{n+1}}$ between the form factors appearing in the expansion (1)

[TABLE]

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 $Z^{\alpha_{2}}_{\alpha_{3},\beta_{3}}$, connecting two configurations of the type $(\eta NN)+N$. Since the nucleons are identical, the condition $\alpha_{n}\neq\beta_{n}$ in Eq. (4) means that the nucleon lines, entering the quasi-particles $\alpha_{n}$ and $\beta_{n}$ and not included into the quasi-particle $\gamma_{n+1}$ should be different. To calculate the form factors $u^{\alpha_{n}(a)}_{\gamma_{n+1}(k)}$ and the propagators $\Delta^{\gamma_{n+1}}_{kl}$ we employed the energy dependent pole expansion (EDPE) method of Ref. EDPE .

II.1 Four-body partitions

Considering nucleons as indistinguishable particles we have only two different types of four-particle partitions:

[TABLE]

The partitions 1 and 2 and the related two-particle subsystems $NN$ and $\eta N$ will further be labeled by the index $\alpha_{4}=1,2$.

In the present calculation, the $NN$ and $\eta N$ $s$-wave interactions were approximated by simplest rank-one separable potentials. For $NN$ we employed

[TABLE]

The corresponding $t$-matrix has the usual form

[TABLE]

with the $NN$ propagator

[TABLE]

where $M_{N}$ is the nucleon mass. The form factors were chosen in the Yamaguchi form

[TABLE]

Since we treat nucleons as spinless particles, the strength $\lambda_{NN}$ was taken as an average of the singlet and the triplet strength

[TABLE]

The singlet and the triplet scattering lengths, $a_{0}$ and $a_{1}$, as well as the cut-off momentum $\beta$ were taken directly from the analysis Yamag of the low-energy $np$ scattering

[TABLE]

It is well known that the Yamaguchi $NN$ potential overestimates attraction at high momenta and yields significant overbinding already in the 3He case (see Table 1). Therefore we also adopted the spin-independent $NN$ potential with exponential form factors

[TABLE]

which yields the same binding energy $E_{NN}$ of two nucleons. The form factors (12) with parameters listed in Table 1 give for the three- and four-nucleon binding energies, $E_{3N}$ and $E_{4N}$, the values which are rather close to those of the 3He and 4He nuclei. At the same time, with the Gauss form factors we have a visibly larger value of the $NN$ effective range $r_{0}$ (see Table 1).

The $\eta N$ $s$-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

[TABLE]

with

[TABLE]

The propagator

[TABLE]

with $W=z+M_{N}+M_{\eta}$, where $M_{\eta}$ is the $\eta$ mass, is determined by the $N(1535)1/2^{-}$ self-energies $\Sigma_{\eta}(W)$ and $\Sigma_{\pi}(W)$. The two-pion channel was included via the $\pi\pi N$ decay width $\Gamma_{\pi\pi}$ parametrized in the form

[TABLE]

The parameters $g_{\eta}$, $\beta_{\eta}$, $g_{\pi}$, $\beta_{\pi}$, $M_{0}$, and $\gamma_{\pi\pi}$ were chosen in such a way that the scattering amplitude $f_{\eta N}$ corresponding to our $t$-matrix $t_{\eta\eta}$ (13) is close to that obtained in the coupled-channel analyses in the energy region from 20 MeV above the $\eta N$ threshold to 100 MeV below the threshold. Here we took the results of two works Wycech and KSW predicting rather different values of $Re\,f_{\eta N}$ (see Fig. 2).

II.2 Three-body partitions

We have four different three-body partitions

[TABLE]

which in the following are numerated by the index $\alpha_{3}=1,\ldots,4$. In the latter two cases there are two pairs of interacting particles propagating independently. The effective potentials $Z^{\alpha_{3}}_{\alpha_{4},\beta_{4}}$ determined by Eq. (4) for $n=4$ are matrix elements of the free resolvent $G_{0}$ between the form factors $g_{\alpha_{4}}$ $(\alpha_{4}=1,2)$

[TABLE]

The functions $g_{\alpha_{4}}(q)$ are given by Eqs. (9) (or (12)) and (14) with $g_{2}(q)\equiv g_{\eta}(q)$. Here we omit the superfluous indices $a,b$, since our separable ansatz for $NN$ and $\eta N$ amplitudes contains in both cases only one term (see Eqs. (7) and (13)).

II.3 Two-body partitions

There are four two-body partitions of the $\eta-4N$ system:

[TABLE]

which will be labeled by $\alpha_{2}=1,\ldots,4$.

The effective potentials $Z^{\alpha_{2}}_{\alpha_{3}a,\beta_{3}b}$ are matrix elements of the ’resolvent’ $\tau_{\alpha_{4}}$ between the form factors $u^{\alpha_{3}(a)}_{\alpha_{4}}$ appearing in the separable expansion (1) for $n=3$:

[TABLE]

The propagators $\tau_{\alpha_{4}}$ ($\alpha_{4}=1,2$) are given by (8) and (II.1) with $\tau_{2}\equiv\tau_{\eta\eta}$.

The calculation of the $NNNN$ ($\alpha_{2}=1$) and $\eta NNN$ ($\alpha_{2}=4$) amplitudes with separable $NN$ potentials may be found, e.g., in Refs. AGS_4N and FiAr3N , and we refer the reader to these works. The effective $(3+2)$ amplitudes ($\alpha_{2}=2,3$) 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 $Z_{\alpha_{2}a,\beta_{2}b}$ (4) as

[TABLE]

The corresponding system of the five-body $\eta-4N$ equations is diagrammatically presented in Fig. 5. After this system is solved, the $\eta^{4}$He scattering amplitude can be calculated as

[TABLE]

Here $N$ is the normalization constant of the 4He wave function, $\mu$ is the $\eta-^{4}$He reduced mass, and the momentum $p$ is fixed by the on-mass-shell condition

[TABLE]

where $E_{4N}>0$ is the four-nucleon binding energy given in Table 1.

In Table 3 we present the value of the $\eta^{4}$He scattering length calculated with different number $N_{\alpha_{2}}$ of terms retained in the separable expansion (1) of the amplitudes $X^{\alpha_{2}}_{\alpha_{3},\beta_{3}}$. 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 $\eta-4N$ the quasi-particle approach based on the EDPE method of Ref. EDPE is very suitable for practical applications. The minimum number of separable terms $N_{\alpha_{2}}$ 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 $\eta^{4}$He scattering length $a_{\eta^{4}\mathrm{He}}=f_{\eta^{4}\mathrm{He}}(0)$. It is given in Table 4 for two versions of the $NN$ potential. For comparison purposes also the $\eta^{3}$He scattering length calculated with the same sets of the $NN$ and $\eta N-\pi N$ parameters is presented.

It is remarkable, that despite the larger number of nucleons in 4He the predicted value of $a_{\eta^{4}\mathrm{He}}$ is smaller than $a_{\eta^{3}\mathrm{He}}$. Direct calculation shows that the main reason of this somewhat unexpected result is rather rapid decrease of the $\eta N$ scattering amplitude in the subthreshold region (see Fig. 2). Because of essentially stronger binding of 4He in comparison to 3He, in the former case the effective in-medium $\eta N$ interaction acts at lower internal $\eta N$ energies, thus leading to general reduction of the attractive $\eta N$ forces (this question was addressed in detail in Refs. WyGrNisk ; HaidLiu ; WycechKrz ). This effective weakening may qualitatively explain why the peculiar slope in the $\eta$ spectrum at low energies seen in the data for $dd\to\eta^{4}$He Adlarson1 and $pd\to\eta^{3}$He Mersmann ; Smyrski becomes less steep, when we turn from $\eta^{3}$He to $\eta^{4}$He.

Summarizing, $\eta^{4}$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 $Re\,a_{\eta^{4}\mathrm{He}}$ is positive and turns out to be smaller than $Re\,a_{\eta^{3}\mathrm{He}}$. This finding should be attributed to effective weakening of the in-medium $\eta N$ interaction. According to our calculation, increase of the attractive forces due to an extra nucleon in 4He is overwhelmed by stronger suppression of the subthreshold $\eta N$ interaction in a more dense nucleus. The resulting attraction in the $\eta-4N$ system is too weak and does not support existence of the $\eta^{4}$He bound state, at least with the $\eta N$ parameters, used in the present calculation. This might be the key reason why no signal of $\eta^{4}$He bound state formation is still revealed, e.g., in the $dd\to^{3}$He $n\pi^{0}$ and $dd\to^{3}$He $p\pi^{-}$ reactions Krzemien ; Adlarson2 .

Finally, we note that although our results obviously suffer from oversimplified treatment of the $NN$ potential, they demonstrate applicability of the quasi-particle formalism to the five-body $\eta^{4}$He problem. The EDPE method provides rather rapid convergence of the separable expansion, so that transition from $\eta-3N$ to the $\eta-4N$ 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.