S-WAVE MOTION IN THE WOODS–SAXON POTENTIAL AND BINDING ENERGIES OF HYPERNUCLEI

V. A. Filimonov

(Presented by the Scientific-Technical Seminar of the High-Energy Laboratory of the Nuclear Physics Research Institute)

1. Introduction

To explain the dependence of the binding energies \(B_\Lambda\) of hypernuclei on the mass number \(A\), Ivanenko and Kolesnikov [1] proposed a model according to which the motion of \(\Lambda\) in a nucleus is considered as motion in a certain potential well with a depth approximately the same for all nuclei and with a radius increasing according to the law \(R \approx r_0 A^{1/3}\).

In [2] it was proposed to regard the depth of the well as an additional parameter that must be satisfied by the \(\Lambda\)-nucleon forces, and to take it equal to the potential energy of \(\Lambda\) in infinite nuclear matter. The first calculations of the depth of the potential well on this basis, with various two-body and three-body \(\Lambda N\) potentials, were carried out in [2–4]. Subsequently, attempts to estimate the depth of the potential well from the binding energies of hypernuclei and to calculate it from \(\Lambda N\) forces were undertaken by many authors [5–8].

The interaction of a particle with a nucleus is in many cases justifiably described with the aid of a Woods–Saxon potential. Motion in this potential has been considered in many works (for example, [10–15]). Recently, analytic expressions have been found for the wave functions of \(S\)-orbital motion [11–15]. In the present article the investigation of \(S\)-wave motion in the Woods–Saxon potential is continued. The results are used to analyze the binding energies of hypernuclei. In Section 2, various representations are obtained for the wave functions of bound \(S\)-states and equations for determining the energy eigenvalues. In the third section, the question of finding dimensional parameters for the \(\Lambda\)-nuclear Woods–Saxon potential is discussed. In Section 4, from the binding energies of heavy hypernuclei, the depth of the potential well for a \(\Lambda\)-particle in nuclei with \(A = 40\text{–}100\) is estimated.

2. Motion in the Woods–Saxon Potential

The Woods–Saxon potential has the form

\[ V(r) = -D/[1+\gamma(r)], \]

\[ \gamma(r) = \exp (r-c)/a . \tag{1} \]

The quantity \(D\) is not equal to the depth of the potential well at \(r=0\) and is related to the latter by the relation

\[ - V(0) = D/[1+\exp(-c/a)] . \tag{2} \]

For \(c/a \gg 1\), the discrepancy between \(-V(0)\) and \(D\) is small, and therefore the depth of the Woods–Saxon potential may be taken to be the quantity \(D\). The parameter \(a\) characterizes the width of the surface region in which the main decrease of the absolute value of the potential from \(D\) to 0 occurs. For \(a=0\), the potential (1) becomes a rectangular potential well of radius \(c\) and depth \(D\). We shall also use the dimensionless parameter \(\lambda=a/c\), which most appropriately characterizes the diffuseness of the nuclear edge.

The radial \(s\)-wave Schrödinger equation is written in the form

\[ (1/2M'_\Lambda)\, d^2u/dr^2 + [E - V(r)]u = 0 . \tag{3} \]

In (3), \(M'_\Lambda\) is the reduced mass of the \(\Lambda\)-hyperon and the residual nucleus, if the potential is used to describe the interaction of the \(\Lambda\) with the nucleus.

Let us introduce the notation

\[ k^2 = -2M'_\Lambda E = d^2 \cos^2 \zeta,\quad d^2 = 2M'_\Lambda D,\quad x = -1/y = -\exp(c-r)/a \]

\[ s = cd,\quad s_x = s\cos\zeta . \tag{4} \]

For bound states \(k\) is real and \(0 \geq \zeta < \pi/2\); we also note that

\[ B_\Lambda = -E = D\cos^2\zeta . \]

By the substitution

\[ u(r)=e^{-kr}\chi(x) \tag{5} \]

(3) is transformed into the equation

\[ x(1-x)\chi'' + (1-x)(2ka+1)\chi' - a^2d^2\chi = 0 . \tag{6} \]

This is a hypergeometric equation with parameters

\[ \alpha = a\bigl(k+i\sqrt{d^2-k^2}\bigr)=\lambda s e^{i\zeta}, \]

\[ \beta = a\bigl(k-i\sqrt{d^2-k^2}\bigr)=\lambda s e^{-i\zeta}, \tag{7} \]

\[ \gamma = 1+2ka = 1+2\lambda s\cos\zeta . \]

Equation (6) has two linearly independent solutions. Under the condition \(\gamma \neq 0,-1,-2,\ldots\) (this condition, according to (7), is always satisfied), the solutions may be chosen as follows [16]: the first solution is regular at the point \(x=0\),

\[ \chi_1(x)=F(\alpha,\beta;\gamma;x); \tag{8} \]

for the second solution, the point \(x=0\) is singular,

\[ \chi_2(x)=x^{1-\gamma}F(\alpha-\gamma+1,\beta-\gamma+1;2-\gamma;x). \tag{9} \]

The general solution of the Schrödinger equation will be

\[ u(r)=e^{-kr}(C_1\chi_1 + C_2\chi_2). \tag{10} \]

A change of \(r\) from 0 to \(\infty\) corresponds to a change of \(x\) from \(x_0=-e^{c/a}<-1\) to 0. The hypergeometric functions entering (8) and (9) can be represented, for \(|x|<1\), by convergent hypergeometric series. As \(r\to\infty\), the hypergeometric functions tend to 1, and the general solution (10) takes the following asymptotic form:

\[ u(r\to\infty)=C_1e^{-kr}-C_2e^{-2kc}e^{kr}. \tag{11} \]

Therefore, for the case of bound states one must set \(C_2=0\).

The solution of equation (3) must also satisfy the boundary condition

\[ u(0)=0 \tag{12} \]

or

\[ \mathrm{F}\left(\alpha,\ \beta;\ \gamma;\ -e^{c/a}\right)=0. \tag{12a} \]

Condition (12) is an equation for determining the discrete energy values of bound states. Since \(e^{c/a}>1\), in (12a) one cannot use the representation of the hypergeometric function by a hypergeometric series. Likewise, the solution (8) cannot be represented by a hypergeometric series over the whole range of variation of \(r\). It is necessary either to perform an analytic continuation of the hypergeometric series from the neighborhood of the point \(x=0\) into the region \(x\geq 1\), or to use some other representation of the hypergeometric functions.

Under the condition

\[ \operatorname{Re}\gamma>\operatorname{Re}\beta>0 \tag{13} \]

Euler’s integral representation is valid for all values of \(x\):

\[ \mathrm{F}(\alpha,\beta;\gamma;x) = \frac{1}{\mathrm{B}(\beta,\gamma-\beta)} \int_{0}^{1} t^{\beta-1}(1-t)^{\gamma-\beta-1}(1-tx)^{-\alpha}\,dt . \tag{14} \]

Condition (13) is satisfied for bound states, except in the case \(a=0\). Expression (14) leads to a complex wave function \(u(r)\). At the same time, it is known that the solution of the radial Schrödinger equation in the case of bound states can have only an arbitrary constant complex phase factor. Therefore it is sufficient to take either the real or the imaginary part of expression (14):

\[ \begin{matrix} \operatorname{Re}\\ \operatorname{Im} \end{matrix} \mathrm{B}(\beta,\gamma-\beta)\mathrm{F}(\alpha,\beta;\gamma;x) = \int_{0}^{1} t^{2ad\cos\zeta} \left(\frac{1-t}{t-t^{2}x}\right)^{ad\cos\zeta} \times \begin{matrix} \cos\\ \sin \end{matrix} \left[ ad\left(\ln\frac{1-t}{t-t^{2}x}\right)\sin\zeta \right]\,dt . \tag{15} \]

Introducing the new variable
\(z=ad(\sin\varphi)\ln[(1-t)/t(1-tx)]\), the integrals (15) are brought to a form more convenient for computation; taking (10) into account, we obtain two representations for the radial wave function:

\[ u(r)=A_s\int_{0}^{\infty} t(\nu-1) \left(\frac{t^{2}\nu}{\gamma}\right)^{\lambda s\cos\zeta} \frac{\sin z\,dz}{1+t^{2}\nu/\gamma}, \tag{16} \]

\[ u(r)=A_c\int_{0}^{\infty} (2-\nu t-t) \left(\frac{t^{2}\nu}{\gamma}\right)^{\lambda s\cos\zeta} \frac{\cos z\,dz}{1+t^{2}\nu/\gamma}, \tag{17} \]

where

\[ \nu=\exp(-z/\lambda s\sin\zeta), \]

\[ t=\frac{2}{\sqrt{(\nu+1)^2+4\nu/\gamma}+\nu+1} \]

and \(A_s,\ A_c\) are normalization constants.

For \(\nu>1\), i.e., for \(r>c\), the wave function can be represented in the form of an expansion in powers of \(\nu^{-1}=\exp(r-c)/a\):

\[ u(r)=A_s e^{-k(r-c)} \left\{ \int_{0}^{\infty} \nu^{\lambda s\cos\zeta} \frac{(\nu-1)\sin z\,dz}{(1+\nu)^{2\lambda s\cos\zeta+1}} - 2(1+\lambda s\cos\zeta)e^{-\frac{(r-c)}{a}} \int_{0}^{\infty} \nu^{1+\lambda s\cos\zeta} \frac{(\nu-1)\sin z\,dz}{(1+\nu)^{2\lambda s\cos\zeta+3}} +\ldots \right\}. \tag{18} \]

a

b

c

Fig. 1. Graphs illustrating the equations for determining the energy eigenvalues of \(s\)-wave motion in the Woods–Saxon potential. Dash-dotted lines are the dependence of the right-hand side of (16) on \(\xi\) for \(\eta = 0\); solid curves are the dependence of the left-hand side of equation (23) on \(\xi\); dashed curves are the same for the approximate equation (26). The numbers next to the curves are the values of the parameter \(\lambda\). The inclined straight lines are the right-hand sides of (25) and (26); the values of the quantum number \(n\) are placed next to the straight lines. For \(\lambda = 0.2\), the solid and dashed lines in the graphs coincide. The indices a, b, c correspond to \(s = 3, s = 6, s = 9\).

Expansion (18) shows that, as was to be expected, integrals (16) and (17) give the correct asymptotic behavior of the wave function represented by them.

Formulas (16)—(18) make it possible to obtain easily the wave functions and eigenvalues of the energy (provided only that \(\lambda s\cos \zeta\) is not too close to zero) of both the ground and excited \(s\)-states, without resorting to numerical integration of the Schrödinger equation. In Fig. 1a—c the dash-dotted lines illustrate the dependence of the left-hand side of (16) at \(r=0\) on \(\zeta\) for various values of \(s\) and \(\lambda\). The energy eigenvalues correspond to the points of intersection of these curves with the abscissa axis.

Let us now consider the analytic continuation of solution (8) in the neighborhood of \(x=0\) into the region of values \(|x|\geqslant 1\) with the aid of hypergeometric series. We shall use the relation

\[ \begin{aligned} F(\alpha,\beta;\gamma;z) &= \Gamma(\gamma)\frac{\Gamma(\beta-\alpha)}{\Gamma(\beta)\Gamma(\gamma-\alpha)} (-z)^{-\alpha} F(\alpha,1-\gamma+\alpha;1-\beta+\alpha;z^{-1}) \\ &\quad + \frac{\Gamma(\alpha-\beta)}{\Gamma(\alpha)\Gamma(\gamma-\beta)} (-z)^{-\beta} F(\beta,1-\gamma+\beta;1-\alpha+\beta;z^{-1}) \biggr\}. \end{aligned} \tag{19} \]

This relation remains valid also in the limit \(a\to 0\) and \(E\to 0\). On the basis of (19), for the wave function in the region \(r\leqslant c\) we obtain

\[ u(r)=-C_1\frac{e^{-\lambda s\sin \zeta}\Gamma(\gamma)}{\sin \zeta} \left[\operatorname{Re}P\sin(rd\sin\zeta)+\operatorname{Im}P\cos kr\right], \tag{20} \]

\[ P= \frac{e^{-i(\zeta+\lambda s\sin\zeta)}\Gamma(1-2i\lambda s\sin\zeta)} {\Gamma^2(1+\lambda s e^{-i\zeta})} F\left(\lambda s e^{i\zeta},-\lambda s e^{-i\zeta};1+2i\lambda s\sin\zeta;-e^{\frac{r-c}{a}}\right). \]

The energy quantization condition is

\[ \operatorname{Im}P=0 \tag{21} \]

or

\[ \arg P=n\pi, \tag{21a} \]

where \(n\) is an integer.

For \(r>c\)

\[ u(r)=C_1 e^{-kr}F\left(\lambda s e^{i\zeta},\lambda s e^{-i\zeta};1+2\lambda s\cos\zeta;-e^{\frac{c-r}{a}}\right). \tag{22} \]

Expressions (20) and (22), together with (21), give a representation of the wave function in terms of hypergeometric series. Equation (21a) can also be written in the form

\[ f(\zeta,\lambda,s)=n\pi-\zeta, \]

\[ \begin{aligned} f(\zeta,\lambda,s) &=\lambda s\sin\zeta -2\arg \Gamma(1+\lambda s e^{i\zeta}) +\arg \Gamma(1+2i\lambda s\sin\zeta) \\ &\quad -\arg F(\lambda s e^{i\zeta},-\lambda s e^{-i\zeta};1+2i\lambda s\sin\zeta;e^{-1/\lambda}). \end{aligned} \tag{23} \]

The equation obtained differs from that given in [13, 14] by the term \(\arg F\). Therefore the remark of the authors of the cited works that they obtained an exact equation for finding the energy eigenvalues of \(s\)-wave motion in the Woods—Saxon potential is not correct. Under the condition \(e^{-c/a}\ll 1\), the term \(\arg F\) may be neglected; as a result one obtains a shortened equation for finding the eigenvalues, into which the hypergeometric function will not enter:

\[ cd\sin\zeta-2\arg \Gamma(1+ad e^{-i\zeta})+\arg \Gamma(1+2iad\sin\zeta)=n\pi-\zeta. \tag{24} \]

As \(a \to 0\), equations (23) and (24) go over into the equation for a rectangular potential well:

\[ s\sin \zeta = n\pi - \zeta . \tag{25} \]

Fig. 1, a—b illustrates equations (23)—(25). The solid curves represent the dependence of the left-hand side of (23), and the dashed curves that of the left-hand side of (24), on \(\zeta\); the straight lines represent the dependence on \(\zeta\) of the right-hand sides of (23)—(25) for various \(n\). The roots of the equations are determined by the points of intersection of the curves with the straight lines. The roots of equation (23) for various values of \(s_x\) and \(\lambda\) are given in Table 1. The numbers in the table are the values of the quantity \(10^4 \cos^2 \zeta\). These values make it possible to find the depth of the well from the binding energy of the \(\Lambda\)-hyperon by dividing \(B_\lambda\) by \(\cos^2 \zeta\) from Table 1.

Table 1

Roots of equation (12) for \(n=1\) for various values of \(s_x\) and \(\lambda\). The quantities \(10^4 \cos^2 \zeta\) are given

3. Construction of the \(\Lambda\)-nuclear potential

We shall assume that, approximately, the form of the \(\Lambda\)-nuclear potential \(V(r)\) can be obtained by folding the distribution of the nucleon density in the nucleus \(\rho(r)\) with the \(\Lambda\)-nucleon potential \(v(r)\):

\[ V(r)=\int \rho(r')\, v(r' - r)\, dr . \tag{26} \]

We assume that in (26) \(\rho(r)\) and \(v(r)\) are normalized to 1. Then the volume integral \(V(r)\) is also equal to 1.

In the general case, carrying out the integration in (26) is difficult. Therefore, in order to obtain the dependence of the \(\Lambda\)-nuclear potential on \(r\), one resorts to approximate estimates of the integral (26). In [7] a relation was obtained between \(V(r)\) and the distribution of the nucleon density in the form

\[ V(r)=\left(1+\frac{\langle r^2\rangle_v}{6}\Delta\right)\rho(r), \tag{27} \]

where \(\langle r^2\rangle_v\) is the mean-square radius of the \(\Lambda N\)-potential and \(\Delta\) is the Laplace operator.

If \(\rho(r)\) is taken in the form of a Fermi distribution,

\[ \rho(r)=\rho_0/[1+\exp(r-c_\rho)/a_\rho], \tag{28} \]

then \(V(r)\) in (27) will not have the Woods--Saxon form. Therefore we shall use another method [17]. We shall describe the interaction of \(\Lambda\) with the nucleus by the potential (1). The parameters \(c\) and \(a\) of this potential will be found from the requirement that the second and fourth moments of the potential (1) coincide with the corresponding moments of \(V(r)\) in (26). The moments of \(V(r)\) in (26) can be related to the moments of the distributions \(\rho(r)\) and \(v(r)\) entering the integrand. We restrict ourselves to moments of even order and adopt for them the notation

\[ \langle r^{2n}\rangle_\varphi=\int r^{2n}\varphi(\mathbf r)\,d\mathbf r . \tag{29} \]

Introduce the three-dimensional Fourier transform

\[ \varphi(\mathbf r)=\frac{1}{(2\pi)^3}\int \widetilde{\varphi}(\mathbf q)e^{i\mathbf q\mathbf r}\,d\mathbf q . \tag{30} \]

The Fourier transforms of several simplest distributions are given in Table 2. Expressions (26) and (29), taking (30) into account, are transformed to the form

\[ \widetilde V(\mathbf q)=\widetilde \rho(\mathbf q)\widetilde v(\mathbf q). \tag{31} \]

and

\[ \langle r^{2n}\rangle_\varphi=(-1)^n\lim_{q\to 0}\Delta_q^n\widetilde{\varphi}(\mathbf q). \tag{32} \]

Table 2

Three-dimensional Fourier transforms and moments of the simplest distributions.
The volume integrals of the spatial distributions are equal to 1

If the distributions \(\rho(r)\) and \(v(r)\) are centrally symmetric, then the potential \(V_1(r)\) in (26) is also centrally symmetric. For centrally symmetric distributions, expression (32) is also simplified:

\[ \langle r^{2n}\rangle_\varphi=(-1)^n(2n+1)\widetilde{\varphi}^{(2n)}(0), \tag{33} \]

where \(\widetilde{\varphi}^{(2n)}(0)\) is the value of the derivative of order \(2n\) with respect to \(q\) at \(q=0\). On the basis of (33) and (31) we obtain the desired relation between the moments of the distributions \(V(r)\), \(\rho(r)\), and \(v(r)\):

\[ \langle r^{2n}\rangle_{V\downarrow}=(2n+1)!\sum_{k=0}^{n} \frac{\langle r^{2(n-k)}\rangle_\rho\,\langle r^{2k}\rangle_v} {(2n-2k+1)!(2k+1)!}. \tag{34} \]

From (34) it follows that

\[ \langle r^2\rangle_{V\downarrow}=\langle r^2\rangle_\rho+\langle r^2\rangle_v, \tag{35} \]

\[ \langle r^4\rangle_{V\downarrow}=\langle r^4\rangle_\rho+\frac{10}{3}\langle r^2\rangle_\rho\langle r^2\rangle_v+\langle r^4\rangle_v. \tag{36} \]

The moments of the Fermi distribution are expressed in terms of the parameters \(c_\rho\) and \(a_\rho\) as follows [18]:

\[ \langle r^{2n}\rangle_\rho = c_\rho^{2n}\frac{F_n(\lambda_\rho)}{F_0(\lambda_\rho)}, \tag{37} \]

where

\[ \begin{aligned} F_n(\lambda) &= \lambda^{2n+3}\int_0^\infty \frac{x^{2n+2}\,dx}{1+e^{(x-1)/\lambda}} \\ &= \lambda^{2n-3} \Biggl\{ 1/\lambda^{2n+3}(2n+3) + (2n+2)! \\ &\quad\times \left[ \sum_{r=0}^{2n+2} \left[1-(-1)^r\right] \left[ 1/\lambda^{2n+2-r}(2n+2-r)! \left(1-2^{-r}\zeta(r+1)\right) \right.\right. \\ &\qquad\qquad\left.\left. - \sum_{m=1}^{\infty} (-1)^m e^{-m/\lambda}/m^{2n+3} \right] \right] \Biggr\}, \qquad \lambda_\rho=a_\rho/c_\rho . \tag{38} \end{aligned} \]

For \(\lambda \ll 1\) we have

\[ F_1/F_0=(3/5)(1+7\pi^2\lambda^2/3), \tag{39} \]

\[ F_2/F_0=(3/7)(1+6\pi^2\lambda^2+3/\pi^4\lambda^4/3). \tag{40} \]

If the second and fourth moments of the potential (1) are known, then the parameters \(c\) and \(\lambda\) can be found with the aid of the following formulas, which follow from expressions (37)—(40):

\[ \lambda = \frac{1}{\pi} \left[ \frac{3\kappa-3+\sqrt{12(4\kappa/7-1)}}{31-7\kappa} \right]^{1/2}, \tag{41} \]

\[ \kappa=49\langle r^4\rangle_V/25\langle r^2\rangle_V^2 . \]

\[ c^2 = \frac{5\langle r^2\rangle_V}{3(1+7y/3)} = \left[ \frac{7\langle r^4\rangle_V}{3(1+6y+31y^2/3)} \right]^{1/2}, \tag{42} \]

\[ y=\pi^2\lambda^2 . \]

Thus, the procedure for finding the parameters of the \(\Lambda\)-nuclear potential in the form (1) reduces to the following. From the known parameters of the nucleon distribution \(c_\rho\) and \(a_\rho\), the second and fourth moments of this distribution are found by means of expressions (37) and (38). Next, by formulas (35) and (36), the moments of the \(\Lambda\)-nuclear potential are determined. The parameters \(c\) and \(a\) of this potential are then calculated with the aid of (41) and (42).

From the data on electron scattering by heavy nuclei [18] it follows that

\[ c_\rho = lA^{1/3} - \frac{\pi^2a_\rho^2}{3l}A^{-1/3} = 1.123A^{1/3} - 0.94A^{-1/3}. \tag{43} \]

Here the parameter \(a_\rho=0.57\) is taken to be the same for all heavy nuclei. It is assumed that the standard deviation for \(l\) is \(10\%\) of its value.

Table 3 gives the parameters of the \(\Lambda\)-nuclear potential for various values of \(A\), obtained with the aid of the procedure described above. The potential of the \(\Lambda\)-nucleon interaction \(v(r)\) was taken in Yukawa form. Values of \(c\) and \(a\) are given, calculated for two ranges of the Yukawa potential, corresponding to exchange by two \(\pi\)-mesons and exchange by one \(K\)-meson. In addition, parameters are given for zero range, which coincide with the parameters of the Fermi distribution of the nucleon density in the nucleus.

Table 3

Parameters of the Woods–Saxon potential for the
\(\Lambda\)-nuclear interaction

It is seen from Table 3 that taking into account the finite range of action of the \(\Lambda N\) force has little effect on the value of \(c\), but substantially changes \(a\). Thus, an increase in the range of action of the forces leads mainly only to a broadening of the region in which the absolute value of the \(\Lambda\)-nuclear potential falls off.

4. Estimation of the depth of the \(\Lambda\)-nuclear potential

To estimate the depth of the \(\Lambda\)-nuclear potential in heavy hypernuclei, we shall use the data available in the literature on the binding energies of hypernuclei with \(A=40\)—100. The production of heavy hypernuclei proves possible owing to the following circumstance. When a \(K^-\)-meson interacts with a heavy nucleus of an Ag or Br emulsion, the \(\Lambda\) and \(\Sigma\) hyperons that are formed may become trapped in the residual nucleus (the \(\Sigma\)’s are then converted into \(\Lambda\)’s). There exist various methods for estimating the mass of the residual nucleus for each particular case of \(K^-\) capture. In this, an ambiguity usually arises, since it is difficult to establish whether the capture occurred on an Ag or Br nucleus.

The binding energy is determined using rare cases of \(\pi^-\)-decays. For decays of the type \(\pi^-—p\)—recoil nucleus, \(B_\Lambda\) is found from the upper boundary of the visible-energy spectrum in the decay. In practice this means that, from the set of hypernuclei with close masses, only one hypernucleus is singled out, the one for which \(B_v\) turned out to be the smallest. The found

Thus obtained, the values \(B_\Lambda\) [19] together with the estimated values of \(A\) are given in Table 3. Note that the smaller value of \(A\) corresponds to the interpretation of the hypernucleus as formed on the Br nucleus, and the larger one on the Ag nucleus.

On the basis of the values of \(B_\Lambda\) given in Table 3 and the parameters of the \(\Lambda\)-nuclear potential, the depths \(D\) were calculated using the formulas of Section 2. The values of \(D\) and the quantities \(\sigma^2=\)

\[ = \left|\frac{\partial D}{\partial l}\right|^2 \Delta^2 l+ \left|\frac{\partial D}{\partial A}\right|^2 \Delta A^2+ \left|\frac{\partial D}{\partial B_\Lambda}\right|^2 \Delta B_\Lambda^2 \]

are given in Table 2.

It is seen from Table 2 that, as the range of the \(\Lambda N\)-potential increases, the well depth required to reproduce the observed binding energy increases. This is connected with the smearing of the edge of the potential, noted above, when the radius of the \(\Lambda N\)-forces is increased, which leads to a weakening of the overall \(\Lambda\)-nuclear interaction. Increasing the well depth compensates for this weakening. The main contribution to \(\sigma^2\) is made by the adopted 10% error in the value of \(l\) in formula (43).

To estimate \(D_\Lambda\), the depth of the potential well for \(\Lambda\) in the nucleus, we shall use the method of maximum likelihood. For the \(i\)-th hypernucleus we form the quantity

\[ \varepsilon_i = D_\Lambda - D_i , \tag{44} \]

where \(D_i\) is the value of \(D\) from Table 3. We shall assume that the quantity \(\varepsilon_i\) has a normal distribution with variance \(\sigma_i^2\), equal to the values \(\sigma^2\) from Table 3. Then the likelihood function for the set of events represented in Table 3 is written in the form

\[ F = C \prod_{i=1}^{5} \left\{ p[\exp(-\varepsilon_i^{(1)2}/2\sigma_i^{(1)2})/\sigma_i^{(1)}] + (1-p)[\exp(-\varepsilon_i^{(2)2}/2\sigma_i^{(2)2})/\sigma_i^{(2)}] \right\}. \tag{45} \]

The two terms in (45) arise because of the ambiguity in the interpretation of the events. In (45), \(p\) is the relative probability of interpreting the hypernucleus as having arisen on the Br nucleus, respectively \((1-p)\) on the Ag nucleus. The two interpretations correspond to the superscripts 1 and 2. The normalization constant \(C\) is usually chosen so that at the maximum \(F\) is equal to 1. As the estimate of the depth of the potential well we shall take the value \(D_\Lambda\) at which \(F\) has a maximum.

Fig. 2. Dependence of the estimates of the depth of the potential well for \(\Lambda\) in the nucleus on the mass of the exchanged meson in the \(\Lambda N\) interaction. The numbers by the curves are the values of \(p\).

Figure 2 gives the obtained estimates of \(D_\Lambda\) as functions of the mass \(m\) of the exchanged meson, which determines the range of the Yukawa potential. For \(m \geqslant 600\) MeV the dependence curves reach a plateau, and the estimate of \(D_\Lambda\) becomes practically independent of the range of the \(\Lambda N\) forces and coincides with the estimate at zero range.

Fig. 3. Dependence of the maximum values of the likelihood function on the mass of the exchanged meson. The numbers by the curves are the values of \(p\).

Figure 3 shows the dependence of the maximum values of the function \(F\) (unnormalized) on the meson mass. The maximum values of \(F\) increase with increasing \(m\). The growth practically ceases at \(m \geqslant 600\) MeV. Therefore, for the mass of the exchanged meson we obtain the estimate \(m \geqslant 600\) MeV. The mass \(m \approx 600\) MeV approximately coincides with the average value of twice the masses of the \(K\) and \(\pi\) mesons: \(m_K + m_\pi = 633\) MeV. Let us also note that this value of the mass coincides with the magnitude of the mass usually assigned to the scalar meson. In [2, 3] it was shown, on the basis of the meson model of the \(\Lambda N\) force, that the main contribution to the potential energy of \(\Lambda\) in nuclear matter is given by the exchange of two \(\pi\)’s and the exchange of two \(K\)-mesons. It is possible that the Yukawa potential with \(m \approx m_K + m_\pi\) reproduces the effect of these two exchange processes. The largest values of \(F\) are reached at \(p=0\), i.e., under the assumption that the hypernuclei presented in Table 3 are formed on the heaviest emulsion nucleus—Ag. Therefore it is preferable to estimate \(D_\Lambda\) at \(p=0\).

Figure 4 presents the normalized likelihood function for \(m=m_K+m_\pi=633\) MeV and \(p=0\) (solid curve). The segment inside the curve determines the interval corresponding to one standard deviation. On the basis of this function, for the depth of the potential well we obtain the estimate \(D_\Lambda=30.6 \pm 0.6\) MeV. The same figure also shows the dependence of \(F\) on \(D_\Lambda\) for \(m=633\) MeV and \(p=1/2\) (dashed curve). The assumption of an equally probable role of the Ag and Br nuclei in the data of Table 3 leads to a somewhat different estimate: \(D_\Lambda=31.0 \pm 0.9\) MeV.

The estimate obtained for the depth of the potential well is based on very limited statistics. In addition, the values of \(B_\Lambda\) given in Table 3 should be regarded only as upper bounds on the binding energy. Therefore, with an increase and refinement of the experimental data, the estimate of the well depth may change somewhat in the future. The method considered here, given a larger body of experimental material, is capable of providing information not only on the depth of the potential well for the \(\Lambda\)-hyperon, but also some information on the range of action

Fig. 4. Normalized probability function as a function of \(D_\Lambda\) for \(p=0\)—solid curve—and for \(p=1/2\)—dashed curve. The segment inside the curve determines the interval corresponding to one standard deviation.

\(\Lambda N\) force, the nuclear size, and other quantities on which the binding energy of the \(\Lambda\)-hyperon in the nucleus may depend.

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