A Nuclear Model with Explicit Mesons

Abstract

A nuclear model is proposed where the nucleons interact by emitting and absorbing mesons, and where the mesons are treated explicitly. A nucleus in this model finds itself in a quantum superposition of states with different number of mesons. Transitions between these states hold the nucleus together. The model—in its simplest incarnation—is applied to the deuteron, where the latter becomes a superposition of a neutron-proton state and a neutron-proton-meson state. Coupling between these states leads to an effective attraction between the nucleons and results in a bound state with negative energy, the deuteron. The model is able to reproduce the accepted values for the binding energy and the charge radius of the deuteron. The model, should it work in practice, has several potential advantages over the existing non-relativistic few-body nuclear models: the reduced number of model parameters, natural inclusion of few-body forces, and natural inclusion of mesonic physics.

Appendix

1.1 Stochastic Sampling of Gaussian Parameters

The Gaussians can be parameterized in the form

$$\begin{aligned} \langle \mathbf {r}\bigm | A\rangle = \exp \left( -\sum _{i<j=1}^N\left( \frac{\mathbf {r}_i-\mathbf {r}_j}{b_{ij}}\right) ^2 \right) \equiv \exp \left( -\mathbf {r}^\mathrm {T} A \mathbf {r} \right) , \end{aligned}$$

(27)

where $\mathbf {r}_i$ is the coordinate of the i-th particle and the matrix A is given as

$$\begin{aligned} A= \sum _{i<j=1}^n \frac{w_{ij}w_{ij}^\mathrm {T}}{b_{ij}^2}, \end{aligned}$$

(28)

where the column-vectors $w_{ij}$ are defined through the equation

$$\begin{aligned} \mathbf {r}_i-\mathbf {r}_j=w_{ij}^\mathrm {T}\mathbf {r}. \end{aligned}$$

(29)

In the laboratory frame $\mathbf {r}=\begin{pmatrix}\mathbf {r}_1&\mathbf {r}_2&\dots&\mathbf {r}_N\end{pmatrix}^\mathrm {T}$ and the $w_{ij}$ are given for a two-body system as

$$\begin{aligned} w_{12}= \begin{pmatrix} 1 \\ -1 \end{pmatrix} \;, \end{aligned}$$

(30)

and for a three-body system as

$$\begin{aligned} w_{12}= \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix},\;\; w_{13}= \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix},\;\; w_{23}= \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}. \end{aligned}$$

(31)

Under a coordinate transformation $\mathbf {r}\rightarrow J\mathbf {r}$ the column-vectors $w_{ij}$ transform as $w_{ij}\rightarrow U^\mathrm {T}w_{ij}$ where $U=J^{-1}$.

The range parameters $b_{ij}$ of the Gaussians are chosen stochastically from the exponential distribution,

$$\begin{aligned} b_{ij}=-\ln (u)b, \end{aligned}$$

(32)

where the quasi-random number $u\in ]0,1[$ is taken from a Van der Corput sequence [17] with the scale $b=3$ fm. Separate sequences with different prime bases are used for each ij-combination.

1.2 Charge Radius

We define the charge radius, $R_c$, of an N-body system as

$$\begin{aligned} R_c^2 = \sum _{i=1}^{N} Z_i\langle \mathbf {r}_i^2\rangle = \sum _{i=1}^{N} Z_i\langle \mathbf {r}^\mathrm {T}w_iw_i^\mathrm {T}\mathbf {r}\rangle , \end{aligned}$$

(33)

where the summation goes over the bodies in the system; $Z_i$ is the charge of the body in unit charges; $\mathbf {r}_i$ is the coordinate of the body (in the center-of-mass frame); the brackets $\langle \rangle$ signify the expectation value in the given state of the system; and the column-vector $w_i$ is defined via the formula

$$\begin{aligned} \mathbf {r}_i=w_i^{\mathrm T}\mathbf {r}. \end{aligned}$$

(34)

In the laboratory frame, for a two-body system

$$\begin{aligned} w_1=\begin{pmatrix} 1 \\ 0 \end{pmatrix}\;,\;\; w_2=\begin{pmatrix} 0 \\ 1 \end{pmatrix}\;,\;\; \end{aligned}$$

(35)

and for a three-body system

$$\begin{aligned} w_1= \left( \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right) , w_2= \left( \begin{array}{c} 0 \\ 1 \\ 0 \end{array} \right) , w_3= \left( \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right) . \end{aligned}$$

(36)

Under a coordinate transformation $\mathbf {r}\rightarrow J\mathbf {r}$ the column-vector $w_i$ transforms with the inverse matrix $U=J^{-1}$ as $w_i\rightarrow U^{\mathrm T}w_i$ (see Sect. 6.3 for a transformation to the center-of-mass frame).

Now the matrix element in (33) between two Gaussians is given as [13],

$$\begin{aligned} \left\langle A\bigm | \mathbf {r}^\mathrm T w_iw_i^\mathrm T \mathbf {r} \bigm | A'\right\rangle = \frac{3}{2} w_i^\mathrm T(A+A')^{-1}w_i \left\langle A\bigm |A'\right\rangle . \end{aligned}$$

(37)

1.3 Coordinate Transformations

Under a linear coordinate transformation to a new set of coordinates,

$$\begin{aligned} \mathbf {r}\rightarrow \mathbf {x}=J\mathbf {r}, \end{aligned}$$

(38)

the matrix elements with correlated Gaussians preserve their mathematical form as long as the determinant of the transformation matrix J equals one (otherwise they have to be divided by the determinant), and as long as one makes the corresponding transformations of the related matrices and column-vectors: the kinetic energy matrix transforms as¹

$$\begin{aligned} K\rightarrow JKJ^\mathrm {T}, \end{aligned}$$

(42)

and the $w_i$ (and $w_{ij}$) column-vectors transform as²

$$\begin{aligned} w_i\rightarrow U^\mathrm {T}w_i, \end{aligned}$$

(39)

where $U=J^{-1}$.

One practical set of coordinates are the Jacobi coordinates defined as

$$\begin{aligned} \mathbf {x}_{i<N} = \frac{\sum _{k=1}^i m_k\mathbf {r}_k}{\sum _{k=1}^i m_k}-\mathbf {r}_{i+1}\;,\; \mathbf {x}_N = \frac{\sum _{k=1}^N m_k\mathbf {r}_k}{\sum _{k=1}^N m_k}\;, \end{aligned}$$

(41)

where the last coordinate, $\mathbf {x}_N$, is the center-of-mass coordinate that can be omitted if no external forces are acting on the system. This is equivalent to simply discarding the last row of the J matrix and the last column of the U matrix.