Scott Bogner, National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
Morten Hjorth-Jensen, National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
Date: 2017
Here we display a typical way to parametrize (non-relativistic expression) the nuclear two-body force in terms of some operators, the central part, the spin-spin part and the central force.
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How do we derive such terms? (Note: no isospin dependence and that the above is an approximation)
To derive the above famous form of the nuclear force using field theoretical concepts, we will need some elements from relativistic quantum mechanics. These derivations will be given below. The material here gives some background to this. I know that many of you have not taken a course in quantum field theory. I hope however that you can see the basic ideas leading to the famous non-relativistic expressions for the nuclear force.
Furthermore, when we analyze nuclear data, we will actually try to explain properties like spectra, single-particle energies etc in terms of the various terms of the nuclear force. Moreover, many of you will hear about these terms at various talks, workshops, seminars etc. Then, it is good to have an idea of what people actually mean!!
| Baryons | Mass (MeV) | Mesons | Mass (MeV) |
|---|---|---|---|
| $p,n$ | 938.926 | $\pi$ | 138.03 |
| $\Lambda$ | 1116.0 | $\eta$ | 548.8 |
| $\Sigma$ | 1197.3 | $\sigma$ | $\approx 550.0$ |
| $\Delta$ | 1232.0 | $\rho$ | 770 |
| $\omega$ | 782.6 | ||
| $\delta$ | 983.0 | ||
| $K$ | 495.8 | ||
| $K^{\star}$ | 895.0 |
But before we proceed, we will look into specific quantum numbers of the relative system and study expectation vaues of the various terms of
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When solving the scattering equation or solving the two-nucleon problem, it is convenient to rewrite the Schroedinger equation, due to the spherical symmetry of the Hamiltonian, in relative and center-of-mass coordinates. This will also define the quantum numbers of the relative and center-of-mass system and will aid us later in solving the so-called Lippman-Schwinger equation for the scattering problem.
We define the center-of-mass (CoM) momentum as
with $\hbar=c=1$ the wave number $k_i=p_i$, with $p_i$ the pertinent momentum of a single-particle state. We have also the relative momentum
and the relative distance
and
we can rewrite the two-particle kinetic energy (note that we use $\hbar=c=1$ as
where $m_n$ is the average of the proton and the neutron masses.
Since the two-nucleon interaction depends only on the relative distance, this means that we can separate Schroedinger's equation in an equation for the center-of-mass motion and one for the relative motion.
With an equation for the relative motion only and a separate one for the center-of-mass motion we need to redefine the two-body quantum numbers.
Previously we had a two-body state vector defined as $|(j_1j_2)JM_J\rangle$ in a coupled basis. We will now define the quantum numbers for the relative motion. Here we need to define new orbital momenta (since these are the quantum numbers which change). We define
where $\hat{l}$ is the orbital momentum associated with the relative motion and $\hat{L}$ the corresponding one linked with the CoM. The total spin $S$ is unchanged since it acts in a different space. We have thus that
which allows us to define the angular momentum of the relative motion
where ${ \cal J}$ is the total angular momentum of the relative motion.
To describe the interaction between the various baryons and mesons of the previous table we choose the following phenomenological lagrangians for spin $1/2$ baryons
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and
for pseudoscalar (ps), scalar (s) and vector (v) coupling, respectively. The factors $g^{v}$ and $g^{t}$ are the vector and tensor coupling constants, respectively.
For spin $1/2$ baryons, the fields $\Psi$ are expanded in terms of the Dirac spinors (positive energy solution shown here with $\overline{u}u=1$)
with $\chi$ the familiar Pauli spinor and $E(k) =\sqrt{m^2 +|\mathbf{k}|^2}$. The positive energy part of the field $\Psi$ reads
with $a$ being a fermion annihilation operator.
Expanding the free Dirac spinors in terms of $1/m$ ($m$ is here the mass of the relevant baryon) results, to lowest order, in the familiar non-relativistic expressions for baryon-baryon potentials. The configuration space version of the interaction can be approximated as
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where $m_{\alpha}$ is the mass of the relevant meson and $S_{12}$ is the familiar tensor term.
We derive now the non-relativistic one-pion exchange interaction.
Here $p_{1}$, $p_{1}'$, $p_{2}$, $p_{2}'$ and $k=p_{1}-p_{1}'$ denote
four-momenta.
The vertices are
given by the pseudovector Lagrangian
From the Feynman diagram rules we can write the two-body interaction as
and
By inserting expressions for the Dirac spinors, we find
In the CM system we have $\mathbf{p}_{2}=-\mathbf{p}_{1}$, $\mathbf{p'}_{2}=
-\mathbf{p'}_{1}$ and so $E_{2}=E_{1}$, $E_{2}'=E_{1}'$.
We can then write down the relativistic contribution
to the NN potential in the CM system:
and then $(p_{1}-p_{1}')^{2}=-\mathbf{k}^{2}$, so we get for the contribution to the NN potential
In coordinate space we have
resulting in
Carrying out the differentation of
we arrive at the famous one-pion exchange potential with central and tensor parts
For the full potential add the exchange part and the $\mathbf{\tau}_1\cdot\mathbf{\tau}_2$ term as well. (Subtle point: there is a divergence which gets cancelled by using cutoffs) This leads to coefficients $C_{\sigma}$ and $C_T$ which are fitted to data.
When we perform similar non-relativistic expansions for scalar and vector mesons we obtain for the $\sigma$ meson
We note an attractive central force and spin-orbit force. This term has an intermediate range. We have defined $1/2(p_{1}+p_{1}')=\mathbf{q}$. For the full potential add the exchange part and the isospin dependence as well.
We obtain for the $\omega$ meson
We note a tensor force with sign opposite to that of the pion. This term has short range. For the full potential add the exchange part and the isospin dependence as well.