Isospin breaking from diquark clustering

Background: Although SU(2) isospin symmetry is generally assumed in the basic theory of the strong interaction, a number of significant violations have been observed in scattering and bound states of nucleons. Many of these violations can be attributed to the electromagnetic interaction but the question of how much of the violation is due to it remains open.

Purpose: To establish the connection between diquark clustering in the two-nucleon system and isospin breaking from the Coulomb interaction between the members of diquark pairs.

Method: A schematic model based on clustering of quarks in the interior of the confinement region of the two-nucleon system is introduced and evaluated. In this model the Coulomb interaction is the source of all isospin breaking. It draws on a picture of the quark density based on the diquark-quark model of hadron structure which has been investigated by a number of groups.

Results: The model produces three isospin breaking potentials connecting the unbroken value of the low-energy scattering amplitude to those of the $pp$, $nn$, and $np$ singlet channels. A simple test of the potentials in the three-nucleon energy difference problem yields results in agreement with the known binding energy difference.

Conclusion: The illustrative model suggests that the breaking seen in the low-energy nucleon-nucleon (NN) interaction may be understood in terms of the Coulomb force between members of diquark clusters. It allows the prediction of the charge symmetry breaking interaction and the $nn$ scattering length from the well measured $np$ singlet scattering length. Values of the $nn$ scattering length around $-18$ fm are favored. Since the model is based on the quark picture, it can be easily extended, in the SU(3) limit, to calculate isospin breaking in the strange sector in the corresponding channels. A natural consequence of isospin breaking from diquark clustering is that the breaking in the strange sector, as measured by the separation energy difference between ${}_{\mathrm{\Lambda }}{}^4\mathrm{H}$ and ${}_{\mathrm{\Lambda }}{}^4\mathrm{He}$, is several times larger than that seen in the comparison of three-nucleon mirror nuclei as observed experimentally.

Figure 1

The dots represent the ratio of the calculation of Blunden and Iqbal [79] based on the $\rho \text{−}\omega$ and $\pi \text{−}\eta$ charge symmetry breaking potentials of [80] to the CSB energies of Sato [81] as a function of the angular momentum of the shell. The solid dots correspond to the SKII results [82], the open circles to DME calculations [83].

Figure 2

Variation with parameters of the model for the full $pp$ Coulomb potential.

Figure 3

Variation with parameters of the model for the $nn$ crypto-Coulomb potential.

Figure 4

Variation with parameters of the model for the $np$ crypto-Coulomb potential.

Figure 5

Typical phase shift fit with the Reid-like potential (curve) (including the CC potential) to the values from Ref. [89] (solid dots). This fit corresponds to the solid line in Fig. 2. The difference between the dash-dot and solid curves reflects the importance of the electric correction. The fitted model has $p=3{\mathrm{fm}}^3$ and $\beta =4{\mathrm{fm}}^{-1}$.

Figure 6

Scattering length vs the strength parameter $p$ for $\beta =8$ ${\mathrm{fm}}^{-1}$. The notation $\mathrm{\Delta }m=0$ means that the difference in pion masses is not taken into account; the neutral pion mass is used in all one-pion exchange calculations. The notation $\mathrm{\Delta }m$ means that the charged and neutral pion masses are used as in Eq. (11).

Figure 7

Scattering length vs the strength parameter $p$ for $\beta =12$ ${\mathrm{fm}}^{-1}$. See Fig. 6 for notation.

Figure 8

Scattering length vs the strength parameter $p$ for $\beta =4{\mathrm{fm}}^{-1}$. See Fig. 6 for notation.

Figure 9

Neutron-neutron and $pp$ potentials resulting from the calculations described in the text. The $pp$ case includes the normal Coulomb interaction as well as the hidden part. The dotted line shows the potential from a uniformly charged sphere of radius 1.56 fm, $V_C\left(r\right)$. In order to find the CC part the differences in these curves is taken.

Figure 10

Crypto-Coulomb potentials with the potential from a uniformly charged sphere of radius 1.56 fm subtracted from the $pp$ interior potential.

Figure 11

CC charge symmetry breaking potential calculated by subtracting the uniformly charged potential and the $nn$ interior Coulomb potential from the $pp$ interior potential (solid line). The dotted line shows the shape of a one-pion-exchange potential with arbitrary normalization. The dash-dot line shows the first term of the $\rho -\omega$ CSB potential [Eq. (8) of Ref. [80] with their parameter $\beta =0]$.

Figure 12

Three examples of $\mathrm{\Lambda }N$ breaking potentials from Eq. (25).