Figure 1

The Landau gauge probability distribution for the $d$ quark of the proton from Eqs. (5, 8), in the plane of the $u$ quarks separated by zero lattice units (left), and by 7 lattice units (right). The $d$ quark is seen to prefer to reside near the $u$ quark which is placed in the scalar pair in the interpolating field of Eq. (4).Reuse & Permissions

Figure 2

The probability distribution for the $d$ quark of the proton in the plane of the $u$ quarks separated by 7 lattice units, in the Landau gauge (left), and the Coulomb gauge (right). Two distinct peaks have formed over the location of the $u$ quarks in the Landau gauge probability distribution, whereas a single, broad peak is visible over the center of mass of the system in the Coulomb gauge. Note: as discussed following Eq. (7) the scale is such that the largest value of all of the fixed quark separations will sit at the top of the grid, with all other points of the probability distribution scaled accordingly.Reuse & Permissions

Figure 3

The probability distribution of the $d$ quark in the proton with the $u$ quarks 7 lattice units apart along the $x$ axis at $x=4$ and 11. To clearly display the double peak structure, uncertainties are reported relative to the distribution at $x=6$.Reuse & Permissions

Figure 4

The probability distribution for the scalar $u$ quark of the proton in the plane of the $u$ and $d$ quarks separated by 7 lattice units, in the Landau gauge (left), and the Coulomb gauge (right). In both gauges, the $u$ quark is seen to prefer to be nearer the $d$ quark. However, in the Coulomb gauge, the scalar $u$ quark is closer to the center of the lattice than in the Landau gauge probability distribution. The scale is as described in Fig. 2.Reuse & Permissions

Figure 5

The probability distribution for the vector $u$ quark of the proton in the plane of the $u$ and $d$ quarks separated by 7 lattice units, in the Landau gauge (left), and the Coulomb gauge (right). The probability distribution is similar to the $d$ quark probability distribution in that strong clustering is seen in the Landau gauge. The Coulomb gauge results here reveal a small amount of preferred clustering with the $d$ quark. Also of note is that these probability distributions show less structure than the others, as can be seen by the height of the smallest values, with the scale as described in Fig. 2.Reuse & Permissions

Figure 6

The probability distribution for an antisymmetrized $u$ quark of the proton in the plane of the remaining quarks which are separated by 7 lattice units, in the Landau gauge (left), and the Coulomb gauge (right). In contrast to the $d$ quark probability distribution, a single peak is visible above the location of the $d$ quark in both the Coulomb and the Landau gauge. Note: as discussed following Eq. (7) the scale is such that the largest value of all of the fixed quark separations will sit at the top of the grid, with all other points of the probability distribution scaled accordingly.Reuse & Permissions

Figure 7

The probability distribution for the $d$ quark cut in the $x-y$ plane of the $u$ quarks, in the presence of a background magnetic field in the Landau gauge, with the first implementation (left), and the second implementation (right) of the vector potential described in Sec. 5. In this image, the field, $\overrightarrow{B}$, is pointing into the page. The red sphere denotes the location of the remaining quarks. There is a clear asymmetry perpendicular to the field that changes with the vector potential, $A_{\mu }$, in spite of the background magnetic field not changing.Reuse & Permissions

Figure 8

The probability distribution for the $d$ quark cut in the $x-z$ plane of the $u$ quarks, after symmetrizing the vector potential, $A_{\mu }$ in the presence of the field in the Landau gauge (left) and Coulomb gauge (right). In this image, the field, $\overrightarrow{B}$, is pointing to the top of the page, and the $u$ quarks are both in the center of the lattice, denoted by the red sphere. In spite of the magnitude of the field, a fairly small deviation from spherical symmetry is seen in both gauges.Reuse & Permissions

Figure 9

The probability distribution for the vector $u$ quark in the presence of the background field, cut in the $x-z$ plane of the remaining quarks in the Coulomb gauge with the spin aligned (left) and antialigned (right) to the field. The direction of the field is down the page, and the red sphere denotes the remaining quarks. The probability distribution appears more spherical and localized when aligned with the field, and a very subtle asymmetry is present in the direction of the field. The smallest value shown for both probability distributions is 10% of the peak value.Reuse & Permissions

Figure 10

The probability distribution of the vector $u$ quark in the presence of the background field, cut in the $x-z$ plane of the remaining quarks in the Landau gauge with the spin aligned (left) and antialigned (right) to the field, and the red sphere denotes the remaining quarks. The direction of the field is down the page. Much like in the Coulomb gauge, the probability distribution appears more spherical and localized when aligned with the field. The smallest value shown for both probability distributions is 10% of the peak value.Reuse & Permissions

Figure 11

The probability distribution of the $d$ quark in the Coulomb gauge cut in the $x-z$ plane of the $u$ quarks which are separated by 7 lattice units in the transverse direction with zero background field (left) and in the presence of the field (right). The direction of the field is up the page and the spheres denote the positions of the $u$ quarks. The smallest value shown for both probability distributions is 20% of the peak value.Reuse & Permissions

Figure 12

The probability distribution of the $d$ quark, in the Landau gauge cut in the $x-z$ plane of the remaining quarks which are separated by 7 lattice units in the transverse direction with zero background field (left) and in the presence of the field (right). The spheres denote the positions of the $u$ quarks. The diquark clustering is barely visible in this view, and disappears completely in the presence of the field. The probability distributions are broader in the Landau gauge and the smallest value shown for both probability distributions is 20% of the peak value.Reuse & Permissions

Figure 13

The probability distribution of the scalar $u$ quark in the Coulomb gauge cut in the $x-z$ plane of the remaining quarks which are separated by 7 lattice units in the transverse direction with zero background field (left) and in the presence of the field (right). The direction of the field is up the page and the $d$ quark is on the right, denoted by the red sphere. In contrast to the $d$ quark probability distribution, there is still a distinct preference for the formation of a scalar diquark. When the field is applied, the probability distribution can be seen to move toward the center of the lattice. The smallest value shown for both probability distributions is 20% of the peak value.Reuse & Permissions

Figure 14

The probability distribution of the scalar $u$ quark, in the Landau gauge which are separated by 7 lattice units in the transverse direction with zero background field (left) and in the presence of the field (right). The direction of the field is up the page and the $d$ quark is on the right, denoted by the red sphere. Preference towards the center of the lattice is also visible in the Landau gauge, but is more subtle than in the Coulomb gauge. The probability distributions are broader in the Landau gauge and the smallest value shown for both probability distributions is 20% of the peak value.Reuse & Permissions

Figure 15

The probability distribution of a $u$ quark in the Coulomb gauge cut in the $x-z$ plane of the remaining quarks which are separated by 7 lattice units in the transverse direction with zero background field (left) and in the presence of the field (right). The direction of the field is up the page and the $d$ quark is on the right, denoted by the red sphere. The symmetrized $u$ quark probability distribution bears close resemblance to the scalar $u$ quark, but is less localized due to the vector $u$ quark contribution. The smallest value shown for both probability distributions is 20% of the peak value.Reuse & Permissions

Figure 16

The probability distribution of a $u$ quark, in the Landau gauge cut in the $x-z$ plane of the remaining quarks which are separated by 7 lattice units in the transverse direction with zero background field (left) and in the presence of the field (right). The direction of the field is up the page and the $d$ quark is on the right, denoted by the red sphere. The contribution to the symmetrized probability distribution from the vector $u$ quark is enhanced in the Landau gauge compared to the Coulomb gauge. The smallest value shown for both probability distributions is 20% of the peak value.Reuse & Permissions

Figure 17

The probability distribution of the vector $u$ quark in the Coulomb gauge cut in the $x-z$ plane of the remaining quarks which are separated by 7 lattice units in the transverse direction with zero background field (left) and in the presence of the field (right). The direction of the field is up the page and the $d$ quark is on the right, denoted by the red sphere. The effect of the field on the vector $u$ quark probability distribution is more pronounced than the $d$ quark and scalar $u$ quark probability distributions. The smallest value shown for both probability distributions is 20% of the peak value.Reuse & Permissions

Figure 18

The probability distribution of the vector $u$ quark, in the Landau gauge cut in the $x-z$ plane of the remaining quarks which are separated by 7 lattice units in the transverse direction with zero background field (left) and in the presence of the field (right). The direction of the field is up the page and the $d$ quark is on the right, denoted by the red sphere. The smallest value shown for both probability distributions is 20% of the peak value.Reuse & Permissions