Baryon transport in color flux tubes

Color flux tubes are a standard perspective from which to understand stopping in high-energy collisions. Mechanisms for baryon transport and polarization within a tube are considered here, both in regards to the transport of baryons from the target and projectile toward midrapidity and in regards to the correlations of baryon-antibaryon pairs created in the tube. The roles of tube merging and gluon radiation within a tube are emphasized.

Figure 1

Graphical representation of $(p,q)$ multiplets. The quark and antiquark color triplets are represented by (1,0) and (0,1), respectively, while the gluon octet is (1,1). Each dot represents one projection of the multiplet, if the dot is on the outer ring. Subsequently, for the next inner-ring ring a dot represents two states, or three states for the next inner-ring, although the increasing degeneracy stops once a ring has either $p=0$ or $q=0$. The net degeneracy of the multiplet is $(p+1)(q+1)(p+q+2)/2$. In SU(2) one can combine multiplets of $j_1$ and $j_2$ to form several multiplets $j^{\prime }$. Similarly, in SU(3) one can combine multiplets of $(p_1,q_1)$ and $(p_2,q_2)$ to form multiplets of several combinations of $(p^{\prime },q^{\prime })$.

Figure 2

A simple flux tube between a quark $(p=1,q=0)$ and an antiquark $(p=0,q=1)$ is illustrated in the upper panel, with the lighter and darker circles representing the quark and antiquark, respectively. Drawing a vertical line through a flux tube divides space into two regions, the left side of the dashed line in a color multiplet $(p=1,q=0)$ and the right side in the multiplet $(p=0,q=1)$. The lower panel illustrates how quark-antiquark pairs can tunnel out of the vacuum so that the space between the tunneling quarks is free of color fields. Tunneling quark-antiquark pairs are denoted by the arrows. Dividing space in such a way that the dividing line does not bisect a tube results in both sides being in color singlets.

Figure 3

(upper illustration) The merging of two tubes into one, where the left-hand side tubes begin with two quarks, and the right-hand side ends with one quark. The two quark multiplets couple to a (0,1) multiplet, which then matches with the quark multiplet on the right. The baryon number, is then located at the point where the two tubes merged. This represents a situation where one quark is scattered far, from its original momentum, and the flux tubes must adjust to form local color singlets. The original quark pair must couple to $(p=0,q=1)$ if the overall configuration is to be a color singlet. (lower illustration) This represents the case where two baryons exchange a soft gluon. In that case both of the baryons are transformed into $(p=1,q=1)$ multiplets. Assuming that for each baryon there is a diquark with $(p=0,q=1)$, the overall color singlet is restored by migrating the baryon number to the two places where the strings merge. Red (or lighter gray) circles represent quarks, while blue (or darker gray) circles represent antiquarks. Ovals represent flux tubes, and arrows describe quark-antiquark pairs that tunneled from vacuum.

Figure 4

The upper illustration show how a Gaussian surface isolates a flux tube into to parts. The color multiplets have interchanged values of $p$ and $q$. The energy density is defined by $\langle Q^{\left(2\right)}\rangle$, where the charge operators cover the left-hand side.

Figure 5

Color multiplets are shown as a function of $(p,q)$ in the left-side panel. Starting from the singlet ($p=0,q=0$), one can combine gluons to reach any of the multiplets denoted by black circles. These “gluon-like” states, can also be reached by combining quarks and antiquarks if the difference of the number of quarks and antiquarks is a multiple of three. To reach a “quark-like state,” one where $(p-q)/3$ has a remainder of $+1$ or $-2$, one can add a quark to a gluon-like state. One can reach an “antiquark-like state,” one where the remainder of $(p-q)/3$ is $-1$ or 2, by adding an antiquark to a gluon-like state. The heavy black lines show the states available after adding either a gluon (heavy black line), an antiquark (light green line) or a quark (light dotted line). If gluon creation is possible, the most efficient way the system can reach a color singlet is to create gluon pairs, or perhaps one quark or antiquark depending on whether the original state is gluon-like, quark-like or antiquark-like. If gluonic steps are not permitted the most efficient decay mechanism is through $p$ quarks and $q$ antiquarks. The right-side panel represents the same states, but the states are plotted as $\langle Q^{\left(2\right)}\rangle$ vs $\langle Q^{\left(3\right)}\rangle$. In this basis the energy per unit length is a function of $\langle Q^{\left(2\right)}\rangle$ and the field driving the baryon current is $\langle Q^{\left(3\right)}\rangle$.

Figure 6

An illustrative example of how a flux tube, whose initial state is characterized by a multiplet $(p=3,q=0)$ to the left, might decay. In the path (a), for any point along the tube, two gluons pass by. This restores the states to the left and right of that point to color singlets. The first gluon alters the left side (as defined from the point) from $(p=3,q=0)$ to $(p=1,q=1)$. The right side also becomes a color octet. The second gluon brings both sides to a color singlet. No baryon number coalesces in this scenario. For path (b), only quarks pass any given point. This requires some combination of three quarks or antiquarks to pass in order to create color singlets. Either three antiquarks pass the point towards the left, three quarks pass to the right, two quarks to the right and one antiquark to the left, or one quark to right and two antiquarks to the left. For any such change, the final state has an antibaryon on the left and a baryon to the right. Finally, in path (c) an example is shown where in one region, three quarks pass by, while in another, two gluons pass. In this case, the final configuration has a baryon and antibaryon located at the opposite ends of the region where quarks and antiquarks were responsible for the $j^{\left(3\right)}$ current.

Figure 7

A compound flux tube driven at one end by three quarks might create a tube by radiating two gluons. As one travels along the tube from the left until the nearest emitted gluon is encountered, the region might be characterized by $(p=3,q=0)$. If this state hadronizes by pair creation, the baryon number on the left will be absorbed by the polarizing pairs, and in effect, transport the baryon number to the position of the first gluon, as illustrated in the lower part of the illustration. The remaining segment can hadronize by creating quark-antiquark pairs where there is no baryon polarization, e.g., one could have two simple flux tubes with opposite directions of baryon polarization. This illustrates how radiated gluons can lead to flux segments characterized by $p\ne q$, which then result in baryon transport.

Figure 8

$N_q=3B$ quarks are randomly coupled in color space, the average of the quadratic Casimir charge, $\langle Q^{\left(2\right)}\rangle$, and the average of the cubic Casimir charge, $\langle Q^{\left(3\right)}\rangle$, grow linearly with the number of quarks being coupled. However, $\langle p-q\rangle$ rapidly plateaus. Because it is $\langle p-q\rangle$ that is most correlated with the transport of baryons, having many quarks at one end of the tube, with their colors assigned randomly, should not induce large baryon currents. In contrast, because $\langle Q^{\left(2\right)}\rangle$ rises linearly with $B$, overlapping flux tubes from individual baryons will result in similar energy density as having all the baryons contribute individually.

Figure 9

For the random color ansatz, three trajectories in color space are displayed as a function of the scaled rapidity $\tilde{y}$. The trajectories all begin and end as color singlets. The intermediate segments have fluctuating values of $Q_{\mathrm{left}}^{\left(2\right)}$, which corresponds to fluctuations of the energy density, and fluctuating values of $p-q$, which drive fluctuating baryon currents.

Figure 10

Values of $p$ and $q$ were averaged over many trajectories for three cases. For the first two cases, $B_{\mathrm{target}}=B_{\mathrm{proj}}=1$ and $N_g=6$ or 12. The energy density, whose proxy is $Q_{\mathrm{left}}^{\left(2\right)}$, rises at midrapidity, with a stronger $\tilde{y}$ dependence for higher $N_g$. This can be thought of as a random walk in color space over $N_g$ steps, where the walk is constrained to begin and end at $(p=0,q=0)$. The greatest distance from the origin will occur at $\tilde{y}=0$ and will increase with the number of steps, i.e., $N_g$. Thus, the maximum of $Q_{\mathrm{left}}^{\left(2\right)}$ for $N_g=12$ is roughly twice that for $N_g=6$, which can be seen by comparing the left-side and center panels. The right-side panel illustrates the asymmetric case, with four baryons on one end and only one baryon on the opposite end. In that case the energy density is asymmetric.

Figure 11

The baryonic balance function as applied to the random color ansatz, with 16 intermediate gluons and no quarks. For any observed baryonic charge, the balancing charge is more readily found in a nearby site, i.e., small $\mathrm{\Delta }{N}_g$. The balance function, which sums to unity, falls off exponentially. The solid line shows a fit to the tail which behaves $\approx e^{-\alpha \mathrm{\Delta }{N}_g}$, with $\alpha =0.576$. This shows that balancing baryonic charge is localized, and that the rapidity separation is roughly the rapidity separation between emitted gluons divided by $\alpha$.