Multiparticle potentials from lightlike Wilson lines in quark-gluon plasmas: A generalized relation of in-medium splitting rates to jet-quenching parameters $\widehat{q}$

A powerful historical insight about the theory of in-medium showering in QCD backgrounds was that splitting rates can be related to a parameter $\widehat{q}$ that characterizes the rate of transverse-momentum kicks to a high-energy particle from the medium. Another powerful insight was that $\widehat{q}$ can be defined (with caveats) even when the medium is strongly coupled, using long, narrow Wilson loops whose two long edges are lightlike Wilson lines. The medium effects for the original calculations of in-medium splitting rates can be formulated in terms of three-body imaginary-valued “potentials” that are defined with three long, lightlike Wilson lines. Corrections due to the overlap of two consecutive splittings can be calculated using similarly defined four-body potentials. I give a simple argument for how such $N$-body potentials can be determined in the appropriate limit just from the knowledge of the values of $\widehat{q}$ for different color representations. For $N>3$, the $N$-body potentials have a nontrivial color structure, which will complicate calculations of overlap corrections outside of the large-$N_{\mathrm{c}}$ or soft bremsstrahlung limits.

Figure 1

(a) A Wilson loop with lightlike edges used to formally define $\widehat{q}$ (subject to caveats mentioned in Sec. 4). $t$ is real (Minkowski) time. In contrast, (b) shows a Wilson loop for static color charges, where $t$ can be real or imaginary (Euclidean) time.

Figure 2

A contribution to the rate for single splitting of a high-energy particle in the medium. Only high-energy particles are shown explicitly; all lines are implicitly interacting with the medium, which is then averaged over. In these diagrams, time runs from left to right, and the amplitude and conjugate amplitude are each implicitly integrated over the time of emission ($t$ and $\overline{t}$, respectively) to get the splitting probability. On the right-hand side is a combined diagram showing the amplitude (blue lines) sewn together with the conjugate amplitude (red lines).

Figure 3

As Fig. 1 but for a three-body potential. Note that the axes are depicted differently than in Fig. 1: in order to be able to show both transverse spatial directions, the $t$ and $z$ axes have been collapsed to $x^{+}=z+t$, with $x^{-}=0$ everywhere. The three-point vertices on the ends are chosen to form the color-neutral combination of the three particles. For example, for the three-gluon potential, the Wilson lines would be adjoint representation and the three-point vertices would each be proportional to the Lie algebra structure constants $f^{abc}$. Note: The constant transverse positions $({\mathit{b}}_1,{\mathit{b}}_2,{\mathit{b}}_3)$ of the three lightlike Wilson lines can be anything; they need not be symmetrically arranged as in this picture.

Figure 4

Similar to the right-hand side of Fig. 2 but for double splitting.

Figure 5

The Wilson loop in the weak-coupling limit. Here, the gluon lines represent two-point correlators of the gauge field in the background of the medium. The particular graph on the left-hand side is just an example. The important point is that, in the weak-coupling limit, localized nonoverlapping two-point correlations dominate and exponentiate as shown on the right-hand side. Each two-point correlation (depicted by a gluon line) is implicitly resummed with in-medium self-energy insertions.

Figure 6

As Fig. 5 but for an $N$-body potential. The blue rectangles indicate some contraction of the lightlike Wilson lines to form a gauge-invariant quantity (and so an overall color-neutral state of the $N$ high-energy particles represented by the Wilson lines).

Figure 7

An example of different ways to contract color for initial and final states in the case of an $N$-body potential with $N>3$. The particular example above is for $N=4$ lightlike adjoint Wilson lines. Here, blue denotes adjoint representation. The two red line segments, however, may each be independently chosen to be in any irreducible representation $8\otimes 8=1\oplus 8\oplus 8\oplus 10\oplus \overline{10}\oplus 27$. Because of the hierarchy of scales discussed in the text, the details of the lengths of the red lines, or the transverse positions of their end points, are not relevant to defining the potential $\underset{\_}{V}$ in the limit of small transverse separations because of the hierarchy of scales discussed in the text.

Figure 8

A contribution where two lightlike Wilson lines are connected by a high-energy ($\omega \gg T$), nearly collinear gluon line. Though not drawn explicitly above, the high-energy Wilson lines and gluon are all interacting repeatedly with the background fields of the plasma.