Renormalization of the Polyakov loop with gradient flow

We use the gradient flow for the renormalization of the Polyakov loop in various representations. Using $2+1$ flavor QCD with highly improved staggered quarks and lattices with temporal extents of $N_{\tau }=6$, 8, 10 and 12 we calculate the renormalized Polyakov loop in many representations including fundamental, sextet, adjoint, decuplet, 15-plet, 24-plet and 27-plet. This approach allows for the calculations of the renormalized Polyakov loops over a large temperature range from $T=116\mathrm{MeV}$ up to $T=815\mathrm{MeV}$, with small errors not only for the Polyakov loop in fundamental representation, but also for the Polyakov loops in higher representations. We compare our results with standard renormalization schemes and discuss the Casimir scaling of the Polyakov loops.

Figure 1

The unrenormalized fundamental Polyakov loop (left) and the renormalized Polyakov loop corresponding to flow time $f=f_0$ (right) as a function of the temperature $T$ for all lattice ensembles.

Figure 2

The free energy $F_3$ obtained from the gradient flow compared to the continuum results for $F_3$ in conventional renormalization scheme from Ref. [23].

Figure 3

The free energy of the static charge in different representations as function of the temperature for $N_{\tau }=6$.

Figure 4

Polyakov loops in various representations scaled by the ratio of the appropriate Casimir operators (see text) for $N_{\tau }=8$ (left), $N_{\tau }=10$ (middle) and $N_{\tau }=12$ (right). The Polyakov loops have been rescaled by $\exp (-{\mathrm{\Delta }}_3/T)$, ${\mathrm{\Delta }}_3=100\mathrm{MeV}$, to match to the conventional scheme for the fundamental Polyakov loop.

Figure 5

The ratios ${\delta }_n$ characterizing the breaking of Casimir scaling for $N_{\tau }=6$ (left) and various representations $n$, and for the octet representation and various $N_{\tau }$ (right).

Figure 6

We compare the renormalized fundamental Polyakov loop free energy $F_3$ obtained with Wilson and Symanzik flow as described in Sec. 3. In addition we show the continuum extrapolated, renormalized free energy from [23] (black triangles). While for lower temperatures the cutoff effects are small, at higher temperatures the Wilson flow shows larger deviations from the conventional result than the Symanzik flow.

Figure 7

The difference $F_3(f)-F_3(f^{\prime })$ for different flow time (upper panels) and the comparison of $F_3(f)$ with the continuum results in the conventional scheme (lower panels). The results in the low and high temperature regions are shown separately in the left and right panels, respectively.

Figure 8

The free energy in the fundamental (triplet) and adjoint (octet) representations calculated for two different quark masses $m_l/m_s=1/20$ and $m_l/m_s=1/40$. The value $3f_0$ is used for the flow time. The upper panels show the fundamental free energy and lower panels show the adjoint free energy.

Figure 9

The measure of the Casimir scaling ${\delta }_n$ shown for $N_{\tau }=6$, 8, 10 and 12 (from top to bottom) and flow times $f=f_0$, $2f_0$ and $3f_0$ (from left to right).