\,\,z=\frac{t}{h^{1/\beta\delta}}.
\label{chiralsuscept}$$

The singular function $f_G$ is well studied in spin models and has been parametrized for $O(2)$ and $O(4)$ groups. For the regular part we consider leading-order (linear) dependence in $H$ and quadratic in $T$: $$f_{M,reg}(T,H) = \left( a_0 + a_1 \frac{T-T_c^0}{T_c^0} + 
a_2 \left(\frac{T-T_c^0}{T_c^0} \right)^2 \right) H.
\label{eq:freg}$$ Then we are left with 6 parameters to be determined from fitting the data, $T_c^0$, $t_0$, $h_0$, $a_0$, $a_1$ and $a_2$.

We perform simultaneous fits to $M_b$ and $\chi_{m,l}$ for the asqtad action on $N_\tau=8$, 12 and the HISQ/tree action on $N_\tau=6$, 8 and 12. An example of such a fit for HISQ/tree, $N_\tau=8$ is shown in Fig. \[pbp\_and\_chi\].

![ An example of a simultaneous fit to the chiral condensate (left) and susceptibility (right) for HISQ/tree on $N_\tau=8$ lattices. Open symbols indicate the range included in the fit. Dotted black line is an extrapolation to the physical light quark mass. []{data-label="pbp_and_chi"}](hisq_pbpO4_Nt8.eps "fig:"){width="48.00000%"} ![ An example of a simultaneous fit to the chiral condensate (left) and susceptibility (right) for HISQ/tree on $N_\tau=8$ lattices. Open symbols indicate the range included in the fit. Dotted black line is an extrapolation to the physical light quark mass. []{data-label="pbp_and_chi"}](hisq_chiO4_Nt8.eps "fig:"){width="48.00000%"}

Then, performing a combined $1/N_{\tau}^2$ extrapolation of $T_c$ values obtained with the asqtad and HISQ/tree action as shown in Fig. \[tc\_comb\] we obtain $$T_c=( 154 \pm 8 \pm 1)\mbox{ MeV},$$ where the first error is from the fit and the second is the overall error on the lattice scale determination. The fits for asqtad and HISQ/tree are constrained to have a common intercept. (See Ref. [@hotqcd2] for more details on the fitting procedure and analysis of systematic errors.) To present a combined error we add the two errors giving our final value $T_c=154\pm9$ MeV.

![image](Tc_phys_O4_extrapQQ.eps){width