Scaling functions for the $O(4)$ model in $d=3$ dimensions

A nonperturbative renormalization group approach is used to calculate scaling functions for an $O(4)$ model in $d=3$ dimensions in the presence of an external symmetry-breaking field. These scaling functions are important for the analysis of critical behavior in the $O(4)$ universality class. For example, the finite-temperature phase transition in QCD with two flavors is expected to fall into this class. Critical exponents are calculated in local-potential approximation. Parametrizations of the scaling functions for the order parameter and for the longitudinal susceptibility are given. Relations from universal scaling arguments between these scaling functions are investigated and confirmed. The expected asymptotic behavior of the scaling functions predicted by Griffiths is observed. Corrections to the scaling behavior at large values of the external field are studied qualitatively. These scaling corrections can become large, which might have implications for the scaling analysis of lattice QCD results.

Figure 1

Scaled RG results $y(x)$ for $H=1.0\times {10}^{-4}$, $1.0\times {10}^{-3}$, and $1.0\times {10}^{-2}{\mathrm{MeV}}^{5/2}$. Errors are estimated from the spread of the data points after rescaling and are due only to scaling corrections. Not all data points are displayed. Shown are further fits to the small-$x$ values $x<1$ (solid black line, small-$x$ fit) and to all $x$-values up to approximately $x=5.3\times {10}^4$ (dashed blue line, asymptotic fit).

Figure 2

Scaled RG results $y(x)$ for $H=1.0\times {10}^{-4}$, $1.0\times {10}^{-3}$, and $1.0\times {10}^{-2}{\mathrm{MeV}}^{5/2}$. Errors are estimated from the spread of the data points after rescaling (so they are mainly due to the appearance of scaling corrections). Not all data points are displayed. The fit to all $x$ values up to approximately $x=5.3\times {10}^4$ is also shown (black dashed line, asymptotic fit).

Figure 3

Comparison of the results for the Widom-Griffiths scaling function $y(x)$, where $x=t/M^{1/\beta }$ and $y=h/M^{\delta }$ ($M>0$). The RG result for $H=1.0\times {10}^{-4}{\mathrm{MeV}}^{5/2}$ is shown as a black solid line. The fit to the $O(4)$ lattice data from 34 (red dashed line, lattice data fit) differs from our results for large $x$ values due to the different values for the critical exponents. The result from the $ϵ$ expansion [33] (green dot-dashed line) is valid only for $x<0$ and displays a significantly smaller curvature than both our RG result and the $O(4)$ lattice simulation result. The result for the scaling function from Parisen Toldin et al. [35] with high-accuarcy values for the critical exponents (orange dot-dash-dashed line) agrees with the lattice results. If we substitute our values for the critical exponents in the parametric expression from 35, we find agreement with our RG result (blue dot-dot-dashed line).

Figure 4

The parametrization of the scaling function $y(x)$ in terms of the scaling variables $x$ and $y$ provides an implicit parametrization of the scaling function $f(z)$. Shown is a comparison of $f(z)$ from the interpolated fit Eq. (59) with the rescaled results for $H=1.0\times {10}^{-4}{\mathrm{MeV}}^{5/2}$ (blue circles, for clarity not all points are shown). The fit is perfect on the scale of the plot. The asymptotic behavior of $y(x)$ for $x\to \infty$ determines the behavior of the fit for $z\gtrsim 0.5$. Using the parametrization in terms of only the small-$x$ behavior underestimates $f(z)$ in this region. In this region, the results from the $ϵ$ expansion show the largest deviation.

Figure 5

Plot of the scaling function $f(z)$ as a function of $z$. Shown are results from our calculation for $H=1.0\times {10}^{-4}{\mathrm{MeV}}^{5/2}$ (black solid line) and from the $ϵ$-expansion calculation to $\mathcal{O}({ϵ}^2)$ of Brézin et al. [33] (green dashed line). For comparison, we also plot the expected asymptotic behavior $f(z)\to (-z{)}^{\beta }$ for $\beta =0.390$ ($ϵ$ expansion) (dot-dashed green line) and for $\beta =0.403$ (our result) (dotted black line). The behavior of the scaling functions for large negative values of $z$ is again determined by the different values of the critical exponent $\beta$. The $ϵ$-expansion result is expected to provide a good description only for $t<0$. We find that indeed the deviation from our scaling function is large for $z\gtrsim 0.5$.

Figure 6

Plot of the scaling function $f(z)$ as a function of $z$. Shown are results from our calculation for $H=1.0\times {10}^{-4}{\mathrm{MeV}}^{5/2}$ (black solid line) and from the calculation of Mendes and Engels [34] (using the parametrization as given in 18) (red dashed line). For comparison, we also plot the expected asymptotic behavior $f(z)\to (-z{)}^{\beta }$ for $\beta =0.380$ (Mendes and Engels) (dot-dashed red line) and for $\beta =0.403$ (our result) (dotted black line). As expected, the behavior of the scaling functions for large negative values of $z$ is determined by the different values of the critical exponent $\beta$.

Figure 7

Plot of the scaling function $f(z)$ as a function of $z$. The RG results from our calculation with $H=1.0\times {10}^{-4}{\mathrm{MeV}}^{5/2}$ are represented by the black solid line, and the result from Parisen Toldin et al. from 35 is shown as a dashed orange line. The asymptotic behavior $f(z)\to (-z{)}^{\beta }$ for $\beta =0.388$ (Parisen Toldin et al.) (dot-dashed orange line) and for $\beta =0.403$ (our result) (dotted black line) is shown as well. Plotting the scaling function from 35 with our values for the critical exponents (blue dot-dot-dashed line), we find substantial agreement with our results.

Figure 8

Order parameter as a function of the reduced temperature $t$ for very small values of the external field $H$ from $1.0\times {10}^{-4}$ to $1.0\times {10}^{-3}{\mathrm{MeV}}^{5/2}$ (left panel), and for the rescaled order parameter $M/h^{1/\delta }$ as a function of $z=t/h^{1/(\beta \delta )}$ (right panel). The $t$ ranges for the different values of $H$ are chosen such that $z$ covers the range $-10\dots 10$ after rescaling. On the scale of the plot in the left-hand panel, a deviation from the scaling behavior is not visible in this range of values.

Figure 9

Results for the order parameter as a function of the reduced temperature $t$ for different values of the field $H$ (left-hand column), and for the rescaled order parameter $M/h^{1/\delta }$ as a function of the scaling variable $z=t/h^{1/(\beta \delta )}$ (right-hand column). For comparison, the scaling function obtained from the result for $H=1.0\times {10}^{-4}{\mathrm{MeV}}^{5/2}$ (solid black line) and the asymptotic behavior of the scaling function $(-z{)}^{\beta }$ (dashed black line) are shown with the rescaled results.

Figure 10

Results for the inverse scaled susceptibility $[H_0{M}^{\delta -1}\chi {]}^{-1}$ for $H=1.0\times {10}^{-4}$, $1.0\times {10}^{-3}$, and $1.0\times {10}^{-2}{\mathrm{MeV}}^{5/2}$ (not all points are shown). We estimate an error due to the scaling violations from the spread of the results. Shown are in addition the fits for small $x$ values $x<1$ (solid black line), and from all values up to approximately $x=5.3\times {10}^4$ (dashed blue line, asymptotic fit).

Figure 11

Results for the inverse scaled susceptibility $[H_0{M}^{\delta -1}\chi {]}^{-1}$ for $H=1.0\times {10}^{-4}$, $1.0\times {10}^{-3}$, and $1.0\times {10}^{-2}{\mathrm{MeV}}^{5/2}$ (not all points are shown). The fit to all data points is also shown (asymptotic fit, blue dashed line). From comparison to the equation of state $y(x)$ in Fig. 2 it is apparent that for large $x$ indeed $\delta y(x)-\frac{1}{\beta }xy(x{)}^{\prime }\simeq y(x)$ as expected from Griffiths’ expansion and the scaling laws (see discussion of the asymptotic behavior below).

Figure 12

Comparison of different fits for the universal function $\delta y(x)-\frac{1}{\beta }x{y}^{\prime }(x)$ which describes the inverse rescaled susceptibility. We show the function with parameters obtained directly from a fit to our results for the rescaled susceptibility (blue dot-dashed line), from our results for the order parameter (black solid line), and from the order parameter in the lattice simulation [34] (note that the values for the critical exponents are different). For all practical purposes, there is complete agreement between our curves with parameters from the order parameter and with those from the susceptibility.

Figure 13

The parametrization $y=y(x)$ of the equation of state in terms of the scaling variables $x$ and $y$ also provides implicitly a parametrization of the scaled susceptibility $H_0{h}^{1-1/\delta }\chi =\frac{1}{\delta }[f(z)-\frac{z}{\beta }{f}^{\prime }(z)]$ in terms of the scaling variable $z$. The scaling function (black solid line) with parameters obtained from the order parameter, not the susceptibility itself, is compared to the results for $H=1.0\times {10}^{-4}{\mathrm{MeV}}^{5/2}$ (blue circles, for clarity not all points are shown). Apart from the slight difference at the peak the fit is almost perfect. Within the linewidth, the fit with parameters obtained from the order parameter and the one with parameters obtained from the susceptibility directly (not shown) are indistinguishable in this plot.

Figure 14

Comparison of the parametrization of the scaling function from the lattice results [34] (red dashed line) to our results for $H=1.0\times {10}^{-4}{\mathrm{MeV}}^{5/2}$ (black solid line). The agreement is very good below the critical temperature, but deviations appear above $T_c$, to the right of the susceptibility peak.

Figure 15

Comparison of the parametrization of the scaling function from Parisen Toldin et al. [35] (orange dot-dashed line) to our results for $H=1.0\times {10}^{-4}{\mathrm{MeV}}^{5/2}$ (black solid line). The largest differences appear to the right of the susceptibility peak, as for the lattice comparison. As we did for the order parameter, we replot the scaling function with the parametrization from 35, but with our values for the critical exponents (blue dot-dot-dashed line). Once again, we find that the agreement is quite good, and that the difference can be explained by the different values of the critical exponents.

Figure 16

Susceptibility as a function of the reduced temperature $t$ for very small values of the external field $H=1.0\times {10}^{-4}{\mathrm{MeV}}^{5/2}$ to $H=1.0\times {10}^{-3}{\mathrm{MeV}}^{5/2}$ (left panel), and rescaled susceptibility $H_0{h}^{1-1/\delta }\chi$ as a function of $z=t/h^{1/(\beta \delta )}$ (right panel). The $t$ ranges for the different values of $H$ are chosen such that $z$ covers the range $-10\dots 10$ after rescaling. Within the width of the symbols, the rescaled curves coincide exactly. Deviations from scaling are not visible at this scale.

Figure 17

Results for the longitudinal susceptibility $\chi$ as a function of the reduced temperature $t$ (left panel), and for the scaled susceptibility $H_0{h}^{1-1/\delta }\chi$ as a function of $z=t/h^{1/(\beta \delta )}$ (right panel) for different values of the field $H$. For comparison, the scaling function obtained from the results for $H=1.0\times {10}^{-4}{\mathrm{MeV}}^{5/2}$ is plotted with the scaled results (black dashed line).

Figure 18

Mass of the transversal fluctuations $m_{\pi }$ (pseudo-Goldstone bosons) vs reduced temperature $t$ (left panel), and mass of the longitudinal fluctuations $m_{\sigma }$ vs $t$ (right panel) for different values of the symmetry-breaking field $H$. For our $O(4)$ model with a scale set by the UV cutoff $\Lambda =1.0\mathrm{GeV}$, large scaling violations appear already for values of the external filed $H$ which correspond to $m_{\pi }$ of less than 10 MeV.