  We measure the local correlation between radio emission and Compton-$y$
signal across two galaxy clusters, Abell~399 and Abell~401, using maps from the
Low-Frequency Array (LOFAR) and the Atacama Cosmology Telescope (ACT) +
\Planck. These datasets allow us to make the first measurement of this kind at
$\sim$arcminute resolution. We find that the radio brightness scales as
$F_{\mathrm{radio}} \propto y^{1.5}$ for Abell~401 and $F_{\mathrm{radio}}
\propto y^{2.8}$ for Abell~399. Furthermore, using \XMM data, we derive a
sublinear correlation between radio and X-ray brightness for both the clusters
($F_{\mathrm{radio}} \propto F_{\rm X}^{0.7}$). Finally, we correlate the
Compton-$y$ and X-ray data, finding that an isothermal model is consistent with
the cluster profiles, $y \propto F_{\rm X}^{0.5}$. By adopting an
isothermal--$\beta$ model, we are able, for the first time, to jointly use
radio, X-ray, and Compton-$y$ data to estimate the scaling index for the
magnetic field profile, $B(r) \propto n_{\mathrm{e}}(r)^{\eta}$ in the
injection and re-acceleration scenarios. Applying this model, we find that the
combined radio and Compton-$y$ signal exhibits a significantly tighter
correlation with the X-ray across the clusters than when the datasets are
independently correlated. We find $\eta \sim 0.6{-}0.8$. These results are
consistent with the upper limit we derive for the scaling index of the magnetic
field using rotation measure values for two radio galaxies in Abell~401. We
also measure the radio, Compton-$y$, and X-ray correlations in the filament
between the clusters but conclude that deeper data are required for a
convincing determination of the correlations in the filament.
