  Lattice QCD with background magnetic fields is used to calculate the magnetic
moments and magnetic polarizabilities of the nucleons and of light nuclei with
$A\le4$, along with the cross-section for the $M1$ transition $np\rightarrow
d\gamma$, at the flavor SU(3)-symmetric point where the pion mass is $m_\pi\sim
806$ MeV. These magnetic properties are extracted from nucleon and nuclear
energies in six uniform magnetic fields of varying strengths. The magnetic
moments are presented in a recent Letter. For the charged states, the
extraction of the polarizability requires careful treatment of Landau levels,
which enter non-trivially in the method that is employed. The nucleon
polarizabilities are found to be of similar magnitude to their physical values,
with $\beta_p=5.22(+0.66/-0.45)(0.23) \times 10^{-4}$ fm$^3$ and
$\beta_n=1.253(+0.056/-0.067)(0.055) \times 10^{-4}$ fm$^3$, exhibiting a
significant isovector component. The dineutron is bound at these heavy quark
masses and its magnetic polarizability, $\beta_{nn}=1.872(+0.121/-0.113)(0.082)
\times 10^{-4}$ fm$^3$ differs significantly from twice that of the neutron. A
linear combination of deuteron scalar and tensor polarizabilities is determined
by the energies of the $j_z=\pm 1$ deuteron states, and is found to be
$\beta_{d,\pm 1}=4.4(+1.6/-1.5)(0.2) \times 10^{-4}$ fm$^3$. The magnetic
polarizabilities of the three-nucleon and four-nucleon systems are found to be
positive and similar in size to those of the proton, $\beta_{^{3}\rm
He}=5.4(+2.2/-2.1)(0.2) \times 10^{-4}$ fm$^3$, $\beta_{^{3}\rm
H}=2.6(1.7)(0.1) \times 10^{-4}$ fm$^3$, $\beta_{^{4}\rm
He}=3.4(+2.0/-1.9)(0.2) \times 10^{-4}$ fm$^3$. Mixing between the $j_z=0$
deuteron state and the spin-singlet $np$ state induced by the background
magnetic field is used to extract the short-distance two-nucleon counterterm,
${\bar L}_1$, of the pionless effective theory for $NN$ systems (equivalent to
the meson-exchange current contribution in nuclear potential models), that
dictates the cross-section for the $np\to d\gamma$ process near threshold.
Combined with previous determinations of NN scattering parameters, this enables
an ab initio determination of the threshold cross-section at these unphysical
masses.
