At the <Gaussian fixed point>, $[\phi_i]=(d-2)/2$, so all three quartic couplings have <engineering dimension> $d-4[\phi_i]=4-d$. Thus
$$
\boxed{\alpha_1=\alpha_2=\alpha_3=\epsilon}.
$$
Take $A>0$ and $\epsilon>0$, as in the perturbative <epsilon expansion>. Define $x=Ag_1/\epsilon$, $y=Ag_2/\epsilon$, $z=A\lambda/\epsilon$. The <renormalization-group fixed point> equations become
$$
x-36x^2-z^2=0,\qquad y-36y^2-z^2=0,\qquad z[1-8z-12(x+y)]=0.
$$
If $z=0$, each of $x,y$ is independently $0$ or $1/36$, giving four <renormalization-group fixed points>. If $z\ne0$, subtracting the first two equations gives
$$
(x-y)[1-36(x+y)]=0.
$$
The branch $x+y=1/36$ forces $z=1/12$; inserting this in the first equation yields a repeated root $x=y=1/72$. Hence every nonzero-$z$ solution has $x=y$. Then $z=1/8-3x$, and
$$
45x^2-\frac74x+\frac1{64}=0
$$
has roots $x=1/40$ and $x=1/72$. The complete list of six <coupled Ising fixed points near four dimensions>, in coordinates $(g_1,g_2,\lambda)$, is
$$
\boxed{\frac{\epsilon}{A}\left\{\begin{gathered}
(0,0,0),\quad\left(\frac1{36},0,0\right),\quad\left(0,\frac1{36},0\right),\\
\left(\frac1{36},\frac1{36},0\right),\quad\left(\frac1{40},\frac1{40},\frac1{20}\right),\quad\left(\frac1{72},\frac1{72},\frac1{12}\right)
\end{gathered}\right\}}.
$$
The internal <symmetries> of these <renormalization-group fixed points> are:

* The <Gaussian fixed point> has an $O(2)$ <orthogonal group> acting on the two identical free massless fields.
* $(\epsilon/(36A),0,0)$ is one <Ising model> <Wilson-Fisher fixed point> and one free field. Its linear internal <symmetry> is $\mathbb Z_2\times\mathbb Z_2$; the two sectors cannot be exchanged. The point $(0,\epsilon/(36A),0)$ has the same <symmetry> with the roles reversed.
* $(\epsilon/(36A),\epsilon/(36A),0)$ describes two identical decoupled <Ising models>. It additionally permits field exchange, giving $(\mathbb Z_2\times\mathbb Z_2)\rtimes S_2$, the eight-element <dihedral group> $D_4$.
* $(\epsilon/(40A),\epsilon/(40A),\epsilon/(20A))$ has $\lambda=2g_1=2g_2$. Its interaction is $g(\phi_1^2+\phi_2^2)^2$, so it is the <O(N) model> <Wilson-Fisher fixed point> with $N=2$ and $O(2)$ <symmetry>.
* $(\epsilon/(72A),\epsilon/(72A),\epsilon/(12A))$ has $\lambda=6g_1=6g_2$. Its <symmetry> is again the eight-element <dihedral group> $D_4$, not $O(2)$.

The last point displays the <field-rotation equivalence of decoupled Ising theories>. With the <orthogonal transformation> $\psi_\pm=(\phi_1\pm\phi_2)/\sqrt2$,
$$
g(\phi_1^4+\phi_2^4+6\phi_1^2\phi_2^2)=2g(\psi_+^4+\psi_-^4),\qquad 2g=\frac{\epsilon}{36A}.
$$
It is the same pair of decoupled <Ising models> written in fields rotated by $45^\circ$, but remains a distinct coordinate solution of the stated <renormalization-group beta functions>. Here $D_4$ denotes the <dihedral group> of order eight; an alternative convention calls it $D_8$. We have described linear internal transformations preserving the <gradient energy>; free massless sectors also have constant <scalar-field shift symmetries>. At $\epsilon=0$, the six coordinates coalesce at the <Gaussian fixed point>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 303 / 3 / b / Solution

The quartic interaction $g_1\phi_1^4+g_2\phi_2^4+\lambda\phi_1^2\phi_2^2$ includes the <Gaussian fixed point>, single and double <Ising model> <Wilson-Fisher fixed points>, and an <O(N) model> fixed point with $N=2$. A rotation of two decoupled identical <Ising models> produces a second coordinate representation of their <renormalization-group fixed point>.

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 Incoming links: Coupled Ising fixed points near four dimensions