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The correlation length is the distance over which fluctuations remain appreciably correlated. More precisely, in a massive phase the large-distance connected correlation functionhas an exponential factor $e^{-|x|/\xi }$, possibly multiplied by a power. Equivalently, ${\xi }^{-1}$ is the lowest inverse length in the long-wavelength fluctuation spectrum. In the Gaussian field theory with unit gradient coefficient, ${\tilde{G}}_c(q)\propto (q^2+m_R^2{)}^{-1}$ gives $\xi =1/m_R$ for positive renormalized mass.

At an ordinary continuous phase transition, the restoring force for long-wavelength order-parameter fluctuations vanishes and there is no finite correlation scale. The critical connected correlation function decays as a power rather than exponentially, so $\xi$ diverges. The correlation-function susceptibility sum rule relates magnetic susceptibility to the integral of this connected correlation function; its divergence is consistent with correlations extending over increasingly large distances. Interactions change the powers, so the renormalized mass and the bare mass need not vanish at the same temperature.

Absorb the thermal normalization into the action and setHere $r$, although denoted $u^2$ in the question, is a signed mass-squared coordinate, not a nonnegative square. This matters because its interacting fixed-point value is negative. Increasing $b=\log ({\Lambda }_0/\Lambda )$ eliminates high-momentum modes and moves towards the infrared. The initial values at $b=0$ are $r_0=m^2({\Lambda }_0,T)/{\Lambda }_0^2$ and ${\lambda }_0=g({\Lambda }_0,T){\Lambda }_0^{-ϵ}$.

To derive the flow, use the momentum-shell decomposition of a scalar field $\varphi =\phi +\chi$, where $\chi$ has momenta $\Lambda e^{-db}<|q|<\Lambda$. Its Gaussian shell covariance is $(q^2+m^2{)}^{-1}$. Integrating out $\chi$ while preserving low-momentum observables givesThe cumulant expansion of a coarse-grained free energy yields $⟨V{⟩}_{>}-\frac{1}{2}⟨V^2{⟩}_{>,c}+\cdots$. By Wick theorem,The last term is field independent, while the quadratic term gives the one-loop shell mass renormalization in scalar quartic theoryThis tadpole is independent of the external momentum, so no gradient renormalization occurs at this order.

For the quartic correction, the mixed vertex is $(g/4)\int {\phi }^2{\chi }^2$. Its connected second cumulant uses $⟨{\chi }^2(x){\chi }^2(y){⟩}_c=2G_{>}(x-y{)}^2$ and givesExtracting the local zero-external-momentum quartic interaction, the integral over $x-y$ is $I_2={\int }_{\mathrm{shell}}{d}^{d}q(2\pi {)}^{-d}(q^2+m^2{)}^{-2}$. Comparing its coefficient with $g{\phi }^4/4!$ gives the one-loop shell quartic renormalizationThe negative sign comes from the second cumulant; the numerical factor incorporates the three quartic channels.

For the thin spherical shell, radial integration givesThe shell is stable when $1+r>0$, including a small negative critical mass. Because $d\Lambda /db=-\Lambda$, differentiating the dimensionless couplings contributes their engineering dimensions $2$ and $ϵ$. Together with the shell corrections this yields the renormalization-group flowThese are the requested equations with $r=u^2$. They express independence of the long-distance theory from the arbitrarily chosen ultraviolet cutoff, to the retained perturbative order; higher interactions and gradient corrections enter at higher orders.

Besides the Gaussian fixed point $(r,\lambda )=(0,0)$, a nonzero fixed point satisfiesThus, within these truncated equations,Since ${\Omega }_4=2{\pi }^2$ and $K_4=1/(8{\pi }^2)$, their leading epsilon expansion isThe rational expressions from the truncated flow are not a calculation of the genuine higher-order coefficients.

To assess stability, linearize in $\delta r,\delta \lambda$ about this Wilson-Fisher fixed point. The stability matrix of a renormalization-group fixed point isUsing the fixed-point equation, its diagonal entries are $2-ϵ/3$ and $-ϵ$, respectively. Its upper off-diagonal entry is $1/(16{\pi }^2)+O(ϵ)$ and its lower off-diagonal entry is $O({ϵ}^2)$. Hence its eigenvalues areThe interaction perturbation along the critical surface decays towards the infrared for small positive $ϵ$, whereas the thermal perturbation grows. Therefore infrared attraction requires tuning to the critical surface. The source's unqualified wording cannot mean attraction in both coupling directions: the positive thermal eigenvalue is precisely what produces the divergence of the correlation length. This is the thermal relevant direction at the Wilson-Fisher fixed point.

Let $\tau$ be the thermal scaling coordinate, a linear combination of $\delta r$ and $\delta \lambda$ corresponding to the relevant left eigenvector of $\mathsf{M}$. To leading order, $\tau =\delta r+[1/(32{\pi }^2)+O(ϵ)]\delta \lambda$ is possible. Its initial value is proportional to $T-T_c$ for a generic thermal path crossing the critical surface; the interaction also shifts the nonuniversal value of $T_c$. In the linear regime,Choose $b_{\ast }$ so that $|\tau (b_{\ast })|$ is of order one. At this scale the dimensionless correlation length is finite. Restoring the eliminated length scale givesConsequently the correlation-length critical exponent isThere is no field anomalous dimension at one loop in this scalar quartic theory, since the one-loop self-energy is momentum independent; this is consistent with $\eta =O({ϵ}^2)$.

Besides the Gaussian fixed point, solve the nonzero-coupling fixed-point equations. The quartic equation gives ${\lambda }_{\ast }=2ϵ(1+{\rho }_{\ast }{)}^2/(3K_D)$. Substitute into the mass equation to get $2{\rho }_{\ast }+ϵ(1+{\rho }_{\ast })/3=0$, hence ${\rho }_{\ast }=-ϵ/(6+ϵ)$ within the displayed one-loop truncation. Since $K_4=2{\pi }^2/(2\pi {)}^4=1/(8{\pi }^2)$,These are the Wilson-Fisher fixed point coordinates in this cutoff scheme. The higher terms obtained by solving the truncated equations exactly should not be regarded as a full higher-order epsilon calculation.

The attraction claim in the PDF needs a thermal qualification. The stability matrix of a renormalization-group fixed point at this solution has entriesThe lower-left entry is $O({ϵ}^2)$, so its eigenvalues areIncreasing $\ell$ means approaching the infrared. The quartic scaling direction is attractive, but a perturbation in the thermal scaling direction grows as $e^{y_t\ell }$. Therefore the Wilson-Fisher fixed point is an infrared attractor on its tuned critical surface, not an attractive fixed point of the entire two-dimensional coupling plane. In the plane it is a saddle; an arbitrary small thermal perturbation is a concrete counterexample to unconditional attraction. This is the thermal relevant direction at the Wilson-Fisher fixed point.

Besides $g=0$, the coupling beta function vanishes atThe mass fixed-point equation isso the root continuously connected to the origin isPerturbation theory requires $b/c\ll 1$, ensuring that the fixed-point coupling is small and the omitted higher powers are suppressed.

For increasing infrared scale $s$, positive $g<g_{\ast }$ flows upward toward $g_{\ast }$ and $g>g_{\ast }$ flows downward toward it. The mass direction remains relevant, so only the tuned critical trajectory reaches the interacting fixed point; trajectories on either side leave toward the two phases.

The relevant thermal eigenvalue is obtained from the stability matrix of a renormalization-group fixed point by differentiating the mass beta function:It follows that

For couplings $g_i(\ell )$ under coarse-graining scale $b=e^{\ell }$, their beta function is ${\beta }_i(g)=d{g}_i/d\ell$. At a renormalization-group fixed point $g_{\ast }$ all beta functions vanish. Linearization givesThe eigenvalues $y_a$ of the stability matrix of a renormalization-group fixed point are the scaling dimensions of the associated coupling directions: $\delta g_a(\ell )=e^{y_a\ell }\delta g_a(0)$.

For $\dot{x}=f(x)$ near an equilibrium point, the stability matrix is the Jacobian matrix of the vector field evaluated at that point. Perturbations obey $\dot{\eta }=J\eta$ to first order. Its eigenvalues determine linear growth and decay: strictly negative real parts imply local asymptotic stability, while a positive real part gives instability. Zero real parts require further analysis. The stability matrix of a renormalization-group fixed point applies this construction to beta-function flows.