Renormalized Trajectories and Counterterm Tuning

Counterterm tuning is a boundary-value problem for an RG recurrence. Relevant coordinates amplify microscopic errors, so their initial values must be chosen as functions of the cutoff and renormalization condition. A theorem needs existence, uniqueness in a stated neighborhood, and uniform control of the accompanying irrelevant coordinate; it does not say that arbitrary bare actions converge to the same limit.

Required background. Polymer Activities and Normed RG Coordinates supplies the remainder norm. Rigorous RG as a Dynamical System supplies the analytic recurrence and its domain.

Helpful background. Interacting Measures, Stability, and Wick Ordering supplies stable bare measures. Existence, Construction, Reconstruction, and Continuum Claims distinguishes a tuned finite-cutoff trajectory from cutoff removal.

Relevant coordinates as boundary data

After a near-identity coordinate change, write the scale map as

$$\begin{aligned} g_{j+1}&=g_j-\beta_jg_j^2+r^g_j,\\ \mu_{j+1}&=\lambda_j\mu_j+r^\mu_j,\qquad \lambda_j>1,\\ K_{j+1}&=\mathcal L_jK_j+r^K_j,\qquad \|\mathcal L_j\|\leq\kappa<1. \end{aligned}$$

The mass-like coordinate $\mu$ is relevant. If one requires $\mu_j$ not to grow at late scales, its initial value is fixed by the future nonlinear forcing. For a constant multiplier $\lambda>1$, the bounded solution is

$$\mu_j=-\sum_{k=j}^{\infty}\lambda^{j-k-1}r^\mu_k.$$

Substitution verifies the recurrence and boundedness. This is the elementary core of the Lyapunov–Perron or mixed-boundary construction: irrelevant coordinates are specified initially, while relevant coordinates are fixed by a terminal or asymptotic condition. In the nonhyperbolic RG system, the sequence-space theorem supplies existence, differentiability, and dependence on initial marginal data Bauerschmidt, Brydges, and Slade 2015, Theorem 1.4 and §§ 2–4, pp. 1040–1056.

A weak three-dimensional φ⁴ tuning problem

On a fine three-dimensional torus lattice, take

$$S_a(\phi)=\frac12\langle\phi,(-\Delta_a+\nu_0(a))\phi\rangle +g_0a^3\sum_x\phi_x^4+u_0(a)|\Lambda_a|,$$

with $g_0>0$ small. At finite cutoff $a=L^{-N}$, choose a renormalization condition on the zero-momentum two-point vertex, for example

$$\Gamma_N^{(2)}(0)=m_R^2, \quad \xi_N=m_R^{-1}.$$

Assume the exact RG map is analytic on $\mathcal D_j$, its mass derivative is bounded away from zero at the target, and $K_j$ obeys the contraction bounds uniformly through scale $N$. The implicit-function theorem then selects a local, unique $\nu_0(a)$; the vacuum counterterm $u_0(a)$ fixes normalization. This is the worked object behind Critical Surfaces, Crossover, and Corrections to Scaling: tune the bare mass so the correlation length hits the declared target while the polymer remainder stays controlled.

Balaban’s ultraviolet method, presented for scalar $\phi^4$ on a three-dimensional torus, controls small and large fields and proves convergence of the expansion and a stability bound Dimock 2013, Part III, §§ 1 and 6, pp. 1–8 and 38–49 of the Open PDF. That result supports the multiscale control mechanism; it should not be paraphrased as the precise renormalization condition above unless the observable derivative and limiting correlation theorem are also supplied.

Proof mechanism and independent checks

Define a Banach space of sequences with weights that compensate $\lambda_j$ in the relevant direction and $\kappa$ in the irrelevant one. The RG recurrence becomes a fixed-point equation: forward Green operators solve stable coordinates, backward Green operators solve relevant coordinates, and the marginal equation is treated with its slow $g_j\sim1/j$ decay. A contraction proves a unique sequence in the chosen ball. Differentiating the fixed-point equation gives dependence on $g_0$ and on the renormalization condition.

Two checks are independent. Differentiate $\Gamma_N^{(2)}(0)$ with respect to $\nu_0$; a vanishing derivative defeats local uniqueness. Then perturb $\nu_0$ by $\delta\nu$. At scale $j$ the linear relevant component grows approximately as

$$\delta\mu_j\simeq\left(\prod_{k<j}\lambda_k\right)\delta\nu,$$

so the allowed tuning window shrinks with the number of scales.

Finite terminal data and the infinite trajectory

For a constant multiplier $\lambda>1$ at cutoff $N$, a terminal condition $\mu_N=b_N$ determines the linear relevant coordinate by backward substitution:

$$\mu_j^{(N)} =\lambda^{j-N}b_N -\sum_{k=j}^{N-1}\lambda^{j-k-1}f_k, \qquad 0\leq j<N.$$

Substitution into $\mu_{j+1}=\lambda\mu_j+f_j$ checks both the exponent and the sign. If $b_N$ is uniformly bounded and $f_k$ decays sufficiently fast, then for fixed $j$ the first term vanishes and the finite sum converges to the infinite-series solution as $N\to\infty$. Without those uniform bounds, solving every finite terminal-value problem does not produce a limiting counterterm.

For the nonlinear RG map, the analogous comparison estimates two trajectories with cutoffs $N$ and $M>N$ in a weighted sequence norm. The theorem must show that the difference of their tuned initial masses tends to zero and that both remainders remain inside the same scale domains. A renormalization condition adds a second ingredient: transversality. If

$$\left|\frac{\partial\Gamma_N^{(2)}(0)}{\partial\nu_0}\right| \geq c_*>0$$

uniformly near the target, the implicit-function theorem converts a controlled error in the vertex into a controlled error in $\nu_0(a)$. If this derivative tends to zero with the cutoff, finite-$N$ uniqueness can become ill-conditioned and gives no uniform continuum conclusion.

Failure and continuum boundaries

The transverse perturbation just displayed eventually leaves $\mathcal D_j$. Irrelevant contraction cannot repair a mistuned relevant mass. Therefore “universality” never means that arbitrary bare mass reaches the same critical theory.

A family $\nu_0(a)$ that keeps finitely many iterates bounded is still not a continuum measure. One needs estimates uniform as $N\to\infty$, tightness or convergence of specified Schwinger functions, removal of volume cutoffs, and identification of the limit. A local trajectory also establishes no ultraviolet completion beyond its proved scale interval.

Exercise

Solve $\mu_{j+1}=\lambda\mu_j+f_j$, $\lambda>1$, subject to $\mu_j\to0$ and $|f_j|\leq Cq^j$ with $0<q<1$.

Solution

The bounded solution is $\mu_j=-\sum_{k=j}^{\infty}\lambda^{j-k-1}f_k$. Hence $|\mu_j|\leq C\lambda^{-1}q^j\sum_{r\geq0}(q/\lambda)^r=Cq^j/(\lambda-q)$. Any other initial value adds $c\lambda^j$ and violates the terminal condition unless $c=0$.

References

• Bauerschmidt, Roland, David C. Brydges, and Gordon Slade. “Structural Stability of a Dynamical System Near a Non-Hyperbolic Fixed Point.” Annales Henri Poincaré 16 (2015): 1033–1065. Open PDF.
• Dimock, Jonathan. “The Renormalization Group According to Balaban—III. Convergence.” 2013. Open PDF.