Constructive quantum field theory — Lecture 6 (slides)

Lecture 4. Interacting field at resolution \(t\) = solution of the FBSDE

\(\displaystyle \Phi_t = \int_0^t \dot{Q}_s F_s \hspace{0.17em} \mathrm{d} s + \int_0^t \dot{Q}_s^{1 / 2} \mathrm{d} W_s, \qquad F_t =\mathbb{E}_t [- DV_T (\Phi_T)] .\)

In the forecast coordinate \(\psi_t =\mathbb{E}_t \Phi_T\) the forward equation has no drift and the trees are integrated exactly; the backward half is the force equation.

Lectures 6–7. Construct \(\Phi^4_3\) on \(\mathbb{T}^3\) with this FBSDE:

1. deterministic renormalisation: an explicit approximate force \(\bar{\mathcal{F}}_q\) and its defect (today);

2. a priori estimates for the exact FBSDE in the coordinates of \(\bar{\mathcal{F}}_q\), uniform in the lattice (Lecture 7);

3. removing the cutoffs and identifying the limit (Lecture 7).

Notes: Lectures 6–7 and the Supplement; Bailleul–Chevyrev–G. (in prep.)

For every coupling \(\lambda > 0\), every reference mass \(q_0 > 0\) and every finite part \(r\) of the counterterm, on a fixed torus, with no smallness, and an explicit counterterm with arbitrary finite part:

• the whole lattice sequence converges to a measure \(\nu\), an explicit density over an explicit SDE;

• the forecasts converge to the solution of an FBSDE from \(Y_{q_0} = 0\), driven by the limit of the lattice innovations, and \(\psi_q = m_q (\Phi_q)\);

• uniqueness in law for its scalar form, a variational formula with no infinite constant, quartic tails, integration by parts.

Not proved: the vector FBSDE alone, strong solutions, OS axioms in full, infinite volume.

Guiding principle. Do not estimate \(\Phi\) and \(F\). Estimate the error of an explicit approximate forecast.

The results of Lectures 6–7 are new: joint work with S.-J. Meyer + AI.

The problem and the exact FBSDE

Lecture 6

\(\Lambda_N = (N^{- 1} \mathbb{Z}/\mathbb{Z})^3\), \(\int f = N^{- 3} \sum_x f (x)\), \(\Delta = \Delta_N\) nearest neighbour. Resolution parameter \(q = q_0 + t^2 \in [q_0, \infty)\):

\(\displaystyle K_q = q - \Delta, \qquad C_q = K_q^{- 1}, \qquad C = C_{q_0} = (q_0 - \Delta)^{- 1} .\)

In Lecture 3 notation (\(T = \infty\), and \(\dot{Q} = \partial_q Q_q\)):

\(\displaystyle Q_q = C - C_q, \quad \dot{Q} = C_q^2, \quad \dot{Q}^{1 / 2} = C_q, \quad S_q = C_q, \quad \mathcal{R}_q = q - q_0 .\)

The bubble \(a_2 (q)\) is the coincident-point variance of the shell at resolution \(q\).

\(U (\varphi) = \frac{\lambda}{12} \int \varphi^4\), \(V_N = U + \frac{\mu_N}{2} \int \varphi^2\), \(\nu_N (\hspace{0.17em} \mathrm{d} \varphi) =\mathcal{Z}_N^{- 1} \mathrm{e}^{- V_N (\varphi)} \mu^C (\hspace{0.17em} \mathrm{d} \varphi)\).

So \(D^2 U (y) = \lambda y^2\): the coupling is the curvature, the same \(\lambda\) as in Lecture 4.

The Wick ordering is now part of \(\mu_N\).

Reference mass \(q_0 > 0\) (constants \(\in \ker \Delta\) on the torus; \(r\) may be negative). Only \(q_0 + \mu_N\) matters: fix \(q_0\), vary \(r\).

These come from bounds on the error \(Z\) of an approximate forecast, uniform in \(N\): UV stability.

By Lecture 4, in the profile \(S_q = C_q\), \(\mathcal{R}_q = q - q_0\); at fixed \(N\) everything is a theorem. The Polchinski process solves

\(\displaystyle \mathrm{d} \Phi_q = C_q^2 F_q \hspace{0.17em} \mathrm{d} q + C_q \hspace{0.17em} \mathrm{d} W_q, \quad F_q =\mathbb{E}_q [- DV_N (\Phi_{\infty})], \quad \Phi_{q_0} = 0,\)

and \(\Phi_q \to \Phi_{\infty} \sim \nu_N\).

• a noisy observation of its endpoint, \(\Phi_q = (q - q_0) C_q \Phi_{\infty} + C_q B_{q - q_0}\), with \(B\) independent of \(\Phi_{\infty}\);

• forecast

  \(\displaystyle \psi_q =\mathbb{E}_q \Phi_{\infty} = \Phi_q + C_q F_q = m_q (\Phi_q),\)

  covariance \(\text{Cov}_q > 0\);

• \(W\) is the innovation: \(\mathrm{d} \psi_q = \text{Cov}_q \hspace{0.17em} \mathrm{d} W_q\);

• noise: \(\omega_q = K_q \Phi_q - (q - q_0) \psi_q =\mathbb{E}_q B_{q - q_0}\), sub-Gaussian whatever the interaction (L5).

One site, \(\nu \propto \mathrm{e}^{- \lambda \varphi^4 / 12} \mu^{1 / q_0}\), \(\lambda = 6\), \(q_0 = 1\). Left: one path of the observation \(\Phi_q\) and of the forecast \(\psi_q\), both converging to the hidden \(\Phi_{\infty}\). Right, over 200 paths: the forecast error \(\sim q^{- 1 / 2}\), the force term \(\psi_q - \Phi_q = C_q F_q \sim q^{- 1}\), and the error \(Z_q\) of the approximate forecast from \(\bar{\mathcal{F}} = \frac{\lambda}{3} y^3\), below the bound \(2 C_{\ast} (\lambda / 3)^{1 / 2} q^{- 3 / 2}\) of Theorem 10.12 (see "Integrating the cube" below).

Since \(\Phi_q \mapsto \psi_q\) is invertible, define \(\mathcal{F}_q\) by

\(\displaystyle \Phi_q = \psi_q + C_q \mathcal{F}_q (\psi_q)\)

and \(H^{\text{ex}}_q (y) = (K_q + D\mathcal{F}_q (y))^{- 1}\). It is the \(\mathcal{F}\) of Lecture 4 (\(x = \psi + S\mathcal{F} (\psi)\)), and \(F_q = -\mathcal{F}_q (\psi_q)\).

Itô on \(\Phi = \psi + C\mathcal{F} (\psi)\). Its integrated form is Wetterich's equation, which we do not need. An identity about the unknown \(\nu_N\), not a construction.

Strategy: an approximate force \(\bar{\mathcal{F}}\) gives approximate forecast coordinates; the error should be driven by the failure of \(\bar{\mathcal{F}}\) to solve this equation.

\(\bar{\mathcal{F}}_q\) odd, \(D \bar{\mathcal{F}}_q\) symmetric, \(K_q + D \bar{\mathcal{F}}_q \geqslant \bar{c}_q I\), \(\bar{c}_q > 0\). Recall \(\psi_q =\mathbb{E}_q [\Phi_{\infty}]\), \(\text{Cov}_q =\operatorname{Var}_q (\Phi_{\infty})\),

\(\displaystyle H_q = (K_q + D \bar{\mathcal{F}}_q)^{- 1}, \quad \mathcal{D}_q = \partial_q \bar{\mathcal{F}}_q + \frac{1}{2} \text{Ctr}_{H_q^2} D^2 \bar{\mathcal{F}}_q, \quad b_q = - H_q \mathcal{D}_q .\) \(\displaystyle Y_q + C_q \bar{\mathcal{F}}_q (Y_q) = \Phi_q, \quad Z_q = \psi_q - Y_q, \quad \Xi_q = \text{Cov}_q - H_q (Y_q) .\)

Proof: Itô on the coordinate equation, with the Polchinski process and \(\mathrm{d} C_q = - C_q^2 \mathrm{d} q\): the martingale part gives \(H\), the drift gives \(HZ + b\).

• \(Y\): forward and explicit once \(Z\) is known; coefficients \(b, H\) built from \(\bar{\mathcal{F}}\).

• \(Z\): backward. The terminal condition \(qZ_q \to 0\) is proved at fixed \(N\) (Lecture 7), because \(\bar{\mathcal{F}}_q \to\) bare force.

• The only source of \(Z\) is the drift \(b\), the force defect, transported by \(\exp \int H\).

Needed: (i) a defect integrable against the backward propagator (renormalisation); (ii) global coercivity of \(\bar{\mathcal{F}}\) (large fields); (iii) a priori estimates (Lecture 7).

\(J\): large-field functional, \(E\): energy (both Lecture 7); \(b^{\text{b}}, b^{\text{l}}\): bounded- and large-field parts of the drift.

Programme: keep \(- \frac{\lambda}{3} \varphi^3\) unclipped, clip the higher orders (result: \(f\)), pay in the source of \(R = F - f\). Test it on one site: \(K_q = q\), \(S = 1 / q\).

\(\displaystyle F_q (x) = - \frac{\lambda}{3} \mathbb{E}_x \Phi_{\infty}^3 = q (m_q (x) - x), \quad |m_q (x) | \leqslant 2 (\lambda / 3)^{- 1 / 4} + 2 (3 q|x| / \lambda)^{1 / 3} .\)

So \(3 R / (\lambda x^3) \to 1\): the remainder cancels the cube (the dip: \(R\) changes sign, Wick term below, cube above). Field equation: \(qx = (q - q_0) m_q + \omega_q\), only linear confinement.

\(f^c = f_1 +\mathcal{C}f_2\), with \(f_1 = - \frac{\lambda}{3} (x^3 + 3 Sx)\), \(f_2\) the first tree, \(\kappa_q\) a smooth clip at \(|x| \sim r_q\), \(\mathcal{C}f_2 = \kappa_q \prime \cdot f_2 \circ \kappa_q\). For \(|x| \geqslant 2 r_q\), the Burgers defect is exactly

\(\displaystyle \mathcal{S}_q = S^2 f_1 f_1 \prime \approx \frac{\lambda^2}{3} S^2 x^5,\)

of order \((q / \lambda)^{1 / 2}\) at the edge of the clip: not integrable in \(q\).

\(\displaystyle \hspace{0.17em} \mathrm{d} R_q = - [\mathcal{S}_q + S^2 (f^c)\prime R_q] \hspace{0.17em} \mathrm{d} q + \Xi^R_q \hspace{0.17em} \mathrm{d} W_q, \quad \text{bracket} \approx \frac{\lambda^2}{3} S^2 x^5 - \lambda S^2 x^2 R.\)

The source and the dissipation cancel for the true \(R \sim \frac{\lambda}{3} x^3\): separate estimates give nothing.

Tails independent of \(R\): on one site from the posterior moment, but not uniform on the lattice (negative bare mass).

More orders do not help: the tree series has radius \(\frac{2}{3} r_q\).

Solve the transport part of the flow exactly:

\(\displaystyle \partial_q f + S^2 ff' = 0, \quad f_{\infty} = - \frac{\lambda}{3} x^3 \quad \Rightarrow \quad f_q (x) = - \frac{\lambda}{3} y^3, \quad qy + \frac{\lambda}{3} y^3 = qx.\) \(\displaystyle \sup_x |F_q (x) - f_q (x) | \leqslant 2 C_{\ast} (\frac{\lambda}{3 q})^{1 / 2}, \quad C_{\ast} < \frac{1}{2} .\)

• \(y_q (\Phi_q) = Y_q\): this is the forecast coordinate. \(F_q - f_q = K_q Z_q\), and \(Z\) is the posterior mean minus the mode.

• The defect is \(qb_q (y)\): bounded for all fields. The trees are in \(x \mapsto y\); the cube does not flow in \(y\), so only loops are clipped.

• The cube is coercive in \(y\): \(qy + \frac{\lambda}{3} y^3 = q \Phi_q\).

Renormalisation: why two of them

Lecture 6

We seek an explicit force with a small defect, not the full solution of the nonlinear flow.

\(\displaystyle \partial_q \mathcal{F}_q = - \frac{1}{2} \text{Ctr}_{(H^{\text{ex}}_q)^2} D^2 \mathcal{F}_q, \qquad H^{\text{ex}}_q = (K_q + D\mathcal{F}_q)^{- 1} .\)

At infinite resolution: \(\mathcal{F}_{\infty} = U\prime + \mu_N y\), with \(U\prime (y) = \frac{\lambda}{3} y^3\).

1. Insert the bare cube into the response and the vertex: a one-loop seed.

2. Feed each approximation back into the equation: two correction steps.

3. Estimate the remaining defect in the backward equation for \(Z\).

Insert \(U\prime\) into the loop, with

\(\displaystyle G_q (y) = (K_q + \lambda y^2)^{- 1} :\) \(\displaystyle - \frac{1}{2} \text{Ctr}_{G_q^2} D^2 U\prime (y) (x) = - \lambda y (x) G_q (y)^2 (x, x) .\)

Its linear Taylor coefficient is

\(\displaystyle - \lambda a_2 (q) y (x), \qquad a_2 (q) = C_q^2 (x, x) .\)

In three dimensions \(a_2 (q)\) has continuum size \(q^{- 1 / 2}\): its integral to infinity diverges. At fixed lattice cutoff it is finite, but grows with \(N\).

For a smooth odd force \(f\), and a translation-invariant operator \(L = \langle L \rangle I + L^{\circ}\) (constant mode plus centred part):

\(\displaystyle (\Pi f) (y) = f (y) -\operatorname{Df} (0) y, \qquad (\Pi' f) (y) = f (y) - \langle \operatorname{Df}(0) \rangle y.\)

• \(\Pi f\) starts at cubic order; \(\Pi' f = \Pi f + (\operatorname{Df}(0))^{\circ} y\) removes only the scalar part of the linear jet.

• The divergences (tadpole, sunset) are constant modes. The centred part of the linear flow is integrable at infinity.

\(\displaystyle \mathcal{F}_q = U\prime + M_q y + \ell_q, \qquad \Pi \ell_q = \ell_q, \quad M_q = \mu_q I + \Theta_q .\)

\(\displaystyle g_q = \lambda q^{- 1 / 2} (\text{dimensionless coupling}),\) \(\displaystyle r_q = (q / \lambda)^{1 / 2} (\lambda r_q^2 = q : \text{nonlinearity} \approx \text{regulator}) .\)

Free field at resolution \(q\): free field \(X_q\), \(\text{Var} X_q (x) \lesssim q^{1 / 2}\), typical size \(q^{1 / 4} = g_q^{1 / 2} r_q \ll r_q\).

Rooted norm of a vertex \(T (x_1, \ldots, x_k)\) and field ball:

\(\displaystyle s (T) = \max_i \sup_{x_i} \int |T| \prod_{j \ne i} \mathrm{d} x_j, \qquad \textsf{S}_k (f) = \sup_{\mathcal{B}_q} s (D^{k - 1} f), \qquad \mathcal{B}_q = \{ \|y\|_{\infty} < 2 r_q \} .\)

Natural size of the \(k\)-leg vertex of the loop corrections:

\(\displaystyle \textsf{S}_k \lesssim g_q \hspace{0.17em} q \hspace{0.17em} r_q^{2 - k} \qquad (k = 4 : \lambda g_q) .\)

Root one leg of a closed loop and follow the loop: two of its propagators are estimated in \(L^2\) (this gives the bubble \(a_2 (q)\)), all the others in operator norm (each \(q^{- 1}\)).

Project the bare loop before integrating:

\(\displaystyle \Pi [- \lambda y (x) G_q (y)^2 (x, x)] = - \lambda y (x) [G_q (y)^2 (x, x) - a_2 (q)] .\)

Use \(\partial_q G_q = - G_q^2\) and \(\partial_q C_q = - C_q^2\). With zero boundary value at infinity:

The leading term is the bubble:

\(\displaystyle - \lambda^2 y (x) \int C_q (x, z)^2 y (z)^2 \mathrm{d} z.\)

Four-leg size: \(\textsf{S}_4 (\ell^{[1]}_q) \lesssim \lambda^2 q^{- 1 / 2} = \lambda g_q\). This is the seed \(\ell^{[1]}\): the superscript counts loops.

The seed as one-loop graphs: lines \(C_q\), a vertex \(\lambda y^2\) with its two external fields at each corner. The \(j\)-th ring has \(2 j\) legs and coefficient \((- 1)^{j + 1} / (2 j)\); \(D\) marks one leg, giving the force. The series sums to a logarithm, with radius of convergence \(r_q\). The quartic ring is the bubble.

For \(\mathcal{F}= U\prime + My + \ell\), the loop map

\(\displaystyle \mathcal{T}_q [M, \ell] = - \frac{1}{2} \text{Ctr}_{R^2} D^2 (U\prime + \ell), \quad R = (K_q + M_q + D (U\prime + \ell))^{- 1} .\)

Three equations of different nature:

\(\displaystyle \partial_q \ell = \Pi \mathcal{T} [M, \ell], \quad \partial_q \Theta = (D\mathcal{T}[M, \ell] (0))^{\circ}, \quad \partial_q \mu = \langle D\mathcal{T}[M, \ell] (0) \rangle .\)

Fix a trial scalar path \(\mu\) (determined at the end as a fixed point, "Closing the mass path"). Level masses \(M^{[j]} = \mu I + \Theta^{[j]}\), sources \(\mathfrak{S}^{[j]} = D\mathcal{T} [M^{[j]}, \ell^{[j]}] (0)\), zero-field response \(R^{[j]}_{0, q} = (K_q + M^{[j]}_q)^{- 1}\):

With \(\ell^{[0]} = 0\), \(M^{[0]} = 0\), the case \(j = 0\) is the seed step without mass, \(\partial_q \ell^{[1]} = \Pi \mathcal{T} [0, 0]\), and \(\Theta^{[1]} = 0\) (the tadpole is local). Output: \((\ell^{[3]}, \Theta^{[3]}) [\mu]\).

A centred kernel \(\Theta (z) = \Theta (0, z)\) is measured by its first moment \(v_1 (\Theta) = \int d (0, z) | \Theta (z) | \mathrm{d} z\), \(d\) the torus distance. For an even kernel, \(\Theta \mathbb{1} = 0\) means that the diagonal cancels the off-diagonal mass. With \((\sigma_z f) (x) = f (x + z)\), \(\delta_z^2 = \sigma_z + \sigma_{- z} - 2 I\),

\(\displaystyle C_q \Theta = \frac{1}{2} \int \Theta (z) \delta_z^2 C_q \mathrm{d} z.\)

The absolute row norm discards the cancellation. For a lattice step \(h\) in direction \(e\), take

\(\displaystyle \Theta_h = \frac{a}{2 h} (\sigma_{\operatorname{he}} + \sigma_{-\operatorname{he}} - 2 I), \qquad v_1 (\Theta_h) = a, \quad \| \Theta_h \|_{\text{row}} = \frac{2 a}{h} .\)

The row norm diverges as \(h \to 0\); the first moment stays bounded.

The resolvent smooths on length scale \(q^{- 1 / 2}\):

\(\displaystyle \| \delta_z^2 C_q \|_{\text{row}} \leqslant Cq^{- 1 / 2} d (0, z) .\)

Scale check: integrate the heat-kernel translation bound \(Cd (0, z) t^{- 1 / 2}\) against \(\mathrm{e}^{-\operatorname{qt}} \mathrm{d} t\).

The iteration (next slides) supplies exactly the compensating moment:

\(\displaystyle v_1 (\Theta_q) \lesssim \lambda^2 q^{- 1 / 2} \quad \Longrightarrow \quad q^{- 1 / 2} v_1 (\Theta_q) \lesssim \lambda^2 q^{- 1} = g_q^2 .\)

A bound by \(d (0, z)^2\) loses a logarithm; one distance power, with \(q^{- 1 / 2}\), suffices (Lemma 19.3 of the Supplement).

The field-dependent response also receives this cancellation. With \(V = \lambda y^2\) and \(G = (K_q + V)^{- 1}\),

\(\displaystyle G \Theta = (I -\operatorname{GV}) C_q \Theta, \qquad \|I -\operatorname{GV}\|_{\infty \to \infty} \leqslant 2.\)

The bounded Neumann series for the full \(R\) then gives

\(\displaystyle \|R \delta M\|_{\infty \to \infty} \leqslant C \rho_q, \quad \rho_q = \frac{| \delta \mu_q |}{q} + q^{- 1 / 2} v_1 (\delta \Theta_q) .\)

Holding the nonlinear correction fixed, \(\delta R = - R \delta M \hat{R}\) (\(\hat{R}\) the response at mass \(M - \delta M\)) has row norm \(\lesssim \rho_q / q\) and row \(L^2\) norm \(\sup_x \| \delta R (x, \cdot)\|_{L^2}\) \(\lesssim \rho_q \sqrt{a_2 (q)}\).

Separate the source at level \(j\) into tadpole and correction vertex:

\(\displaystyle \mathfrak{S}^{[j]}_q = - \lambda \text{Tr} ((R^{[j]}_{0, q})^2) I + E_q, \quad E_q = - \frac{1}{2} \text{Ctr}_{(R^{[j]}_{0, q})^2} T^{[j]}_q, \quad T^{[j]}_q = D^3 \ell^{[j]}_q (0) .\)

Write \(E_q \mathbb{1} = e_q \mathbb{1}\). The constant mode contains the whole local contribution:

\(\displaystyle \langle \mathfrak{S}^{[j]}_q \rangle = - \lambda \text{Tr} ((R^{[j]}_{0, q})^2) + e_q, \qquad (\mathfrak{S}^{[j]}_q)^{\circ} = E_q - e_q I.\)

The tadpole cancels; so does any term \(c_q I\). Off-diagonal entries stay unchanged. Since \(d (0, 0) = 0\),

\(\displaystyle v_1 ((\mathfrak{S}^{[j]}_q)^{\circ}) = v_1 (E_q) .\)

Four-leg vertex \(T^{[j]}_q\) of \(\ell^{[j]}\), proved below level by level (\(\Theta^{[2]}\) needs only the bubble \(T^{[1]}\), \(\Theta^{[3]}\) only \(T^{[2]}\), at \(M^{[1]} = \mu I\)):

\(\displaystyle s (T^{[j]}_q) + \sqrt{q} v_1 (T^{[j]}_q) \lesssim \lambda^2 q^{- 1 / 2}, \quad \text{hence} \quad v_1 (T^{[j]}_q) \lesssim \lambda^2 q^{- 1} .\)

Keep the distance between the uncontracted legs of \(T^{[j]}_q\); since \(\sup_{u, v} | (R^{[j]}_{0, q})^2 (u, v) | \lesssim q^{- 1 / 2}\),

\(\displaystyle v_1 ((\mathfrak{S}^{[j]}_q)^{\circ}) = v_1 (E_q) \lesssim q^{- 1 / 2} v_1 (T^{[j]}_q) \lesssim \lambda^2 q^{- 3 / 2} .\)

Integrate with zero boundary value at infinity:

\(\displaystyle v_1 (\Theta^{[j + 1]}_q) \leqslant \int_q^{\infty} v_1 ((\mathfrak{S}^{[j]}_s)^{\circ}) \mathrm{d} s \leqslant C \lambda^2 q^{- 1 / 2} .\)

Uniformly in \(N\): \(\eta_q (\Theta^{[2]}) \lesssim g_q^2\), and one more \(g\) for the difference, \(\eta_q (\Theta^{[3]} - \Theta^{[2]}) \lesssim g_q^3\), with \(\eta_q (\Theta) = q^{- 1 / 2} v_1 (\Theta)\).

Set \(R^{[j]} = (K + M^{[j]} +\operatorname{DU}\prime + D \ell^{[j]})^{- 1}\) and \(\delta^{[2]} = \ell^{[2]} - \ell^{[1]}\). Subtract the bare loop from the loop containing the seed:

\(\displaystyle \delta^{[2]}_q = \frac{1}{2} \int_q^{\infty} \Pi [\text{Ctr}_{(R^{[1]})^2 - G^2} D^2 U\prime + \text{Ctr}_{(R^{[1]})^2} D^2 \ell^{[1]}]_s \mathrm{d} s.\)

Two changes: the propagators and the vertex.

\(\displaystyle R^{[1]} - G = - R^{[1]} (\mu + D \ell^{[1]}) G, \qquad M^{[1]} = \mu I.\)

The insertion \((\mu + D \ell^{[1]}) / q\) is \(O (g_q)\). One extra loop gives

\(\displaystyle \textsf{S}_4 (\delta^{[2]}_q) \lesssim \int_q^{\infty} g_s s^{1 / 2} r_s^{- 4} \mathrm{d} s = \lambda^3 q^{- 1} = \lambda g_q^2 .\)

All factors inside the integral are evaluated at scale \(s\).

Put \(\delta^{[3]} = \ell^{[3]} - \ell^{[2]}\). Subtract the first two Picard equations:

\(\displaystyle \delta^{[3]}_q = \frac{1}{2} \int_q^{\infty} \Pi [\text{Ctr}_{(R^{[2]})^2 - (R^{[1]})^2} D^2 (U\prime + \ell^{[1]}) + \text{Ctr}_{(R^{[2]})^2} D^2 \delta^{[2]}]_s \mathrm{d} s.\)

The marked insertion is now the first correction itself, nonlinear and centred:

\(\displaystyle R^{[2]} - R^{[1]} = - R^{[2]} (\Theta^{[2]} + D \delta^{[2]}) R^{[1]},\)

where \(\Theta^{[2]}\) costs \(\eta_q (\Theta^{[2]}) \lesssim g_q^2\) (above).

\(\displaystyle \textsf{S}_4 (\delta^{[3]}_q) \lesssim \int_q^{\infty} g_s^2 s^{1 / 2} r_s^{- 4} \mathrm{d} s = \frac{2}{3} \lambda^4 q^{- 3 / 2} .\)

For squared propagators use \(A^2 - B^2 = A (A - B) + (A - B) B\); no commutativity is assumed.

At zero field, set \(W_q (T) = s (T) + \sqrt{q} v_1 (T)\). Mark two external legs and follow their distance around the loop.

• A displacement inside a vertex is controlled by that vertex's first moment.

• A displacement along a propagator is controlled by its weighted \(L^1\) or \(L^2\) bound. The triangle inequality distributes the external distance over these pieces.

For \(p\) vertices and \(p\) propagators:

\(\displaystyle W_q (\text{loop}) \lesssim q^{- 1 / 2} q^{- (p - 1)} \prod_{i = 1}^p W_q (T_i) .\)

Keep within-vertex moments: smoothing the propagators cannot supply spatial localisation for an arbitrary nonlocal vertex.

A loop difference with relative size \(\varepsilon_s\) has a marked vertex or propagator. With \(\delta \mathcal{T}_s =\mathcal{T}_s [M, \ell] -\mathcal{T}_s [M, \hat{\ell}]\), the weighted comparison gives

\(\displaystyle W_s (\mathrm{D}^{k - 1} \delta \mathcal{T}_s (0)) \lesssim \varepsilon_s s^{1 / 2} r_s^{- k} .\)

For \(s \geqslant q\), \(\sqrt{q} \leqslant \sqrt{s}\):

\(\displaystyle W_q \left( \int_q^{\infty} A_s \mathrm{d} s \right) \leqslant \int_q^{\infty} (s (A_s) + \sqrt{q} v_1 (A_s)) \mathrm{d} s \leqslant \int_q^{\infty} W_s (A_s) \mathrm{d} s.\)

\(\Pi\) subtracts only a linear force, so it leaves all \(k \geqslant 4\) vertices unchanged. The field supremum also propagates, since \(\mathcal{B}_q \subseteq \mathcal{B}_s\).

Write \(T^{[j]}_q = \mathrm{D}^3 \ell^{[j]}_q (0)\). The seed is the bubble:

\(\displaystyle W_q (T^{[1]}_q) \lesssim \lambda^2 \int (1 + \sqrt{q} d (0, z)) C_q (z)^2 \mathrm{d} z \lesssim \lambda^2 q^{- 1 / 2} .\)

For the increments, integrate the weighted loop bounds:

\(\displaystyle W_q (T^{[2]}_q - T^{[1]}_q) \lesssim \lambda^3 \int_q^{\infty} s^{- 2} \mathrm{d} s = \lambda^3 q^{- 1},\) \(\displaystyle W_q (T^{[3]}_q - T^{[2]}_q) \lesssim \lambda^4 \int_q^{\infty} s^{- 5 / 2} \mathrm{d} s = \frac{2}{3} \lambda^4 q^{- 3 / 2} .\)

The \(10 \to 8 \to 6\) leg budget supplies the higher vertices needed by both comparisons. Thus \(v_1 (T^{[j]}_q) \lesssim \lambda^2 q^{- 1}\).

Impose the scalar part of the linear flow for the final approximation \(M = M^{[3]} = \mu I + \Theta^{[3]}\), with \(R^{[3]}_{0, q} = (K_q + M_q)^{- 1}\):

\(\displaystyle \partial_q \mu_q = \beta_q = \langle \mathfrak{S}^{[3]}_q \rangle, \quad \mathfrak{S}^{[3]}_q = - \lambda \text{Tr} ((R^{[3]}_{0, q})^2) I - \frac{1}{2} \text{Ctr}_{(R^{[3]}_{0, q})^2} D^3 \ell^{[3]}_q (0),\)

with \(\mu_{q_{\ast}} = b_{\ast}\), \(|b_{\ast} | \leqslant B_0 \lambda q_{\ast}^{1 / 2}\), at an auxiliary scale \(q_{\ast}\). The centred part of \(\mathfrak{S}^{[3]}\) is not imposed. Bare mass \(\mu_N = \text{lim}_{q \to \infty} \mu_q\): an output of the flow. \(\beta\) depends on the whole path \(\mu\) through \(\ell^{[3]} [\mu]\): a fixed point ("Closing the mass path").

The bounded-field approximation:

\(\displaystyle \mathcal{F}^o_q = U\prime + M_q y + \ell^{[3]}_q, \qquad \partial_q \mu = \langle D\mathcal{T}_q [M, \ell^{[3]}] (0) \rangle .\)

The approximations use the preceding level, \(\partial_q \ell^{[3]} = \Pi \mathcal{T} [M^{[2]}, \ell^{[2]}]\), \(\partial_q \Theta^{[3]} = (\mathfrak{S}^{[2]})^{\circ}\). Hence

\(\displaystyle \mathcal{D}^o_q = \Pi' (\mathcal{T}_q [M^{[2]}, \ell^{[2]}] -\mathcal{T}_q [M^{[3]}, \ell^{[3]}]) :\)

nonlinear part and a centred linear part \(\mathcal{E}^c_q y\); the scalar part vanishes by the mass equation.

The constant and scalar linear jets of the defect vanish, so on the field ball

\(\displaystyle |b_s | \lesssim s^{- 1} (\textsf{S}_4 (\mathcal{D}_s) r_s^3 +\|\mathcal{E}^c_s \|_{\text{row}} r_s), \qquad Z_q \approx \int_q^{\infty} \frac{s}{q} b_s \mathrm{d} s.\)

Both terms have the same size; \(s / q\) is the backward transport. The cubic source:

corrections	\(\textsf{S}_4 (\mathcal{D}_s)\)	backward source integral
0: seed	\(\lambda^3 s^{- 2}\)	\(\lambda^{3 / 2} q^{- 1} \int_q^{\infty} s^{- 1 / 2} \mathrm{d} s = \infty\)
1	\(\lambda^4 s^{- 5 / 2}\)	\(\lambda^{5 / 2} q^{- 1} \int_q^{\infty} s^{- 1} \mathrm{d} s = \infty\)
2	\(\lambda^5 s^{- 3}\)	\(\lambda^{7 / 2} q^{- 1} \int_q^{\infty} s^{- 3 / 2} \mathrm{d} s = 2 \lambda^{7 / 2} q^{- 3 / 2}\)

Each step costs two field derivatives: keep the iteration finite. Full estimates: Lecture 7.

Each step inserts the previous correction into the loop: one more loop, one more \(g_q\). Two steps give a defect of order \(g_q^3\).

Given a trial scalar path \(\mu\), compute the two steps of the iteration, corrections and centred kernels, then

\(\displaystyle \beta_q [\mu] = \langle D\mathcal{T}_q [M^{[3]} [\mu], \ell^{[3]} [\mu]] (0) \rangle, \qquad \mathcal{A} [\mu]_q = b_{\ast} + \int_{q_{\ast}}^q \beta_s [\mu] \mathrm{d} s.\)

This is an equation for the whole path: the corrections and centred kernels at \(q\) depend on \(\mu\) at later scales.

For \(q_{\ast}\) large, uniformly in \(N\):

• a unique scalar path in the ball \(\| \mu \|_{\ast} \leqslant R\) (norm on the next slide), so \(| \mu_q | \lesssim \lambda q^{1 / 2}\);

• centred kernels: \(v_1 (\Theta^{[j]}_q) \lesssim \lambda^2 q^{- 1 / 2}\);

• increments: \(\mathcal{N}_{k, q} (\ell^{[j]} - \ell^{[j - 1]}) \lesssim g_q^j qr_q^{2 - k}\), \(j = 1, 2, 3\), \(4 \leqslant k \leqslant 12 - 2 j\), where

  \(\displaystyle \mathcal{N}_{k, q} (f) = \textsf{S}_k (f) + \sqrt{q} v_1 (D^{k - 1} f (0)) ;\)

• defect: \(\textsf{S}_4 (\mathcal{D}^o_q) \lesssim \lambda^5 q^{- 3}\), \(\|\mathcal{E}^c_q \|_{\text{row}} \lesssim g_q^4\).

Inner recursion: for each trial scalar path \(\mu\), start from the seed and compute exactly two steps.

\(\displaystyle \mu \longmapsto (\ell^{[2]}, \Theta^{[2]}) [\mu] \longmapsto (\ell^{[3]}, \Theta^{[3]}) [\mu] \longmapsto \beta [\mu] \longmapsto \mathcal{A} [\mu] .\)

Outer contraction: start at \(\mu^{(0)}_q = b_{\ast}\); iterate \(\mu^{(n + 1)} =\mathcal{A} [\mu^{(n)}]\), in the norm \(\| \mu \|_{\ast} = \sup_{q \geqslant q_{\ast}} | \mu_q | / (\lambda \sqrt{q})\):

\(\displaystyle \|\mathcal{A}[\mu] -\mathcal{A}[\hat{\mu}]\|_{\ast} \leqslant Cg_{q_{\ast}} \| \mu - \hat{\mu} \|_{\ast} .\)

Compare two scalar paths with the same \(b_{\ast}\), \(d = \| \mu - \hat{\mu} \|_{\ast}\); \(\delta\) denotes output differences, \(\delta M^{[j]} = \delta \mu I + \delta \Theta^{[j]}\).

\(\displaystyle R^{[j]} - \hat{R}^{[j]} = - R^{[j]} (\delta M^{[j]} + D \delta \ell^{[j]}) \hat{R}^{[j]} .\)

The relative scalar and centred insertions obey

\(\displaystyle \frac{| \delta \mu_q |}{q} \leqslant g_q d, \qquad \eta_q (\delta \Theta^{[j]}) \lesssim g_q^3 d,\)

the latter by integrating the differences of the centred sources from infinity. Loop comparisons through the two levels give \(\mathcal{N}_{k, q} (\delta \ell^{[j]}) \lesssim g_q^2 qr_q^{2 - k} d\), \(j = 2, 3\), \(4 \leqslant k \leqslant 12 - 2 j\) (the seed does not depend on \(\mu\)).

Scalar update: the common boundary value \(b_{\ast}\) cancels. With \(x = q / q_{\ast} \geqslant 1\),

\(\displaystyle | \delta \mathcal{A}[\mu]_q | \leqslant C \lambda^2 d \int_{q_{\ast}}^q \frac{\mathrm{d} s}{s} = C \lambda^2 \log xd,\) \(\displaystyle \frac{| \delta \mathcal{A}[\mu]_q |}{\lambda \sqrt{q}} \leqslant Cg_{q_{\ast}} \frac{\log x}{\sqrt{x}} d \leqslant \frac{2 C}{\mathrm{e}} g_{q_{\ast}} d.\)

The centred kernels are not part of the unknown: their variation is already contained in \(\delta \beta\).

A loop needs \(k + 2\) legs to control a \(k\)-leg vertex. Every evaluation of \(\mathcal{A}\) uses the same budget:

\(\displaystyle \ell^{[1]} \text{: } 10 \longrightarrow \ell^{[2]} \text{: } 8 \longrightarrow \ell^{[3]} \text{: } 6 \longrightarrow \text{defect: } 4.\)

A \(k\)-leg force vertex has \(k - 1\) field derivatives. The mass source uses only the four-leg vertex of \(\ell^{[3]}\), the centred sources those of \(\ell^{[1]}\), \(\ell^{[2]}\).

• A path difference inserts \(\delta M^{[j]}\) in a resolvent. It is field-independent: \(D_y \delta M^{[j]} = 0\). The comparison uses the same field-derivative ranges as the construction.

• \(\mathcal{A}\) returns a scalar path. At the next outer step, restart from the original seed and perform the same two steps. We do not use the previous \(\ell^{[3]}\) as a new seed.

The two divergent self-energies of Appendix A.2 of Lecture 5:

• Tadpole: \(- \lambda \int_{q_{\ast}}^{\infty} \text{Tr} R_{0, q}^2 \hspace{0.17em} \mathrm{d} q \approx - \lambda \text{Tr} C \sim - \lambda N\). Wick ordering: \(\frac{\lambda}{12} \varphi^4 - \frac{1}{2} \lambda c \varphi^2 = \frac{\lambda}{12} H_4 (\varphi ; c) +\)const.

• Sunset: contracting the bubble in \(D^3 \ell^{[3]} (0)\) with \(R_0^2\) and integrating gives the two-loop self-energy,

  \(\displaystyle \frac{2}{3} \lambda^2 \int C^3 \approx \frac{\lambda^2}{24 \pi^2} \log N \quad (= 2 \gamma_T \text{ of Lecture 5}) .\)

Power counting: nothing else. In the flow: the integrals from \(q = \infty\) of all vertices with \(k \geqslant 4\) legs converge; only the constant mode of the linear jet needs a boundary condition at a finite scale.

\(\mu_q\) at fixed \(q\) has a limit as \(N \to \infty\); \(\mu_N\) does not. Never interchange the two limits.

The cross is \(\partial_q K_q = 1\), as before; with the two lines through it, it gives \((R^{[3]}_{0, q})^2\). The bubble in \(D^3 \ell^{[3]} (0)\) produces the sunset.

The flow at large fields

Lecture 6

All bounds hold on \(\mathcal{B}_q\) only. The free field exceeds \(r_q\) with small, nonzero probability, and nothing a priori controls the interacting forecast. We need \(\bar{\mathcal{F}}_q\) defined and uniformly coercive for all fields.

Perturbative control for \(|y| \lesssim r_q\); outside, a non-perturbative replacement that keeps the structure the estimates use: coercivity, carried by the bare cube.

One site: \(qy + \frac{\lambda}{3} y^3 = q \Phi_q\), noise coefficient \(H_q (y) = (q + \lambda y^2)^{- 1} \leqslant q^{- 1} \wedge (\lambda y^2)^{- 1}\). Large fields are more elliptic and less noisy.

\(\tau\) smooth odd, \(\tau (u) = u\) on \([- 1, 1]\), constant outside \([- 2, 2]\), \(0 \leqslant \tau \prime \leqslant 1\);

\(\displaystyle \kappa_q (y) = r_q \tau (y / r_q), \quad \tilde{\ell}_q = D \kappa_q \cdot \ell^{[3]}_q \circ \kappa_q, \quad \bar{\mathcal{F}}_q = U\prime + M_q y + \tilde{\ell}_q .\)

The cubic and linear parts are not clipped, and \(D \kappa_q\) keeps the force a gradient. Globally: \(s (D^{j - 1} \tilde{\ell}_q) \leqslant Cg_q qr_q^{2 - j}\) (\(j = 2, 3, 4\)), \(s (D \partial_q \tilde{\ell}_q) \leqslant Cg_q\).

Neumann series for \(H = (I + G (M + D \tilde{\ell}))^{- 1} G\), \(G = (K_q + \lambda y^2)^{- 1} \leqslant C_q\).

\(b = b^{\text{b}} + b^{\text{l}}\), \(b^{\text{b}} = - H \hspace{0.17em} D \kappa \hspace{0.17em} \mathcal{D}^{\text{o}} (\kappa y)\), \(t_q (y) = y \mathbb{1}_{|y| > r_q}\).

Proposition. (uniformly in \(y\), \(N\), \(p\))

\(\displaystyle \|b^{\text{b}}_q \|_p \leqslant C \lambda^{7 / 2} q^{- 5 / 2}, \qquad \|b^{\text{l}}_q \|_p \leqslant C \lambda q^{- 3 / 2} (\|t_q \|_p + r_q^{- 1} \|y^2 \mathbb{1}_{|y| > r_q} \|_p),\)

\(\displaystyle \|b^{\text{l}}_q \|_2^2 \leqslant C \lambda^4 q^{- 5} \|t_q \|_6^6 .\)

Sources of \(b^{\text{l}}\) (four groups): the mass term \(M' y - JM' \kappa y\) (\(J = D \kappa\), \(M' = \partial_q M\); linear part unclipped, flow computed at \(\kappa y\)); the motion of the clip; cubic vertex at \(y\) vs correction at \(\kappa y\); chain-rule terms with \(\kappa'', \kappa'''\). All vanish if \(\|y\|_{\infty} \leqslant r_q\).

The point

\(b^{\text{l}}\) is as large as the whole drift, but supported on large fields. It is controlled by how often the actual dynamics visits them: \(\int q^{- 3 / 2} \lambda \|t_q (Y_q)\|_6^6 \hspace{0.17em} \mathrm{d} q\) (Lecture 7).

With \(U' = \frac{\lambda}{3} y^3\), up to a small Lipschitz perturbation, the forecast solves the cubic elliptic equation driven by the field at resolution \(q\):

\(\displaystyle K_q \Phi_q = K_q (Y_q + \varrho_q (Y_q)) + \frac{\lambda}{3} Y_q^3,\) \(\displaystyle \varrho_q (y) = C_q (M_q y + \tilde{\ell}_q (y)), \quad \|D \varrho_q \|_{p \to p} \leqslant C (g_q + g_q^2) .\)

The expansion needs \(g_q\) small, so \(\bar{\mathcal{F}}_q\) is built for \(q \geqslant q_{\ast}\). Below \(q_{\ast}\) nothing needs constructing: freeze it,

\(\displaystyle \bar{\mathcal{F}}_q (y) = \bar{\mathcal{F}}_{q_{\ast}} (y) + (q_{\ast} - q) y.\)

Then \(K_q + D \bar{\mathcal{F}}_q = K_{q_{\ast}} + D \bar{\mathcal{F}}_{q_{\ast}}\): coercivity is automatic, \(H_q = H_{q_{\ast}}\), and \(y_q (x) = y_{q_{\ast}} (C_{q_{\ast}} K_q x)\).

The defect is explicit: \(\mathcal{D}_s (y) = - y + \frac{1}{2} \text{Ctr}_{H_{q_{\ast}}^2} D^2 \bar{\mathcal{F}}_{q_{\ast}} (y)\), with \(\|\mathcal{D}_s (y)\|_p \leqslant C \|y\|_p\).

So the system holds on \([q_0, \infty)\) with \(Y_{q_0} = 0\): the initial value is deterministic, with no large mass. \(q_{\ast}\) stays only in the proofs, where the expansion and the closure of the loop need \(g_q\) small.

At every lattice:

• \(\mu_N\), \(M_q\) and a globally coercive \(\bar{\mathcal{F}}_q\), all deterministic;

• the exact FBSDE for \((Y, Z)\);

• a source whose bounded-field part is integrable (two steps), and whose large-field part is controlled by sixth powers of \(Y\) on large fields.

Next: estimates on \((Y, Z)\) uniform in \(N\).