Scale cocycles, finite scheme changes and controlled RG equations

Assume Countable Choice.

Let $\mathbb{M}={\mathbb{R}}^4$ have Minkowski metric $\eta =\mathrm{diag}(-1,1,1,1)$. A scalar local polynomial functional is a finite sum $F(\varphi )=c+{\sum }_{j=1}^N{\int }_{\mathbb{M}}{f}_j(x)P_j(j^{s_j}\varphi (x))d^{4}x,$ where $N$ is a finite nonnegative integer, $\varphi \in C^{\mathrm{\infty }}(\mathbb{M};\mathbb{R})$, $c\in \mathbb{C}$, and $f_j\in C_c^{\mathrm{\infty }}(\mathbb{M};\mathbb{C})$. For a finite integer $s_j\ge 0$, the $s_j$-jet is $j^{s_j}\varphi (x)=({\mathrm{\partial }}^{\alpha }\varphi (x){)}_{\alpha \in \{0,1,2,\dots {\}}^4,\mathrm{\mid }\alpha \mathrm{\mid }\le s_j}$; each $P_j$ is a complex polynomial in these finitely many jet coordinates. This is the local functional class used by the supplied scalar time-ordered system.

For every mass label $m\ge 0$, suppose a normalized scalar time-ordered system $T_n^{(m)}$ with the same local scalar contract is supplied. Define its coefficientwise formal S-map by ${\mathcal{S}}_m(V)={\sum }_{n=0}^{\mathrm{\infty }}\frac{1}{n!}{T}_n^{(m)}(V,\dots ,V),T_0^{(m)}=1,T_1^{(m)}=\mathrm{id}\mathrm{.}$ The series is formal in the interaction degree; no analytic convergence is asserted.

Use units $c=\mathrm{\hslash }=1$, so the mass label $m$ has mass dimension one. The scale parameter $\rho >0$ is dimensionless. On scalar field configurations put $(C_{\rho }\varphi )(x)={\rho }^{-1}\varphi ({\rho }^{-1}x),$ the engineering scaling of a four-dimensional scalar field. For a mass-indexed family of functionals $F=(F_m{)}_{m\ge 0}$, define $({\sigma }_{\rho }F{)}_m=C_{\rho }^{\ast }{F}_{{\rho }^{-1}m},(C_{\rho }^{\ast }F)(\varphi )=F(C_{{\rho }^{-1}}\varphi )\mathrm{.}$ Thus the component at mass $m$ is pulled back from the component at mass ${\rho }^{-1}m$; equivalently the mass label scales as $m\mapsto \rho m$. If $A=(A_m{)}_{m\ge 0}$ is instead a family of maps from local functionals (or their formal series) to local or microcausal functionals (or their formal series), define its induced scale conjugate, denoted ${\sigma }_{\rho }A{\sigma }_{\rho }^{-1}$, by $({\sigma }_{\rho }A{\sigma }_{\rho }^{-1}{)}_m=C_{\rho }^{\ast }\circ A_{{\rho }^{-1}m}\circ (C_{\rho }^{\ast }{)}^{-1},(C_{\rho }^{\ast }{)}^{-1}=C_{{\rho }^{-1}}^{\ast }\mathrm{.}$ Here the rightmost pullback acts on the input, the leftmost pullback acts on every output coefficient, and formal indeterminates are fixed. In particular the mass-$m$ component of the scale-conjugated formal S-map is $({\sigma }_{\rho }\mathcal{S}{\sigma }_{\rho }^{-1}{)}_m=C_{\rho }^{\ast }\circ {\mathcal{S}}_{{\rho }^{-1}m}\circ (C_{\rho }^{\ast }{)}^{-1}\mathrm{.}$ These functional and map actions are extended coefficientwise to formal series. They obey ${\sigma }_1=\mathrm{id}$ and ${\sigma }_{\rho \tau }={\sigma }_{\rho }{\sigma }_{\tau }$; the dilation sends compact supports to compact supports and normalizes the proper orthochronous Poincare action.

For $m\ge 0$, $\rho >0$ and each $n\ge 0$, define the conjugated sequence of multilinear maps componentwise by $T_{n,\rho }^{\mathrm{\prime }(m)}(F_1,\dots ,F_n):=C_{\rho }^{\ast }\text{\hspace{0.17em}⁣}\left[T_n^{({\rho }^{-1}m)}\right((C_{\rho }^{\ast }{)}^{-1}{F}_1,\dots ,(C_{\rho }^{\ast }{)}^{-1}{F}_n\left)\right]\mathrm{.}$ For $n=0$ this means $T_{0,\rho }^{\mathrm{\prime }(m)}=C_{\rho }^{\ast }1=1$; for $n=1$ it gives $T_{1,\rho }^{\mathrm{\prime }(m)}=\mathrm{id}$. Assume for every $m\ge 0$ and $\rho >0$ that this sequence, as well as the supplied sequence $(T_n^{(m)}{)}_{n\ge 0}$, satisfies the normalized local scalar time-ordered-system contract for the same free data at mass $m$, after transporting the free data at mass ${\rho }^{-1}m$ by the stated pullback. This is the scale-admissibility condition on the sequences. Their formal S-maps are distinct objects: by multilinearity and coefficientwise pullback, the S-map of $(T_{n,\rho }^{\mathrm{\prime }(m)}{)}_{n\ge 0}$ is exactly $({\sigma }_{\rho }\mathcal{S}{\sigma }_{\rho }^{-1}{)}_m$ displayed above. Define $Z_{\rho }=(Z_{\rho ,m}{)}_{m\ge 0}$ by the componentwise equations $C_{\rho }^{\ast }\circ {\mathcal{S}}_{{\rho }^{-1}m}\circ (C_{\rho }^{\ast }{)}^{-1}={\mathcal{S}}_m\circ Z_{\rho ,m}(m\ge 0),$ where each $Z_{\rho ,m}$ is the unique formal local map with identity linear term, supplied by the finite-renormalization comparison proposition under these hypotheses. The notation $Z_{\rho ,m}(V)=V+O(V^2)$ means that all terms after the linear term have interaction degree at least two. This is a conditional definition: no existence of a time-ordered system for any particular model is asserted.

Local scalar time-ordered product contract

Renormalization maps form a formal composition group

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The time-ordered systems and scale-admissibility condition are supplied inputs. Their comparison map is determined uniquely by a triangular coefficient recursion under Countable Choice from the actual contraction contract. Constructing time-ordered systems is a separate theorem with its own Choice hypotheses; no existence of a system for a particular model is asserted here.

Assume Countable Choice.

Let $Z_{\rho }$ be the mass-indexed formal renormalization map defined for the scale action ${\sigma }_{\rho }$, $\rho >0$. For all $\rho ,\tau >0$, $Z_{\rho \tau }=Z_{\rho }\circ \left({\sigma }_{\rho }{Z}_{\tau }{\sigma }_{\rho }^{-1}\right)\mathrm{.}$ If ${\sigma }_{\rho }{Z}_{\tau }{\sigma }_{\rho }^{-1}=Z_{\tau }$ for every $\rho ,\tau >0$, this reduces to $Z_{\rho \tau }=Z_{\rho }\circ Z_{\tau }$. No one-parameter subgroup law is asserted without this condition.

Formal scaling and renormalization cocycle

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The mass label is part of the scale action. Suppressing it in the notation does not turn the cocycle into a group homomorphism; the required action on the renormalization maps remains explicit in the formula.

Let $\rho >0$ be a dimensionless scale and put $t=\log \rho \in \mathbb{R}$. Take $I\subseteq \mathbb{R}$ to be an interval containing $0$. Let $g:I\to \mathbb{R}$ be a differentiable real coupling and let $b,g_0\in \mathbb{R}$ be dimensionless constants; require $g$ to satisfy the truncated scalar initial-value problem $\frac{dg}{dt}=b{g}^2,g(0)=g_0\mathrm{.}$ A solution through $(0,g_0)$ is maximal if there is no strictly larger interval $J⊋I$ and solution $G:J\to \mathbb{R}$ of the same initial-value problem with $G{\mathrm{\mid }}_I=g$. If $c:=b{g}_0=0$, the unique maximal real solution is $g(t)=g_0$ on $\mathbb{R}$. If $c\mathrm{\ne }0$, the unique maximal solution through $t=0$ is $g(t)=\frac{g_0}{1-b{g}_{0}t}$ on $(-\mathrm{\infty },c^{-1})$ when $c>0$ and on $(c^{-1},\mathrm{\infty })$ when $c<0$.

For the formal beta-series calculation, let $u$ be an indeterminate and work in $\mathbb{R}[[u]]$. Write $\beta (u)=b{u}^2+O(u^3)$ and let $\mathrm{\Phi }(u)=u+a{u}^2+O(u^3)$ be an invertible formal change of dimensionless coupling, with dimensionless $a\in \mathbb{R}$. The notation $O(u^k)$ means an element of $u^k\mathbb{R}[[u]]$, so its terms have degree at least $k$; the real trajectory $g(t)$ is distinct from the formal indeterminate $u$. Put $\tilde{u}=\mathrm{\Phi }(u)$, let ${\mathrm{\Phi }}^{\mathrm{\prime }}$ be the termwise formal derivative, and define the transformed formal vector field in $\mathbb{R}[[\tilde{u}]]$ by $\tilde{\beta }(\tilde{u})=({\mathrm{\Phi }}^{\mathrm{\prime }}(u)\beta (u)){\mid }_{u={\mathrm{\Phi }}^{-1}(\tilde{u})}\mathrm{.}$ The quadratic coefficient of $\tilde{\beta }$ is $b$. The real ODE is exact only for the displayed truncation; neither it nor the formal scheme-change statement determines $b$ from a QFT graph or bounds omitted higher-order terms. Its pole is a property of this truncated equation, not a theorem about an ultraviolet completion.

First-order systems, initial value problems, and solutions on intervals

The adjacent Low-momentum matching has a controlled remainder is a separate spectral remainder estimate under its low-energy band hypothesis. It supplies no estimate for omitted terms in this beta truncation. This theorem derives no numerical QED beta coefficient and identifies no scheme parameter with a measured observable. The context of Formal scaling and renormalization cocycle motivates the logarithmic scale convention; the present theorem uses that convention directly and assumes no existence of a renormalization map or time-ordered system.