Contents

# Contents

## Idea

In perturbative quantum field theory the construction of the scattering matrix $\mathcal{S}$, hence of the interacting field algebra of observables for a given interaction $g S_{int}$ perturbing around a given free field vacuum, involves choices of normalization of time-ordered products/Feynman diagrams (traditionally called "re"-normalizations) encoding new interactions that appear where several of the original interaction vertices defined by $g S_{int}$ coincide.

Whenever a group $RG$ acts on the space of observables of the theory such that conjugation by this action takes ("re"-)normalization schemes into each other, then these choices of ("re"-)normalization are parameterized by – or “flow with” – the elements of $RG$. This is called renormalization group flow (prop. below); often called RG flow, for short.

The archetypical example here is the group $RG$ of scaling transformations on Minkowski spacetime (def. below), which induces a renormalization group flow (prop. below) due to the particular nature of the Wightman propagator resp. Feynman propagator on Minkowski spacetime (example below). In this case the choice of ("re"-)normalization hence “flows with scale”.

Now the main theorem of perturbative renormalization states that (if only the basic renormalization condition called “field independence” is satisfied) any two choices of ("re"-)normalization schemes $\mathcal{S}$ and $\mathcal{S}'$ are related by a unique interaction vertex redefinition $\mathcal{Z}$, as

$\mathcal{S}' = \mathcal{S} \circ \mathcal{Z} \,.$

Applied to a parameterization/flow of renormalization choices by a group $RG$ this hence induces an interaction vertex redefinition as a function of $RG$. One may think of the shape of the interaction vertices as fixed and only their (adiabatically switched) coupling constants as changing under such an interaction vertex redefinition, and hence then one has coupling constants $g_j$ that are parameterized by elements $\rho$ of $RG$:

$\mathcal{Z}_{\rho_{vac}}^\rho \;\colon\; \{g_j\} \mapsto \{g_j(\rho)\}$

This dependendence is called running of the coupling constants under the renormalization group flow (def. below).

One example of renormalization group flow is that induced by scaling transformations (prop. below). This is the original and main example of the concept (Gell-Mann & Low 54)

In this case the running of the coupling constants may be understood as expressing how “more” interactions (at higher energy/shorter wavelength) become visible (say to experiment) as the scale resolution is increased. In this case the dependence of the coupling $g_j(\rho)$ on the parameter $\rho$ happens to be differentiable; its logarithmic derivative (denoted “$\psi$” in Gell-Mann & Low 54) is known as the beta function (Callan 70, Symanzik 70):

$\beta(g) \coloneqq \rho \frac{\partial g_j}{\partial \rho} \,.$

Notice that this is related to, but conceptually different from, Polchinski's flow equation in the context of Wilsonian RG.

The running of the coupling constants is not quite a representation of the renormalization group flow, but it is a “twisted” representation, namely a group 1-cocycle (prop. below). For the case of scaling transformations this may be called the Gell-Mann-Low renormalization cocycle (Brunetti-Dütsch-Fredenhagen 09).

For more see at

## Definition

###### Proposition

(renormalization group flow)

Let

$vac \;\coloneqq\; (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$

be a relativistic free vacuum (according to this def.) around which we consider interacting perturbative QFT.

Consider a group $RG$ equipped with an action on the Wick algebra of off-shell microcausal polynomial observables with formal parameters adjoined (as in this def.)

$rg_{(-)} \;\colon\; RG \times PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g, j ] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ \hbar, g, j ] ] \,,$

hence for each $\rho \in RG$ a continuous linear map $rg_\rho$ which has an inverse $rg_\rho^{-1} \in RG$ and is a homomorphism of the Wick algebra-product (the star product $\star_H$ induced by the Wightman propagator of the given vauum $vac$)

$rg_\rho( A_1 \star_H A_2 ) \;=\; rg_\rho(A_1) \star_H rg_\rho(A_2)$

such that the following conditions hold:

1. the action preserves the subspace of off-shell polynomial local observables, hence it restricts as

$rg_{(-)} \;\colon\; RG \times LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g,j\rangle \longrightarrow LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g,j\rangle$
2. the action respects the causal order of the spacetime support (this def.) of local observables, in that for $O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]$ we have

$\left( supp(O_1) \,{\vee\!\!\!\wedge}\, supp(O_2) \right) \phantom{A} \Rightarrow \phantom{A} \left( supp(rg_\rho(O_1)) \,{\vee\!\!\!\wedge}\, supp(rg_\rho(O_2)) \right)$

for all $\rho \in RG$.

Then:

The operation of conjugation by this action on observables induces an action on the set of S-matrix renormalization schemes (this def., this remark), in that for

$\mathcal{S} \;\colon\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g, j] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})( (\hbar) )[ [ g, j] ]$

a perturbative S-matrix scheme around the given free field vacuum $vac$, also the composite

$\mathcal{S}^\rho \;\coloneqq\; rg_\rho \circ \mathcal{S} \circ rg_{\rho}^{-1} \;\colon\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g, j] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})( (\hbar) )[ [ g, j] ]$

is an S-matrix scheme, for all $\rho \in RG$.

More generally, let

$vac_\rho \;\coloneqq\; (E_{\text{BV-BRST}}, \mathbf{L}'_\rho, \Delta_{H,\rho} )$

be a collection of gauge fixed free field vacua parameterized by elements $\rho \in RG$, all with the same underlying field bundle; and consider $rg_\rho$ as above, except that it is not an automorphism of any Wick algebra, but an isomorphism between the Wick algebra-structures on various vacua, in that

(1)$rg_{\rho}( A_1 \star_{H, \rho^{-1} \rho_{vac}} A_2 ) \;=\; rg_{\rho}(A_1) \star_{H, \rho_{vac}} rg_{\rho}(A_2)$

for all $\rho, \rho_{vac} \in RG$

Then if

$\{ \mathcal{S}_{\rho} \}_{\rho \in RG}$

is a collection of S-matrix schemes, one around each of the gauge fixed free field vacua $vac_\rho$, it follows that for all pairs of group elements $\rho_{vac}, \rho \in RG$ the composite

(2)$\mathcal{S}_{\rho_{vac}}^\rho \;\coloneqq\; rg_\rho \circ \mathcal{S}_{\rho^{-1}\rho_{vac}} \circ rg_\rho^{-1}$

is an S-matrix scheme around the vacuum labeled by $\rho_{vac}$.

Since therefore each element $\rho \in RG$ in the group $RG$ picks a different choice of normalization of the S-matrix scheme around a given vacuum at $\rho_{vac}$, we call the assignment $\rho \mapsto \mathcal{S}_{\rho_{vac}}^{\rho}$ a re-normalization group flow.

###### Proof

It is clear from the definition that each $\mathcal{S}^{\rho}_{\rho_{vac}}$ satisfies the axiom “perturbation” (in this def.).

In order to verify the axiom “causal additivity”, observe, for convenience, that by this prop. it is sufficient to check causal factorization.

So consider $O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$ two local observables whose spacetime support is in causal order.

$supp(O_1) \;{\vee\!\!\!\wedge}\; supp(O_2) \,.$

We need to show that the

$\mathcal{S}_{\rho_{vac}}^{\rho}(O_1 + O_2) = \mathcal{S}_{\rho_{vac}}^\rho(O_1) \star_{H,\rho_{vac}} \mathcal{S}_{vac_e}^\rho(O_2)$

for all $\rho, \rho_{vac} \in RG$.

Using the defining properties of $rg_{(-)}$ and the causal factorization of $\mathcal{S}_{\rho^{-1}\rho_{vac}}$ we directly compute as follows:

\begin{aligned} \mathcal{S}_{\rho_{vac}}^\rho(O_1 + O_2) & = rg_\rho \circ \mathcal{S}_{\rho^{-1} \rho_{vac}} \circ rg_\rho^{-1}( O_1 + O_2 ) \\ & = rg_\rho \left( {\, \atop \,} \mathcal{S}_{\rho^{-1}\rho_{vac}} \left( rg_\rho^{-1}(O_1) + rg_\rho^{-1}(O_2) \right) {\, \atop \,} \right) \\ & = rg_\rho \left( {\, \atop \,} \left( \mathcal{S}_{\rho^{-1}\rho_{vac}}\left(rg_\rho^{-1}(O_1)\right) \right) \star_{H, \rho^{-1} \rho_{vac}} \left( \mathcal{S}_{ \rho^{-1} \rho_{vac} }\left(rg_\rho^{-1}(O_2)\right) \right) {\, \atop \,} \right) \\ & = rg_\rho \left( {\, \atop \,} \mathcal{S}_{\rho^{-1} \rho_{vac}}\left(rg_{\rho^{-1}}(O_1)\right) {\, \atop \,} \right) \star_{H, \rho_{vac}} rg_\rho \left( {\, \atop \,} \mathcal{S}_{\rho^{-1} \rho_{vac}}\left( rg_\rho^{-1}(O_2)\right) {\, \atop \,} \right) \\ & = \mathcal{S}^\rho_{\rho_{vac}}( O_1 ) \, \star_{H, \rho_{vac}} \, \mathcal{S}_{\rho_{vac}}^\rho(O_2) \,. \end{aligned}
###### Definition

(running coupling constants)

Let

$vac \coloneqq vac_e \;\coloneqq\; (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$

be a relativistic free vacuum (according to this def.) around which we consider interacting perturbative QFT, let $\mathcal{S}$ be an S-matrix scheme around this vacuum and let $rg_{(-)}$ be a renormalization group flow according to prop. , such that each re-normalized S-matrix scheme $\mathcal{S}_{vac}^\rho$ satisfies the renormalization condition “field independence”.

Then by the main theorem of perturbative renormalization (this prop.) there is for every pair $\rho_1, \rho_2 \in RG$ a unique interaction vertex redefinition

$\mathcal{Z}_{\rho_{vac}}^{\rho} \;\colon\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ] \longrightarrow LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]$

which relates the corresponding two S-matrix schemes via

(3)$\mathcal{S}_{\rho_{vac}}^{\rho} \;=\; \mathcal{S}_{\rho_{vac}} \circ \mathcal{Z}_{\rho_{vac}}^\rho \,.$

If one thinks of an interaction vertex, hence a local observable $g S_{int}+ j A$, as specified by the (adiabatically switched) coupling constants $g_j \in C^\infty_{cp}(\Sigma)\langle g \rangle$ multiplying the corresponding interaction Lagrangian densities $\mathbf{L}_{int,j} \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})$ as

$g S_{int} \;=\; \underset{j}{\sum} \tau_\Sigma \left( g_j \mathbf{L}_{int,j} \right)$

(where $\tau_\Sigma$ denotes transgression of variational differential forms) then $\mathcal{Z}_{\rho_1}^{\rho_2}$ exhibits a dependency of the (adiabatically switched) coupling constants $g_j$ of the renormalization group flow parameterized by $\rho$. The corresponding functions

$\mathcal{Z}_{\rho_{vac}}^{\rho}(g S_{int}) \;\colon\; (g_j) \mapsto (g_j(\rho))$

are then called running coupling constants.

## Properties

###### Proposition

(running coupling constants are group cocycle over renormalization group flow)

Consider running coupling constants

$\mathcal{Z}_{\rho_{vac}}^{\rho} \;\colon\; (g_j) \mapsto (g_j(\rho))$

as in def. . Then for all $\rho_{vac}, \rho_1, \rho_2 \in RG$ the following equality is satisfied by the “running functions” (3):

$\mathcal{Z}_{\rho_{vac}}^{\rho_1 \rho_2} \;=\; \mathcal{Z}_{\rho_{vac}}^{\rho_1} \circ \left( \sigma_{\rho_1} \circ \mathcal{Z}_{\rho^{-1} \rho_{vac}}^{\rho_2} \circ \sigma_{\rho_1}^{-1} \right) \,.$
###### Proof

Directly using the definitions, we compute as follows:

\begin{aligned} \mathcal{S}_{\rho_{vac}} \circ \mathcal{Z}_{\rho_{vac}}^{\rho_1 \rho_2} & = \mathcal{S}_{\rho_{vac}}^{\rho_1 \rho_2 } \\ & = \sigma_{\rho_1} \circ \underset{ = \mathcal{S}_{\rho_1^{-1} \rho_{vac}}^{\rho_2} = \mathcal{S}_{\rho_1^{-1} \rho_{vac}} \circ \mathcal{Z}_{\rho_1^{-1} \rho_vac}^{\rho_2} }{ \underbrace{ \sigma_{\rho_2} \circ \mathcal{S}_{\rho_2^{-1}\rho_1^{-1}\rho_{vac}} \circ \sigma_{\rho_2}^{-1} }} \circ \sigma_{\rho_1}^{-1} \\ & = \underset{ = \mathcal{S}_{\rho_{vac}} \circ \mathcal{Z}_{\rho_{vac}}^{\rho_1} \circ \sigma_{\rho_1} }{ \underbrace{ \sigma_{\rho_1} \circ \mathcal{S}_{\rho_1^{-1} \rho_{vac}} \circ \overset{ = id }{ \overbrace{ \sigma_{\rho_1}^{-1} \circ \sigma_{\rho_1} } } }} \circ \mathcal{Z}_{\rho_1^{-1} \rho_{vac}}^{\rho_2} \circ \sigma_{\rho_1}^{-1} \\ & = \mathcal{S}_{\rho_{vac}} \circ \mathcal{Z}_{\rho_{vac}}^{\rho_1} \circ \underbrace{ \sigma_{\rho_1} \circ \mathcal{Z}_{\rho_1^{-1} \rho_{vac}}^{\rho_2} \circ \sigma_{\rho_1}^{-1} } \end{aligned}

This demonstrates the equation between vertex redefinitions to be shown after composition with an S-matrix scheme. But by the uniqueness-clause in the main theorem of perturbative renormalization the composition operation $\mathcal{S}_{\rho_{vac}} \circ (-)$ as a function from vertex redefinitions to S-matrix schemes is injective. This implies the equation itself.

## Examples

### Scaling transformations

We discuss (prop. below) that, if the field species involved have well-defined mass dimension (example below) then scaling transformations on Minkowski spacetime (example below) induce a renormalization group flow (def. ). This is the original and main example of renormalization group flows (Gell-Mann& Low 54).

###### Example

(scaling transformations and mass dimension)

Let

$E \overset{fb}{\longrightarrow} \Sigma$

be a field bundle which is a trivial vector bundle over Minkowski spacetime $\Sigma = \mathbb{R}^{p,1} \simeq_{\mathbb{R}} \mathbb{R}^{p+1}$.

For $\rho \in (0,\infty) \subset \mathbb{R}$ a positive real number, write

$\array{ \Sigma &\overset{\rho}{\longrightarrow}& \Sigma \\ x &\mapsto& \rho x }$

for the operation of multiplication by $\rho$ using the real vector space-structure of the Cartesian space $\mathbb{R}^{p+1}$ underlying Minkowski spacetime.

By pullback this acts on field histories (sections of the field bundle) via

$\array{ \Gamma_\Sigma(E) &\overset{\rho^\ast}{\longrightarrow}& \Gamma_\Sigma(E) \\ \Phi &\mapsto& \Phi(\rho(-)) } \,.$

Let then

$\rho \mapsto vac_\rho \;\coloneqq\; (E_{\text{BV-BRST}}, \mathbf{L}'_{\rho}, \Delta_{H,\rho} )$

be a 1-parameter collection of relativistic free vacua on that field bundle, according to this def., and consider a decomposition into a set $Spec$ of field species (this def.) such that for each $sp \in Spec$ the collection of Feynman propagators $\Delta_{F,\rho,sp}$ for that species scales homogeneously in that there exists

$dim(sp) \in \mathbb{R}$

such that for all $\rho$ we have (using generalized functions-notation)

(4)$\rho^{ 2 dim(sp) } \Delta_{F, 1/\rho, sp}( \rho x ) \;=\; \Delta_{F,sp, \rho = 1}(x) \,.$

Typically $\rho$ rescales a mass parameter, in which case $dim(sp)$ is also called the mass dimension of the field species $sp$.

Let finally

$\array{ PolyObs(E) & \overset{ \sigma_\rho }{\longrightarrow} & PolyObs(E) \\ \mathbf{\Phi}_{sp}^a(x) &\mapsto& \rho^{- dim(sp)} \mathbf{\Phi}^a( \rho^{-1} x ) }$

be the function on off-shell polynomial observables given on field observables $\mathbf{Phi}^a(x)$ by pullback along $\rho^{-1}$ followed by multiplication by $\rho$ taken to the negative power of the mass dimension, and extended from there to all polynomial observables as an algebra homomorphism.

This constitutes an action of the group

$RG \coloneqq \left( \mathbb{R}_+, \cdot \right)$

of positive real numbers (under multiplication) on polynomial observables, called the group of scaling transformations for the given choice of field species and mass parameters.

###### Example

(mass dimension of scalar field)

Consider the Feynman propagator $\Delta_{F,m}$ of the free real scalar field on Minkowski spacetime $\Sigma = \mathbb{R}^{p,1}$ for mass parameter $m \in (0,\infty)$; a Green function for the Klein-Gordon equation.

Let the group $RG \coloneqq (\mathbb{R}_+, \cdots)$ of scaling transformations $\rho \in \mathbb{R}_+$ on Minkowski spacetime (def. ) act on the mass parameter by inverse multiplication

$(\rho , \Delta_{F,m}) \mapsto \Delta_{F,\rho^{-1}m}(\rho (-)) \,.$

Then we have

$\Delta_{F,\rho^{-1}m}(\rho (-)) \;=\; \rho^{-(p+1) + 2} \Delta_{F,1}(x)$

and hence the corresponding mass dimension (def. ) of the real scalar field on $\mathbb{R}^{p,1}$ is

$dim(\text{scalar field}) = (p+1)/2 - 1 \,.$
###### Proof

By (this prop.) the Feynman propagator in question is given by the Cauchy principal value-formula (in generalized function-notation)

\begin{aligned} \Delta_{F,m}(x) & = \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu x^\mu} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k \,. \end{aligned}

By applying change of integration variables $k \mapsto \rho^{-1} k$ in the Fourier transform this becomes

\begin{aligned} \Delta_{F,\rho^{-1}m}(\rho x) & = \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu \rho x^\mu} }{ - k_\mu k^\mu - \left( \rho^{-1} \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k \\ & = \rho^{-(p+1)} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu x^\mu} }{ - \rho^{-2} k_\mu k^\mu - \rho^{-2} \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k \\ & = \rho^{-(p+1)+2} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu x^\mu} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k \\ & = \rho^{-(p+1) + 2} \Delta_{F,m}(x) \end{aligned}
###### Proposition

(scaling transformations are renormalization group flow)

Let

$vac \coloneqq vac_m \coloneqq (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_{H,m})$

be a relativistic free vacua on that field bundle, according to this def. equipped with a decomposition into a set $Spec$ of field species (this def.) such that for each $sp \in Spec$ the collection of Feynman propagators the corresponding field species has a well-defined mass dimension $dim(sp)$ (def. )

Then the action of the group $RG \coloneqq (\mathbb{R}_+, \cdot)$ of scaling transformations (def. ) is a renormalization group flow in the sense of this prop..

###### Proof

It is clear that rescaling preserves causal order and the renormalization condition of “field indepencen”.

The condition we need to check is that for $A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]$ two microcausal polynomial observables we have for any $\rho, \rho_{vac} \in \mathbb{R}_+$ that

$\sigma_\rho \left( A_1 \star_{H, \rho^{-1} \rho_{vac} c} A_2 \right) \;=\; \sigma_\rho(A_1) \star_{H,\rho_{vac}} \sigma_\rho(A_2) \,.$

By the assumption of decomposition into free field species $sp \in Spec$, it is sufficient to check this for each species $\Delta_{H,sp}$. Moreover, by the nature of the star product on polynomial observables, which is given by iterated contractions with the Wightman propagator, it is sufficient to check this for one such contraction.

Observe that the scaling behaviour of the Wightman propagator $\Delta_{H,m}$ is the same as the behaviour (4) of the correspponding Feynman propagator. With this we directly compute as follows:

\begin{aligned} \sigma_\rho (\mathbf{\Phi}(x)) \star_{F, \rho_{vac} m} \sigma_\rho (\mathbf{\Phi}(y) & = \rho^{-2 dim } \mathbf{\Phi}(\rho^{-1} x) \star_{F, \rho_{vac} m} \mathbf{\Phi}(\rho^{-1} y) \\ & = \rho^{-2 dim } \Delta_{F, \rho_{vac} m}(\rho^{-1}(x-y)) \\ & = \Delta_{F, \rho^{-1}\rho_{vac}m }(x,y) \mathbf{1} \\ & = rg_{\rho}\left( \Delta_{F, \rho^{-1}\rho_{vac}m }(x,y) \mathbf{1} \right) \\ & = rg_{\rho} \left( \mathbf{\Phi}(x) \star_{F, \rho^{-1} \rho_{vac} m} \mathbf{\Phi}(y) \right) \end{aligned} \,.

## References

The original informal discussion for RG-flow along scaling transformations is due to

Formulation in the rigorous context of causal perturbation theory/pAQFT, via the main theorem of perturbative renormalization, is due to

reviewed in

In the context of factorization algebras, this is given by the book Renormalization and Effective Field Theory

Last revised on August 5, 2019 at 18:08:23. See the history of this page for a list of all contributions to it.