Dimensionless vs. dimensional RG-Flow equations

Question

When one writes down RG-Flow equations for any theory, at some point one encounters statements like

"It is useful to properly rescale the above exact flow equations and rewrite them in dimensionless form."

or

"The rescaled form of the RG-Flow equations is also most convenient to discuss fixed points."

or

"The advantage of working with the rescaled flow equations is that we can directly read off the canonical dimensions of the couplings and thus classify all couplings according to their relevance w.r.t. a given fixed point."

(These sentences are more or less literally from the Kopietz-book.)

I don't really understand this logic; I don't really understand why one has to go to dimensionless flow-equations; Can't I just calculate the critical exponents (by diagonalizing the stability matrix and reading off it's eigenvalues, assuming that this is possible) for the RG-flow equations of the dimensional couplings?

An explanation that I could deal with is, that it is just an standard-fact that has been verified experimentally that the only "interesting" critical exponents are those, that come with the flow-equations of the dimensionless couplings.

Can anybody elaborate a good explanation please?

EDIT:

An example for dimensional flow equations:

$$ \left\{ \, \begin{aligned} \frac{ \mathrm{d} }{ \mathrm{d} \Lambda } \gamma_{1} \quad & \approx \quad 0 \\ \frac{ \mathrm{d} }{ \mathrm{d} \Lambda } \gamma_{2} \quad & \approx \quad 0 \\ \frac{ \mathrm{d} }{ \mathrm{d} \Lambda } \gamma_{3} \quad & \approx \quad + {\gamma}_{1} \cdot \left( 4 \, D_{ \Lambda } \right) \cdot {\gamma}_{2} \\ \frac{ \mathrm{d} }{ \mathrm{d} \Lambda } \gamma_{4} \quad & \approx \quad + {\gamma}_{2} \cdot \left( -16 \, D_{ \Lambda } \right) \cdot {\gamma}_{4} \end{aligned} \right. $$

An example for dimensionless flow equations:

$$ \left\{ \, \begin{aligned} \beta_{1} ( \tilde{\gamma}_{1} , \tilde{\gamma}_{2} , \tilde{\gamma}_{3} , \tilde{\gamma}_{4}) \quad & \approx \quad - 2 \tilde{\gamma}_{1} / \Lambda \\ \beta_{2} ( \tilde{\gamma}_{1} , \tilde{\gamma}_{2} , \tilde{\gamma}_{3} , \tilde{\gamma}_{4}) \quad & \approx \quad - 2 \tilde{\gamma}_{2} / \Lambda \\ \beta_{3} ( \tilde{\gamma}_{1} , \tilde{\gamma}_{2} , \tilde{\gamma}_{3} , \tilde{\gamma}_{4}) \quad & \approx \quad - 2 \tilde{\gamma}_{3} / \Lambda + \tilde{\gamma}_{1} \cdot \left( 4 \, D_{ \Lambda } \, \Lambda^2 \right) \cdot \tilde{\gamma}_{2} \\ \beta_{4} ( \tilde{\gamma}_{1} , \tilde{\gamma}_{2} , \tilde{\gamma}_{3} , \tilde{\gamma}_{4}) \quad & \approx \quad - 2 \tilde{\gamma}_{4} / \Lambda + \tilde{\gamma}_{2} \cdot \left( -16 \, D_{ \Lambda } \, \Lambda^2 \right) \cdot \tilde{\gamma}_{4} \end{aligned} \right. $$

where $D_{ \Lambda }$ is the dominant contribution for the flow of the couplings; Mode-elimination (in the FRG, analog to Wilson's RG) corresponds to decreasing $\Lambda$ from $\Lambda = \infty$ to $\Lambda = 0$ and $\tilde{\gamma }_{ \bullet } = \gamma_{ \bullet } / \Lambda^2$

Answer 1

I will limit my answer to the situation in Wilson's RG approach with the functional renormalization group (FRG) as explicit implementation. Lets begin with an obvious observation: considering a set the flow equation of the couplings $\{\lambda_i\}$ $$ \partial_k\lambda_i(k)=\mathrm{Flow}_i(k,\{\lambda_i(k)\}),\tag{1} $$ where $k$ denotes the RG scale ($\Lambda$ in Antihero question) and $\mathrm{Flow}_i(k,\{\lambda_i\})$ is a not further specified function which depends on RG scale and all couplings. Let $\lambda_i$ be of dimension $d_i$ which gives rise to the dimensionless coupling $\bar\lambda_i(k)\equiv \lambda_i(k)/k^{d_i}$. Inserting the definition of the dimensionless coupling into Eq. (1) yields $$ k\,\partial_k \bar\lambda_i(t)= -d\, \bar\lambda_i(k)+k^{1-d}\,\mathrm{Flow}_i(k,\{k^{d_i}\bar\lambda_i(k)\}),\tag{2} $$ where we multiplied both sides with $k^{1-d}$. For convenience we introduce "RG time" as the dimensionless parameter $t\equiv\ln(k/\Lambda)$, with an at the moment arbitrary reference scale $\Lambda$. With $\partial t/\partial k=1/k$ we rewrite Eqs. (1) and (2): $$ \partial_t\lambda_i(t) = k\,\mathrm{Flow}_i(k,\{\lambda_i(t)\})\equiv\,\mathrm{Flow}_i(t,\{\lambda_i(t)\})\tag{4} $$ $$ \partial_t \bar\lambda_i(t) = -d\, \lambda_i(k)k^{-d}+k^{-d}\,\mathrm{Flow}_i(t,\{\lambda_i(t)\})=-d\bar\lambda_i(t)+\mathrm{Flow}_i'(t,\{\bar\lambda_i(t)\}).\tag{5} $$ Now to the aforementioned observation: demanding $\partial_t\lambda_i(t)=0$ for a "fixed" point or demanding $\partial_t\bar\lambda_i(t)=0$ for a fixed point are two completely different requirements/definitions of fixed points since: $$ 0\stackrel{!}{=}\mathrm{Flow}_i(t,\{\lambda_i(t)\})\neq -d\, \lambda_i(k)k^{-d}+k^{-d}\,\mathrm{Flow}_i(t,\{\lambda_i(t)\})\stackrel{!}{=}0. $$ $\bar\lambda_i(t)$ carries a canonical running based on the dimension of the coupling $\lambda_i(t)$, see also the related question "Are fixed points of RG evolution really scale-invariant?". In Literature RG fixed points are usually refereed to as fixed points ($\partial_t\bar\lambda_i(t)=0$) of the dimensionless couplings $\{\bar\lambda_i(t)\}$. In connection to the classical picture of Wilson's RG these dimensionless couplings stay constant at a fixed point under RG transformations which include a change in scale ($k$ or equivalently $t$)/mode elimination, a rescaling of momenta, and a rescaling of fields. The latter two properties might require the inclusion of the anomalous dimension of the fields of the theory which modify the canonical scaling on the r.h.s of (5): $-d\rightarrow -d +\eta_{\lambda_i}$ with $\eta_{\lambda_i}$ as a sum of field anomalous dimensions $\eta_i=-(\partial_t Z_{\phi_i})/Z_{\phi_i}$. The r.h.s of Eq. (5) with $\mathrm{Flow}_i'(\ldots)$ is then completely independent of $t$ and $k$, which means a fixed point solution $\{\bar{\lambda}_i^*\}$ with $\forall i \ \partial_t\bar{\lambda}_i^*=0$ naturally holds at all scales $k$ -- which is a defining property in the canonical notion of RG fixed points. "Fixed" points with $\partial_t{\lambda}_i=0$ are associated with a scale $k^*$ and are not scale-independent (since the r.h.s. of Eq. (1) or equivalently (4) depends on the scale). In the sense of invariance under RG transformations at all scales it does not sense to discuss solutions of $\partial_t{\lambda}_i=0$. Fixed points of $\partial_t\bar\lambda_i(t)=0$ are very important in understanding theories and their RG flows. I would rephrase the second quote of the original question

The rescaled form of the RG-Flow equations is used to define and discuss specific fixed points (scale invariant solutions of the RG flow equations).

Coming to the first quote and the comment of bbrink I again rephrase

For certain applications it is useful to rescale the above exact flow equations and rewrite them in dimensionless form.

There is more to (quantum) field theories than fixed points (using the canonical notion). While they are undoubtedly very important they are not the only aspect of interest. For certain applications especially involving numerical solutions of explicit systems of flow equations it can be quite disadvantageous to compute in dimensionless couplings by pulling out the canonical scaling.

Regarding "fixed" points of (1) and equivalently (4): these flow equations have "fixed"/stationary points at certain scales/couplings. I strongly disagree with bbrink's comment. The construction principle of the functional renormalization group (as an exact implementation of Wilson's RG approach) is based on such a stationary points/the associated couplings and scales: the (F)RG flow is initialized at a for a given theory asymptotically large scale $\Lambda$ (for certain theories $\Lambda\rightarrow\infty$ is considered). This scale has to be larger than all internal and external scales which means that quantum fluctuations above the scale are not relevant for the theory and the theory can be considered classical at $\Lambda$, meaning that we can initialize the RG flow for the effective average action with the classical action of the theory (potentially including gauge-fixing or anomaly related terms). One way to quantity the scale $\Lambda$ for a theory specified at $\Lambda$ (with, e.g., a set of couplings $\{\lambda_i(k=\Lambda)\}$) is to require $\partial_k\lambda_i(k)|_{k=\Lambda}\stackrel{!}{=}0$ or at least $\approx 0$ to a sufficient degree. This requirement is refereed to in FRG literature as "RG consistency". Some theories additionally have $\partial_k\lambda_i(k)=0$ or $\approx 0$ when approaching the infrared/small $k$. It is true that the RG and also FRG have an inherent diffusive character but this does not mean the absence of "fixed"/stationary points in $\{\lambda_i(k)\}$. The argument given by bbrink in his comment about the absence of fixed point solutions in the heat equation $\partial_t c(t,x)=\partial_{xx}c(t,x)$ is wrong: $c(x)=a+m x$ for arbitrary constants $a$ and $m$ is a fixed point solution due to its vanishing second derivative. In that regard fixed points of $\partial_t c(t,x)$ are "thermal" equilibrium solutions while fixed points of the rescaled equation resemble steady state solutions. This might in fact be a good picture for the RG fixed points of $\{\lambda_i(t)\}$ and $\{\bar\lambda_i(t)\}$. RG flows can often be understood and discussed using such fluid dynamic notions. (The people who formulated the (F)RG did not call the underlying equations "flow" equations for no reason).

Coming to the last quote:

"The advantage of working with the rescaled flow equations is that we can directly read off the canonical dimensions of the couplings and thus classify all couplings according to their relevance w.r.t. a given fixed point."

One can classify directions in the space of couplings $\{\bar\lambda_i(t)\}$ by inearizing Eq. (5) around the fixed point using $$ \bar\lambda_i(t)=\bar\lambda_i^*+\delta\bar\lambda_i(t) \Rightarrow \partial_t\bar\lambda_i(t) = B_{i,j}\delta\bar\lambda_i(t)+O(\delta\bar\lambda_i(t)^2),\tag{6} $$ where we used the fact that at the fixed point the flows/beta functions vanish. Computing the eigenvalues $b_i$ of the stability matrix w.r.t. normalized eigenvectors one may classify directions/couplings: $b_i<0$ is UV attractive/IR repulsive, $b_i>0$ is UV repulsive/IR attractive, and $b_i=0$ is a marginal direction. This might be meant by "relevance w.r.t. a given fixed point" but if that is the case the choice of the word "relevance" is quite poor since the canonical notion of relevant, irrelevant and marginal couplings during RG flow is not directly relate to the notion of UV repulsive/IR attractive and flow around a fixed point (I think but maybe I am wrong on this one). The canonical dimensions are needed for a proper rescaling. In the spirit of what comes first: starting a dimensionful flow eq. one needs to figure out the canonical dimensions (dimensional analysis and proper additions of $\eta_i$) to formulate the dimensionless flow equation.

I hope this answer helps in understanding the differences in the flow equations of $\{\bar\lambda_i\}$ and $\{\lambda_i\}$. I however have to admit that I am no expert when it comes to fixed points in the (F)RG: I usually work with dimensionful couplings $\{\lambda_i\}$ and my limited knowledge of fixed points/dimensionless flows comes from lecture notes and talks and not from practical experience (which at least for me is somewhat necessary to really understand this kind of stuff). For an explicit but relatively simple application I would recommend looking for FRG discussions of the O(N) model specifically (but not necessarily exclusively) in the infinite-N limit, see, e.g., the papers "Critical O(N) models in the complex field plane" and "Critical exponents from optimised renormalisation group flows", where the r.h.s. of the flow equation is known exactly without truncations in a simple and closed form. Allowing for relatively simple computations of critical points and related quantites.