Today I’ll be reviewing P.M. Stevenson, “Dimensional Analysis in Field Theory,” Annals of Physics 132, 383 (1981). It’s a cute paper that helps provide some insight for the renormalization group.

A theory is a black box that we can shake to make predictions of physical observables.

We’ve already said a few cursory words on dimensional analysis and renormalization. It turns out that we can use simple dimensional analysis to yield some insight on the nature of the renormalization group without having to think about the technical ‘heavy machinery’ required to do actual calculations.

First let us define a theory as a black box that is characterized by a Lagrangian and its corresponding parameters: coupling constants, masses, fields, etc. All these things, however, are contained within the black box and are in some sense abstract objects. One can ask the black box to predict physical observables, which can then be measured experimentally. Such observables could be cross sections, ratios of cross sections, or potentials, as shown in the image above.

Let’s now restrict ourselves to the case of a naively-scale-invariant’ or naively-dimensionless’ theory, i.e. one where there are no couplings with mass dimensions. For example, $\lambda\phi^4$ theory or massless QCD. We shall further restrict to dimensionless observables, such as the ratio of cross sections. Let’s call a general observable $\rho(Q)$, where we have inserted a dependence on the energy $Q$ with foresight that such things renormalize with energy scale.

Dimensional Analysis

But one can immediately take a step back and realize that this is ridiculous. How could a dimensionless observable from a dimensionless theory have a nontrivial dependence on a dimensionful quantity, $Q$? Stevenson makes this more explicit by quoting a theorem of dimensional analysis:

Thm. A function $f(x,y)$ which depends only on two massive variables $x,y$ and which is

1. dimensionless
2. uniquely defined
3. defined without any dimensionful constants

must then be a function of the ratio of $x/y$ only, $f(x,y)=f(x/y)$.

Cor. If $f(x,y)$ is independent of $y$, then $f(x,y)$ is constant.

Then by the corollary, $\rho(Q)$ must be constant. This is a problem, since our experiments show a $Q$ dependence.

The answer is that the theorem doesn’t hold: the theory inside the black box is not uniquely defined,’ violating condition 2. This is what we meant by the stuff inside the black box being abstract,’ the Lagrangian is actually a one-parameter family of theories with different bare couplings. That is to say that the black box is defined up to a freedom in the renormalization conditions.

Now that we see that it is possible to have $Q$-dependence, it’s a bit of a curiosity how our dimensionless theory manages to define a dimensionful dependence of $\rho$ without any dimensonful quantities to draw upon. The simplest way to do this is to have the theory define the first derivative:

$\frac {d\rho}{dQ} = \frac{1}{Q} \beta(\rho)$,

where $\beta$ is the usual beta function calculated in perturbation theory. It is dimensionless and is uniquely defined by the theory. Another way one can define $Q$ dependence is to do so recursively; one can read Stevenson’s paper to see that this is equivalent to defining the $\beta$ function.

One can integrate the equation for $\beta$ to write,

$\log Q +$ constant $= \int^\rho_{-\infty} \frac{d\rho'}{\beta(\rho)} \equiv K(\rho)$.

The constant of integration now characterizes the one-parameter ambiguity of the theory. (The ambiguity can be mapped onto the lack of a boundary condition.) We may parameterize this ambiguity by writing

constant = $K_0 - \log \mu$,

for some arbitrary $\mu$ of mass dimension 1. (This form is necessary to get a dimensionles logarithm on the left-hand side.) The appearance of this massive constant is something of a virgin birth’ for the naively-dimensionless theory and is called dimensional transmutation. By setting $Q=\mu$ we see that $K_0 = K(\rho(\mu))$. Thus we see finally that the integral of the $\beta$ equation is

$\log(Q/\mu) + K(\rho(\mu)) = K(\rho(Q))$.

All of the one-parameter ambiguity of the theory is now packaged into the massive parameter $\mu$. $K$ is an integral that comes from the $\beta$ function, which is in turn specified by the Lagrangian of the theory. On the left-hand side we have quantities which depend on the arbitrary scale $\mu$ while the right-hand side contains only quantities that depend on the energy scale $Q$.

If $K(\rho(\mu))$ vanishes for some $\mu=\Lambda$, then we can write our observable in terms of this scale,

$\rho(Q) = K^{-1}(\log(Q/\Lambda))$.

Note that $\mu$ is arbitrary, while $\Lambda$ is fixed for a particular theory. This latter quantity is rather interesting because even though it is an intrinsic property of the black box, it is not predicted by the black box, it must be fixed by explicit measurement of an observable.

Connection to familiar calculations

Let’s pause to connect this to massless QCD. We may take the observable $\rho$ to be the strong coupling $\alpha$. (Some subtleties of this choice are discussed in Stevenson.) Then the well-known beta function to leading order is

$\beta(\alpha) = - b\alpha^2 + \cdots$

and so

$K(\alpha) = \int^\alpha_{-\infty} \frac{d\alpha'}{-b\alpha'^2} = \frac{1}{b\alpha}$.

Integrating we get the usual results

$\alpha(Q) = \frac{\alpha(\mu)}{1+b\alpha(\mu)\log(Q/\mu)}$
$\alpha(Q) = \frac{1}{b\log(Q/\Lambda)}$,

and we see that $\Lambda$ is the scale where the theory becomes nonperturbative.

Determining other physical quantities

Thus far we haven’t really done anything. We’ve written out a dimensionless physical quantity in terms of an unphysical scale $\mu$. We would now like to determine the form of other physical quantities, $\sigma$, of nontrivial mass dimension $D$. If $\sigma$ is a function of massive variables $x_1, \cdots, x_n$, then naive dimensional analysis tells us it may be written in terms of a dimensionless function via

$\sigma(x_1, \cdots, x_n) = x_1^D S\left(\frac{x_2}{x_1},\cdots, \frac{x_n}{x_1}\right)$.

This, however, doesn’t account for the one-parameter ambiguity of the theory that we’ve been going on about. One might think that we should then go through the same rigamarole as before, but this is not the case. We’ve already encapsulated the entire ambiguity in writing down the observable $\rho(\mu)$. Thus we should write our observable as

$\sigma(x_1, \cdots, x_n) = x_1^D S\left(\frac{x_2}{x_1},\cdots, \frac{x_n}{x_1}; \frac{x_1}{\mu},\rho(\mu)\right)$.

The function $S$ must be independent of $\mu$, i.e. the $\mu$ dependences must cancel. We would now like to extract the RG equations. As a trick, we can write a more general function

$\tilde S\left(\frac{x_2}{x_1},\cdots, \frac{x_n}{x_1}; \frac{x_1}{\mu},r\right)$

This new function is only independent of $\mu$ when $r = \rho(\mu)$, and so we may write the differentual equation

$\frac{d}{d\mu} \tilde S_{r=\rho(\mu)} = \left(\frac{\partial \tilde S}{\partial\mu} + \frac{\partial \tilde S}{\partial r} \rho'(\mu)\right)_{r=\rho(\mu)}$.

Recalling from the defining equation for the $\beta$ function (with $Q = \mu$) that we may write $\mu \rho'(\mu) = \beta(\rho)$, we end up with

$\left(\mu\frac{\partial}{\partial\mu}+\beta(\rho)\frac{\partial}{\partial r}\right)\tilde S_{r=\rho(\mu)} = 0$

from which we ma then write the RG equation

$\left(\mu\frac{\partial}{\partial\mu}+\beta(\rho)\frac{\partial}{\partial r}\right) \sigma = 0$.

Thus the RG equation is just the expression of the one-parameter ambiguity of the theory’s black box. One will notice that the above equation appears to be missing an anomalous dimension. This is because a physical observable (as we assumed $\sigma$ to be) don’t have anomalous dimensions and scale according to engineering’ dimensions.

Determining unphysical quantities

Anomalous dimensions occur when we study the RG equation for unphysical quantities, i.e. when some theorist decides to peek inside the black box and see how all the little pieces work. We could, for example, ask how a quantum field renormalizes. The meaning of an unphysical quantity is deeply entrenched in the details of the gears and knobs within the black box — renormalization scheme, gauge choice, etc. But we won’t worry too much about meaning.

An unobservable quantity $\Gamma(x_1,\cdots,x_n;\mu)$ of mass dimension $D$ needn’t be independent of the arbitrary scale $\mu$. Let us assume that the $\mu$ dependence takes the simplified form

$\Gamma(x;\mu) = \mu^{\gamma} x^{D-\gamma} G(\frac{x_2}{x_1},\cdots,\frac{x_n}{x_1};\mu,\rho(\mu))$.

This is the analog of the previous equation for $\sigma$. We can now go through all of the same machinery and use the same trick to get the RGE for this unphysical quantity. The only difference is that when one takes the derivative $\frac{d}{d\mu}$, there is an extra term due to the explicit factor of $\mu^\gamma$. The RGE is then

$\left(\mu\frac{\partial}{\partial\mu}+\beta(\rho)\frac{\partial}{\partial r} - \gamma\right) \Gamma = 0$.

More generally the $\mu$ dependence is a little trickier, but one ends up with an RGE of the same form with $\gamma = \gamma(\rho(\mu))$. The $\gamma$ dependence of various unphysical quantities cancel in the calculation of a physical observable.

More general theories

We started off by talking about a dimensionless’ or naively-scale-invariant’ theory. We know now that this scale invariance is broken by dimensional transmutation, but we used the fact that the black box had no dimensionful parameters to draw upon and hence we had to define dimensionful dependence through the beta function. What happens in Lagrangians with mass parameters?

One way argue that this does not affect the previous arguments since bare parameters are ill-defined and cannot be used to determine the dimensionful dependence of physical observables. Alternately, the theories have to still make sense in the limit that the masses go to zero.

The addition of a mass parameter, however, introduces another ambiguity into the theory, i.e. the theory now has a two-parameter ambiguity. We can then use the same analysis as above for both the coupling $\alpha(\mu)$ and the running mass $m(\mu)$. Each of these must be determined by a physical observation, and further physical observables can then be expressed in terms of those two observations.