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.
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, theory or massless QCD. We shall further restrict to dimensionless observables, such as the ratio of cross sections. Let’s call a general observable , where we have inserted a dependence on the energy with foresight that such things renormalize with energy scale.
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, ? Stevenson makes this more explicit by quoting a theorem of dimensional analysis:
Thm. A function which depends only on two massive variables and which is
- uniquely defined
- defined without any dimensionful constants
must then be a function of the ratio of only, .
Cor. If is independent of , then is constant.
Then by the corollary, must be constant. This is a problem, since our experiments show a dependence.
Evading Dimensional Analysis
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 -dependence, it’s a bit of a curiosity how our dimensionless theory manages to define a dimensionful dependence of without any dimensonful quantities to draw upon. The simplest way to do this is to have the theory define the first derivative:
where is the usual beta function calculated in perturbation theory. It is dimensionless and is uniquely defined by the theory. Another way one can define dependence is to do so recursively; one can read Stevenson’s paper to see that this is equivalent to defining the function.
One can integrate the equation for to write,
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 = ,
for some arbitrary 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 we see that . Thus we see finally that the integral of the equation is
All of the one-parameter ambiguity of the theory is now packaged into the massive parameter . is an integral that comes from the 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 while the right-hand side contains only quantities that depend on the energy scale .
If vanishes for some , then we can write our observable in terms of this scale,
Note that is arbitrary, while 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. (more…)