Setting of renormalization scale in field theory calculations

Question

In dimensional regularization an arbitrary mass parameter $\mu$ must be introduced in going to $4-\epsilon$ dimensions. I am trying to understand to what extent this parameter can be eliminated from physical observables.

Since $\mu$ is arbitrary, physical quantities such as pole masses and scattering amplitudes must be independent of it. Nevertheless at any fixed order in perturbation theory these quantities contain residual $\mu$-dependence. One expects this dependence to decrease at higher orders in perturbation theory.

For concreteness, consider dimensional regularization with minimal subtraction of $\phi^4$ theory, which has bare Lagrangian

$\mathcal{L}_B = \frac{1}{2}(\partial \phi_B)^2 - \frac{1}{2}m_B^2 \phi_B^2 - \frac{\lambda}{4!}\phi_B^4$

Here are the 1-loop expressions for the physical mass $m_P$ and 4-point coupling $\lambda_P \equiv (\sqrt{Z})^4 \Gamma^{(4)}$ after minimal subtraction of poles and taking $\epsilon \to 0$:

\begin{equation} m_P^2 = m_R^2 \left\{1 + \frac{\lambda_R}{2(4\pi)^2}\left[\log\left(\frac{m_R^2}{4\pi\mu^2}\right)\right] + \gamma - 1 \right\} \end{equation}

$$\lambda_P = \lambda_R + \frac{3\lambda_R^2}{2(4\pi)^2}\left[\log\left(\frac{m_R^2}{4\pi\mu^2}\right) + \gamma - 2+\frac{1}{3}A\left(\frac{m_R^2}{s_E},\frac{m_R^2}{t_E},\frac{m_R^2}{u_E}\right) \right] $$

where $A\left(\frac{m_R^2}{s_E},\frac{m_R^2}{t_E},\frac{m_R^2}{u_E}\right) = \sum_{z_E = s_E,t_E,u_E} A\left(\frac{m_R^2}{z_E}\right)$ and $A(x) \equiv \sqrt{1+4x}\log\left(\frac{\sqrt{1+4x}+1}{\sqrt{1+4x}-1}\right) $.

Both of these quantities ($m_P$ and $\lambda_P$) are physically observable.

Suppose we conduct an experiment at a reference momentum $p_{E0}\equiv(s_{E0},t_{E0},u_{E0})$ and make measurements of the pole mass and 4-point coupling with the result $\lambda_{P0}, m_{P0}$. We now have a system of two equations in the three unknowns ($\lambda_R,m_R,\mu$). This means that in principle I can solve for $\lambda_R = \lambda_R(\mu)$ and $m_R = m_R(\mu)$.

I would now like to make a prediction for the 4-point amplitude at a different momentum $p_{E}' \neq p_{E0}$. Since I have two equations in three unknowns I need to guess a suitable value for $\mu$ (say $\mu' = \sqrt{s_{E}'}$) which allows me to fix $\lambda_R$ and $m_R$ and then calculate $\lambda_P (p_E')$. This procedure seems quite ad hoc to me because a different (arbitrary) choice of $\mu$ (e.g. $\mu'/2$ or $2\mu'$) will lead a different physical answer (albeit only logarithmically different).

From what I can gather from the literature, the problem of determining the renormalization scale I described above is a genuine problem in actual calculations of QCD (e.g. http://arxiv.org/abs/1302.0599) which leads theorists to introduce so-called "systematic uncertainties".

What concerns me is that I haven't been able to find any mention of this problem in any textbook on quantum field theory that deals with QED or QCD (anyone know of a reference?). Since this problem appears already in arguably the simplest QFT of $\lambda\phi^4$, I would expect it also to occur in e.g. 1-loop calculations of Bhabha scattering but I haven't been able to find any mention of it in this context.

Does anyone know how this problem is dealt with in real loop calculations (e.g. at LEP or LHC?)

Also, I would be interested to know if there is any analogue of this problem in condensed matter theory.

Answer 1

The choice of $\mu$ is indeed arbitrary, and leads to uncertainties in the final result when working at any finite loop order. Quantities like $\lambda_R$ inherently depend on the renormalization scheme and scale. Therefore, expanding the physical result in $\lambda_R$ introduces this dependence into the computed result. If renormalization is unnecessary (as in finite theories or in 1 space-time dimension) one can expand in $\lambda_0$ (the unrenormalized coupling) and this dependence will not arise. In realistic theories, however, $\lambda_0$ contains divergences and is therefore not suitable as an expansion parameter.

Choosing a different scale $\mu$ (or more generally a different renormalization scheme altogether) will change the definition of the expansion parameter $\lambda_R$, leading to different expansion coefficients, and thus to a different result at any finite order in the expansion. The way one usually determines the scale $\mu$ is by demanding that the higher-order correction be as small as possible. If we do that, the computed result (at any finite order in perturbation) will be as close as possible to the physical result.

In order to minimize the higher-order corrections without actually computing them, one can invoke the fact that all corrections contain stuff of the form $\log s^2/\mu^2$ where $s$ is some scale present in the problem. In the expressions you wrote above, $s^2$ would be $m_R^2/4\pi$. If we put $\mu = s$, a recurrent term in the higher-order corrections vanishes, and we can hope our approximation to be better.

Answer 2

It seems that the problem of $\mu$-selection can be avoided altogether in QED and $\lambda\phi^4$ simply by using the on-shell renormalization scheme, which in contrast to minimal subtraction, soaks up all $\mu$ dependence in the counterterms, leading to unambiguous expressions for amplitudes in terms of physical observables.