# Physical meaning of mass renormalization

In the case of charge renormalization, one can present a neat and nice physical idea that brings a physical ground to it called "Vacuum Polarization". Which can be even extended to non-abelian gauge groups(as Coleman did)

I want to know if there's any mechanism that gives some physical sprit to mass renormalization, since I can't understand how mass can be screened with virtual electron-positron pairs.

• A common idea in the context of condensed matter is to interpret the renormalization of mass as an effect of interactions. In particular, people often call the mass-shift as "self interaction energy". This idea is more or less straightforward to see in the Wilsonian picture of RG. Perhaps someone who understands this point-of-view well can write a full-blown answer. Jul 6, 2020 at 1:05

## 3 Answers

Mass renormalization (in QED) does not involve virtual electron-positron pairs. It involves the emission and reabsorption of a virtual photon by an electron in motion. This is a quantum-mechanical interaction between the electron and the electron’s own electromagnetic field.

The intuitive idea of mass renormalization actually can be explained without quantum mechanics, however. A charged electron produces an electromagnetic field around itself. That field makes contributions to the energy-momentum tensor, so it possesses energy and inertia. If you want to accelerated the electron, you also have to accelerate the electromagnetic field attached to it, and this changes the effective, observable mass of the electron. This is the essence of mass renormalization, with the electromagnetic field adding to mass of a charged particle (adding an infinite amount, if the calculation is not regularized).

• Nice answer. But the only change in QFT is that the dependence of the mass(energy) on the cut-off is logarithmic rather than linear. True? But I don't know where this change actually happens? Why linear dependence is substituted by a logarithmic one? (I mean the physical reason, I know that the linear cut-off dependence is not "natural" but I need some more physical reasoning to be convinced.) Thanks Jul 6, 2020 at 9:54

It is very good and deep question. Let me try to answer and clarify some subtleties.

In QED we have:

$$e^2_{phys}(\Lambda) = e_0^2\left(1-\frac{e_0^2}{6\pi^2}\ln \frac{\Lambda}{m_0}+\dots\right) = \frac{e^2_0}{1+\frac{e_0^2}{6\pi^2}\ln \frac{\Lambda}{m_0}}$$

$$m_{run}(\Lambda) = m_0\left(1+ \frac{3e_0^2}{8\pi^2}\ln \frac{\Lambda}{m_0}+\dots\right) = m_0 \left(\frac{e_0^2}{e^2_{phys}}\right)^{9/4}$$

1. Here $$m_0$$ is bare mass, $$e_0$$ is bare charge (physical charge on scale $$\Lambda = m_0$$). So running mass is directly related to value of physical charge. Physical interpretation: energy of electromagnetic field around electron give contribution to mass.

2. Note, that on scale $$\Lambda>m_0$$:

$$e^2_{phys} < e_0^2$$ $$m_{run} > m_0$$

1. $$e^2_{phys}$$ is real experimentally observable quantity, $$e^2_{phys}$$ change with scale, and has an interpretation as "vacuum polarization", as you said in the question.

2. 'Running mass' $$m_{run}$$ is a parameter in the Lagrangian. It depends on the regularisation scheme and can be reproduced from physical (pole) mass.

3. There is also notion of pole mass -- pole in full propagator. Pole mass is the physical mass, scale-independent and independent of any renormalization scheme we use to subtract any infinite parts of the loop corrections. It is what we observe.

• Suggestion about notation: since there are so many mass variables here, it's probably better to call the renormalization scale $\mu$ or something else instead of $m$, in order to prevent any confusion. Jul 6, 2020 at 0:58
• @Arkya, yes, it is valuable comment. Jul 6, 2020 at 1:03
• It's the best answer. But if I'm not wrong the running mass runs with $\mu$. Since the electromagnetism contribution to mass has a "unique" amount that only depends on the hard cut-off $\Lambda$(Which I like to interpret as the inverse size of the electron) and the charge of the charged particle, amd since the charge itself is scale dependent, the mass becomes scale dependent too. True? If I translate it in a naive an classical manner, I should say that if I assign each scale to an sphere around the localized charge, only the internal field configuration can modify the bare mass. Am I right? Jul 6, 2020 at 10:36
• @BastamTajik, I edited answer. I don't need $\mu$, it was incorrect. Jul 6, 2020 at 11:07
• @BastamTajik, I think that yes. But I think better to understand, that there are different notions of mass. Jul 6, 2020 at 18:52

Let us start from QED.

In this theory, vacuum polarization in the lowest order is given by one-fermion loop correction to bare photon propagator (note that photon has zero mass). It is quite straightforward (or I can provide a derivation) that this correction modifies charge but not mass. For me, physical ground becomes clear after calculation.

If you consider electron propagator renormalization, one loop correction shifts the pole in propagator and in this sense electron mass changes. But there is nothing relating to electron-positron pairs and vacuum polarization.

• Thanks. But the question is still unanswered I guess. Can we justify intuitively why mass is renormalized? In fact We can understand the physical meaning of running of the charge by assigning an intuitive picture to it that can be extended to Non-Abelian case(Coleman). It's vaccum polarization as we both know well. What baffles me is how can one construct such intuitive picture for running of the mass!? Jul 5, 2020 at 16:45
• In other words one can argue if mass-running can make sense at all!? Actually it can also depend on renormalization scheme that might lead to a running mass or an absolutely asymptotic pole mass without running! Jul 5, 2020 at 16:52