The definition of mass of an electron (after the renorm group)

[metroplex021]

Hi there,
I have a question about the rest mass of an electron. As we all know, the charge of an electron is a function of the energy at which the system is probed. When defining the charge, we typically use as our reference scale the charge measured in Thompson scattering at the orders of the electron mass. However, the mass of the electron is itself a function of the energy -- partly because, in an interacting theory, the mass measures the strength of self-interactions. So how do we define *it*? We can't say 'the mass of an electron at the order of the mass of an electron', because that would be circular. Can anyone help me here?! In particular, is there a lowest energy scale that is appropriate for defining the mass of an electron?

Any help much appreciated!

 

[nrqed]

Hi there,
I have a question about the rest mass of an electron. As we all know, the charge of an electron is a function of the energy at which the system is probed. When defining the charge, we typically use as our reference scale the charge measured in Thompson scattering at the orders of the electron mass. However, the mass of the electron is itself a function of the energy -- partly because, in an interacting theory, the mass measures the strength of self-interactions. So how do we define *it*? We can't say 'the mass of an electron at the order of the mass of an electron', because that would be circular. Can anyone help me here?! In particular, is there a lowest energy scale that is appropriate for defining the mass of an electron?

Any help much appreciated!

The mass is defined in terms of the pole of the propagator. by definition, the propagator has a pole at p^2 = m^2. So the value of p^2 at which there is a pole define the value of m^2. At tree level, this occurs at the bare mass squared. With loops, this value is modified.

 

[metroplex021]

Thank you!
First, just to clarify: Depending on what renormalization scheme you use, the 'bare mass' is thought of as either infinite or the measured mass of an electron. So I take it you must have the latter scheme in mind, is that right?

So now two questions:
(1) Is the mass of the electron at tree level the same as the mass of a free electron (ie, with all interactions 'turned off')? And is that a scale-invariant quantity?

(2) Since the mass of the electron 'runs' with energy (since we can think of the loops as like 'counterterms' with a coupling given by the mass parameter, and the couplings generically run), do we regard the poles of the amplitude as 'running' as well?

Thanks so much!

 

[The_Duck]

(1) Is the mass of the electron at tree level the same as the mass of a free electron (ie, with all interactions 'turned off')? And is that a scale-invariant quantity?

Yes, at tree level the electron mass is just the bare parameter in the Lagrangian and there are no corrections to mess this up or cause running.

(2) Since the mass of the electron 'runs' with energy (since we can think of the loops as like 'counterterms' with a coupling given by the mass parameter, and the couplings generically run), do we regard the poles of the amplitude as 'running' as well?

No. These poles occur when the external momenta take on definite values. It doesn't make sense to speak of the poles as running. The locations of these poles are independent of the renormalization scheme.

There's some potential for confusion here because there are different renormalization schemes one can use to define the renormalized mass. In the "on-shell" scheme we define the renormalized electron mass to be the location of the pole in the electron propagator. In this scheme the electron mass *doesn't* run. We call the mass defined in this way the "pole mass." The pole mass is closest to what we intuitively mean by the mass of the particle; e.g. it is equal to the energy of an electron at rest. Real electrons always satisfy $E^2 = \vec{p}^2 + m_{\rm pole}^2$ where $m_{\rm pole}$ is the pole mass-squared.

However in the MS-bar scheme the renormalized electron mass is defined differently. In the MS-bar scheme the renormalized electron mass is a function of the renormalization scale $\mu$--it "runs." The MS-bar renormalized electron mass is close to the pole mass if $\mu$ is small, but becomes different for very large $\mu$. The advantage of this is that at high energies the MS-bar scheme perturbation series may converge better than the on-shell scheme perturbation series if we choose $\mu$ of order the energy of the process we are considering. We just have to keep in mind that the MS-bar renormalized mass $m^{\bar{\rm MS}}(\mu)$ is not quite the same as the energy of a real electron at rest. If we like we can calculate the rest energy of an electron--the pole mass--in the MS-bar scheme and we get a result that looks like $m^{\bar{\rm MS}}(\mu) + O(e^2)$.

We can't say 'the mass of an electron at the order of the mass of an electron', because that would be circular.

Actually we can define such a quantity. For example the http://pdg8.lbl.gov/rpp2014v1/pdgLive/DataBlock.action?node=Q004M the MS-bar charm quark mass renormalized at the scale of the charm quark mass, $m_c(\mu = m_c)$. Just imagine plotting $m(\mu)$ vs $\mu$ and then finding the intersection with the line $m = \mu$.

 

[metroplex021]

Thank you very much The_Duck. That's tremendously useful.

One last thing (if you're still around!). At times you make it sound as if the mass 'running' is just an artefact of the renormalization scheme. However, am I right in thinking that we have actually measured that the masses (sometimes called the 'effective' masses) of particles are different at different scales -- just as the effective charge of an electron is? If so, then presumably the effect isn't just an artefact of a choice of definition.

Thanks again though, especially for the PDG link.

 

[The_Duck]

Well, in some sense all running is an "artifact of the renormalization scheme." You get the same physical predictions in any renormalization scheme, so you don't have to ever invoke running couplings and masses if you don't want to.

But you are right that the running of the parameters does represent a real physical effect. For example the increase of $\alpha(\mu)$ with $μ$ means that electromagnetic interactions are stronger at higher energies than you might naively expect from dimensional analysis.

 

[metroplex021]

Thank you very much TheDuck. A final question. You phrased your claim regarding the physicality of running couplings in terms of alpha, the charge parameter. But just to confirm, would you regard the running of mass as as 'real' as the running of charge?

Thanks again though -- really helpful as ever!

 

[The_Duck]

Sure, though to be honest I'm not quite sure what the running of the mass means intuitively.

 

[ohwilleke]

Just to add re the importance of the running of the mass -- one of the things that phenomenologists do is try to discern if there is any formula or theoretical structure that can explain the mass of the Standard Model particles.

When doing so, one of the non-trivial things one has to do is to guess which energy scale and mass definition is a sensible place to look for those relationships. In the case of lepton masses, like the electron mass, and the masses of the heavier quarks (top, bottom and charm) for example, a "pole mass" is well defined and produces a sensible number.

But, the "conventional" way to describe the masses of the up, down and strange quarks is to use the masses of the MS renormalization scheme evaluated at 2 GeV (roughly the mass-energy of two nucleons at rest, rather than their own masses), because the renormalized pole masses of these quarks produce values that are ill defined and don't make a lot of sense.

Extreme running of the Higgs boson mass to the GUT scale, where it runs to values approaching zero, have been used to make statements about the stability of the Standard Model vacuum and about a possible source of the Standard Model mass scale derived from this asymptotic value.

 

[metroplex021]

Sure, though to be honest I'm not quite sure what the running of the mass means intuitively.

The_Duck, Susskind discusses mass renormalization in a way that aims to be intuitive here: . (In case that's of interest!)

 

[metroplex021]

Thanks for the video! What a great resource.