# What is the difference between 'running' and 'current' quark mass?

When looking at the PDG, there is a difference between the 'running' and the 'current' quark masses.

Does anyone know which is the difference between these two?

• It's because there is a 'bare mass' which is not observable, and a 'running mass' that will vary depending on how you actually make the measurement. This wiki page on running couplings (en.wikipedia.org/wiki/Coupling_constant#Running_coupling) doesn't appear to be very good so I'll post a better answer later when I have more time - unless someone else gets there first and saves me the trouble of course! ;) Commented Apr 18, 2011 at 18:42

It's possible I have the terminology wrong, because this isn't my field, but as far as I can tell "current quark" masses are "running" masses. The distinction isn't between "current" masses and "running" masses, it's between "current quark" masses and "constituent quark" masses.

Let's back up and start from some basics. The reason quark masses are hard to define compared to, for example, lepton masses is that free quarks don't exist in the IR limit. QFT says that the "bare" masses in the Lagrangian are infinite for all particles, but divergent loop contributions to the propagator cancel them out to give finite "dressed" masses that you actually measure. So in the case of an electron, for example, you plug in the experimentally measured electron mass as an input parameter to your theory, and QFT tells you that, well, the bare electron mass must really be infinite, but there's a nice well-defined way that the mass "runs" from infinity at very small length scales, to a constant at very large length scales ("IR fixed point"). So when someone just says "electron mass" they're talking about this IR-limit value, which is the same as the experimentally measured value (although I remember hearing that in some high-energy collisions of electrons they could actually measure an electron mass significantly different from the IR-limit value).

For quarks masses the same kind of "running" happens, but instead of converging to a constant, they diverge at the energy scale called $\Lambda_\text{QCD}$. So if you ask the question "what should the quark masses be for me, a macroscopic observer?", the answer is undefined, because they already became infinite at a much smaller length scale. This makes perfect sense because quarks are confined into hadrons and can't be observed macroscopically.

The masses that PDG gives, then, are the values of the "running" masses at some energy scale greater than (length scale smaller than) $\Lambda_\text{QCD}$, defined in some specific renormalization scheme. Specifically, it says they're the values in the $\overline{MS}$ scheme at renormalization scale $\mu = 2\, \text{GeV}$. Without this information the mass numbers are pretty much meaningless, because at a renormalization scale closer to $\Lambda_\text{QCD}$ the masses will of course be much bigger. The way you can convert between different renormalization scales is to use the renormalization group equation, which can be calculated in perturbation theory.

So that's what "running"/"current quark" masses are. It's a pretty subtle concept but it's rigorously defined because the $\overline{MS}$ renormalization scheme is. If somebody else calculates the mass at a different scale or with a different renormalization scheme, they'll get a different number, but you can compare them by using the renormalization group equation (as long as all the scales are far to the UV side of $\Lambda_\text{QCD}$).

As for "constituent quark" masses, I don't really know how those are defined, so I'll just quote this paragraph from the PDG explanation:

"The quark masses for light quarks discussed so far are often referred to as current quark masses. Nonrelativistic quark models use constituent quark masses, which are of order 350 MeV for the u and d quarks. Constituent quark masses model the effects of dynamical chiral symmetry breaking, and are not related to the quark mass parameters $m_k$ of the QCD Lagrangian Eq. (1). Constituent masses are only defined in the context of a particular hadronic model." —"Note on Quark Masses": http://pdg.lbl.gov/2010/reviews/rpp2010-rev-quark-masses.pdf

So that about sums it up. The "constituent quark" masses (which are the only other finite quark masses I've heard of) are not directly related to the mass parameters in the QCD Lagrangian, but are supposed to represent all this nonperturbative stuff that happens as you cross $\Lambda_\text{QCD}$ and quarks become very significantly "dressed" and confined into hadrons.

• not your fault obviously, but that constituent quark wiki page could do with a rewrite! Commented Nov 23, 2013 at 19:10

Until recently, quark masses were thought to be one-third of a protons' (612 "electrons"). Now, experimentalists state "bare quark mass" to be about one percent of this at any given moment (based upon their observations). But a difference exists between how positive and negative charges carry mass because the more massive quarks (e.g. "top") are positive and, of course, protons are too (compared to electrons). Perhaps this is why our universe is dominated by "positive mass"...

• I think this contains quite a few errors, especially e.g. "... a difference exists between how positive and negative charges carry mass because the more massive quarks (e.g. "top") are positive". The anti-top is negatively charged but just as massive as the top. Commented Nov 23, 2013 at 19:03
• Also, "until recently" refers to the last 40 years? Commented Nov 23, 2013 at 19:09