QCD scale and massless limit of u & d quarks

[GIM]

Hello!
Could anybody help me?
My wondering seems so trivial, but I can't skip it.
They say that since u and d quarks are much lighter than QCD scale(~200MeV), in reality we can consider the QCD Lagrangian has an approximate global chiral symmetry with respect to these two flavors. At first, it sounded plausible, but now I wonder what relation between the QCD scale and the quark masses there is and how. Is this problem related to the renormalization scale?

 

[RGevo]

Hello,

The renormalisation scale is unphysical, but appears as an artefact when we do perturbation theory only to a finite order. So I don't think that's important.

The question is a tricky one, since the light quarks are never free and are in bound hadrons where strong (non-perturbative) qcd effects are ever present... Maybe a lattice person would give a clearer answer.

On the other hand, what is the qcd scale? The value at which the strong coupling becomes non-perturbative. It's not clear to me at which scale this is (below a gev for sure).

For practical purposes, it seems like ignoring the u,d quark masses is reasonable. Even the strange could be considered massless. You can see how good/bad an approximation this is by comparing pion and kaon massed.

 

[RGevo]

Having read the below thread. The quark review on the light quarks which vanhees posted is more informative!

 

[GIM]

Thank you, @RGevo.
I just copied the value of ~200MeV from the literature. In many literatures(I've seen), which treat the chiral symmetry, most of them mention the similar statements. Is this really tricky?

 

[vanhees71]

It's very tricky. A more pragmatic answer is that you can use chiral symmetry to build effective hadronic models. As it turns out, indeed chiral symmetry is a very good approximate symmetry of the strong interactions since the typical current-light-quark masses are small compared to the scale $4 \pi f_{\pi} \simeq 1 \; \text{GeV}$, where $f_{\pi} \simeq 92 \; \mathrm{MeV}$ is the pion-decay constant that can be measured through the weak decay of the pions. Note that chiral symmetry is as good a symmetry of the strong interaction as is isospin symmetry (which is violated, because the $u$ and $d$ quark masses are not the same, and the difference is also a few MeV).

 

[GIM]

Thank you, @vanhees71.
What kind is the scale $4\pi {f_\pi }$?
Now, you changed my question into the relation between quark masses and pion decay constant. Then, again, what reasoning makes me ignore the masses?
(For example, when I learned the pion decay, my professor used the formula $\left\langle 0 \right|\bar u{\gamma ^5}d\left| {{\pi ^ - }} \right\rangle = \frac{{\sqrt 2 {f_\pi }m_\pi ^2}}{{\left( {{m_u} + {m_d}} \right)}}$. In this case, the massless limit gives "unacceptable result" and I think I have no hope to find the relation between the pion decay constant and the quark masses. I guess my example is beyond the main direction of my question and if so, then please ignore this. Thank you.)