How do we known that $\langle \bar{\psi}_i \psi_j\rangle=(250 MeV)^3\delta_{ij}$?

Question

I have started to read the phenomenology of QCD in low energy regime. I understand that, from the QCD renormalization group equation, the QCD becomes nonperturbative theory when energy scale is below $\Lambda_\text{QCD}\simeq 250$ MeV.

The quark mass terms give the explicit chiral symmetry breaking, which should give \begin{align} \langle \bar{\psi}_i \psi_j\rangle\neq 0. \end{align} However, I do not understand that why $\langle \bar{\psi}_i \psi_j\rangle=(\Lambda_\text{QCD})^3\delta_{ij}$?. Can it be $\langle \bar{\psi}_i \psi_j\rangle=(\Lambda_{\chi SB})^3\delta_{ij}$ or another value in a few GeV?

Answer 1

The quark condensate at zero temperature sets the scale for the square of the mass of pseudoscalar mesons, in what is often referred to as the Gell-Mann—Oakes—Renner fundamental relation of chiral dynamics, and turns out to be something like $$ \langle \bar q q\rangle \sim -(242 MeV)^3, $$ when estimated at the scale of 1 GeV, where perturbation theory has failed. Cf here.

It can be computed/estimated from first principles QCD on the lattice, e.g. in Full result for the QCD equation of state with flavors, Physics Letters B730 (2014) pp 99-104, S Borsányi, Z Fodor, C Hoelbling, S Katz, S Krieg, K Szabó for the light quarks. (It varies with the type of quarks included.)

It is a pure QCD phenomenon, so it's not a surprise it crucially depends on the QCD scale $\Lambda_{QCD}$ which controls it (in fact, makes it possible!) in the strong coupling regime. Browse the "Related" links.

It quantifies χSB which happens at a slightly higher energy scale than the inverse confinement radius, so there is a subtle "zone" inside hadrons where chiral symmetry breaking has already occurred, so quarks are heavy/constituent, and hence couple to pions/pseudoscalars, the effective pseudogoldstons of the SSB.

Answer 2

I agree with the answer given by @Cosmas Zachos but I want to comment/elaborate a bit more. Quark masses introduce explicit chiral symmetry breaking in QCD -- in case of the light quarks quite a small one at that. The formation of a chiral condensate $\langle\bar\psi_i\psi_i\rangle\neq 0$ signaling chiral symmetry breaking of the vacuum ground state is not caused by quark masses. It is caused by quantum fluctuations (especially of quarks). Why do we know that? Because we find $\langle\bar\psi_i\psi_i\rangle\neq 0$ in the chiral limit (limit of vanishing quark masses) of QCD. The masses of the light quarks (for this discussion lets assume $m_u=m_d\equiv m$ have only a rather small effect on the value of $\langle\bar\psi_i\psi_i\rangle$.

Another rather obvious fact which one can read of the statement $\langle\bar\psi_i\psi_j\rangle= -\Lambda_\mathrm{QCD}^3 \delta_{ij}$ is that this quantity is obviously RG scheme dependent and in a way stating it sets part of the scheme. So I would make the case that $\langle\bar\psi_i\psi_j\rangle= -\Lambda_\mathrm{QCD}^3 \delta_{ij}$ is an established choice -- or lets say a consequence of the established choice -- to set $\Lambda_\mathrm{QCD}$ but not necessarily a requirement. As for $\Lambda_{\chi\mathrm{SB}}$ or $\Lambda_\mathrm{conf}$ what are those? They are again (RG) scales defined by some relation -- diverging of effective four fermi couplings, first formation of a condensate during RG flow, some value for the Polyakov loop, ... -- but their value and even what they really are again depends on the chosen scheme. That being said they are not directly connected to $\langle\bar\psi_i\psi_i\rangle$ in the sense $\Lambda_\mathrm{QCD}$ is. I do not want to give the impression that those scales are not important: the fact that they emerge is very important for the phenomenology of QCD and the strong interaction. Their precise definitions in the framework of choice and the corresponding numerical values however not so much.

The value of $\langle\bar\psi_i\psi_i\rangle$ (in what ever scheme or with what ever method we obtained it) is of extreme importance for the low energy sector of QCD and its vacuum. In fact it is directly related to one ($B_0$) of only two parameters for the leading order of one of the prime low energy effective theory of QCD: chiral perturbation theory. The other parameter being the pion decay constant $F_0$. Considering chiral perturbation theory at the lowest-order effective Lagrangian for three flavors one obtains $$ 3F_0^2 B_0=-\langle\bar\psi\psi\rangle, $$ where summation over the three quark flavors is implied with $0\neq \langle\bar\psi\psi\rangle=3\langle\bar{u}u\rangle=3\langle\bar{d}d\rangle=3\langle\bar{s}s\rangle$, see, e.g., S. Scherer, 2002, Introduction to Chiral Perturbation Theory. To get to the more common form of the Gell-Mann—Oakes—Renner relation one can use the LO result for the pion mass $$ M_\pi^2=2B_0 m. $$ Old estimates (I am sure there are newer ones but I do not have references at hand) in the chiral limit are $$ \begin{align} F_0&=88.3\pm1.1\,\mathrm{MeV},\\ \langle\bar\psi\psi\rangle_0^{1/3}&=-(225\pm25)\,\mathrm{MeV}, \end{align} $$ where we notice that both are a bit smaller in magnitude than the usual values out of the chiral limit coming back to the small effect the (light) quark masses have on those quantities. The main contribution to those quantites are fluctuations not the small explicit chiral symmetry breaking of the light quarks. Regarding the underling renormalization procedure in chiral perturbation theory I refer to the literature (I my self do not know the details...).