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In low-energy QCD, there are several dimensionful quantities that come up. Writing the chiral condensate as $$\langle \Omega | \bar{q}_{Ri} q_{Lj} | \Omega \rangle = - v^3 \exp \left(\frac{2 i \pi^a(x) \sigma^a}{f_\pi} \right)$$ the two scales are the pion decay constant and the chiral condensate, $$f_\pi \simeq 93 \, \text{MeV}, \quad v \simeq 250 \, \text{MeV}.$$ There are two further energy scales: $$\Lambda_{\text{QCD}} \simeq 250 \, \text{MeV}, \quad \Lambda_{\chi \text{SB}} \simeq 1 \, \text{GeV}$$ which are the scales at which color confinement and chiral symmetry breaking occur.

I know that according to "naive dimensional analysis", we expect $\Lambda_{\chi \text{SB}} \simeq 4 \pi f_\pi$. Furthermore, if the light quarks have mass $m_q$, then the QCD vacuum energy has amplitude $v^3 m_q$, which I've seen estimated as $4 \pi f_\pi^3 m_q$. Apparently, it also follows by naive dimensional analysis that $$v^3 \simeq 4 \pi f_\pi^3.$$ However, I haven't been able to find any source explaining why this is. Where does this result come from? Is there a resource that spells out the relation between these four quantities in more detail? Are there further relations that are important here?

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  • $\begingroup$ A slightly different topic, but for QCD condensates in general you should look into SVZ sum rules. $\endgroup$ – Mitchell Porter May 16 at 18:24
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We should not expect many rigorous relations involving the four quantities $\{f_{\pi}, v, \Lambda_{\chi SB}, \Lambda_{QCD}\}$, since most of these (especially the $\Lambda$ parameters) are not so rigorously defined.

The naive dimensional analysis statement $\Lambda_{\chi SB} = 4\pi f_{\pi}$ is a fine definition of $\Lambda_{\chi SB}$. In my work I've never tried to treat $\Lambda_{\chi SB}$ as anything more than a rough scale around which chiral perturbation theory ($\chi$PT) is likely to become unreliable, so I've never needed anything more rigorous (and can't recall seeing anything more rigorous).

A common definition of $\Lambda_{QCD}$ (discussed in chapter 83 of Srednicki's textbook) is the energy scale at which the perturbative gauge coupling diverges. This of course depends on the choice of renormalization scheme for that coupling, including the number of loop orders in the perturbative $\beta$ function used to run the coupling to lower energies. Even if you use lattice calculations instead of loop calculations, the dependence on the renormalization scheme remains.

Finally, that $v^3 \simeq 4\pi f_{\pi}^3$ does not look sensible, since the left-hand side is also scheme-dependent while the right-hand side is not. This looks almost like somebody has taken the Gell-Mann--Oakes--Renner (GMOR) relation (i.e., leading-order $\chi$PT), $$m_{\pi}^2 f_{\pi}^2 = 4m_q v^3,$$ and replaced $m_{\pi}^2 \propto m_q f_{\pi}$ (where $m_q$ is scheme-dependent while $m_{\pi}$ and $f_{\pi}$ are not). I would suggest just sticking with the GMOR relation itself, since this emphasizes that the scheme dependence of $v^3$ 'cancels out' with the scheme dependence of $m_q$.

PS. The factor of 4 in the GMOR relation above comes from Srednicki's Eq. 83.18 with $m_u+m_d=2m_q$. Leutwyler's Scholarpedia article instead has a factor of 2 (with the same convention $f_{\pi} \approx 93$ MeV). Perhaps they are using different normalizations for the Pauli matrices $\sigma^a$ vs. $\tau_i$. Recall that the $f_{\pi}$ and $v$ appearing in the GMOR relation are evaluated in the chiral limit $m_q \to 0$.

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  • $\begingroup$ Thanks for the answer! I think I follow what you're saying. The statement I asked about comes from this paper. More precisely, they state that the height of barriers in the vacuum energy is $\Lambda^4 = 4 \pi f_\pi^3 m_q$ and that this just follows from naive dimensional analysis. I guess you would conclude this is not rigorous/reliable? $\endgroup$ – knzhou May 18 at 15:59
  • $\begingroup$ That may be reasonably reliable, but if I'm looking at the right place (the paragraph containing Eq. 18) it doesn't seem particularly rigorous, as hinted by the "$\simeq$" they use instead of "$=$". My intuition would be to treat this as a fine definition (or "estimate" as they call it) of their "barrier coefficient" $\Lambda^4$ in the model they're building. The scheme dependence at least looks consistent on both sides of this relation. $\endgroup$ – David Schaich May 19 at 16:30

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