I was looking at Peskin and Schroeder (Section 19.3, page $667-668$). They talk about $\rm U(2)\times U(2)$ symmetry for the QCD Lagrangian in the limit of massless $u$ and $d$ quarks. However, this must be an approximate symmetry because $u$ and $d$ are not actually massless. If this is so, why does it not talk about $\rm U(3)\times U(3)$ symmetry including $s$ quark as well? Or even better, why not talk about $\rm U(6)\times U(6)$ symmetry involving all six flavours of quarks? What is so special about $\rm U(2)\times U(2)$ which too is approximate?
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It is because the $\rm U(2)\times U(2)$ symmetry breaking is much smaller than the $\rm U(3)\times U(3)$ symmetry breaking, not to speak about $\rm U(6)\times U(6)$ symmetry which is badly violated by large masses of heavy quarks $(c,b,t)$. The characteristic scale of strong interactions is $1$ GeV (the proton mass), while the u,d quark masses are of order a few MeV. Thus the predictions of $\rm U(2)\times U(2)$ symmetry for hadrons must be valid to an accuracy of about $1\%$. The s-quark mass is much larger (around 100 MeV), so $\rm U(3)\times U(3)$ symmetry predictions are not nearly as accurate. Of course, chiral anomalies destroy the axial $\rm U(1)$ symmetry, so we should really be talking about $\rm SU(2)\times SU(2)\times U(1)$ instead of $\rm U(2)\times U(2)$ and $\rm SU(3)\times SU(3)\times U(1)$ instead of $\rm U(3)\times U(3)$.
