# [Physics] Derivation of Gell-Mann Okubo relation for mesons

baryonsgroup-theorymesonsparticle-physicsquarks

In SU(3) quark model of hadronic structure one assumes that mass splitings between hadrons is due to difference between masses of $s$ quark and $u,d$. This is modeled by perturbation Hamiltonian $$\delta H=\frac{m_s-m}{3}(1-3 Y),$$ where $m$ is mass of $u,d$ and $Y$ is hypercharge. In particular in fundamental representation in basis $u,d,s$ this matrix has the form $$\delta H =\mathrm{diag(0,0,m_s-m)}.$$
In eigenbasis of hypercharge one immedietely gets the expected values of this operator and from that corrections to energy to first order in perturbation theory. This yields correct formulas for baryon multiplets: mass differences are approximately proportional to differences in hypercharge, $$M=a+b Y,$$ with $a,b$ some constants. However in lecture notes from the class in particle physics I attended different approach is used for mesons. My teacher uses the fact that $Y$ is an eighth element of $(1,1)$ irreducible representation of SU(3) and then claims to have used Clebsch-Gordan coefficients for SU(3) to obtain the following formula: $$M= a'+ b' Y + c' \left( I(I+1) -\frac{1}{4} Y^2 \right),$$
with $a',b',c'$ some constants. From that using assumption that $b'=0$ because mass is the same for particle-antiparticle pairs it is quite easy to get the celebrated Gell-Mann Okubo relation (actually one gets this for masses rather than their squares, but it is closer to truth if we put squares by hand) $$4 M_K ^2 = M_{\pi}^2 +3 M_{\eta}^2.$$
I don't understand why in this case we can't just explicitly evaluate the $Y$ operator to get the usual relation which holds for baryons. In Perkins it is written that this GMO relation is empirical rather than derived from SU(3) model. How should I understand this?

The basic idea of Dashen's formula (often also referred to as Gell-Mann-Oakes-Renner (1968) doi:10.1103/PhysRev.175.2195 in the sloppy shorthand of chiral perturbation theory. It is a blending of a current algebra Ward identity with PCAC, $m_\pi^2 f_\pi^2=-\langle 0|[Q_5,[Q_5,H]]|0\rangle$) is that the square of the mass of the pseudoglodstone boson is proportional to the explicit breaking part of the effective lagrangian, here linear in the quark masses, as you indicated.
That is, for example, naively, the pion mass, which should have been zero for massless quarks, now picks up a small value $m_\pi^2 \sim m_q \Lambda^3/f_\pi^2$, where $m_q$ is the relevant light quark mass in the real world QCD Lagrangian, which explicitly breaks chiral symmetry; $f_\pi$ is the spontaneously broken chiral symmetry constant, about 100MeV; and Λ the fermion condensate value ~ 250MeV. That is to say, the square of the mass of the pseudogoldston is the coefficient of the second derivative of the effective lagrangian (it pulls two powers of the goldston out of the chiral vacuum with strength $f_\pi^2$) and so the commutator of the QCD lagrangian w.r.t. two chiral charges. Normally, that would be zero, but if there is a small quark mass term, it snags, so you get the quark mass term provide a quark bilinear times a quark mass, the v.e.v. of the bilinear amounting to Λ cubed.