# Motivation for the $\text{SU}_f(3)$ flavour symmetry of Gell-Mann and why did it work?

Historically, the $\text{SU}_f(3)$ flavour symmetry was implemented as $$\begin{pmatrix}u^\prime \\ d^\prime\\ s^\prime\end{pmatrix}=U\begin{pmatrix}u\\d\\s\end{pmatrix}\tag{1}$$ where $U\in \text{SU}_f(3)$. But in the context of Standard Model (SM), if the quarks were massless, there seems to be only a $SU(2)$ flavour symmetry given by $$\begin{pmatrix}u^\prime \\ d^\prime\end{pmatrix}=U\begin{pmatrix}u\\d\end{pmatrix}\tag{2}$$ where $\begin{pmatrix}u & d\end{pmatrix}^T$ is a quark doublet in the SM.

1. What was the motivation for implementing a $\text{SU}_f(3)$ flavour symmetry as implemented in (1) and not as (2)?

2. Can the QCD symmetry (2) exist for other quark doublets $\begin{pmatrix}c & s\end{pmatrix}^T$ and $\begin{pmatrix}t & b\end{pmatrix}^T$?

3. What was the triumph of the symmetry (1) and why did it work when the Standard model shows that the symmetry is something else as given in (2)?

• Do you mean SU(3) colour? – ZeroTheHero Jan 15 '17 at 20:52
• Well, if , with E. Piaf, "Je me fous du passé", flavor SU(3) is still spectacularly useful, in low energy physics. The masses of the 3 light quarks are still lower than Λ(QCD) and the χSB scale thereof, so flavor-symmetric QCD is the starting point of Chiral perturbation theory. No adequate description of low-energy HEP can ignore it. – Cosmas Zachos Jan 17 '17 at 16:00
• @ZeroTheHero I've edited the question. Hope it makes sense now? – SRS Mar 25 '18 at 15:56

Some history

Is it because three more quarks $(c,b,t)$ were not discovered at that time?

People even didn't know about quarks when Gell-Mann was discovering his "eightfold way" (and long-long time after that). They only knew about pions, nucleons and other strongly interacting particles, and characterize them by the spin, electric charge, isospin. The number of these particles was very large, and people wanted to describe this strongly-interacting zoo in a simple way.

What was the motivation for implementing a $SU_{f}(3)$ flavour symmetry

There were underlying reasons for this.

One of them was $\sigma$-model constructed by Gell-Mann and Levy in 1960 in order to describe nucleon-meson interactions. It explains many phenomenological things like vector current conservation, PCAC and Goldberger-Treiman relation. The fundament of $\sigma$-model is approximate $G \simeq SU_{L}(2)\times SU_{R}(2)$ global symmetry group spontaneously broken down to $SU(2)$. This $SU(2)$ group is called the isospin group. The isospin symmetry was later extended on all strongly interacting particles, since it was realized that the strong interaction (approximately) respects it; other particles formed isospin multiplets.

Another reason was observation of some strongly interacting very massive (in compare to pions) particles which have relatively long life-time (even if their masses are large): precisely, they are decayed through weak interactions, which characteristic life-time is much larger than the strong interactions characteristic time. Gell-Mann proposed to explain this phenomenon by introducing new quantum number called strangeness; the strangeness is preserved by the strong interaction but is violated by the weak interaction.

And another important reason was the Gell-Mann-Nishijima formula $$Q = I_{3}+ (B+S)/2,$$ which relates the electric charge of stringly interacting particle to the linear combination of the isoapin projection $I_{3}$, the baryon number $B$ and strangeness $S$. This phenomenological expression describes hundreds particles!

So, to conclude: at the time of implementing the flavor symmetry all strongly interacting particles were characterized by their spin, isospin (and isospin projection), strangeness and electric charge. There was not enough for describing them in a simple but systematic way. But above reasons signaled that the true (approximate) global symmetry group of the strong interaction is some group larger than the isospin group $SU(2)$.

Gell-Mann then proposed to extend the global symmetry group to $SU(3)$. This group is what we know called the strong interaction flavor symmetry group. In his approach, different strongly interacting particles form multiplets with definite spin and similar masses. These multiplets are contained in the tensor product of (some) fundamental representations $3, \bar{3}$. Gell-Mann identified, for example, the pseudo-scalar mesons as the octet contained in the $3\otimes \bar{3} = 8 \oplus 1$ product, while the nucleons in the octet contained in the $3\otimes 3 \otimes 3 = 10 \oplus 8 \oplus 8 \oplus 1$ product.

What was the triumph of the symmetry (1)

So, the triumph of this approach was that it provided systematical way to describe all strongly interacting particles by representing them in a very simple way based on a universal property of strong interactions - the flavor symmetry. It also predicted the new particle called $\Omega^{-}$ and relations of masses of particles from given multiplet. But from the modern perspective it was the main theoretical reason to introduce quarks. It also explains the Gell-Mann-Nishijima formula, since three quarks $(u,d,s)$ can have only three independent quntum numbers, and hence if there are $m$ extra quantum numbers, then there must be $m$ relations between all $3+m$ quantum numbers leaving only 3 numbers independent.

The modern view

From the modern point of view (the QCD one), these fundamental representations $3, \bar{3}$ are physically recognized as quarks triplets $(u,d,s)$, $(\bar{u}, \bar{d},\bar{s})$. The underlying global symmetry of the QCD is $SU_{L}(3)\times SU_{R}(3)$ (here I "neglected" the $U_{A}(1)\times U_{B}(1)$ group, which is not relevant here). It is approximate symmetry, since quarks have relatively small masses breaking it explicitly. This symmetry is spontaneously broken down to $SU_{V}(3)$. The $SU_{V}(3)$ is what You can call the flavor symmetry. The fact that mesons (for example) are contained in $3\otimes \bar{3}$ means that they consist of two quarks (some quark and some anti-quark); the nucleons are contained in $3\otimes 3 \otimes 3$, and this means that they consist of 3 quarks.

and why did it work when the Standard model shows that the symmetry is something else as given in (2)?

Of course, there are three other quarks - $c,t,b$, and formally You may think about $SU(6)$ symmetry. But their masses are so large (more than $\text{ GeV}$) that they don't make the contribution in the observed particles spectrum: the $SU(6)$ symmetry isn't even approximate symmetry at most interesting scales. Even $SU_{f}(3)$ symmetry works bad below $100 \text{MeV}$ because of large mass of $s$-quark.

I am not sure what you mean by the $SU(3)_L$ symmetry of the standard model. The SM only has a $SU(2)_L$ gauge symmetry, and no global flavor symmetries.

QCD, without the weak interaction, has a global $SU(3)_L\times SU(3)_R$ symmetry in the limit of massless $u,d,s$ quarks. The $SU(3)_{L-R}$ is spontaneously broken in QCD, and only $SU(3)_{L+R}$ is a symmetry of the spectrum.

The $SU(3)_{L+R}$ flavor symmetry is broken by the small differences between quark masses, and electroweak effects. As a consequence, Gell-Mann's $SU(3)$ is an approximate symmetry.

• Minor quibble: the 3 L-R axial symmetries broken by χSB don't actually close into an SU(3)... – Cosmas Zachos Jan 15 '17 at 20:47
• SU(3)_A is a group, isn't it? The coset is not in general a group, but here even that is the case $SU(3)_V \times SU(3)_A/SU(3)_A = SU(3)_V$. – Thomas Jan 16 '17 at 2:44
• There is no such thing as SU(3)_A. Two axial generators close to a vector under commutation and cannot close to an axial. Just never write $SU(3)_A$ or $SU(3)_{L-R}$. The unbroken diagonal group is $SU(3)_V=SU(3)_{L+R}$, while the 8 sbroken generators of the coset space $(SU(3)_L\times SU(3)_R)/SU(3)_V$ do not close in a group. Pls look at the algebra in a book. – Cosmas Zachos Jan 16 '17 at 12:23
• Oops, easy to forget .. – Thomas Jan 17 '17 at 4:57
• @Thomas I've edited the question. Hope it makes sense now? – SRS Mar 25 '18 at 15:57

Your equation (2) is essentially malformed: The SM weak isospin doublet structure (which I am guessing you are trying to evoke by choosing massless doublets) only applies to left-handed quarks. The right-handers are singlets. There are three of these doublets, (u,d),(c,s),(t,b), and when the Higgs Yukawa couplings give them masses by connecting them with the right-handers, these masses are very different, so no degeneracy is apparent (--or else, all 6 of them are relatively degenerate compared to the 1/4 TeV EW breaking scale!) In fact, weak isospin is a distraction and a spoiler in degeneracy patters, as you appear to suspect.

• The motivation for (1) was practical: "approximate degeneracy" of the three light quarks, (u,d,s). (The charmed and heavier quarks were not known, but, more importantly, they are all above the scale of the strong interactions, that make SU(3) spectroscopy easy and systematic, so there is no point in using flavor symmetry except as a classification scheme.) This, then, is the triumph of the Eightfold Way: the tractable systematics of symmetry breaking. (In addition, it encouraged particle physicists to take Lie groups seriously, thereby making future developments easier.)

• Since you are asking about the reason this works ("why did it work"?), it is, of course, what good particle physics texts rush to clarify: The masses of (u,d,s) are quite smaller than the mass scales controlling QCD and its χSB, which confine quarks and endow hadrons with masses.

• Specifically, the masses of u,d,s are (2.3,4.8,95)MeVs, contrasted to 218MeV for $\Lambda_{QCD}$ and 250MeV for the chiral condensate. So u,d,s are not degenerate (far from it!), but their effects are approximately so, since the essential scales of most hadrons are not that sensitive to them, but, instead, are controlled by QCD. (Most hadrons, except for the pseudoscalar mesons, which are very sensitive to these masses as they are involved in the chiral symmetry breaking.)

The takeaway is that the EW interactions do not affect low-mass hadron masses, except through Yukawa couplings.

For the 3 heavier quarks, by sharp contrast, they completely overwhelm QCD effects and flavor SU(4), SU(5), SU(6) are mere state-counting classification, not degeneracy, schemes. (There are actually heavy quark symmetries of the heavier quarks, BJ's "brown muck" symmetries, where QCD hadronic features effectively decouple from the much larger quark masses, but let us not complicate matters here.)