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Before introducing top bottom and charm quarks,Strong Hypercharge is defined in the following two ways---

$1.\,\,\,Y=B+S$ where $Y,B,S$ are the hypercharge, baryon number and strangeness respectively.

$2.\,\,\,SU(3)$ can be decomposed into its maximal subgroup---$SU(2) \times U(1)$.Let in a basis, $SU(2)$ generators be the first three Gell-Mann matrices $\lambda_1,\lambda_2,\lambda_3$ and $U(1)$ generator be the 8th Gell-Mann matrix $\lambda_8$. Hypercharge is often defined as the U(1) quantum number, just like isospin is the $SU(2)$ quantum number. Therefore hypercharge is the eigenvalue of $\lambda_8$.

How can we get from 2 to 1 or the reverse, i.e. how are these definitions equivalent?

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    $\begingroup$ Your second "definition" is not a definition. There are infinitely many ways to choose a $\mathrm{SU}(2)\times \mathrm{U}(1)\subset\mathrm{SU}(3)$ - you've hidden all of the actual definition in your "Let in a basis". Obviously if you pick the basis such that $\lambda_8 = B+S$ this is equivalent to the first definition, otherwise it is not. $\endgroup$
    – ACuriousMind
    Oct 3, 2020 at 9:01
  • $\begingroup$ Then how do we relate $U(1)$ to hypercharge of the correct definition (1)? $\endgroup$ Oct 3, 2020 at 11:38

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You can construct representations of $\text{SU}(3)$ from tensor products of the standard and dual actions of its complexified Lie algebra $\mathfrak{su}(3)\otimes\mathbb{C}\simeq\mathfrak{sl}(3,\mathbb{C})$ on $\mathbb{C}^3$, corresponding to the number of quarks and antiquarks respectively, which we will call the $(p,q)$ representations. (All I mean by complexification is taking complex linear combinations of the generators of $\text{SU}(3)$ so we can define ladder operators. The isomorphism to $\mathfrak{sl}(3,\mathbb{C})$ is a standard result that you can ignore if you don't know it). These end up giving us the decuplet, the different octets and singlets when combined with spin and requiring symmetry under permutations. These are representations of a flavor symmetry (that is broken by mass).

Like you said, $\mathfrak{sl}(3,\mathbb{C})$ contains some $\mathfrak{sl}(2,\mathbb{C})\oplus \mathbb{C}$ subalgebra. It is made up of a spin algebra $\mathfrak{sl}(2,\mathbb{C})\simeq\mathfrak{su} (2)\otimes\mathbb{C}) $ which mixes up and down quarks and under which the strange quark is a singlet, and a hypercharge algebra, as $\mathbb{C}\simeq\mathfrak{u}(1)\otimes\mathbb{C}$, and these algebras commute with eachother. And like ACuriousMind commented, you have to choose a basis and therefore define what a strange quark is (and so the notion of strangeness) for the formula to make sense.

We take a basis as

$$ I_3=\frac{1}{2}\lambda_3,\quad I_\pm = \frac{\lambda_1\pm i\lambda_2}{2},\quad Y=\frac{\lambda_8}{\sqrt{3}}$$

$$ u = \pmatrix{1\\0\\0}\quad d = \pmatrix{0\\1\\0}\quad s = \pmatrix{0\\0\\1}$$

where we have that $[Y,I_3]=0=[Y,I_\pm]$ as needed. We are of course omitting mention of the other ladder operators in $\mathfrak{sl}(3,\mathbb{C})$ that mix the strange quark with the others.

Using the explicit form of these matrices, we see that the strange quark has zero isospin, the up and down quark have $\pm\frac{1}{2}$ isospin respectively. The strange quark has hypercharge $-\frac{2}{3}$, and the up and down quark have hypercharge $\frac{1}{3}$. This is reversed for the antiparticles in the dual representation, since the action of a Lie algebra element $\lambda$ on some $\overline{q}\in\left(\mathbb{C}^3\right)^\ast$ is

$$ \lambda (\overline{q})=-\lambda^t\cdot \overline{q}$$

where $\cdot$ represents the regular matrix multiplication. You can calculate the charges of the tensor products by the usual operators $F = f\otimes 1 \otimes\ldots\otimes 1+\ldots$, etc, where each $f$ that is tensored is the single particle operator. Then the hypercharge of a vector in some representation is then given by $$Y=\frac{n_u+n_d-2n_s}{3}-\frac{n_\overline{u}+n_\overline{d}-2n_\overline{s}}{3}$$

Now define the baryon number as $B=\frac{p-q}{3}=\frac{1}{3}\left(n_u+n_d+n_s-n_\overline{u}-n_\overline{d}-n_\overline{s}\right)$, and the strangeness as $S=-(n_s-n_\overline{s})$, and your result follows.

You can also get the charges of the quarks as $Q = I_3 + \frac{Y}{2}$. This just relates the $\text{SU}(3)$ flavor symmetry charges to the electric charge under which the quarks couple to the electromagnetic field.

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