Tensor Product of two doublets

Question

What will be the tensor product of two doublets $$ (x_1,x_2) ~\text{and}~ (y_1,y_2)? $$ I am very much confused in determining this.

Answer 1

Let me give a simpler (and, surely, more naiive) answer.
Given two n-tuplets $x_i$ and $y_j$, their tensor product is a matrix: $$a_{ij} = x_iy_j$$ So, in your case: $$a_{ij} = \left(\begin{array}{cc}x_1y_1&x_1y_2\\x_2y_1&x_2y_2\end{array}\right)$$

Answer 2

The doublets – I assume that you mean 2-dimensional representations of $SU(2)\equiv Spin(3)$ – are spin-1/2 representation. Tensor products mean the addition of the angular momentum. The tensor product is 4-dimensional and under $SU(2)$, it decomposes to a $j=0$ multiplet, a scalar or singlet, and a $j=1$ multiplet, a vector: $${\mathbf 2}\otimes {\mathbf 2} = {\mathbf 1}\oplus {\mathbf 3} $$ In your notation, you may write $$(x_1,x_2)\otimes (y_1,y_2)\mapsto ((v_1,v_2,v_3),(s)) $$