The following is an about a Left-Right Symmetric model.

$SU(2)\otimes SU(2)$ $(2\otimes 2=3\oplus 1)$ will generate a triplet, which in Left-Right Symmetric model is $$\vec{\Delta}=\begin{pmatrix}\delta_{1}\\ \delta_{2} \\ \delta_{3} \end{pmatrix}$$. The $SU(2)$ quantum number/charge on above is $$\begin{pmatrix} 1\\0\\-1 \end{pmatrix}$$. We can write this in $2\times 2$ representation as $\Delta=\frac{1}{2}\vec{\tau}\cdot \vec{\Delta}$, where $\vec{\tau}$ is vector made out of pauli matrices.

  1. What is the complete mathematical explanation of $2\otimes 2=3\oplus 1$?

  2. How to make a scalar triplet from a scalar doublet?

    Answer. There is some procedure like $H^{T} i \tau_{2} \tau H$. (PS. I don't know exact form of this at this point of time).

  3. What is the exact form? and why it is correct or how to make a cross product out of two doublet scalar fields?

  4. How to calculate the charges of the components $\Delta$?

    Answer. There is some mechanism to calculate the $T_{3L}+T_{3R}$ using commutator as $T_{3L}+T_{3R}=\frac{1}{2}[\tau_{3},\Delta]$. Or we can say the formula to calculate $Q$ is $Q=T_{3L}+T_{3R}+\frac{1}{2}(B-L)=\frac{1}{2}[\tau_{3},\Delta]+ \frac{1}{2}(B-L)$.

    Main Question. What is the explanation to this? I tried to solve this but fails. There is also some charge distribution which is unexplained to me and canbe written as: $$\begin{pmatrix}\delta_{1}^{++}\\ \delta_{2}^{+}\\ \delta_{3}^{0}\\ \end{pmatrix}$$

  5. Why is the $B-L$ charge for $\Delta_{L}$ is not $0$ like higgs doublet?

    Answer. We need to break the $U(1)_{B-L}$ symmetry so we need $B-L$ charge in Triplets.

  6. What is the other explanation to this? We can have this charge on Bi-doublet too, but we are not choosing that for some reason.

    Answer. Bi-Doublet is responsible for symmetry breaking of Intermediate Salam-Weinberg Model gauge group which needs zero charge of $B-L$.

  7. How come Bi-doublet is balancing left-handed and right-handed scalar fields i.e., We need a doublet in minimal Standard model now we should need another doublet instead we are defining a bidouble. Doesn't the left handed part will disturb the right handed fields, If not, how?

  8. How to distinguish between $\Delta_{L}$ & $\Delta_{R}$?


closed as too broad by ACuriousMind, AccidentalFourierTransform, Qmechanic Apr 19 '16 at 0:05

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Please ask a single specific question per post, not a slew of related-but-different questions, as that makes giving complete answers hard. Also, you seem to be answering some of your questions already in the post, what is the point of including them? Lastly, don't use MathJax to format text. Use *text* for italic, and **text** for bold. If you want to do a numbered list, just put n. as the enumerator, the system will automatically format that as a list. $\endgroup$ – ACuriousMind Apr 18 '16 at 12:22
  • $\begingroup$ @ACuriousMind The answers need some corrections that's why I asked. I want to know if the analysis is right or not? $\endgroup$ – Love Grover Apr 18 '16 at 12:27