This question comes from Srednicki's textbook "Quantum Field Theory". On page 532, the left-handed Weyl fields $\ell$ (a single lepton family, electron and its neutrino) and $\overline{e}$ are in the representations $(2, -\frac{1}{2})$ and $(1, +1)$ of $SU(2) \times U(1)$. It is stated:

We cannot write down a mass term involving $\ell$ and/or $\overline{e}$ because there is no gauge-group singlet contained in any of the products \begin{equation} (2, -\frac{1}{2}) \otimes (2, -\frac{1}{2}), \\ (2, -\frac{1}{2}) \otimes (1, +1), \\ (1, +1) \otimes (1, +1) .\tag{88.4} \end{equation}

I calculate the first product as follows:

1. Using Young tableaux to do the calculation for the first entry in $SU(2)$ gives \begin{equation} 2 \otimes 2 = 1 \oplus 3 \end{equation}
2. For the second entry, I simply use addition \begin{equation} -\frac{1}{2} - \frac{1}{2} = -1 \end{equation}
3. Combining 1. and 2., I get the result \begin{equation} (2, -\frac{1}{2}) \otimes (2, -\frac{1}{2}) = (1, -1) \oplus (3, -1) \end{equation} Is this correct? If yes, isn't $(1, -1)$ a singlet, which will refute the statement in the text?

This question comes from Srednicki's textbook Quantum Field Theory. On page 532, the left-handed Weyl fields $\ell$ (a single lepton family, electron and its neutrino) and $\overline{e}$ are in the representations $(2, -\frac{1}{2})$ and $(1, +1)$ of $SU(2) \times U(1)$. It is stated:

We cannot write down a mass term involving $\ell$ and/or $\overline{e}$ because there is no gauge-group singlet contained in any of the products \begin{equation} (2, -\frac{1}{2}) \otimes (2, -\frac{1}{2}), \\ (2, -\frac{1}{2}) \otimes (1, +1), \\ (1, +1) \otimes (1, +1) .\tag{88.4} \end{equation}

I calculate the first product as follows:

1. Using Young tableaux to do the calculation for the first entry in $SU(2)$ gives \begin{equation} 2 \otimes 2 = 1 \oplus 3 \end{equation}
2. For the second entry, I simply use addition \begin{equation} -\frac{1}{2} - \frac{1}{2} = -1 \end{equation}
3. Combining 1. and 2., I get the result \begin{equation} (2, -\frac{1}{2}) \otimes (2, -\frac{1}{2}) = (1, -1) \oplus (3, -1) \end{equation} Is this correct? If yes, isn't $(1, -1)$ a singlet, which will refute the statement in the text?

This question comes from Srednicki's textbook "Quantum Field Theory". On page 532, the left-handed Weyl fields $\ell$ (a single lepton family, electron and its neutrino) and $\overline{e}$ are in the representations $(2, -\frac{1}{2})$ and $(1, +1)$ of $SU(2) \times U(1)$. It is stated:

We cannot write down a mass term involving $\ell$ and/or $\overline{e}$ because there is no gauge-group singlet contained in any of the products \begin{equation} (2, -\frac{1}{2}) \otimes (2, -\frac{1}{2}), \\ (2, -\frac{1}{2}) \otimes (1, +1), \\ (1, +1) \otimes (1, +1) .\tag{88.4} \end{equation}

I don't know how to calculate these products. I try to calculate the first product as follows:

1. Using Young tableaux to do the calculation for the first entry in $SU(2)$ gives \begin{equation} 2 \otimes 2 = 1 \oplus 3 \end{equation}
2. For the second entry, I simply use addition \begin{equation} -\frac{1}{2} - \frac{1}{2} = -1 \end{equation}
3. Combining 1. and 2., I get the result \begin{equation} (2, -\frac{1}{2}) \otimes (2, -\frac{1}{2}) = (1, -1) \oplus (3, -1) \end{equation} Is this correct? If yes, isn't $(1, -1)$ a singlet, which will refute the statement in the text?

This question comes from Srednicki's textbook "Quantum Field Theory". On page 532, the left-handed Weyl fields $\ell$ (a single lepton family, electron and its neutrino) and $\overline{e}$ are in the representations $(2, -\frac{1}{2})$ and $(1, +1)$ of $SU(2) \times U(1)$. It is stated:

We cannot write down a mass term involving $\ell$ and/or $\overline{e}$ because there is no gauge-group singlet contained in any of the products \begin{equation} (2, -\frac{1}{2}) \otimes (2, -\frac{1}{2}), \\ (2, -\frac{1}{2}) \otimes (1, +1), \\ (1, +1) \otimes (1, +1) .\tag{88.4} \end{equation}

I calculate the first product as follows:

1. Using Young tableaux to do the calculation for the first entry in $SU(2)$ gives \begin{equation} 2 \otimes 2 = 1 \oplus 3 \end{equation}
2. For the second entry, I simply use addition \begin{equation} -\frac{1}{2} - \frac{1}{2} = -1 \end{equation}
3. Combining 1. and 2., I get the result \begin{equation} (2, -\frac{1}{2}) \otimes (2, -\frac{1}{2}) = (1, -1) \oplus (3, -1) \end{equation} Is this correct? If yes, isn't $(1, -1)$ a singlet, which will refute the statement in the text?

This question comes from Srednicki's textbook "Quantum Field Theory". On page 532, the left-handed Weyl fields $\it{l}$ (a single lepton family, electron and its neutrino) and $\overline{e}$ are in the representations $(2, -\frac{1}{2})$ and $(1, +1)$ of $SU(2) \times U(1)$. It is stated:

We cannot write down a mass term involving $\it{l}$ and/or $\overline{e}$ because there is no gauge-group singlet contained in any of the products \begin{equation} (2, -\frac{1}{2}) \otimes (2, -\frac{1}{2}), \\ (2, -\frac{1}{2}) \otimes (1, +1), \\ (1, +1) \otimes (1, +1) .\tag{88.4} \end{equation}

I don't know how to calculate these products. I try to calculate the first product as follows:

1. Using Young tableaux to do the calculation for the first entry in $SU(2)$ gives \begin{equation} 2 \otimes 2 = 1 \oplus 3 \end{equation}
2. For the second entry, I simply use addition \begin{equation} -\frac{1}{2} - \frac{1}{2} = -1 \end{equation}
3. Combining 1. and 2., I get the result \begin{equation} (2, -\frac{1}{2}) \otimes (2, -\frac{1}{2}) = (1, -1) \oplus (3, -1) \end{equation} Is this correct? If yes, isn't $(1, -1)$ a singlet, which will refute the statement in the text?

This question comes from Srednicki's textbook "Quantum Field Theory". On page 532, the left-handed Weyl fields $\ell$ (a single lepton family, electron and its neutrino) and $\overline{e}$ are in the representations $(2, -\frac{1}{2})$ and $(1, +1)$ of $SU(2) \times U(1)$. It is stated:

We cannot write down a mass term involving $\ell$ and/or $\overline{e}$ because there is no gauge-group singlet contained in any of the products \begin{equation} (2, -\frac{1}{2}) \otimes (2, -\frac{1}{2}), \\ (2, -\frac{1}{2}) \otimes (1, +1), \\ (1, +1) \otimes (1, +1) .\tag{88.4} \end{equation}

I don't know how to calculate these products. I try to calculate the first product as follows:

1. Using Young tableaux to do the calculation for the first entry in $SU(2)$ gives \begin{equation} 2 \otimes 2 = 1 \oplus 3 \end{equation}
2. For the second entry, I simply use addition \begin{equation} -\frac{1}{2} - \frac{1}{2} = -1 \end{equation}
3. Combining 1. and 2., I get the result \begin{equation} (2, -\frac{1}{2}) \otimes (2, -\frac{1}{2}) = (1, -1) \oplus (3, -1) \end{equation} Is this correct? If yes, isn't $(1, -1)$ a singlet, which will refute the statement in the text?