Suppose that there exist several representations (i= 1, …, N) of Higgs scalars whose charge-zero

Step 1: **Understand the Parameters and Definitions** The parameter $rho$ is defined as $rho =

Question

Suppose that there exist several representations $(i=$ $1, \ldots, N)$ of Higgs scalars whose charge-zero members acquire vacuum expectation values $v_i$. Show that $$ \rho=\left(\frac{M_W}{M_Z \cos \theta_W}\right)^2=\frac{\sum v_i^2\left[T_i\left(T_i+1\right)-\frac{1}{4} Y_i^2\right]}{\sum \frac{1}{2} v_i^2 Y_i^2}, $$ where $T_i$ and $Y_i$ are, respectively, the weak isospin and hypercharge of representation $i$. Show that $\rho=1$ if only Higgs doublets with $Y_i=+1$ exist.

   Suppose that there exist several representations $(i=$ $1, \ldots, N)$ of Higgs scalars whose charge-zero members acquire vacuum expectation values $v_i$. Show that
$$
\rho=\left(\frac{M_W}{M_Z \cos \theta_W}\right)^2=\frac{\sum v_i^2\left[T_i\left(T_i+1\right)-\frac{1}{4} Y_i^2\right]}{\sum \frac{1}{2} v_i^2 Y_i^2},
$$
where $T_i$ and $Y_i$ are, respectively, the weak isospin and hypercharge of representation $i$. Show that $\rho=1$ if only Higgs doublets with $Y_i=+1$ exist.

Quarks and leptons: introductory course in modern particle physics

Francis Halzen, Alan… 1st Edition

Chapter 15, Problem 4

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Step 1

The quantities $v_i$, $T_i$, and $Y_i$ represent the vacuum expectation values, the weak isospin, and the hypercharge of the $i$-th Higgs scalar representation, respectively. Show more…

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Suppose that there exist several representations $(i=$ $1, \ldots, N)$ of Higgs scalars whose charge-zero members acquire vacuum expectation values $v_i$. Show that $$ \rho=\left(\frac{M_W}{M_Z \cos \theta_W}\right)^2=\frac{\sum v_i^2\left[T_i\left(T_i+1\right)-\frac{1}{4} Y_i^2\right]}{\sum \frac{1}{2} v_i^2 Y_i^2}, $$ where $T_i$ and $Y_i$ are, respectively, the weak isospin and hypercharge of representation $i$. Show that $\rho=1$ if only Higgs doublets with $Y_i=+1$ exist.

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