mW constraint

We measure \(p_\mu^\mu\) and \(\vec{p}_T^\nu\).

We will impose \({p_\mu^{\rm{comb.}}}^2 = m_W^2\)

Assume \(p_\nu^2 = p_\mu^2 = 0\). Only unknown is \(p_z^\nu\)

\(p_\mu^{\rm{comb.}} = \begin{pmatrix} E_\mu + E_\nu \\ \vec{p}_T^\mu + \vec{p}_T^\nu \\ {p_z}_\mu + {p_z}_\nu \end{pmatrix}\)

\({p_\mu^{\rm{comb.}}}^2 = (E_\mu + E_\nu)^2 - \left[ (\vec{p}_T^\mu + \vec{p}_T^\nu)^2 + ({p_z}_\mu + {p_z}_\nu)^2 \right]\)

\(= \cancel{|p_\mu|^2} + 2|p_\mu| |p_\nu| + \cancel{|p_\nu|^2} - \left[ \cancel{{\vec{p}_T^\mu}^2} + {\cancel{\vec{p}_T^\nu}^2} + 2 \vec{p}_T^\mu \cdot \vec{p}_T^\nu + \cancel{{{p_z}_\mu}^2} + \cancel{{{p_z}_\nu}^2} + 2{p_z}_\mu{p_z}_\nu \right]\)

OR

\(m_W^2 = 2 E_\mu \sqrt{ {\vec{p}_T^\nu}^2 + {{p_{z}}_\nu}^2 } - 2 \left[ \vec{p}_T^\mu \cdot \vec{p}_T^\nu + {p_z}_\mu {p_z}_\nu \right]\)

\(\frac{m_W^2}{2} + \left[ \vec{p}_T^\mu \cdot \vec{p}_T^\nu + {p_z}_\mu {p_z}_\nu \right] = E_\mu \sqrt{ {\vec{p}_T^\nu}^2 + {{p_{z}}_\nu}^2 }\)

Squaring

\(\frac{m_W^4}{4} + m_W^2 \left[ \vec{p}_T^\mu \cdot \vec{p}_T^\nu + {p_z}_\mu {p_z}_\nu \right] + \left[ \vec{p}_T^\mu \cdot \vec{p}_T^\nu + {p_z}_\mu {p_z}_\nu \right]^2 = E_\mu^2 ( {\vec{p}_T^\nu}^2 + {{p_{z}}_\nu}^2 )\)

\(\frac{m_W^4}{4} + m_W^2 \left[ \vec{p}_T^\mu \cdot \vec{p}_T^\nu + {p_z}_\mu {p_z}_\nu \right] + \left[ (\vec{p}_T^\mu \cdot \vec{p}_T^\nu)^2 + 2(\vec{p}_T^\mu \cdot \vec{p}_T^\nu) {p_z}_\mu {p_z}_\nu + ({p_z}_\mu {p_z}_\nu)^2 \right] = E_\mu^2 ( {\vec{p}_T^\nu}^2 + {{p_{z}}_\nu}^2 )\)

OR …

\(\left( \frac{m_W^4}{4} + m_W^2 \vec{p}_T^\mu \cdot \vec{p}_T^\nu + (\vec{p}_T^\mu \cdot \vec{p}_T^\nu)^2 - E_\mu^2 {\vec{p}_T^\nu}^2 \right) + \left(m_W^2{p_z}_\mu + 2(\vec{p}_T^\mu \cdot \vec{p}_T^\nu){p_z}_\mu \right){p_z}_\nu + \left( {p_z}_\mu^2 - E_\mu^2 \right){{p_z}_\nu}^2 =0\)

which is

\(C + B \cdot {p_z}_\nu + A \cdot {{p_z}_\nu}^2 = 0\)

with

\(A = \left( {p_z}_\mu^2 - E_\mu^2 \right)\)

\(B = \left(m_W^2{p_z}_\mu + 2(\vec{p}_T^\mu \cdot \vec{p}_T^\nu){p_z}_\mu \right)\)

\(C = \left( \frac{m_W^4}{4} + m_W^2 \vec{p}_T^\mu \cdot \vec{p}_T^\nu + (\vec{p}_T^\mu \cdot \vec{p}_T^\nu)^2 - E_\mu^2 {\vec{p}_T^\nu}^2 \right)\)

Then have two solutions for \({p_z}_\nu\).

\({p_z}_\nu = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\)