Unitarity Constraints

Consider the scattering of two longitudinally polarised gauge bosons $W_L^{+}{W}_L^{-}\to W_L^{+}{W}_L^{-}$:

Recall that

${\Phi }^{†}\Phi =\frac{1}{2}({\varphi }_1^2+{\varphi }_2^2+(v+h{)}^2+{\varphi }_4^2)={\varphi }^{+}{\varphi }^{-}+\frac{1}{2}{\varphi }^0{\varphi }^0+\frac{1}{2}(v+h{)}^2$,

where we abbreviate ${\varphi }^{\pm }=({\varphi }_1\pm i{\varphi }_2)/\sqrt{2}$ and relabel ${\varphi }^0={\varphi }_4$. Inserting this into the Higgs potential, we obtain

$V(\Phi )=\frac{m_h^2}{2}{\Phi }^{†}\Phi +\frac{m_h^2}{2v^2}({\Phi }^{†}\Phi {)}^2=\frac{m_h^2}{8v^2}[(v+h{)}^2+2{\varphi }^{+}{\varphi }^{-}+{\varphi }^0{\varphi }^0{]}^2+\text{irrelevant terms}$

$=\frac{m_h^2}{2v^2}{\varphi }^{+}{\varphi }^{-}{\varphi }^{+}{\varphi }^{-}\frac{m_h^2}{v}h{\varphi }^{+}{\varphi }^{-}+\dots$

From this, we can obtain the Feynman rules for the interaction of the Nambu-Goldstone bosons.

Note: There is no direct coupling of the three Nambu-Goldstone bosons.

Consequently, for the highly-boosted $W^{+}{W}^{-}$ scattering, we obtain:

where the wavy-dashed lines represent the Nambu-Goldstone bosons ${\varphi }^{\pm }$.

The scattering amplitude is given by

$\mathcal{A}(W_{\lambda =0}^{+}{W}_{\lambda =0}^{-}\to W_{\lambda =0}^{+}{W}_{\lambda =0}^{-})=\mathcal{A}({\varphi }^{+}{\varphi }^{-}\to {\varphi }^{+}{\varphi }^{-})+\mathcal{O}\left(\frac{M_W^2}{s}\right)$

$=-\left[\frac{2m_h^2}{v^2}+{\left(\frac{m_h^2}{v}\right)}^2\frac{1}{s-m_h^2}+{\left(\frac{m_h^2}{v}\right)}^2\frac{1}{t-m_h^2}\right]$

$=-\frac{m_h^2}{v^2}\left[2+\frac{m_h^2}{s-m_h^2}+\frac{m_h^2}{t-m_h^2}\right]$,

where $s=(p_1+p_2{)}^2$ and $t=(p_1-p_3{)}^2$ are the Mandelstam variables.

Sidenote: Partial Wave Amplitudes

Using the $J=0$ partial wave amplitude as an example, we find and apply the unitarity constraint to $W^{+}{W}^{-}$ scattering. The $J=0$ partial wave amplitude is given by

$A_{J=0}(s)=-\frac{m_h^2}{16\pi v^2}\left[2+\frac{m_h^2}{s-m_h^2}-\frac{m_h^2}{s}\ln \left(1+\frac{s}{m_h^2}\right)\right]$

In the limit $s\gg m_h^2$,

${\mathcal{A}}_{J=0}(s)\to -\frac{m_h^2}{8\pi v^2}=-\frac{{\lambda }^2}{2\pi }$.

This result implies a bound on the Higgs mass of $m_H\lesssim 870\text{ GeV}$, known before the discovery of the Higgs boson at the LHC. A similar constraint can be obtained from $W^{+}{W}^{-}\to ZZ$ scattering, which yields a slightly more stringent upper bound of $m_H\lesssim 710\text{ GeV}$.

But, in the limit $m_h^2\ll s$, we obtain

${\mathcal{A}}_0(s)\to -\frac{s}{32\pi v^2}$

, from which we extract $s_c<(1.8\text{ TeV}{)}^2$. Using different channels, this can be further refined to $s_c<(1.2\text{ TeV}{)}^2$.