Higgs Mechanism

We now consider a slight variation of the argument from last day, where we use a slightly different field equation for the Higgs mode,

$\begin{equation}\label{klein2} (\partial_t-i A_t)^2\phi -(\partial_x-i A_x)^2\phi-(\partial_y-i A_y)^2\phi-m^2\phi+\lambda^2 |\phi|^2 \phi=0. \end{equation}$ Now in addition to the stationary point at

$\phi=0$ , this equation has a static uniform solution

$\phi= (m/\lambda)e^{i\chi}$ for any phase

$\chi$ . The mental picture is that

$\phi$ is in a "mexican hat" potential, with a local maximum at the center, and a ring of local minima. We are going to repeat our previous argument, but this time linearize about

$\phi=\phi_0=(m/\lambda)$ . That is we write

$\begin{equation} \phi=\frac{m}{\lambda}+a+i b \end{equation}$ where

$a$ and

$b$ are small. The punch line will be that before we had 4 modes: an unphysical gauge transformation, a single ``photon" mode, and two massive excitations of the Higgs. When we expand about this new location these will become: an unphysical gauge transformation, one massive ``longitudinal" Higgs mode, and two massive "photon" modes,. Thus coupling to the Higgs mode with this sort of potential leads to massive gauge fields. This is how in the electroweak theory the W and Z bosons are given mass. Now back to the mathematics. We linearize and take the real and imaginary parts to get

$\begin{eqnarray} (\partial_t^2-\partial_x^2+2 m^2)a &=&0\\ (\partial_t^2 b-\phi_0 \partial_t A_t)-(\partial_x^2 b-\phi_0 \partial_x A_x)-(\partial_y^2 b-\phi_0 \partial_y A_y)&=&0 \end{eqnarray}$ The real component of the fluctuations (corresponding to changing the length of

$\phi$ decouples, and acts as a massive relativistic particle. This mode is the analog of the "Higgs boson" which was recently seen at the LHC. The imaginary component of the fluctuations are coupled to the Gauge field. Linearizing Maxwell's equations written in terms of the potential, (see last lecture), we get

$\begin{eqnarray} \partial_t [\partial_x A_x+\partial_y A_y]-(\partial_x^2+\partial_y^2)A_t &=& \phi_0\partial_t b-\phi_0^2 A_t\\ \partial_x [-\partial_y A_y+\partial_t A_t]-(-\partial_y^2+\partial_t^2)A_x &=& -\phi_0\partial_x b+\phi_0^2 A_x\\ \partial_y [\partial_t A_t-\partial_x A_x]-(\partial_t^2-\partial_x^2)A_y &=& -\phi_0\partial_y b+\phi_0^2 A_y \end{eqnarray}$ The four coupled equations can be combined into

$\begin{equation} \left( \begin{array}{cccc} \omega^2-k_y^2-\phi_0^2&k_xk_y&\omega k_x& i k_x \phi_0\\ k_x k_y &\omega^2-k_x^2-\phi_0^2&\omega k_y&i k_y \phi_0\\ \omega k_x&\omega k_y& k_x^2+k_y^2+\phi_0^2&i\omega\phi_0\\ -i k_x \phi_0&-i k_y \phi_0&-i\omega\phi_0&\omega^2-k^2 \end{array} \right) \left( \begin{array}{c} A_x\\ A_y\\ A_t\\ b \end{array} \right)=0 \end{equation}$ Looking at the eigenvalues, we find

$(k_x \omega ,k_y \omega,k^2+\phi_0^2,-i\phi_0\omega)$: is an eigenvector with eigenvalue $\omega^2+k^2+\phi_0^2$ . Since this is always positive, it is never a mode.
$(k_x,k_y,-\omega,-i \phi_0)$: is an eigenvector which always has eigenvalue $0$ . This is again a gauge transformation -- and has no physical significance.
$(-k_y,k_x,0,0)$ and $(\phi_0 k_y,\phi_0 k_x,0,2 i k_x k_y )$: are eigenvectors with eigenvalue $k^2+\phi_0^2-\omega^2$ . These are the new "massive" gauge fields.

Beyond Scalar QED3