Elliptically oscillating classical solution in Higgs potential and the effects on vacuum transitions

We investigate oscillating solutions of the equation of motion for the Higgs potential. The solutions are described by Jacobian elliptic functions. Classifying the classical solutions, we evaluate a possible parameter space for the initial conditions. To construct the field theory around the oscillating solutions, quantum fluctuations are introduced. This alternative perturbation method is useful to describe the nontrivial quantum theory around the oscillating state. This perturbation theory reduces to the standard one if we take the solution at the vacuum expectation value. It is shown that the transition probability between the vacuum and multiquanta states is finite as long as the initial field configuration does not start from the true vacuum.

Figure 1

Sketch of the oscillations for each solution. The elliptic functions dn, $1/\mathrm{dn}$ ($0<|{\varphi }_0|<\sqrt{2}v$) show the oscillation around $\varphi =v$ (correct vacuum); on the other hand, the function, cn ($\sqrt{2}v<|{\varphi }_0|$), shows the oscillation around $\varphi =0$ (unstable state).

Figure 2

The behavior of (a) the moduli $k$, $\kappa$ and (b) $k^{\prime }$, ${\kappa }^{\prime }$ as the function of ${\varphi }_0/v$. The moduli $k$ and $\kappa$ appear in dn ($v<{\varphi }_0<\sqrt{2}v$) and $1/\mathrm{dn}$ ($0<{\varphi }_0<v$), respectively.

Figure 3

The behavior of the norm $q$ (a) and the Fourier coefficient $c_n$ (b) as the function of ${\varphi }_0/v$.

Figure 4

The behavior of the zero mode, ${\varphi }_{cl,0}/v$, as the function of ${\varphi }_0/v$.

Figure 5

The comparison of three normalized masses $\{m_{\mathrm{cl}}/m_{\mathrm{cl}},m_{\mathrm{EOM}}/m_{\mathrm{cl}},M/m_{\mathrm{cl}}\}$ for the allowed parameter space (a) in the classical theory ($0<{\varphi }_0<\sqrt{2}v$) and (b) in the quantum theory ($\frac{v}{\sqrt{3}}<{\varphi }_0<\frac{\sqrt{5}}{\sqrt{3}}v$) as the function of ${\varphi }_0/v$.

Figure 6

Feynman diagrams for ${\tilde{h}}^2$ interactions. The left (right) vertex is proportional to ${\tilde{\varphi }}_{\mathrm{cl}}({\tilde{\varphi }}_{\mathrm{cl}}^2)$.

Figure 7

Feynman diagrams for ${\tilde{h}}^3$ interactions. The left vertex is standard one, and the right vertex is proportional to ${\tilde{\varphi }}_{\mathrm{cl}}$.

Figure 8

Feynman diagram for ${\tilde{h}}^4$ interaction.

Figure 9

The velocity ${\beta }_{(n)}$ of the produced particle in the process $\mathrm{vac}\to \tilde{h}\tilde{h}$ as the function of ${\varphi }_0/v$. Note that ${\beta }_{(1)}$ cannot be a real number, so the lowest mode is $n=2$.

Figure 10

The logarithmic plot of the normalized number density $\rho (\mathrm{vac}\to \tilde{h}\tilde{h})$ for the production process $\mathrm{vac}\to \tilde{h}\tilde{h}$ as the function of ${\varphi }_0/v$. Note that $\rho$ vanishes at ${\varphi }_0/v\to 1$.