Kink in a phi-six model (${\varphi }^6$ kink)

Kinks in a phi-six model commonly arise from a nonnegative degree-six potential with three degenerate vacua. For the unit-vacuum normalizationthe kink joining $0$ to $1$ isand has mass $1/4$. A rescaled model with $U(\varphi )={\varphi }^2({\varphi }^2-4{)}^2$ has the elementary kinkand its mass is $4\sqrt{2}$ in the normalization $E=\int [({\varphi }^{\prime }{)}^2/2+U]dx$.

For $U={\varphi }^2({\varphi }^2-b^2{)}^2/2$ and $b>0$, the three scalar-field vacua are $-b,0,b$. The static first integral is ${\varphi }_x^2=2U$. In either adjacent interval, $y={\varphi }^2/b^2$ obeys $y_x=\pm 2b^{2}y(1-y)$, giving the displayed four oriented kink profiles with $\sigma ,\eta \in \{1,-1\}$. Each has rest energy $b^4/4$. The intermediate-vacuum obstruction to a kink excludes a direct static connection from $-b$ to $b$: reaching zero would give zero derivative, and uniqueness of the smooth second-order field equation forces the zero solution. A Lorentz boost replaces $x-x_0$ by $(x-vt-x_0)/\sqrt{1-v^2}$.

// Destination: relativistic-quantum-field.bigb

If a proposed kink must cross a third degenerate vacuum, its first integral makes both the field derivative and potential vanish there. ODE uniqueness prevents it from crossing at finite distance, so the two elementary kink sectors can be joined only at infinite separation.