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Solutions for the $\phi^3$ constraint for several values of $K_3\equiv \kappa/(m\lambda^{1/2})$.

Constrained instantons for the $\phi^3$ constraint at different values of $K_3\equiv \kappa/(m\lambda^{1/2})$. The red and green dashed lines show the short and long distance fits given by eq.~\eqref{eq:small-large-fits}

Constrained instantons for the $\phi^3$ constraint at different values of $K_3\equiv \kappa/(m\lambda^{1/2})$. The red and green dashed lines show the short and long distance fits given by eq.~\eqref{eq:small-large-fits}

Constrained instantons for the $\phi^3$ constraint at different values of $K_3\equiv \kappa/(m\lambda^{1/2})$. The red and green dashed lines show the short and long distance fits given by eq.~\eqref{eq:small-large-fits}

Constrained instantons for the $\phi^3$ constraint at different values of $K_3\equiv \kappa/(m\lambda^{1/2})$. The red and green dashed lines show the short and long distance fits given by eq.~\eqref{eq:small-large-fits}

Left: The action $S$ of our solutions as a function of $K_3\equiv \kappa/(m\lambda^{1/2})$ for the $\phi^3$ constraint. The dashed red line denotes the value of the massless instanton action \eqref{eq:mlessact}. The solid black line represents the solutions that contribute to the tunnelling rate, while the grey dashed line shows the ones that do not. Right: The constraint $\xi$ as a function of $K_3$ for the same solutions.

Left: The action $S$ of our solutions as a function of $K_3\equiv \kappa/(m\lambda^{1/2})$ for the $\phi^3$ constraint. The dashed red line denotes the value of the massless instanton action \eqref{eq:mlessact}. The solid black line represents the solutions that contribute to the tunnelling rate, while the grey dashed line shows the ones that do not. Right: The constraint $\xi$ as a function of $K_3$ for the same solutions.

The action as a function of the constraint $\bar\xi$ for the $\phi^3$ constraint. The black solid line corresponds to the solutions that contribute to the tunnelling rate. The dashed red line shows the massless instanton action.

The projection prefactor $\nu$ as a function of $K_3\equiv \kappa/(m\lambda^{1/2})$ for the $\phi^3$ constraint. The red dashed line marks $\kappa = \kappa_\mathrm{crit}$.

Variation in the shape of the modified potential with the $\phi^6$ constraint for different values of $K_6\equiv (m^2/\lambda^2)\kappa$.

Solutions for several values of $K_6\equiv (m^2/\lambda^2)\kappa$ with the $\phi^6$ constraint.

Solutions for different values of $K_6\equiv (m^2/\lambda^2)\kappa$ with the $\phi^6$ constraint, together with the short-range massless instanton fits (dashed red) and long-range exponential fits (dashed green) (eq.~\eqref{eq:small-large-fits}).

Solutions for different values of $K_6\equiv (m^2/\lambda^2)\kappa$ with the $\phi^6$ constraint, together with the short-range massless instanton fits (dashed red) and long-range exponential fits (dashed green) (eq.~\eqref{eq:small-large-fits}).

Solutions for different values of $K_6\equiv (m^2/\lambda^2)\kappa$ with the $\phi^6$ constraint, together with the short-range massless instanton fits (dashed red) and long-range exponential fits (dashed green) (eq.~\eqref{eq:small-large-fits}).

Solutions for different values of $K_6\equiv (m^2/\lambda^2)\kappa$ with the $\phi^6$ constraint, together with the short-range massless instanton fits (dashed red) and long-range exponential fits (dashed green) (eq.~\eqref{eq:small-large-fits}).

Action and constraint values for the $\phi^6$ constraint. Left: The action as a function of $K_6\equiv (m^2/\lambda^2)\kappa$. The red line denotes the value of the corresponding massless instanton action. The solid black line indicates the constrained instantons, that contribute to the vacuum decay rate, while the grey dashed line indicates those solutions that do not. Right: The constraint as a function of $K_6$.

Action and constraint values for the $\phi^6$ constraint. Left: The action as a function of $K_6\equiv (m^2/\lambda^2)\kappa$. The red line denotes the value of the corresponding massless instanton action. The solid black line indicates the constrained instantons, that contribute to the vacuum decay rate, while the grey dashed line indicates those solutions that do not. Right: The constraint as a function of $K_6$.

Action as a function of the constraint value for the $\phi^6$ constraint. The red line represents the corresponding value of the massless instanton action. The black solid line corresponds to the configurations that contribute to the tunnelling rate, while the grey dashed line denotes those that do not.

The projection prefactor $\nu$ as a function of $K_6\equiv (m^2/\lambda^2)\kappa$ for the $\phi^6$ constraint. The red dashed line marks $\kappa=\kappa_{\rm crit}$.

A comparison between the absolute value of the derivative of the unconstrained action with respect to the Lagrange multiplier (black) and the calculated values of the constraint (red dashed). Left: $\phi^3$ constraint. Right: $\phi^6$ constraint.

A comparison between the absolute value of the derivative of the unconstrained action with respect to the Lagrange multiplier (black) and the calculated values of the constraint (red dashed). Left: $\phi^3$ constraint. Right: $\phi^6$ constraint.

$\Delta I_1$ (blue) and $\Delta I_2$ (red) of eq.~\eqref{eq:integralchecks} as a function of $\kappa$. Left: $\phi^3$ constraint. Right: $\phi^6$ constraint.

$\Delta I_1$ (blue) and $\Delta I_2$ (red) of eq.~\eqref{eq:integralchecks} as a function of $\kappa$. Left: $\phi^3$ constraint. Right: $\phi^6$ constraint.

$\phi^3$ constraint. The difference between the value of the action computed at $R_{\min}=10^{-7}$ and $R_{\max}=50$ for $K_3 = -1$, and the action computed for different values of $R_{\min}$ (left) and different values of $R_{\max}$ (right).

$\phi^3$ constraint. The difference between the value of the action computed at $R_{\min}=10^{-7}$ and $R_{\max}=50$ for $K_3 = -1$, and the action computed for different values of $R_{\min}$ (left) and different values of $R_{\max}$ (right).

$\phi^6$ constraint. The difference between the value of the action computed at $R_{\min}=10^{-7}$ and $R_{\max}=50$ for $K_6 = 10^{-6}$, and the action computed for different values of $R_{\min}$ (left) and $R_{\max}$ (right).

$\phi^6$ constraint. The difference between the value of the action computed at $R_{\min}=10^{-7}$ and $R_{\max}=50$ for $K_6 = 10^{-6}$, and the action computed for different values of $R_{\min}$ (left) and $R_{\max}$ (right).

The difference between the action computed at a working precision of 40 digits and the action computed for different values of the working precision. Left: $\phi^3$ constraint, $K_3 = -1$. Right: $\phi^6$ constraint, $K_6 = 10^{-6}$.

The difference between the action computed at a working precision of 40 digits and the action computed for different values of the working precision. Left: $\phi^3$ constraint, $K_3 = -1$. Right: $\phi^6$ constraint, $K_6 = 10^{-6}$.