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Regions in $N_c, N_s$ for constant $N_x$ for which the theory is CAF. The solid lines show the borders of the region with two fixed flow solutions (Region I) for $N_x \in \{0,1,2,3,4\}$ The blue boundary $N_x=0$ represents a limit and does not satisfy CAF. The light gray (gray) region marks the region where two (four) real distinct sets of pseudo fixed points exist. The dashed black lines are the asymptotic behavior of the borders of the two fixed flow solution region.

UV behavior around the fixed flow solutions in spherical coordinates $\left( \theta, \phi\right)$. Upper left: Full phase space in case of two fixed flow solutions. Upper right: Close-up at the region close to the two fixed points. Lower left: Full phase space in case of four fixed flow solutions. Lower right: Close-up at the region close to the four fixed points. Flows not connected to the UV fixed point are in grey regions. Flows between fixed flows that do not cross the tree level symmetry breaking lines are in white regions; those that do cross are in red or blue regions as discussed in Sec.~\ref{tree-analysis}.

UV behavior around the fixed flow solutions in spherical coordinates $\left( \theta, \phi\right)$. Upper left: Full phase space in case of two fixed flow solutions. Upper right: Close-up at the region close to the two fixed points. Lower left: Full phase space in case of four fixed flow solutions. Lower right: Close-up at the region close to the four fixed points. Flows not connected to the UV fixed point are in grey regions. Flows between fixed flows that do not cross the tree level symmetry breaking lines are in white regions; those that do cross are in red or blue regions as discussed in Sec.~\ref{tree-analysis}.

UV behavior around the fixed flow solutions in spherical coordinates $\left( \theta, \phi\right)$. Upper left: Full phase space in case of two fixed flow solutions. Upper right: Close-up at the region close to the two fixed points. Lower left: Full phase space in case of four fixed flow solutions. Lower right: Close-up at the region close to the four fixed points. Flows not connected to the UV fixed point are in grey regions. Flows between fixed flows that do not cross the tree level symmetry breaking lines are in white regions; those that do cross are in red or blue regions as discussed in Sec.~\ref{tree-analysis}.

UV behavior around the fixed flow solutions in spherical coordinates $\left( \theta, \phi\right)$. Upper left: Full phase space in case of two fixed flow solutions. Upper right: Close-up at the region close to the two fixed points. Lower left: Full phase space in case of four fixed flow solutions. Lower right: Close-up at the region close to the four fixed points. Flows not connected to the UV fixed point are in grey regions. Flows between fixed flows that do not cross the tree level symmetry breaking lines are in white regions; those that do cross are in red or blue regions as discussed in Sec.~\ref{tree-analysis}.

IR flow behavior around the IR fixed points in spherical coordinates $\left( \theta, \phi\right)$ with $\alpha$ kept fixed at the IR fixed point, $\alpha_* = B/C$. Left: Close-up of the case with two IR fixed points. Right: Full phase space in the case of four IR fixed points. Each fixed point (magenta dot), has its eigendirections superimposed. Color coding: Red is IR attractive, Blue IR repulsive. The red dashed lines are the projections of the third eigen-directions (IR attractive) of the fixed point in $\left(r, \theta, \phi\right)$ onto the $\left( \theta, \phi\right)$-subspace. The shaded white region marks the region in $\left( \theta, \phi\right)$, where the quartic couplings become comparable to the gauge coupling at the fixed point, $\lambda_1^2 +\lambda_2^2 = 5 \alpha_*^2$, and higher order corrections are expected.

IR flow behavior around the IR fixed points in spherical coordinates $\left( \theta, \phi\right)$ with $\alpha$ kept fixed at the IR fixed point, $\alpha_* = B/C$. Left: Close-up of the case with two IR fixed points. Right: Full phase space in the case of four IR fixed points. Each fixed point (magenta dot), has its eigendirections superimposed. Color coding: Red is IR attractive, Blue IR repulsive. The red dashed lines are the projections of the third eigen-directions (IR attractive) of the fixed point in $\left(r, \theta, \phi\right)$ onto the $\left( \theta, \phi\right)$-subspace. The shaded white region marks the region in $\left( \theta, \phi\right)$, where the quartic couplings become comparable to the gauge coupling at the fixed point, $\lambda_1^2 +\lambda_2^2 = 5 \alpha_*^2$, and higher order corrections are expected.

Renormalisation group running of all couplings from the fully IR attractive fixed point to the fully UV repulsive fixed flow. All couplings are normalised in units of $\alpha^*$. We use the one-loop beta functions for the quartic couplings together with the one-loop (dotted), two-loop (dashed) and three-loop (solid) gauge beta function.

Phase diagram for the case of two (four) fixed flows are shown in the left (right) panel in spherical coordinates $\theta$ and $\phi$. The initial radial coordinate, $r$, is chosen close to the Gaussian fixed point, i.e. $r \ll 1$. The phases are then determined by analyzing the numerical solutions to the beta functions at three-loop order in the gauge coupling, together with the one-loop for the quartic couplings. The diagrams illustrate three IR phases connected to the Gaussian UV fixed point. Long distance conformality (white), two different spontaneous symmetry breaking patterns (blue and red). Directions with trajectories not originating from the Gaussian UV fixed point are colored gray. The light red and light blue regions correspond to initial conditions with unbounded tree level scalar potentials.

Phase diagram for the case of two (four) fixed flows are shown in the left (right) panel in spherical coordinates $\theta$ and $\phi$. The initial radial coordinate, $r$, is chosen close to the Gaussian fixed point, i.e. $r \ll 1$. The phases are then determined by analyzing the numerical solutions to the beta functions at three-loop order in the gauge coupling, together with the one-loop for the quartic couplings. The diagrams illustrate three IR phases connected to the Gaussian UV fixed point. Long distance conformality (white), two different spontaneous symmetry breaking patterns (blue and red). Directions with trajectories not originating from the Gaussian UV fixed point are colored gray. The light red and light blue regions correspond to initial conditions with unbounded tree level scalar potentials.

Projected RG flow of the couplings onto the $(\alpha, \lambda_i)$-planes (upper left and right) and $(\lambda_1, \lambda_2)$-plane with the third coupling fixed to $\alpha = 0$ (lower left) and $\alpha=\alpha^*$ (lower right). In each panel we show the lines where spontaneous symmetry breaking occurs (red and blue) including one-loop effects. The gray regions here are the broken phases. Solid dots are fixed points in the full system, while circles mark fixed points in the reduced systems with one coupling kept fixed. Dashed lines mark the fixed flow lines for the reduced systems.

Projected RG flow of the couplings onto the $(\alpha, \lambda_i)$-planes (upper left and right) and $(\lambda_1, \lambda_2)$-plane with the third coupling fixed to $\alpha = 0$ (lower left) and $\alpha=\alpha^*$ (lower right). In each panel we show the lines where spontaneous symmetry breaking occurs (red and blue) including one-loop effects. The gray regions here are the broken phases. Solid dots are fixed points in the full system, while circles mark fixed points in the reduced systems with one coupling kept fixed. Dashed lines mark the fixed flow lines for the reduced systems.

Projected RG flow of the couplings onto the $(\alpha, \lambda_i)$-planes (upper left and right) and $(\lambda_1, \lambda_2)$-plane with the third coupling fixed to $\alpha = 0$ (lower left) and $\alpha=\alpha^*$ (lower right). In each panel we show the lines where spontaneous symmetry breaking occurs (red and blue) including one-loop effects. The gray regions here are the broken phases. Solid dots are fixed points in the full system, while circles mark fixed points in the reduced systems with one coupling kept fixed. Dashed lines mark the fixed flow lines for the reduced systems.

Projected RG flow of the couplings onto the $(\alpha, \lambda_i)$-planes (upper left and right) and $(\lambda_1, \lambda_2)$-plane with the third coupling fixed to $\alpha = 0$ (lower left) and $\alpha=\alpha^*$ (lower right). In each panel we show the lines where spontaneous symmetry breaking occurs (red and blue) including one-loop effects. The gray regions here are the broken phases. Solid dots are fixed points in the full system, while circles mark fixed points in the reduced systems with one coupling kept fixed. Dashed lines mark the fixed flow lines for the reduced systems.

The blue line marks the lower value of $N_c$, as a function of $N_x$, above which there are solutions to Eq.~(\ref{eq:reducedflow}). The red solid line is the value of $N_c$ for which the curve, $C=0$, intersects $N_s = 1$. Below this line there is a region with $N_s >= 1$ for which $C<0$. For $N_x < 8/\sqrt{3}$ the curve with $C=0$ does not intersect $N_s = 1$ (marked with gray dashed line). The dashed red line is straight line $N_c = N_x$ below which the suppression of higher loop contributions to the IR fixed points are of order one.

Morphology of the quartic phase diagram. Each gray dashed curve represent the zero-contour of one of the rescaled quartic beta functions. The blue solid lines (red dashed lines) mark the curves where the flow is pointing towards (away from) the origin. The grey dots are the pseudo fixed points. Right: For the case of two solutions. Left: For the case of four solutions.

Morphology of the quartic phase diagram. Each gray dashed curve represent the zero-contour of one of the rescaled quartic beta functions. The blue solid lines (red dashed lines) mark the curves where the flow is pointing towards (away from) the origin. The grey dots are the pseudo fixed points. Right: For the case of two solutions. Left: For the case of four solutions.

This shows the corresponding beta function of the flows along the lines shown in Fig.~\ref{SchematicGeoFlow}, parametrized by $\lambda_{1s}$. The dashed line is where the $c_s = 2 B$. The grey dots are the pseudo fixed points, and the blue dots are solutions to the fixed flow. Right: For the case of two solutions. Left: For the case of four solutions.

This shows the corresponding beta function of the flows along the lines shown in Fig.~\ref{SchematicGeoFlow}, parametrized by $\lambda_{1s}$. The dashed line is where the $c_s = 2 B$. The grey dots are the pseudo fixed points, and the blue dots are solutions to the fixed flow. Right: For the case of two solutions. Left: For the case of four solutions.

Estimate of the relative size of the three-loop contribution. The dots mark integer values of $N_c$, $N_s$ and $N_f$ such that $N_x = 1/4$ in the left panel and $N_x = 10$ on the right. The excluded regions does not satisfy the CAF conditions. The dashed contours are values of $N_f$, while colored regions show the relatica size of the three-loop contribution as $\log_{10} \left| \alpha^*_{3}-\alpha^*_{2}\right| / \alpha^*_{2}$, with $\alpha^*_{2}$ given by Eq.~(\ref{alpha2}), while $\alpha^*_{3}$ is the three loop result assuming for simplicity that $\lambda_{1} = \lambda_{2} = \alpha$.

Estimate of the relative size of the three-loop contribution. The dots mark integer values of $N_c$, $N_s$ and $N_f$ such that $N_x = 1/4$ in the left panel and $N_x = 10$ on the right. The excluded regions does not satisfy the CAF conditions. The dashed contours are values of $N_f$, while colored regions show the relatica size of the three-loop contribution as $\log_{10} \left| \alpha^*_{3}-\alpha^*_{2}\right| / \alpha^*_{2}$, with $\alpha^*_{2}$ given by Eq.~(\ref{alpha2}), while $\alpha^*_{3}$ is the three loop result assuming for simplicity that $\lambda_{1} = \lambda_{2} = \alpha$.

In this figure we choose a particular slice at $g=0.1$ perpendicular to the gauge coupling direction. The blue line corresponds to the symmetry breaking boundary line while the green and red lines come from two vacuum stability conditions. The blue shaded region represent the broken phase $SU(N_c)\times U(N_s) \to SU(N_c-N_s) \times SU(N_s) \times U(1)$.

In this figure we choose a particular slice at $g=0.1$ perpendicular to the gauge coupling direction. The black line corresponds to the symmetry breaking boundary line while the orange and purple lines come from two vacumm stability conditions. The blue shaded region represent the broken phase $SU(N_c) \times U(N_s) \rightarrow SU(N_c-1) \times U(N_s -1) \times U(1)$.

In this figure we choose a particular slice at $g=0.1$ perpendicular to the gauge coupling direction. The blue and black lines correspond to the symmetry breaking boundary lines while the orange, purple, red and green lines come from four vacumm stability conditions respectively. The two shaded regions represent the broken phases $SU(N_c)\times U(N_s) \to SU(N_c-N_s) \times SU(N_s) \times U(1)$ and $SU(N_c) \times U(N_s) \rightarrow SU(N_c-1) \times U(N_s -1) \times U(1)$ respectively.

In this figure we choose a particular slice $g=0.044$ which is the coupling value of the Banks-Zaks fixed points. The two shaded regions represent the broken phases $SU(N_c) \times U(N_s) \rightarrow SU(N_s) \times SU(N_c - N_s)\times U(1)$ and $SU(N_c) \times U(N_s) \rightarrow SU(N_c-1) \times U(N_s -1) \times U(1)$ respectively.

In this figure we choose a particular slice $g=0.1$. The shaded region represents the broken phases $SU(N_c)\times U(N_s) \to SU(N_c-N_s) \times SU(N_s) \times U(1)$.