Since the model has three marginal couplings we now investigate its ultraviolet behavior and establish the conditions under which it can be completely asymptotically free. We are interested in characterizing the flow behavior around the Gaussian fixed point, and therefore one can use one-loop expressions for the beta functions.

Using the rescaled couplings, i.e. α=g2/(4π)2, λ1=v/(4π)2, λ2=u/(4π)2, the one-loop beta functions (first calculated in [33], with general RG functions to two-loop order provided in [41]) are βα=-13(22Nc-4Nf-Ns)α2βλ1=4(NcNs+4)λ12+12λ22+λ1[8(Nc+Ns)λ2-6(Nc2-1)Ncα]+3(Nc2+2)4Nc2α2βλ2=4(Nc+Ns)λ22+λ2[24λ1-6(Nc2-1)Ncα]+3(Nc2-4)4Ncα2.(2.2)We note that, to this order, the gauge beta function only depends on the gauge coupling, and it is of the form βα=-Bα2. Requiring asymptotic freedom (AF), is equivalent to restricting the coefficient B to be positive. The critical number of fermion flavors, Nf*, for which asymptotic freedom is lost in the gauge coupling, B=0, is Nf*=(22Nc-Ns)/4. The gauge coupling will be AF for models with non-negative integer values Nf<Nf*. We thus have an upper bound on Ns, i.e. Ns≤22Nc.

The two beta functions for the quartic couplings depend on all three couplings but only on Nc and Ns. At sufficiently high energies, the scalar potential of a CAF model is well approximated by the tree-level terms V=λ1(TrS†S)2+λ2Tr(S†S)2(2.3)This potential is positive definite for values of the couplings satisfying λ1+rλ2>0 for all r=Tr(S†S)2/(TrS†S)2, which ranges from 1/min(Nc,Ns) and 1. Note that this allows one of the two couplings to be negative, but not both simultaneously.

For CAF to exist we first need to find solutions to the following fixed flow relation, (βα,βλ1,βλ2)=c(α,λ1,λ2),(2.4)for a nonzero scale-dependent coefficient c. In Appendix A, we discuss how Eq. (2.4) is related to other, equivalent CAF requirements. When the gauge coupling is zero, α=0, the quartic couplings are only asymptotically free along lines that correspond to unstable scalar potentials, Eq. (2.3).

We will now outline the criteria under which solutions exist and count the number of solutions. The fixed flow solution for α>0 will necessarily satisfy c=-Bα, which can be substituted into the remaining equations. Since βλ2 depends linearly on λ1, we can easily solve for λ1(λ2,α) and substitute this into the equation for βλ1. The result is a quartic equation in λ2. The coefficients of this equation depend on Nc, Ns, B. Introducing the quantity Nx=Nf*-Nf, we can express B=4Nx/3. The number of fixed flow solutions corresponds to the number of real roots of the quartic polynomial, which can be calculated using the discriminant method. The full expression for the quartic polynomial and the expressions for classifying the nature of the roots can be found in Appendix B. For a fixed value of Nx, we find a region with two distinct real solutions (region I) and a smaller region with four distinct real solutions (region II). The upper border of region II is shared with region I.

In Fig. 1, we show how the borders of these regions change when varying the number of fermion flavors (Nx∈{0,1,2,3,4}). It should be noted, that for small values of Nx, the effect of increasing Nx is to move the upper border of region I downward, while the borders of region II largely remains unchanged. Furthermore, we note that in the limit Nx=0 (Nf=Nf*), Eq. (2.4) no longer describes fixed flows, since the one-loop beta function for the gauge coupling is vanishing and higher-order terms will dictate the running. Nonetheless, solving the quartic function with Nx=0 corresponds to finding fixed points in the subsystem βλ1, βλ2 and is closely related to finding infrared fixed points for the full model (see more details in later section). The region with two distinct sets of solutions for Nx=0 is marked with light gray in Fig. 1, while the region with four distinct sets of solutions is marked with gray. From Fig. 1, we see that for fixed values of Ns≥2 and Nx>0, there exists a lower bound of Nc above which the models have two fixed flow solutions For even higher values of Nc, two additional fixed flow solutions appear. Increasing either Nx (i.e. lowering Nf) or Ns will push the lower bound on Nc towards higher values, whereas the transition from two to four fixed flow solutions is only mildly dependent on Nx. In Table II, the values of Nc, Ns, Nf for models with CAF are tabulated.

Existence of fixed flows only implies that specific directions flow out of the Gaussian fixed point. However, to complete the picture, we need information about the behavior of the RG trajectories in the neighborhood of the fixed flow lines. To investigate this, we parametrize the couplings (α,λ1,λ2) using spherical coordinates, α=rsinθcosϕ,λ1=rsinθsinϕ,λ2=rcosθ,(2.5)and obtain expressions for the RG beta functions of (r,θ,ϕ), i.e. (βr,βθ,βϕ).

At one loop, the beta functions, βθ, βϕ, depend only multiplicatively on the radial coordinate, i.e. βθ,ϕ=rfθ,ϕ(θ,ϕ), and the direction (θ,ϕ), therefore, does not depend on r. Factoring out the radial coordinate r, we can examine the UV behavior in a reduced space of only two parameters. This is shown in Fig. 2 for a model in region I (top panels) and a model in region II (bottom panels), with the arrows pointing from UV to IR.

In order for a trajectory to be connected to the Gaussian UV fixed point, the radial coordinate has to go to zero in the UV. Within the one-loop approximation, the change in r is of the form βr=r2fr(θ,ϕ). The radial coordinate is, thus, only decreasing in regions of (θ,ϕ) where fr(θ,ϕ)<0. In Fig. 2, the transition (fr(θ,ϕ)=0) is shown as dashed grey lines. The regions with flows that cross this line and are, thus, not connected to the Gaussian UV fixed point, are colored grey. Similarly, we mark the regions with red and blue, where the flows cross the conditions for a bounded tree-level potential (dashed red and blue lines) depending on which quartic coupling is negative.

We show the UV behavior for the full phase space in the case of two solutions in the upper left panel of Fig. 2 and a close-up of the region around the two fixed flow points in the upper right panel. We note that one point is completely repulsive, while the other has one repulsive direction and one attractive direction (mixed). For the mixed case, the repulsive directions (the red and blue solid lines) separate the flows with UV asymptotic freedom (the red and blue shaded regions) from the ones that are UV divergent, i.e. the grey region. The attractive direction (the black line between two fixed flow points) separates the two tree-level symmetry-breaking regions. In the lower two panels of Fig. 2, we show the case of four fixed flow solutions. In this case, the two additional fixed points for the fixed flow open up a two-dimensional region where the flows are between fixed points without crossing the tree-level symmetry-breaking lines. This region is marked with white. One of the two additional fixed points for the fixed flows is fully attractive, while the other is with mixed properties. For the fully attractive one, only this exact relation of the three couplings is connected to the UV. This case therefore offers full predictability in the IR. This will be further discussed in Sec. II C.

In the results discussed above, we factor out the r-dependence of the couplings, which is valid within the one-loop approximation of the beta functions. Therefore Fig. 2 is only adequate for describing the behavior close to the Gaussian UV fixed point. Starting from a point where r≪1, such that the beta functions are well approximated by the one-loop expressions, we trust the flows in the backward direction (towards higher energies) outside the grey regions, since r is decreasing. On the contrary, we cannot follow the flows too far forwards, since the approximation is getting worse (r increasing). However, we note that the change in r is generally suppressed by a factor of r compared to the change in θ and ϕ, except at points close to the fixed flow directions, where fθ,ϕ(θ,ϕ)∼0. This means that the relations among the three couplings can change substantially while the one-loop approximation is still valid. Seen from the UV perspective of the Gaussian fixed point (one-loop approximation), we expect the four-solution case to offer more possibilities to flow from the UV Gaussian fixed point to a possible IR fixed point because of the fully attractive fixed flow solution, which provides a region that seemingly does not cross the symmetry-breaking lines.

Here we summarize our conclusions from restricting the model to be CAF: (i)

In Fig. 1, we show the region in Ns and Nc within which there exists a window in Nf (with upper endpoint given by Nf*), for which the models are CAF. We note that Nc>Ns in the whole region.

In Fig. 1, we see that the whole region in Ns and Nc corresponding to CAF models is contained in the region with Nx=0. In Appendix B, we show that the existence of solutions to Eq. (2.4) for Nx=0, guaranties the existence of fixed points to the one-loop (βλ1,βλ2)-subsystem, where α is treated as any nonzero, positive scale-dependent value.

In other words, if the gauge coupling has a fixed point (βα=0), then we already know that there exist values λ1, λ2 satisfying βλ1=0, βλ2=0 at one loop in the quartic subsystem. Following the ordering from the Weyl-consistency conditions [42–44], we should treat the system at three loops in the gauge beta function together with the one-loop quartic beta functions. To keep the analysis in the main section light, we study here the two-loop gauge beta function together with the one-loop quartic beta functions, while the full result is presented in the Appendix D. At this order, the running of the gauge coupling still decouples from the quartic couplings. We derive the gauge fixed point, and compute the accompanying fixed points values of the quartic couplings. We find that in most cases all three couplings at the fixed points are perturbatively small, and we therefore do not expect the three-loop contributions to the running of the gauge couplings to quantitatively change the preliminary findings of the this section. In fact, we find that each higher-loop order contribution will be suppressed by a factor of Nx/Nc. This, on the other hand, requires us to restrict our IR analysis to models close to losing AF, such that Nx≪Nc, to achieve perturbative control of the loop expansion. In the generalized Veneziano limit, where Nc, Ns, Nf→∞ in such a way that all ratios are kept constant, the loop suppression can be arbitrarily small, whereas for finite values of Nf, Ns and Nc the smallest value for Nx is Nx=1/4. Further details are found in Appendix D.

If we write the two-loop gauge beta function as βα=-Bα2+Cα3, then the nontrivial fixed point occurs for α*=B/C. For B>0 and C>0, this is an IR fixed point. The two-loop coefficient for the gauge beta function takes the form C=83NcNs-68Nc23-2NfNc+26NcNf3-2NsNc(2.6)The critical number of fermion flavors for this coefficient to be positive is N¯f=34Nc3-4Nc2Ns+3Ns13Nc2-3(2.7)and since in our case, where Nc>2, we always have that Nf*>N¯f, the functions N¯f and Nf* define the window of existence for the infrared fixed point in the gauge coupling. From requiring the model to be CAF, we already know from the UV analysis, that if the gauge beta function has a nontrivial fixed point, then so does the quartic coupling subsystem. Therefore, the window for the existence of IR fixed points is the grey region in Fig. 1 for Nf within Nf∈[N¯f,Nf*]. The result is summarized in Table III, where the fixed points are determined numerically, and nonperturbative fixed points (α,λi>1) have been discarded. Comparing Table II with Table III, we conclude that the CAF condition is stronger than the condition for the existence of IR fixed points, meaning that any CAF model of this kind always possesses IR fixed points to this order in perturbation theory.

In order for the results not to be significantly altered by higher-order contributions, we need to show that these higher-loop contributions are suppressed. This is shown in Appendix D. Here it is sufficient to say that a Banks-Zaks-like analysis [45,46] is possible.

In the following, we will characterize the IR fixed points. To find eigendirections of the IR fixed points, we need to study the following matrix: M=(∂βα∂α∂βα∂λ1∂βα∂λ2∂βλ1∂α∂βλ1∂λ1∂βλ1∂λ2∂βλ2∂α∂βλ2∂λ1∂βλ2∂λ2)|α=α*,λ1=λ1*,λ2=λ2*,(2.8)where (α*,λ1*,λ2*) corresponds to the coupling solutions at the IR fixed points. In the convention that the RG flow runs from UV to IR, the positive (negative) eigenvalues represent IR attractive (repulsive) directions. In the region, where we have perturbative control of our IR fixed points, they inherit their characteristics from the corresponding fixed flow solutions. The third eigendirection, which is not part of the fixed flow picture (Fig. 2), is dominated by the gauge coupling, and is always IR attractive. Within the region with two IR fixed points, we have one, which has two repulsive eigendirections, while the other fixed point has one repulsive and one attractive eigendirection. Combined with the third eigendirection, we can conclude that the one with two repulsive directions is only connected to the UV along a single trajectory, while the other one is connected through a one parameter family of trajectories. These trajectories all originate from the fully repulsive fixed flow direction. There is also a trajectory connecting the two IR fixed points. In Fig. 3, we show the flow behavior around the IR fixed points on the plane of constant α=α* in spherical coordinates. This allows us to see the similarities with the UV picture (Fig. 2). In the case with four IR fixed points, we have additionally one fixed point with mixed properties and one which is fully IR attractive. Unlike from the previous case, the fully IR attractive fixed point has a three dimensional basin of attraction, implying IR conformal stability in all directions. In Fig. 3, this is illustrated by a two dimensional region, since α is kept fixed. In Table IV, we provide a summary of the IR fixed points.

We learn that CAF models with spin zero and spin half quarks also feature IR interacting fixed points. In Appendix C, we explore the connection in detail. In short, we can show that for fixed flows to exist with both λ1 and λ2 positive, there must be fixed points in the quartic subsystem of beta functions. These fixed points correspond fixed points for the full system since the gauge coupling has an IR fixed point as well (C>0 in the region with fixed points in the subsystem). The case where both λ1 and λ2 are negative can be discarded by demanding a bounded scalar potential in the UV. The cases with one of the quartic coupling negative, could be realized without the existence of fixed points in the subsystem. However, for the particular Nc and Ns dependence of the coefficients for this model, this does not occur.

From the previous section, we know there exist two kinds of phase structures; one with two IR fixed points and another with four IR fixed points. On one hand, the four IR fixed points case requires quite large number of colors Nc and flavors Nf even for small values of Ns. On the other hand, this scenario provides a fully IR predictive case. In other words, this group of models possesses a fully UV repulsive fixed flow, for which all relations among couplings are fixed, and we can thus fully determine the IR fate of the model at the highest known order in perturbation theory. We will in the following do this for the minimal choice of colors, i.e. Nc=26, which in order to satisfy the CAF conditions requires Ns=2 and Nf=138. Afterwards, we will focus our attention on the general phases of the two types of phase structures. Note that, for simplicity, we assume all the particles to be massless. In general, the RG functions will be modified after symmetry breaking and a more sophisticated treatment will be required (e.g. threshold contributions need to be considered). In this sense, we only consider what happens until spontaneous symmetry breaking occur. A detailed study of the IR phases after symmetry breaking is beyond the scope of this work.

In Fig. 4, for our particular choice of Nc, Ns and Nf, we show the running of the couplings from the UV (with coupling ratios fixed by the fully repulsive fixed flow) towards the IR. We show the result from using both the one-, two- and three-loop gauge beta function together with the one-loop beta function for the quartic couplings. It is evident, that both in the two-loop and three-loop gauge case, the IR fate of the model is long distance conformality.

Similarly, we can solve the differential equations for each direction out of the Gaussian UV fixed point, and determine the IR fate anticipating and using the symmetry-breaking conditions derived in Sec. III. In this way, we distinguish between three IR phases connected to the Gaussian UV fixed point. Long distance conformality (white) along with two kinds of spontaneous symmetry breaking (blue and red). Gray regions are not connected to the Gaussian UV fixed point.

In the left panel of Fig. 5, we show the result of this analysis for a representative model with the two IR fixed point phase structure. We see that practically all directions connected to the UV, lead to spontaneous symmetry breaking. On the separatrices of the two symmetry-breaking regions, we find fine-tuned solutions reaching the IR fixed points, and solutions crossing the intersection of the two symmetry-breaking lines.

In the right panel of Fig. 5, the phase diagram for the other phase structure with four IR fixed points is shown. Here we have all three IR phases present.

We notice that the phase diagrams are basically identical to the UV picture in Fig. 2. From this we conclude, that the lowest-order approximations to the symmetry-breaking conditions together with the one-loop beta functions are good indicators for the IR fate of CAF models. This is tightly connected to the relation between fixed flows in the UV and fixed points in the IR together with the fact that βr is r-suppressed (in the UV) compared to βθ and βϕ. However, for directions close to the fixed flows or the separatrices (where βr∼βθ,ϕ) we need higher-order terms to correctly describe the RG evolution. Even so, for r≪1 the affected regions are not directly visible, except for the phase of long distance conformality in between the four fixed flows.

For values of the gauge coupling larger than the one at the IR fixed points, other phases exists. However, these are not UV free and are therefore outside the focus of this work.

In Fig. 6, we show the RG flow of the couplings projected onto the (α,λi)-planes and (λ1,λ2)-plane with the third coupling fixed. In each panel, we show the lines where spontaneous symmetry breaking occurs using the one-loop result from Sec. III B. In the upper left panel, it is clear that the flows above the fixed flow line are not originating from the UV Gaussian fixed point (blue dot), whereas the flows below (unless in the broken phase, marked with gray) all emanate from the fixed point along the other fixed flow line. The same features are seen in the upper right panel. In the lower left panel, we see that only flows in the broken phase originate from the Gaussian UV fixed point, when the gauge coupling is kept fixed, α=0. The lower right panel, shows the two IR fixed points on the plane α=α*. We see that the symmetry-breaking lines both with α=0 and α=α* seem to be identical. This illustrates that for couplings less than or comparable to α*, the tree-level result is still a good approximation. From these diagrams it is clear that the fixed points are all in the unbroken phase.

We have so far uncovered the conditions for the models to be completely asymptotically free and shown numerically that these conditions are stronger than the conditions for the existence of IR fixed points. Furthermore, we have described the phase structure of the models anticipating three infrared phases (two types of radiative symmetry breaking and a phase of long distance conformality) using conditions which will be derived in the following section.

The tree-level analysis is the limiting case when the gauge contributions are turned off (i.e. α=0) and higher-order terms proportional to λn(n>1) are ignored. Thus, the boundary of the broken phase is a line in the λ1-λ2 coupling space rather than a two-dimensional surface in the λ1-λ2-α space.

The tree-level potential has the following form: V=m2TrS†S+λ1(TrS†S)2+λ2Tr(S†S)2,(3.1)where the Nc×Ns scalar field matrix S is invariant under SU(Nc)×U(Ns) rotations. For the scalar potential to be bounded from below, at sufficiently high-energy scales, the quartic couplings have to satisfy λ1+rλ2>0 for all r=Tr(S†S)2/(TrS†S)2, which ranges from 1/Ns and 1. To illustrate the symmetry-breaking pattern, it will be convenient to replace the traces with sums over the eigenvalues and subsequently find the minimum of the potential in the eigenvalue space. To that end, it will be useful to write the matrix S in the form S(x)=Uc†(x)D(x)Us(x),(3.2)where Uc and Us are unitary matrices and D is a matrix which is diagonal in a Ns×Ns block (with real entries) and zero everywhere else, assuming that Nc>Ns (which is true for the CAF conditions to be satisfied). Although this is a well-known result, we summarize the proof in Appendix E. We emphasize that this is not a local flavor and color transformation, but simply rewriting the matrices S in terms of different degrees of freedom. In the symmetry-breaking section, we are just interested in finding the vacuum configurations which are spacetime independent. Once the minimum is found, the matrix D can be rotated back to give the field value corresponding to the minimum in the original basis. As a result, this procedure does not give rise to additional terms from the kinetic part of the Lagrangian. Thus, the potential can be rewritten as V=m2TrD†D+λ1(TrD†D)2+λ2Tr(D†D)2,(3.3)i.e. all the dependence on the unitary matrices Uc and Us vanishes. The one-loop effective potential will likewise depend only on the components of D and so only the components of D can obtain nonzero vacuum expectation values.

The relevant degrees of freedom relevant in understanding the behavior of the effective potential are the diagonal part diag (ρ1,ρ2,…,ρNs) of D. The tree-level potential can, thus, be simplified into the following form, V=m2∑i=1Nsρi2+λ1(∑i=1Nsρi2)2+λ2∑i=1Nsρi4,(3.4)where m2 could be positive, zero or negative. For a positive mass term, i.e. m2>0, the potential has minimum at ρi=0, and this excludes symmetry breaking, while for m2<0, we will have spontaneous symmetry breaking as long as the potential is bounded. One can further show that the nontrivial vacuum configurations for the m2<0 case at tree level are the same as in the massless cases at tree level (a detailed proof is in [47]).

Restricting to the m2=0 case,²

The massless case corresponds to classically conformal models which possess many interesting features such as providing naturally light Higgs in asymptotically safe or free scenarios [17] (see also earlier works [48–50]).

In order for the direction to be flat, we require the V|min=V(ρi=0)=0 to obtain a ray on which loop effects can induce spontaneous symmetry breaking. These rays exist under two conditions: For  λ2>0:  Nsλ1+λ2=0(3.11)For  λ2<0:  λ1+λ2=0.(3.12)These lines are summarized in Fig. 7. In the figure, we overlap two sets of coordinates (λ1,λ2) and (λ1s,λ2s), with λis=λi/α, for which the lines coincide. Requiring the potential to be bounded from below, implies the RG flow in the UV (t→∞) to be in region I. However for a completely asymptotically free model, λi=0. To plot these models in the diagram, it is useful to use the alternative parameters λis, for which the UV fixed point is not at the origin. This condition was used in Sec. II A to correctly find the CAF models. For spontaneous symmetry breaking to occur, the RG flow (from UV to IR) must run from region I to region II regardless of the chosen set of coordinates.

We will now discuss the symmetry-breaking patterns. These can be directly read from the vacuum configurations as follows.

For λ2<0, V is minimized when there is only one nonzero ρa, which leads to the following form of the vacuum configuration: ⟨SAa⟩=ρδA1δa1.(3.13)The corresponding symmetry-breaking pattern is SU(Nc)×U(Ns)→SU(Nc-1)×U(Ns-1)×U(1).(3.14)For λ2>0, V is minimized when there are Ns nonzero ρa, providing ⟨SAa⟩=ρδAa.(3.15)In this case, the symmetry-breaking pattern is SU(Nc)×U(Ns)→SU(Nc-Ns)×SU(Ns)×U(1).(3.16)These symmetry-breaking patterns are worked out in detail in Appendix F.

This concludes the tree-level analysis, naturally leading to the question as to whether or not higher orders can affect these findings. This will be discussed momentarily.

At the quantum level, there are two interesting cases classified according to how quartic coupling λ scaling with gauge coupling α. (i)

The Gildener Weinberg case [39]: λ scale linearly with α (i.e. λ∼α)

The Gildener Weinberg case works where the scalar couplings λi scale linearly with α (i.e. λ∼α). In this region, the classical (tree-level) symmetry-breaking lines Eq. (3.11) or (3.12) derived in the previous section remains intact. This is because quantum corrections will provide extra term ∼α2 in the tree-level lines Eq. (3.11) or (3.12) which is negligible since we are studying in the region α≪1 and thus α2≪λ. In Gildener Weinberg case, the curvature of the effective potential generally will be proportional to λ∼α and we need to choose a flat direction (to make curvature ∼α2), along which quantum corrections can induce radiative symmetry breaking when the RG flows cross the lines Eq. (3.11) or (3.12).

At the quantum level, interesting possibilities for the vacuum structure of the model may emerge when scalar and gauge couplings start competing (i.e. λi∼α2). As it can be seen from the UV picture in Fig. 2, there are UV free trajectories which run arbitrarily close to the origin in (λ1,λ2)-space (indicated by a black dot), where the condition for the second case λi∼α2 holds. In these cases, the tree-level potential is approximately flat in all directions, and the symmetry breaking is dominated by the gauge loop contributions. The symmetry breaking is therefore no longer restricted to the symmetry-breaking patterns discussed in the previous section. This possibility was first pointed out in [40] and we will explore the possibilities of alternative symmetry-breaking patterns by using RG improvement in the following. This scenario is phenomenologically interesting, since when all directions are flat, the model could radiatively generate a spectrum for all the scalar masses (see e.g. [50–52]), as compared to the Gildener-Weinberg scenario [39], where the only radiatively generated scalar mass is the mass of the scalon.

There are two main ways to implement the one-loop effects: explicit logarithmic summation (see e.g. [51–54]) or implicit logarithmic summation (see e.g. [55]). In this work, we focus mainly on the method of implicit RG improvement because the explicit calculations shown in Appendix H require diagonalization of the mass matrices which quickly becomes impractical for Nc≥4, Ns≥3 in the fundamental representation. The RG improvement method developed in this section could be generalized to arbitrary symmetry groups and representations.

The renormalization group improved effective potential is: V({ρi})=[λ1(t)(∑i=1Nsρi2)2+λ2(t)∑i=1Nsρi4]⁢exp(4∫0tdt′γ(t′)),(3.17)where t is defined as t=log[∑i=1Nsρi2/μ2] with renormalization scale μ and γ(t) is the anomalous dimension of the scalar field. The model at hand generally does not allow for a Yukawa coupling, and therefore the anomalous dimension γ(t) will, at one-loop level, be of order α. In the case where λ(t)∼α2, the anomalous dimension will give rise to terms of the form λ(t)γ(t)∼α3 in the minimization condition which can be regarded as higher-order effects. We can therefore further simplify the RG improved effective potential, Eq. (3.17), to V({ρi})=λ1(t)(∑i=1Nsρi2)2+λ2(t)∑i=1Nsρi4≡λ1(t)f1(ρi)+λ2(t)f2(ρi),(3.18)where we introduced functions f1 and f2 for convenience.

The RG improved minimization condition is given by: Vρi(1)≡∂V∂ρi=∑j=12(dλjdt∂t∂ρifj+λj∂fj∂ρi)=∑j=12(βλj∂t∂ρifj+λj∂fj∂ρi).(3.19)Now it is clear that although the effective potential is of the form of the tree level potential, the one-loop information is encoded through the RG functions (βλ1,βλ2).

In order for alternative vacuum configurations to exist, we need a number of distinct values of ρi to satisfy Eq. (3.19), which is different from the tree level assignments of ρi [i.e. Eqs. (3.13) and (3.15)]. For each distinct nonzero value of ρi, the minimization condition corresponds to a nontrivial constraint on the couplings. Since we have three marginal couplings (α,λ1,λ2), we therefore expect that at most there could be two distinct vacuum expectation values to fully determine symmetry-breaking lines in (α,λ1,λ2)-space. In the case with three distinct vacuum expectation values, all three couplings will be fully determined and the system is already overdetermined. An exception to this is when λ2=0; then there is only a single constraint on the remaining couplings which depends on f1 and f2.

We assume in the following that the vacuum configuration to be such that n1 values of ρi are ρ, n2 are κρ, with κ positive and different from unity, and Ns-n1-n2 values equal to zero. From Eq. (3.19), we get two constraints on the couplings.

By solving Eq. (3.19), we obtain λ2=0 at the broken scale. In this special case, the boundary sheets between the unbroken phase and broken phases become boundary lines; for each κ, this corresponds to a particular curve on the (λ1,α) plane. It seems we could have alternative vacuum configurations and symmetry-breaking patterns since κ could be any non-negative value. As we explore further, we find in this case one of the eigenvalues of the mass matrix will always be negative, violating the vacuum stability conditions discussed below. Hence the alternative symmetry-breaking patterns scenario yields unphysical solutions.

Furthermore, the effective potential needs to be stable at the vacuum configuration, we therefore derive the eigenvalues of the Hessian matrix Vρiρj(2)≡∂2V∂ρi∂ρj|vacuum,(3.20)where all three RG functions (βλ1,βλ2,βα) are encoded. A stable vacuum and physical scalar masses requires the eigenvalues to be non-negative.

We find that these requirements cannot be met [two distinct VEVs will always lead to one negative mass eigenvalue throughout Eq. (3.20)] unless n2=0, and n1 is either 1 or Ns. These are exactly the tree level vacuum configurations, implying no alternative symmetry-breaking patterns are found in this model beyond the two discussed at the tree level analysis, i.e. Eq. (3.14) and Eq. (3.16).

In Appendix G, we carry out the analysis of the case with (Nc=6, Ns=3, Nf=31). We find that the RG improved boundary lines (G3) and (G5) for the broken phases actually shift the tree level boundary lines when λi∼α2≪1. While, when λi∼α≪1 the RG improved reduces to the tree level result. In this way, Fig. 14 provides a detailed view of region near the origin of Fig. 7.

Similarly, in Appendix H, we perform the same analysis based on the explicitly calculated one-loop effective potential in the Coleman-Weinberg renormalization scheme. There are differences in the exact conditions on the couplings for spontaneous symmetry breaking, but the corresponding vacuum configurations, and thus symmetry-breaking patterns, are identical and the regions satisfying the vacuum stability conditions (comparing Fig. 16 with Fig. 12) are similar and consistent in both renormalization schemes.