Stable Asymptotically Free Extensions (SAFEs) of the Standard Model

We consider possible extensions of the standard model that are not only completely asymptotically free, but are such that the UV fixed point is completely UV attractive. All couplings flow towards a set of fixed ratios in the UV. Motivated by low scale unification, semi-simple gauge groups with elementary scalars in various representations are explored. The simplest model is a version of the Pati-Salam model. The Higgs boson is truly elementary but dynamical symmetry breaking from strong interactions may be needed at the unification scale. A hierarchy problem, much reduced from grand unified theories, is still in need of a solution.


I. INTRODUCTION
We start by considering an elementary Higgs boson in a world without low energy supersymmetry. In this world there are two conflicting demands on the nature of new physics on higher mass scales. Naturalness strongly constrains the new physics to prevent unwanted contributions to the Higgs mass. Either the new physics mass scale cannot be much higher than the Higgs mass or the Higgs coupling to the new physics must be extremely weak. The other demand on the new physics is that it must significantly alter the running of couplings, including the quartic coupling of the Higgs. This is because the Landau poles in the quartic coupling and the U(1) hypercharge coupling would signal new mass scales of the dangerous type. To avoid this requires new massive degrees of freedom that do couple to standard model fields and thus are also dangerous for naturalness. These two demands are suggesting that if there is new physics to cure the Landau problem then it must enter at as low a scale as possible to minimize the naturalness problem.
The absence of Landau poles is a requirement for the theory to be UV complete, or in other words that there is a description of the theory on arbitrarily high energy scales in terms of elementary fields. The fermions and gauge bosons of asymptotically free gauge theories are prime examples of truly elementary fields. The standard model is not of this type, but it often thought that there is no reason it should be given the presence of gravity. The onset of gravitational effects at Planckian energies is usually taken to mean that the theory experiences a complete change of character on these scales. But once again this is at odds with naturalness.
It is only if gravity somehow exerts only a very minimal effect on the scalar sector in a UV complete theory is there is any hope of naturalness.
There have been recent attempts to show how the effects of gravity in UV complete quantum field theories could be consistent with naturalness. Ref. [1] illustrated a proposed mechanism in a 2D model of quantum gravity. These authors introduce the concept of "gravitational dressing" of a QFT, where Planck mass effects modify the S-matrix directly without inducing any physical mass scales. Ref. [2] (see also [3]) suggests that the pure gravitational action in the high energy regime just contains two terms, an R 2 term and the Weyl term 1 3 R 2 − R 2 µν . The Einstein-Hilbert term is induced via the VEV of a new scalar field with non-minimal coupling to R. The point is that the gravitational interactions may then be both renormalizable and asymptotically free [4][5] [6] . Ref. [2] argues that such a gravity sector could be arranged to couple sufficiently weakly to the standard model fields to preserve naturalness. The gravity sector here is not quite complete because of a ghost and a tachyon in the spectrum.
Our interest here is the other half of the problem, how to build UV complete quantum field theories containing truly elementary scalar fields. We approach this by searching for gauge theories containing both fermions and scalars where all couplings run to zero in the UV. This could provide a completely asymptotically free extension (CAFE) of the standard model. A nice study of this type was conducted long ago in [7]. There the constraints were found on theories with a simple gauge group with varying numbers of scalar fields in various representations and with fermions. Gauge, quartic and Yukawa couplings were considered. CAFEs were found and described in terms of UV fixed points (UVFPs) where ratios of couplings approached fixed values. The fixed points were also required to be UV attractive from all directions in coupling space. Thus these are CAFEs that also have complete UV stability, and we denote such an extension of the standard model as a SAFE. That such theories were found in [7] may have been of interest to the construction of grand unified theories. But the study showed that it was difficult for the scalars that were allowed to sufficiently break down the original gauge theory via the Higgs mechanism. For this reason and perhaps also because it was thought that gravity would nevertheless provide an ultraviolet cutoff, it appears that SAFEs were never considered to be of particular importance in GUTs.
Our work can be considered to be a continuation of this old work. Since we need to embed the standard model into a gauge group without a U(1) factor at the lowest possible scale we are here dealing with low scale unification. Thus we must extend the original work to semi-simple gauge groups. A minimal requirement is that the scalar content of the theory must yield the Higgs doublet after symmetry breaking. We don't require that the scalars be entirely responsible for gauge symmetry breaking, other than electroweak symmetry breaking, since we leave open the possibility that strong interactions could dynamically break some symmetries.
After the work [7] there were attempts to find other realistic CAFEs, not necessarily grand unified. From our point of view these attempts were not completely successful since UV stability was dropped (see review [8] and references therein and in particular [9]). The fixed point was allowed to be UV repulsive in some directions in coupling space. In this case the space of couplings that do flow to the fixed point has reduced dimensionality. This amounts to constraints (sometimes called predictions) on the low energy couplings that are also affected by higher order corrections. Satisfying the constraints would require fine tuning the couplings order by order in perturbation theory. In our work we shall insist on complete UV stability.
Much more recently there has been another attempt to find UV complete theories with elementary scalars, but this time the search was for nontrivial UVFPs [10]. Unlike the case of asymptotic freedom, here the fixed point requires knowledge of the β-functions beyond lowest order. Interesting examples were found but here again complete UV stability was not attained. Also, in this context the work in [11] suggests that the transition from a regime of running couplings to a nontrivial UVFP is sufficient to cause a contribution to the Higgs mass.
So in this case as well, the corresponding mass scale must be as low as possible.
The prototype of low scale unification is the Pati-Salam model [12], based on the gauge group SU(4) × SU(2) L × SU(2) R , with the fermions of one family in the (4, 2, 1) L + (4, 1, 2) R representation. Our study will answer the question as to whether scalars can be added such that a SAFE results. But we shall set up our study in a more general context where we consider products of various SU(N ) gauge groups with various scalars that may transform simultaneously under two or three of these gauge groups. We only consider scalars in the fundamental representation since then we can expect a Higgs doublet to emerge after symmetry breaking. These results may be of more general interest for model building.
Since we are discussing theories that are UV complete above the Planck scale, one might wonder about the effect of gravity on the running couplings of the matter fields. This was discussed in the quadratic higher derivative gravity theories of [2,3]. The coupling f 2 2 , appearing as 1/ f 2 2 times the Weyl term, is always asymptotically free with both gravity and matter fields contributing with the same sign to the β-function. This means that f 2 2 is typically much smaller than the gauge couplings in the deep UV, and so its effect can be neglected.
The coupling f 2 0 appearing in the R 2 term will be asymptoticallly free only if the ratio f 2 0 / f 2 2 becomes negative in the UV. Depending on the matter content it is possible that f 2 0 could run relatively slowly and thus play a more significant role. Here we note a discrepancy in the calculated f 2 0 contribution to the scalar quartic β-functions in [2] and [3]. In the following we shall ignore the possible effect of gravity on the matter β-functions.
This paper is organized as follows. In Sec. II we first review the basic idea to realize SAFEs with a simple Lie group. Then we generalize the study to a semi-simple gauge group in Sec. III, as motivated by low scale unification. For quantitative study we choose several benchmarks for gauge groups and scalar representations. In Sec. IV we present and discuss the numerical results. Based on these studies we consider the simplest example of a SAFE with low scale unification in Sec. V. We conclude in Sec. VI.

II. SAFEs WITH SIMPLE LIE GROUP
In this section we review the basic idea to realize SAFEs in [7]. This reference systematically studied the simple group SU(N ) or O(N ) case with fermions and scalars in various representations. Here we supplement their work with some numerical results for comparison with our later analysis.
Since we study UV asymptotic freedom, the one loop β-functions are sufficient to study the UV behavior. At one loop the coupled β-functions of gauge, Yukawa and quartic couplings can be solved sequentially. For the gauge coupling, its β-function only depends on itself and with t = ln(µ/Λ). b < 0 gives asymptotic freedom with an infrared Landau pole at t = 0 (µ = Λ). The β-function coefficient b gives the running speed of gauge coupling at large t.
For the Yukawa coupling y, its β-function has the generic form (4π) 2 β y = a y y 3 − a g g 2 y, where a y , a g > 0. The dependence on g can be eliminated with a change of variablesȳ ≡ y 2 /g 2 , and this gives where dependence on b has appeared. To have asymptotically free y amounts to finding a UVFP forȳ. When a g + b ≤ 0 and sinceȳ ≥ 0 by definition the only UVFP isȳ = 0, which is UV repulsive. A stable UVFP requires that a g + b > 0 in which caseȳ = 0 is the stable UVFP.
The result is thatȳ decreases asymptotically as As clarified in [7], the same conclusion applies to the more complicated case when the Yukawa couplings are described by a matrix. So in SAFEs, the contribution of Yukawa couplings is negligible in the β-functions of quartic couplings in the deep UV.
The one loop β-function of a scalar quartic coupling is in general a function of both gauge and Yukawa couplings, but as we have just explained we ignore the latter. We may illustrate the general features with one scalar Φ i in the fundamental representation of a SU(N A ) gauge group. The gauge invariant scalar potential at dim = 4 has only one term, The one loop β-function for λ is This β-function is composed of three pieces: the positive pure quartic terms, negative gaugequartic terms and positive pure gauge terms. To have β λ = 0 the three contributions should be comparable and so this disfavors a large hierarchy between quartic and gauge couplings.
In particular quartic couplings must also run as 1/t in the deep UV.
We may again eliminate the dependence on g by a change of variablesλ ≡ λ/g 2 , and this with b again appearing in the linear term. Defining pure gauge boson contribution, the regions with 2b + 6(N − 1 N ) > 0 and 2b + 6(N − 1 N ) < 0 meet at the value r 0 = 9 11 (1 − 1/N 2 ). These two regions correspond to the slow gauge running (r s < r 0 ) and fast gauge running (r f > r 0 ) cases respectively, and there is a one-to-one mapping with 2r s b M + 6(N − 1 N ) = −(2r f b M + 6(N − 1 N )) and λ → −λ. βλ = 0 is simply a quadratic equation forλ and there are two real roots when This inequality sets an upper (lower) bound on r in the slow (fast) running region with solutionsλ > 0 (λ < 0). For the present example the lower bound on r in the branch r > r 0 is always above one and so this cannot be realized with any matter assignment. Also this region is disfavored due to the upper bound on r from the UV stability of Yukawa coupling and the vacuum stability condition for the quartic couplingλ > 0. So we need only consider the slow running region, where the inequality (8) sets N ≥ 3. For each N ≥ 3, we present the upper bound on 0 ≤ r ≤ 1 for various N in Fig. 1. We can determine the number of Dirac fermions n F to satisfy this bound from The minimum n F basically grows with N , and it is shown for the fundamental representation b F = 2/3 in the last row in Fig. 1.
The values ofλ = λ/g 2 at stable (red) and unstable (blue) UVFPs as r varies over the allowed range.
For each N , b that satisfy (8) and r < r 0 there are two positive real rootsλ 1 <λ 2 . Given the positive contribution from the pure quartic and pure gauge terms, it is the smaller root λ 1 that is stable, i.e. dβ/dλ < 0 atλ =λ 1 . For each N we depictλ 1 ,λ 2 for all possible b in Fig. 2, where red and blue label stable and unstable UVFPs respectively. b → 0 at the ends of each line. In large N 1 limit, the stable and unstable UVFPs become insensitive to N and these end values approach 0.14 and 1.3 respectively. For a stable UVFP,λ is always smaller than one. Also, the stable UVFPλ 1 is UV attractive with respect to all quartic couplings λ <λ 2 . By increasing the size and/or number of scalar representations, a larger N may be required to achieve a SAFE. This generally does not allow sufficient scalar fields to break the simple gauge group in some realistic manner [7]. For example SU(5) grand unification typically requires two scalars, in the adjoint and fundamental representations, to break SU(5) down to the SM. But with this set of scalars the theory is a SAFE only if N ≥ 7.
For a given gauge group, the larger the total number of scalar degrees of freedom, the tighter is the constraint on b [7]. This general feature will carry over to our generalizations and it is another motivation to restrict ourselves to scalars in the fundamental representation.

III. GENERALIZATION TO SEMI-SIMPLE LIE GROUP
Motivated by low scale unification we shall focus on scalar fields transforming under the following two types of gauge groups with N i ≥ 2.
We first discuss the behavior of Yukawa couplings for the semi-simple case. In the simplest case of a single Yukawa coupling y, as a generalization of the β-function in (3) we find whereȳ = y 2 /g 2 j and with g j one of the gauge couplings. The a i depend on the scalar and fermion representations. In the deep UV the gauge coupling g i approaches its asymptotic form and becomes insensitive to its initial value. So we may replace the ratio of gauge couplings in (11) by their β-functions coefficients, i.e.
then there is a stable UVFP and it is atȳ = 0.
We have checked various fermion and scalar representations for the gauge groups in (10).
It turns out that (12) is easy to satisfy since a i ∼ N i and b i is negative. In some cases (12) may put a upper bound on b i , but as we shall see below, in the parameter space of interest the constraint is much weaker than constraints from the quartic couplings. For a matrix of Yukawa couplings we expect these features will continue to hold, as in [7]. Therefore in our study of SAFEs for semi-simple gauge group we will focus on the quartic couplings and neglect the contribution of Yukawa couplings in their β-functions.
We now build four benchmarks for semi-simple Lie groups in (10).
For the gauge group SU(N A ) × SU(N B ) the simplest nontrivial setup is to have one scalar field Φ ik that transforms in the fundamental representation of both groups, i.e. (N A , N B ). The most general dim = 4 scalar potential is when at least one N i > 2. λ d and λ s denote double trace and single trace couplings respectively. In the deep UV, the β-functions for these quartic couplings are It is straightforward to verify that (14) reduces to (6) in the single gauge group case with

case corresponds to the bidoublet in the
left-right symmetric model and it has a larger set of couplings [13].
In the second benchmark we consider the same gauge group with two scalars. We don't expect to learn much by considering two copies of (N A , N B ), especially since the replication of scalars was considered in [7]. For the combination (N A , 1) + (1, N B ) there is a limit where the two scalars decouple and so this case is also of not much interest. So we will study two different scalars that share a common gauge group.
N A specifies the common gauge group. The most general scalar potential when at least one N i > 2 has five terms, Here there are two mixing couplings λ d12 , λ s12 . The one loop β-functions are presented in (A1) in Appendix A. Due to the presence of the common gauge group we shall find that there is no UVFP solution where the mixing couplings vanish and the two scalars decouple.
, the next interesting scalar content starts with two scalars. It is again interesting to study the case with two different scalars sharing a common gauge group. The case different from Case B is the following.
We set N A > 2 for the common gauge group. In the context of the Pati-Salam model, this setup may correspond to left-right symmetric scalars (4, 2, 1) and (4, 1, 2). The scalar potential is where λ d12 , λ s12 are mixing couplings. We may consider a simplified version of this theory by imposing a Z 2 symmetry, the analogy of left-right symmetry in the Pati-Salam model.
This Case C1 amounts to picking a special slice in the whole parameter space, with only two gauge couplings and four quartic couplings. The β-functions are presented in (A2).
We denote by case C2 the general case with six quartic couplings. The β-functions are in (A3).
In the case of the Pati-Salam model with Φ L = (4, 2, 1) and Φ R = (4, 1, 2) we may construct a gauge invariant quartic term with the Levi-Civita symbol, In the last benchmark we study a scalar representation charged under all three groups.
In particular we consider the fundamental representation Φ ika : (N A , N B , N C ). This type of scalar field is less studied in literature since its VEV breaks all gauge symmetries at the same scale. But in view of finding SAFEs it is intriguing to ask whether it helps to have a scalar transforming under more gauge groups. The scalar potential is There are now three single trace couplings. The one loop β-functions are presented in (A6), and they are symmetric under interchanges between (N A , λ s1 ), (N B , λ s2 ) and (N C , λ s3 ). One can verify that (A6) reduces to (14) with In the Pati-Salam model with one (4, 2, 2) scalar we may construct another Levi-Civita term, The β-functions involving λ ε are presented in (A7) and (A8).

IV. NUMERICAL RESULTS AND ANALYSIS
In this section we present the numerical results and analysis of the four benchmarks. As before we change variablesλ i = λ i /g 2 j where g j is one of the gauge couplings. Then we replace the ratios of different gauge couplings by their asymptotic values, g 2 i /g 2 j → b j /b i . This leaves us with coupled quadratic equations of theλ i . Taking case A as an example, the β-functions in (14) become With these we can solve for the UVFP of {λ i } as functions of N i and b i . Since the coupled quadratic equations are usually difficult to solve analytically, we find numerical solutions for a parameter scan over N i , b i . To illustrate the pattern, we choose 2 ≤ N i ≤ 8. The β-function coefficients b i depend on the matter and are model dependent. For convenience we use UVFPs should be real but need not be positive.
To find UV stability we study the RG flows in vicinity of the UVFP. At linear order it is characterized by the matrix The UVFP is absolutely stable as long as all eigenvalues κ k of D i j (λ 0,i ) are negative. The UVFP for theλ i 's is approached along the directions of the eigenvectors as t −κ k /2b A . In each panel the black dot line denotes b A = b B . This figure highlights the fact that it is a large hierarchy between N A and N B that helps most to achieve a SAFE. And when there is a hierarchy it is the r i of the larger gauge group that is bounded from above.

A. Constraints on
We present the upper bounds on r i for all our benchmark models in Fig. 4 and Fig. 5. This information can be used to constrain the matter content to achieve SAFEs. To illustrate the number fraction of viable points for each N i set, we use dark (light) blue for more (less) viable We may briefly consider the fate of the fast running solutions, as we did for the simple gauge group. The vanishing of the linear terms in (23) defines a boundary on the r A − r B plane as follows, The region below (above) the boundary features slow (fast) running, and the UVFP solutions in the two regions are related by a rescaling of the r i and λ i → −λ i . In this Case A we find that the boundary (25) and thus all fast running UVFP solutions are outside of the physical region 0 < r i ≤ 1.
For Case B in Fig. 4     Finally we turn to Case C2. It depends on all three N A , N B , N C and the results cannot be summarized in one 2D plane. But we do find that the constraints when N B = N C are quite similar to Case C1 in Fig. 4(c). The general upper bounds on r A , r B , r C for various N A are displayed in Fig. 5 as functions of N B (row) and N C (column). We present these limits using the notation (r A ) r C r B . From the four tables one can see that solutions tend to appear when some hierarchy develops between the three values N A , N B , N C . Among the possibilities, a hierarchy with a large common gauge group is the most efficient. And it can be seen that the upper bound on r i is typically relaxed or nonexistent (= 1) in those cases where the associated N i is small relative to some other N j .

B.λ j values from the parameter scan
Next we show results for the values of the quartic couplings at the UVFPs. We definē λ j ≡ λ j /g 2 i where g i is the coupling of the largest gauge group. We saw in previous section that this coupling runs most slowly in the UV (has the smallest b i ) and thus is the largest gauge coupling.
FIG. 6. The projection of the parameter scan on theλ d −λ s (first row) and r A -r B (second row) planes for different {N A , N B }. The quartic couplings are normalized by the largest gauge coupling. The red and blue dots represent stable and unstable UVFPs respectively. Note that some characteristics of these plots are determined by the step size of the parameter scan.
We start from the simplest Case A with only two quartic couplings. In Fig. 6, for some typical (N A , N B ), the first row shows the projection of the parameter scan on theλ d -λ s plane, while the second row shows the r A -r B projection for comparison. Among all UVFP of (23) we depict the stable and unstable solutions by red and blue dots respectively. The situation is clearest for the left plots where the ratio N A /N B is the greatest . For each (r A , r B ), there are always a pair of solutions, one stable and one unstable with smaller and largerλ d respectively.
With decreasing r A we go through different arcs from inside out, where the arc length depends on the number of viable r B . In r A → 0 limit, the solutions become independent of r B and reach the corners of the red and blue regions that possess the largest distance between stable and unstable UVFPs. When N A , N B are similar both gauge couplings play significant roles and the solution pattern becomes more involved.
The unstable solution in each case is actually a saddle point, with one direction UV attractive and the other one repulsive. Also, at least for 2 ≤ N i ≤ 8, we find that the quartic couplings at the UVFPs are positive and typically of order 0.1 or 0.2 times the largest gauge coupling. The stability of tree level potential demands the conditions but here they put no further constraint.
In comparison to these slow running UVFPs the unphysical fast running UVFPs again come With a large positiveλ d1 we find that the mixing couplingλ d12 can be negative, but then the coefficient ofλ d12 in βλ d12 is positive and the solution becomes unstable. Mixing couplings are usually positive for stable UVFPs, but a new feature we see here is that they can be close to zero. This is due to the suppressed pure gauge terms in the β-functions of the mixing couplings, which only receives a small contribution from the common gauge group (it is 0 for N A = 2 case). Finally the general picture of UVFPs for Case C2 without the Z 2 symmetry is similar to Case C1. Case D has four quartic couplings, one double trace and three that are single trace. We depict the projectionsλ s1 -λ d andλ s2 -λ d in Fig. 9 for N A = 8, N B = N C = 2. The typical feature is reflected on the range of single trace couplings at UVFPs. We find that the coupling with a single trace associated with the largest gauge groupλ s1 has comparable size with other couplings at UVFPs, while those associated with small gauge groups,λ s2 orλ s3 , could be close to zero or even slightly negative. Again this is determined by the dominant pure gauge terms in the β-functions.

V. THE SIMPLEST MODEL
As a general feature of the previous results, when a hierarchy in the sizes of the different gauge groups helps to achieve SAFEs, the gauge coupling associated with the largest group is constrained to run quite slowly. A small ratio with n F copies of Ψ L + Ψ R and n Q copies of chiral fermions Q L . To be anomaly free when N B > 2 we need an integer ratio n F /n Q = N B /N A . (For N B = 2 we only need n F N A + n Q N B to be even [14].) The β-function coefficients of two gauge couplings are [15] and the other is the Pati-Salam model SU(4) × SU(2) L × SU(2) R [12]. Our results show that the former cannot be SAFE and so we turn to the latter. In this case of all the SAFEs that we have found there is only one that is of relevance. From the results for Case A we find that we can add a single scalar Φ transforming as (4, 2, 1). We choose (4, 2, 1) rather than (4,1,2) to ensure that Φ will yield the SM Higgs doublet.
As we have just discussed, the constraint on the SU(4) β-function from  Table I. Upon the breakdown SU(4) → SU (3) we see that the model predicts a colored scalar doublet in addition to the Higgs doublet.  The one loop β-functions are where n F and n f are defined in Table I. As shown in Fig. 10 that give SAFEs. The corresponding fixed point values of the coupling ratios, for both the stable and unstable cases, are shown in Fig. 10(b). We note that the fermion content includes the right-handed neutrino, and a right-handed neutrino condensate does break the Pati-Salam gauge group down in the desired manner. 1 Lepton number is violated, but baryon number and proton stability is preserved.
Here we see the remaining tension in a low scale unification model because there is still some hierarchy between the unification scale and the Higgs mass that remains unexplained.
In our case the neutrino condensate would give rise to a massive SU(4) gauge boson which in turn will contribute to the Higgs mass via the diagram in Fig. 12. Some other peculiar property of the strong interactions would be needed to explain the suppression of K → eµ and the small Higgs mass simultaneously.  [16] we find that perhaps the best compromise is 2n F + n f = 15. Then the SU(4) IRFP is at α 4 ∼ 0.43 while the SU(3) IRFP is at α s ∼ 0.12.
The difference between these two numbers is interesting but it is not certain that it is large enough.

VI. CONCLUSIONS
In this paper we explore the construction of UV complete quantum field theories containing truly elementary scalar fields without UV Landau poles. We extend the old study in [7] to search for SAFEs for semi-simple gauge groups, which is well motivated to achieve low scale unification. The UV property of gravity is far from clear and we restrict ourselves to study β-functions of the coupled system of gauge, Yukawa and quartic couplings. We review the basic idea of a SAFE in Sec. II and present numerical results of simple gauge group for comparison with latter analysis. In Sec. III we generalize the analysis to the semi-simple gauge groups in (10), which includes the Pati-Salam model and other low scale unification models as examples. We only consider scalars in fundamental representations, both to incorporate the SM Higgs and to minimize the number of scalar degrees of freedom.
We build up four benchmarks for quantitative study and the β-functions are presented in (14) and Appendix A. Our main numerical results and analysis are presented in Sec. IV. We search for solutions by parameter scan over gauge group size N i and β-function coefficients b i . For each N i set, we find the upper bounds on To provide a guide for model building, we present these upper bounds in Fig. 4 and Fig. 5 for all benchmarks. In Sec. V we consider the simplest model that illustrates some of the issues to be faced in SAFE model building.
We list here the properties of the UVFP in SAFEs that we have observed.
• The gauge couplings and typically most of the quartic couplings are running as 1/t in fixed ratios.
• Stability demands that the Yukawa couplings vanish more rapidly, 1/t α with α > 1, as do those quartic couplings that have vanishingλ i .
• Fewer scalar degrees of freedom helps to achieve SAFEs.
• A hierarchy in the sizes of the different gauge groups helps to achieve SAFEs.
• Among all UVFPs there is always one that is UV stable.
• SAFEs with negative quartic couplings are rare.
• The gauge coupling associated with the largest group is typically constrained to run the slowest of all the couplings. Since its associated b is the smallest, it is the largest coupling in the UV.
• To achieve this small b the theory typically needs some number of vector-like fermions that are only charged under the largest gauge group(s).
If the coupling ratios remain anywhere in the vicinity of the fixed point as the couplings themselves grow larger, then it will be the case that the largest gauge group grows strong first in the infrared. This situation may be related to the real world where the quartic couplings and the gauge couplings of the small electroweak gauge groups are observed to be small.
In fact in our simplest model we saw that the IR flow of couplings was such that a linear combination of the quartic couplings was bounded from above.
Yukawa Note Added: As we were finalizing this paper we saw the new paper [17]. This paper discusses CAFEs that are not SAFEs, since nonvanishing Yukawa couplings at unstable fixed points are utilized. We also noticed a particular quartic term (third term in their (A.3f)) that we missed that would be present in our Case C with (4, 2, 1) and (4, 1, 2) scalars. This term has the same property we discussed for the Levi-Civita term and it does not change the absence of a SAFE in this case. Otherwise our β-functions agree where they overlap up to the normalization of the quartic couplings.
For Case B, we deduce the five one loop β-functions from the potential (16), where N A denotes the common gauge group.
Case C is split into two benchmarks. In Case C1, by imposing Z 2 symmetry as in (18), we deduce one loop β-functions for the four quartic couplings from (19).
For N A = 4, N B = N C = 2 case, the modification of β-functions from the Levi-Civita term in (22) is quite similar to that in Case C. We find The β-function of this new coupling is