In search of a UV completion of the Standard Model - 378.000 models that don't work

Asymptotically safe extensions of the Standard Model have been searched for by adding vector-like fermions charged under the Standard Model gauge group and having Yukawa-like interactions with new scalar fields. Here we study the corresponding renormalization group beta functions to next and next-to-next to leading order in the perturbative expansion, varying the number of extra fermions and the representations they carry. We test the fixed points of the beta functions against various criteria of perturbativity to single out those that are potentially viable. We show that all the candidate ultraviolet fixed points are unphysical for these models: either they are unstable under radiative corrections, or they cannot be matched to the Standard Model at low energies.

29 4. A model that almost works 32 5. Five benchmark models studied in the literature 33 6. Two more benchmark models studied in the literature 35 7. Combining more than one representation 36 V. Conclusions 38

I. INTRODUCTION
In the early days of quantum field theory, renormalizability was used as a criterion to select physically viable models. It was later understood that effective field theories can be useful and predictive in their domain of validity even if not renormalizable.
Current particle physics is largely based on this paradigm. It leaves an enormous freedom. One would like to have a more restrictive framework to guide the search for physics beyond the Standard Model (BSM). Non-perturbative renormalizability, also known as asymptotic safety (AS), provides such a framework. A quantum field theory is AS if all its couplings, running along the renormalization group (RG) flow, reach a fixed point in the ultraviolet (UV) limit [1,2]. The fixed point can be interacting or free (Gaussian). In the latter case, AS reduces to asymptotic freedom (AF). In both cases, the theory is well behaved and predictive at all energies. UV completeness is by itself a rather abstract notion, being untestable in practice. The real bonus of AS is that when a suitable fixed point exists, typically there are only a finite number of relevant directions that can be used to reach it in the UV. This greatly restricts the infrared physics.
While AF theories have been studied in great detail and for a long time, work on AS models for particle physics has only begun quite recently. For some early references based on the use of the functional renormalization group see [3][4][5][6][7][8]. A breakthrough came with the work of Litim and Sannino, who constructed gauge-Yukawa systems admitting interacting fixed points that are under perturbative control [9].
In these models the fixed point arises from a cancellation between one-and two-loop terms in the β-functions. The crucial ingredient is the Veneziano limit, providing the small expansion parameter where N c and N f are the numbers of colors and flavors respectively. It is reasonable to expect that there may exist AS models also for finite values of . General conditions for the existence of such fixed points have been discussed in [9,10]. Ap-plications of these ideas to BSM physics have appeared [11][12][13][14][15][16][17][18][19].
The Standard Model (SM) by itself is not AS because of the Landau pole in the U (1) gauge coupling [20,21] and the uncertain fate of the Higgs quartic interaction [22]. The Landau pole can only be avoided by assuming that the gauge coupling is identically zero at all energies. This is known as the triviality problem.
Can the SM be turned into an AS theory by extending its matter content? The simplest (and most studied) extensions consist of multiple generations of vectorlike fermions carrying diverse representations of the SM gauge group. The choice of vector-like fermions is motivated by their not giving rise to gauge anomalies and their masses being technically natural. The authors of [15] have studied the β-functions to two-loop order in the simplified case of SU (3) × SU (2) gauge interactions and a Yukawa-like interaction among the vector-like fermions. They find several UV fixed points, which they match to the low-energy SM in a number of benchmark cases.
In a parallel development, the authors of [16,17] studied AS for the full SM gauge group, again extended by vector-like fermions, by means of a resummation of the perturbative series of the β-functions. They find several UV fixed points, which however cannot be matched to the low-energy SM in a consistent manner [17].
To move forward in this program, we report our results for a large class of mod- In contrast to [16,17] we do not use resummed β-functions. Instead, we compare the results of the two-loop (NLO) gauge β-functions considered so far in the literature with the three loop results (NNLO). As explained in section II, the β-functions for the Yukawa and scalar couplings are retained always at one-and two-loops less than the gauge couplings, respectively. By comparing the results at these two dif-ferent approximation schemes, we are able to assess quantitatively the impact of radiative corrections and therefore to decide whether a given fixed point is within the perturbative domain or not. This selection is supported by the use of other tests of perturbativity that the fixed points must satisfy, as discussed in sections II F and II G.
The core of our work consists of a systematic search of reliable fixed points in a large grid parameterized by the value of N f (the number of vector-like fermions) and their SU (3) c × SU (2) L × U (1) Y quantum numbers (this grid is defined precisely in section IV). We first find all the zeroes of the β-functions for each model in the grid. We then test each fixed point thus found against two conditions: • The fixed point must occur in a region in which the perturbative expansion is reliable. At the very least, this implies that it must be possible to reasonably trace its value at the NNLO back to that of the NLO. We see a posteriori • The fixed point can be connected to the SM at low energy. In general this would require a delicate numerical analysis of the trajectories emanating from it. However, we find that a rough necessary condition is sufficient for our purposes: the fixed point must not have any coupling that is zero and irrelevant, because such couplings must be identically zero at all scales to avoid Landau poles.
As we shall see, these two requirements taken together, while quite reasonable, are very restrictive. As a matter of fact, we are not able to identify any choice for the group representations and number of generations of the vector-like fermions that would make the extension of the SM reliably AS. This does not mean that such an extension does not exist: it only means that if such an AS extension of the SM exists, it must either be different from those that we have considered, or else it must have a fixed point that lies outside the reach of perturbation theory.

II. METHODS
In this section we describe the general procedure that we follow in the rest of the paper. This allows us to motivate better the requirements (introduced in section I and further elaborated here) that we impose on the fixed points in order for them to be considered as physical. We recommend [23] as a general reference on RG flows.

A. The fixed points of the β-functions
Consider a theory with generic (gauge, fermion or scalar) fields and (generally dimensionful) couplingsḡ i of the interactions among them. In the study of the RG it is customary to use dimensionless couplings g i , related to the dimensionful couplings by g i = µ −d iḡ i , d i being the mass dimension ofḡ i . The renormalization of the theory is completely characterized by its β-functions where µ is the sliding scale of the quantum theory. A fixed point of this theory, denoted g * j , is defined by the vanishing of the β-functions of all couplings: When the couplings g j assume the values g * j , the renormalization of the quantum theory stops. In general, a given fixed point can be reached either in the UV or in the IR limit, depending on the direction of the approaching trajectory. Notice that the familiar distinction between UV and IR fixed points is only meaningful when there is a single coupling in the theory.
The β-function of a single coupling is independent of the gauge choice in dimensional regularization. It is regularization scheme-independent up to NLO. If there are several couplings running together, their β-functions depend on the scheme already at the NLO [24]. There is therefore a degree of ambiguity in the position of the fixed points we are going to discuss because their position could be moved by changing the scheme. We assume that these changes are small if the fixed point is found within the perturbative regime. One should however bear in mind this problem of scheme dependence in all the discussions to follow.
In general, there are no conditions on the values of the fixed point g * i and they could take any value. However, when we work in perturbation theory, we have to remain within its range of validity. Therefore, we demand all the coupling at the fixed point g * i to be sufficiently small. In practice this means that going to the next order of the expansion should not change appreciably the position of the fixed point as well as its other properties. We will see that this implies that the numerical value of the fixed points must satisfy the condition in addition to being positive. Notice that in eq. (2.3), and in what follows, the definition of the coupling α follows a convention widely adopted in the AS literature which however differs from the usual one by an additional factor 4π in the denominator.
The condition in eq. (2.3) would suffice to keep the perturbative expansion within its limits of validity if the coefficients in the perturbative expansions were of the same order and not too large. If they are not, the condition in eq. (2.3) should be strengthened and only smaller values allowed to prevent terms of higher order to be more important than those at the lower order.
As we shall see, this is the case for many of the fixed points we discuss. As a matter of fact, many of the fixed points discussed in the literature are due to a cancellation between the first two orders in the perturbative expansion of the βfunctions. This is acceptable only if the higher terms in the perturbative expansion are then more and more suppressed. This is the main motivation for going to threeloop order in the gauge β-functions.

B. Linearized flow
Once we have a candidate fixed point, we can study the flow in its immediate neighborhood. We move away from the fixed point and study what happens when we shift the couplings by a small amount y i ≡ g i − g * i . To this end, we linearize the β-functions as where M ij ≡ ∂β i /∂g j is referred to as the stability matrix. Next, we diagonalize the linear system by going to the variables z i = (S −1 ) ij y j , defined by the equation so that the β-functions and their solutions are in the simplified form From the expression of z i as functions of µ, we see that there are different situations depending on the sign of ϑ i : • For ϑ i > 0, as we increase µ we move away from the fixed point and z i increases without control; the direction z i is said to be irrelevant.
• If ϑ i < 0, as we increase µ we approach the fixed point; the direction z i is called a relevant direction.
• If ϑ i = 0, we do not know the fate of z i and we have to go beyond the linear order as explained below; the direction z i is called marginal in this case.
The notion of relevance/irrelevance is independent of the direction of the flow and of the choice of basis. AS theories correspond to trajectories lying on the surface whose tangent space at the fixed point is spanned by the relevant directions. This tangent space is shown in Figure 1 as a white plane. In the same figure we depict the full UV critical surface in blue. The eigenvalues ϑ i have the property of being universal quantities-meaning that they are invariant under a general transformation in the space of couplings. In perturbation theory, they cannot take any arbitrary value, there are restrictions on their size. We know that in general the β-function for g i has the form where β q i encodes the pure quantum contributions to the β-functions. Therefore, the stability matrix is given by which is equal to the classical scaling exponent plus quantum corrections. Then, the quantity −ϑ i represents the full scaling dimension of the couplingḡ i . If we want to remain in perturbation theory, we should demand that the scaling dimension be small. In the cases we consider, where all the couplings have d i = 0, this means that There is a degree of arbitrariness about where exactly one should set this bound. In our study, we look at the scaling dimensions for the models under examination and set the bound in the first gap in the distribution of their O(1) values.

C. Marginal couplings
If one of the eigenvalues is equal to zero, the linear approximation does not give us information about the RG behaviour in the direction associated to it. Then we have to go further in the expansion [25]. At second order in the couplings y i , the β-functions take the form (2.10) The structure of these quadratic flows is quite complicated to describe in full generality, with the fate of a specific trajectory depending strongly on the position of the initial point in the neighbourhood of the fixed point.
However, marginal couplings do not generally occur for a fully interacting fixed point: they can always be identified with some coupling that is itself zero at the fixed point. We show in Appendix A that the structure of the β-functions is such that the flow of the marginal couplings near the fixed point is of the form (no summation implied). Our flows will be written always in terms of the α i , which are bound to be positive. Therefore, marginal directions y i = α i with P iii < 0 are UV attractive and are called marginally relevant (a well-known example being the QCD gauge coupling) while those with P iii > 0 are UV repulsive and are called marginally irrelevant. Altogether, the UV critical surface is thus spanned by the relevant and marginally relevant directions.

D. Standard Model matching
Once we have an understanding of the fixed point structure-and the conditions on the couplings α i and the scaling exponents ϑ i are satisfied-there remains to find the trajectory connecting a given fixed point to the SM value of the coupling at some IR scale. This is accomplished in the following manner.
First, all the SM couplings are run to a common RG scale, which we take to be 1.83 TeV, using the SM β-functions, to the values This defines the target for the flow to the IR from the UV fixed point. Then, the RG flow is started from a point belonging to the UV critical surface, infinitesimally close to the fixed point (red circle in Figure 1). This guarantees that, to high precision, the flow towards the UV ends at the fixed point. The system is then allowed to flow by means of the full β-functions of the theory towards the IR. The initial point of the flow is varied until, ideally, the trajectory hits exactly the target SM values.
The scale at which one starts the flow, µ 0 , is not known a priori. If one reaches the target values of the couplings after some RG time t = log(µ/µ 0 ) < 0 (in our conventions t increases towards the UV), the corresponding scale µ is identified with 1.83 TeV and the starting scale is identified as µ 0 = µe −t .
For most of the models that we consider, this laborious procedure is not necessary.
For all their fixed points that can be regarded as being in the perturbative domain, the hypercharge is zero at the fixed point and is also a marginally irrelevant coupling.
This means that in order to reach the fixed point in the UV limit, the hypercharge must be zero at all energies. All other trajectories have a Landau pole. These models are thus excluded by a version of the triviality problem.

E. Approximation schemes
The perturbative β-functions of the SM and its extensions have a natural hierarchy originating from the Weyl consistency conditions [26][27][28][29][30]: Other approximation schemes are also possible, for example retaining all βfunctions at the same order or keeping only the gauge β-functions one order higher than the others. These different choices do not satisfy eq. (2.12). They are analysed in [31] where they respective merits (and shortcomings) are discussed. The perturbative study of the β-functions of the SM, together with some of its possible extensions, has been a collective endeavor covering many years. We collect here the main stepping stones in this ongoing computation.
The one-loop (LO) β-function for a non-abelian gauge group was computed in the classic papers [32] and [33] where AF was discovered. The LO β-function for the Yukawa coupling was presented in [34] and that for the quartic Higgs interaction in [35]. The two-loop (NLO) β-functions for the gauge groups have been calculated in [36][37][38][39], those for the Yukawa couplings in [40][41][42] and that for the quartic Higgs interaction in [42][43][44]. The case of the SM has been discussed in [45]. Mistakes in some of these results were corrected in [46,47] where they were also generalized to arbitrary representations of non-simple groups. The three-loop (NNLO) β-functions of a gauge theory with simple groups were given partially in [48], then in [49]. The full NNLO β-functions for the SM were presented in [50] and those for generic representations of non-simple gauge groups in [51]. In this last paper, some contributions from the Yukawa and quartic Higgs interactions were not included. For these terms we have used currently unpublished results of L. Mihaila [52]. The NNLO β-functions for the Yukawa and quartic Higgs couplings were partially computed in [53] and fully in [54,55]. We will not need them here.

F. Another test of perturbativity
Besides the smallness of the couplings themselves, there is another simple test that we use to assess whether a fixed point is in the perturbative domain.
Let us write the β-functions of the gauge couplings α i in the schematic form where A, B and C are the one-, two-and three-loops coefficients. At a fixed point we can split each beta function in the following way where A rs α r * α s * α 2 i * . When we insert the values of the fixed point calculated in the 321 approximation scheme, we expect the three contributions to be ordered as |C Note that in this case ρ + σ = 1. In priciple it might also happen that the sum of the one-and three-loop terms cancel the two-loop term, i.e. σ − ρ = 1. We shall see that this does not happen. Alternatively, if we insert in (2.13) the fixed point values of the 210 approximation scheme, β i will not be zero. The first two terms in (2.14) will cancel, but we still expect ρ i < 1. In the following, when we report results in the 210 approximation scheme, we give the values of ρ i defined at the 210 fixed point and when we report results in the 321 approximation scheme, we give the values of ρ i and σ i defined at the 321 approximation scheme fixed point.

G. Testing fixed points with central charges
At a fixed point the theory is a conformal field theory (CFT). As explained in appendix B, one can estimate the size of the relative changes of the central charges of the CFT to decide whether a fixed point is within the domain of perturbation theory. These relative changes are obtained in terms of the function a = a f ree + a q (a q refers to the contribution of quantum corrections) and of the c-function as If δa or δc become smaller than −1 the fixed point is unphysical because it cannot correspond to a CFT (since c > 0 and a > 0 are guaranteed for CFT). A fixed point for which δc or δa is of order 1 should be discarded as well since quantum corrections are then comparable in size to the free-theory contribution.
The central charges in the 210 approximation scheme can be easily computed by embedding the models in the general gauge-Yukawa Lagrangian of [56]. Computation in the 321 approximation scheme is significantly more complicated due to a major increase in complexity of the Zamolodchikov metric. We do not pursue the 321 computation for that reason and also because the results in the 210 ap-proximation scheme are enough to confirm that our other perturbativity criteria are compatible with the CFT tests.

H. Procedure summary
Given a model, we first look for all the fixed points of the β-functions. Since the β-functions are given in the form of a Taylor expansion, they will have several zeroes that are mere artifacts of the expansion, and we have to select those that have a chance of being physical. The criteria we apply are: stability under radiative corrections and matching to the SM at low energy.
We begin by analyzing the fixed points of the 210 approximation scheme. In the first step, we retain only those fixed points that can be reasonably assumed to be within the perturbative regime, that is, those for which the couplings and the scaling exponents satisfy the bounds in eq. (2.3) and eq. (2.9). We use the criteria discussed in sections II F and II G to confirm that these bounds are indeed reasonable indicators of radiative stability.
We then compare with the results of the same analysis in the 321 approximation scheme. We retain only those fixed points that can be reasonably identified in both approximations. Their number is quite small. We find that the identification is only possible if the couplings and scaling exponents are sufficiently small.
Finally, for the fixed points that are radiatively stable in the sense just described, we look for the possibility of matching to the SM at low energy. If all these conditions are satisfied, we have a fixed point that can be considered as physical. Otherwise, the fixed point should be rejected and deemed unphysical.

III. THE FATE OF THE STANDARD MODEL COUPLINGS
The running of the SM couplings, when extended to high energies, presents two important features: partial gauge coupling unification and a Landau pole in the abelian gauge coupling. Since this singularity appears beyond the Planck scale, where gravitational effects are important, it might well happen that there will be no divergence and that all couplings are well-behaved once we consider a full theory of gravity and matter. Nevertheless, it is interesting to investigate whether such infinities could be avoided within the matter sector. This study will nicely illustrate our procedure by means of the familiar case of the SM.
Throughout this paper, we shall consider a simplified version of the SM where only the top-Yukawa coupling y t is retained. The remaining Yukawa couplings are set to zero. For simplicity we will keep calling this the SM. However, we stress that the degrees of freedom that enter the flow are not only those of the top quark but the full SM matter content (i.e., the number of fermions that enters into the 1-loop coefficient of the gauge β-functions counts all the quarks and leptons).
The first question is whether the β-functions of the SM have fixed points. This does not happen with the LO beta functions. In Figure 2 we show the running of the couplings toward the (quasi)-unification point and the beginning of the ascent of the coupling α 1 toward the Landau pole.

A. The 210 approximation scheme
We then consider the beta functions in the 210 approximation scheme (NLO in the gauge couplings), which are given by where, following the convention (2.3), we use the variables The set of β-functions in eq. (3.1) admits several zeroes. They are given by the last column of Table XII in Appendix C. However, only two of them (solutions P 16 and P 17 ) have all α i positive. Their properties are summarized in Table I.  Although less than 1, the values for the couplings constants α * i are quite sizeable and we may suspect that they may lie outside the perturbative domain. This suspicion is substantiated by looking at the linearized flow. Considering that these exponents are classically zero, we see that the quantum correction are quite large.
The fate of the marginal directions (z 3 and z 4 ) is determined by looking at the quadratic approximation to the flow, as discussed in section II C and in appendix A. We find that the third direction is marginally irrelevant while the last one is marginally relevant.
Even if we had decided to ignore the breaking of the perturbative regime and insisted on looking for trajectories connecting one of the fixed points to the IR regime, the requirement of lying on the UV critical surface would have implied that there is always a coupling that vanishes at all scales. Namely, given that α * 1 = 0, and that the β-function for α 1 is proportional to a power of α 1 itself, this coupling cannot run at all. In other words, the coupling α 1 is frozen at zero at all scales and the U (1) gauge interaction is trivial. Clearly there are no physical fixed point within the SM in the 210 expansion: the problem of the Landau pole is still present even when the gauge couplings are taken at NLO.

B. The 321 approximation scheme
To check the perturbative stability of the two fixed points of the previous section, we now study the β-functions to the next order. In the 321 approximation scheme (NNLO in the gauge couplings), the β-functions take the form [30] β NNLO where the quartic Higgs coupling is no longer unrenormalized.
Due to the higher order of the equations, there are more fixed points than the two found in the 210 approximation scheme. They are listed in Table II. That there is a problem is confirmed by looking at the criteria of perturbativity introduced in section II F. In the 210 approximation scheme, for the two fixed points of Table I, we have B (2) * = 1.87 and B (2) * = 2.46, respectively, while C (2) * = 32.7 and C (2) * = 53.9, respectively. For both fixed points the ratio ρ 2 is of order 10, grossly violating the bound (2.15). It therefore appears that we are outside the domain where perturbation theory can be trusted.
We conclude that the SM (at least in the simplified form considered here) does not have a physical fixed point within perturbation theory. In the next section, we study a family of models that represents the simplest extension to the SM content with the potential of generating new fixed points.

IV. STANDARD MODEL EXTENSIONS
In this section, we consider (minimal) extensions of the SM by adding new matter In eq. (4.1), L SM stands for the SM lagrangian, y is the BSM Yukawa coupling, which we assume to be the same for all fermions, the trace sums over the SM representation indices as well as the flavour indices, and we have decomposed ψ as ψ = ψ L + ψ R with ψ R/L = 1 2 (1±γ 5 )ψ. We neglect the role of quartic self interactions of the scalars S ij as well as portal couplings of the latter to the Higgs sector. Casimir invariants C and Dynkin indices S, which are defined in general as Here, = 0, 1/2, 1, 3/2, . . . denotes the highest weight of R 2 , and (p, q) (with p, q = 0, 1, 2 . . .) the weights of R 3 .
In the 210 approximation scheme, the β-functions are given by [39,41,43,46] β NLO where we have included the gauge and matter contributions in the coefficients B i , Y and N f as follows. For the diagonal and mixing gauge contributions to the gauge β-functions we have , For the Yukawa contribution to the gauge β-functions we have whereas the running of the new coupling α y is characterized by the coefficients  Table XII and can be split in two categories according to whether they depend on the hypercharge Y or not. All the latter have α * 1 = 0.
For the gauge couplings, the β-functions in the 321 approximation scheme, are given, using the variables in eq. (3.2), as follows For the Yukawa and quartic Higgs couplings, the β-functions are given by In eqs.

Colorless vector-like fermions
Colorless vector-like fermions are the least phenomenologically restricted and therefore the most attractive candidates for a successful extension of the SM. In the 210 approximation scheme we find that only the Y -independent set of solutions contains fixed points fulfilling the required conditions (α < 1, |ϑ| < O(1)).
To set the precise bound on |ϑ|, we plot in Figure 3 the largest eigenvalues of the stability matrix. For the Y -independent solutions there is a gap between 2.21 and 62.6; for the Y -dependent solutions there are no eigenvalues less than 9.63.
Accordingly, we decide to consider fixed points with |ϑ| < 3. In this way we probably include some fixed points that are not within perturbation theory,but we prefer to err on this side than to miss potentially interesting fixed points. In this way we discard all the Y -dependent fixed points since there is always an eigenvalue which is at least of order 10. After having applied all the criteria discussed in section II we find that, for any value of the hypercharge Y , the only representations producing satisfactory candidate fixed points are those collected, together with the corresponding eigenvalues, in Table III. The eigenvalues of the stability matrix turn out to be Y -independent as well. We also show in Table III the ratio ρ 2 . As discussed in section II F, this shows how large the three-loop contribution is with respect to the two-loop contribution.
The bounds on N f and come from the behavior of the eigenvalues as functions of these parameters. If we plot one of the eigenvalues as a function of N f for several values of l, we observe that it increases very fast. From Figure 4, we see that only  Table  XII. We show in the last column the ratio ρ 2 defined in Eq. (2.14) knowing that in 210 A  Table III according to the size of their eigenvalues and the ratio ρ.
Now that we have isolated the candidates to study, we check whether these fixed points can be connected to the SM via the RG flow. We find that β 1 is proportional to α 2 1 and so, in order to avoid Landau poles, α 1 has to vanishes at all energy scales. In conclusion, although we have perturbative fixed points, these cannot be matched to the SM because we know that g 1 is different from zero at the TeV scale.
We then perform a similar search in the 321 approximation scheme. Since we see in Table III that the fixed point with |ϑ| > 1 produce a rather large ρ 2 ratio, we stick to solutions having |ϑ| < 1. We find that the same combinations of N f and that provide perturbative fixed points in the 210 case also give viable solutions here. Moreover, the solutions turn out to be Y -independent as well.
In Table IV we show the fixed point solutions satisfying the criteria in eq. (2.3) and eq. (2.9). All the fixed points in Table IV can be traced back to fixed points that were already present in the 210 approximation scheme and listed in Table III. Notice that for a given pair (N f , ), not all the fixed points in 210 persist. For those that do, the values of α * and ϑ change by relatively small amount. We can then claim that the solutions given in Table IV are radiatively stable fixed points. Unfortunately, when we look at trajectories lying on the UV critical surface, we find again that the coupling α 1 must be zero at all scales in all the models. The abelian interactions suffer from the triviality problem and no matching to the SM is possible if asymptotic safety is assumed.
All these colorless models are therefore ruled out.

Vector-like fermions in the fundamental of SU c (3)
For the fundamental representation (p = 1 and q = 0 or vice-versa) we follow the same procedure as before and generate 126,000 models by scanning the same grid in the (N f , , Y ) space. We split the solutions in two families depending on whether they depend on the value of their hypercharge Y or not. The distribution of the largest eigenvalues given in Figure 5 shows that there are no fixed points with |ϑ| < 52.1 for the Y -dependent solutions, whereas for the Y -independent solutions there is a gap between 10.8 and 372. Accordingly, we eliminate all Y -dependent solutions and impose the bound |ϑ| < 11 for those that are Y -independent. In this way, even more than in the preceding section, we include models that are probably unreliable, but these can be eliminated at a later stage. For the Y -independent solutions, we find the combinations of N f and in Tables V and VI that generate satisfactory candidate fixed points. This selection is confirmed by the study of the central charges for these models.
Among the 49 distinct Y -independent fixed points with eigenvalues up to ±10, all have positive c-function, but 6 of them have a negative a-function (with one more being borderline acceptable). The CFT test seems to work well here: all fixed points with reasonable critical exponents pass it, whereas the ones with relatively large exponents do not. An unexpected fact is that the separation between large and small exponents seems to be around a maximum value of |ϑ| around 3. For these perturbative and "semi-perturbative" fixed points, we also notice that the a-function is generically pushed toward 0 (a q < 0) whereas the c function is generically shifted to larger values (c q > 0). This is why the fixed points with negative a-function still seem to pass the c-function test. If one considers δc instead, then for most of these fixed points δc > 1, but apparently not for all. Finally, if one also studies the collider bounds one finds that ten more fixed points are excluded, usually those which just barely satisfied one or both of the a and c tests. The collider bounds tests seem to be the most stringent.
When one tries to match these fixed points to the SM at low energies, it turns out that the abelian gauge coupling α 1 must again be zero at all scales. None of these fixed points is physically viable.
In the 321 approximation scheme, there exist fixed points that can be reasonably traced back to those in the 210 approximation scheme. These solutions are shown in Table VII, where we have included only fixed points with |ϑ| < 1 in order to get small ratios ρ i and σ i . However, they all have at least one coupling that has to be zero at all scales, thus preventing a proper matching to the SM.
We conclude that also all the models with the vector-like fermions in the fundamental representation of SU c (3) cannot provide an AS extension to the SM.  Table XII. The last column gives the values of the ratio ρ for α 2 or α 3 depending on the case (see 2.15).

Vector-like fermions in higher representations of SU c (3)
For the adjoint representation (with p = q = 1), the search over the same grid of values for (N f , , Y ) (and thus 126,000 further models) does not produce any fixed  point within the perturbative domain. This is true both in the 210 and in the 321 approximation scheme.
In Figure 6, we show the distribution the largest eigenvalues of the stability matrix for representative couplings of the fixed points for the 210 approximation scheme. We clearly see that the eigenvalues are rather large. In fact, the minimum eigenvalue in the Y -independent set of solutions is 1342, while in the Y -dependent set is 426.

A model that almost works
Having ruled out all possible candidates, one may wonder if the criteria in (2.3) and (2.9) might be too stringent and make us miss some potentially interesting models. In the case at hand, we can indeed find additional fixed points that naively seem to be good candidates for an asymptotically safe extension of the SM. This is achieved if we allow for larger values of ϑ and relinquish the condition (2.9).
As an example, consider the case with the vector-like fermions in the representations with N f = 3, = 1/2 and Y = 3/2. Its fixed points and eigenvalues are given in Table VIII. We see that some of the scaling exponents ϑ i are large and the criterion (2.9) is accordingly violated. Nonetheless, let us momentarily suspend disbelief and apply the formula in (2.3). We do not find any coupling frozen to zero and therefore a SM matching seems plausible. In fact we find a good matching to the SM couplings in the IR, with an error of the order of per mille, see Figure 7.  Table VIII). This running provides a trajectory in the theory space connecting the fixed point and the physics at a matching scale around 2 TeV.
This model seems to provide (but for the large scaling exponents) a very promising candidate for an AS extension of the SM. Yet it is not radiatively stable-a fact that vindicates the role of criteria in (2.9) as a filter for the physical fixed points. The 321 approximation scheme β-functions generate very different fixed points that cannot be easily traced back to those in the 210 approximation scheme. Moreover, all these fixed points have a trivial coupling and cannot provide a viable extension to the SM.

Five benchmark models studied in the literature
The authors of [15] find that it is possible to generate asymptotically safe extensions to the SM in the subsystem (α 2 , α 3 , α y ) of the couplings. The five benchmark models discussed in [15] (labelled as A, B, C, D and E) are not among those in our scan because they do not include hypercharge, top Yukawa and quartic interaction.
We analysed them separately.
The hypercharge Y can easily be added to these models.   [15] for the 210 approximation scheme.  If we take the fixed points in the 321 approximation scheme at their face value and try to match them to the SM, we always encounter a coupling, α 2 in almost all the cases (see Table X), that is frozen to its vanishing value: the theory is trivial in the coupling α 2 and it cannot be matched to the SM. In other words, the benchmark models in [15] suffer from the same pathology of the models in our scan. Unlike those models, in this case it is a non-abelian coupling that is trivial. This is worrisome and should be born in mind if one were to entertain the idea of embedding U Y (1) in a non-abelian group in order to find an AS extension of the SM. It is difficult to compare these results to ours because of the non-perturbative nature of the resummation procedure. In our approach these models are all ruled out because of the larger scaling dimensions we expect for the given large values of N f (see Figure 4 and the discussion in section IV). The fixed points obtained in the approach of [16] and [17] are probably linked to essential singularities in the complete resummation [58] and assume no mixing between the two families. Since many of the BSM extensions attempt to describe dark matter, we take one of the possible minimal models discussed in [59] and identify some of the vector-like fermions with dark matter. This exercise makes clear the potential relevance of AS in selecting physics BSM.
We take N f 2 vector-like fermions with quantum numbers p = q = 0, = 2 and Y = 0. That is, we take colorless quintuplets with no hypercharge. Within the 210 approximation scheme, for the combination (1, 2, 3/2) ⊕ (1, 5, 0), we realize that fixed points split in two categories: fixed points that depend on the number of quintuplets N f 2 and fixed points that do not. Clearly, the latter have α y = 0 so that the vector-like fermions enter only via loops in the gauge β-functions. Consequently, the conditions to lie on the critical surface of those fixed points imply that α 2 = 0.
This feature makes the corresponding fixed points uninteresting.
For the remaining fixed points, we find that in order to have α i < 1 for all Nevertheless, we can find a matching with the SM. The only difference with respect to the model in section IV B 4 is that, in the present case, two matching scales are needed-the reason being that the large number of quintuplets makes α 2 decrease fast so that these fields must be decoupled at very high energies. In We conclude that it is not possible, at least with the 378,000 models we have examined, to extend the SM up to arbitrarily high energies in perturbation theory.
The same happens with the models discussed in section IV B 4 and IV B 7, and the five models proposed as benchmarks in [15]: the 321 approximation scheme βfunctions generate very different fixed points that cannot be easily traced back to those in the 210 approximation scheme. All the fixed points in 321 approximation scheme have at least one trivial coupling.
Our search for AS extensions of the SM has returned a negative answer and no viable candidate. This result might indicate that the search must be enlarged to include models with BSM fields more complicated than vector-like fermions. Because the vector-like fermions are actually just a proxy for more general matter fields, this does not seem too promising a line of enquiry. Another possibility is that the SM gauge groups must be embedded in larger groups before AS becomes manifest. All in all, the most plausible scenario seems that in which the fixed points making the SM AS, if they exist at all, lay outside the perturbative regime and accordingly are inherently invisible to our approach. Such a non-perturbative computation has been partially explored with negative results in [17]. Over and above that, we should always bear in mind the possibility that there exists no AS extension to the SM or, if it exists, that it requires the inclusion of gravity [61][62][63].

ACKNOWLEDGMENTS
We are very grateful to Luminita Mihaila for giving us access to her results on the 3-loop β-functions for the gauge couplings in non-simple groups prior to their publication. We also thank Daniel Litim and Francesco Sannino for useful discussions.

Appendix A: Analysis of marginal couplings
Here we prove the statement, made in Section II C, that when the marginal couplings are associated to vanishing gauge couplings, the behavior of the flow at quadratic order is determined by the coefficients P iii .
The general form of the gauge β-functions is where A i , B i r and C i rs represent the one, two and three-loops coefficients. Their contribution to the stability matrix is given by We see that if α * i = 0, the row i will have zeros in all the entries. This does not happen for the Yukawa interactions, whose NLO β-functions have the form β Y i = (D i r α r + F i rs α r α s )α i . Then, the contribution to the stability matrix reads where we see that if α * i = 0, the last piece will be in general different from zero. Consequently, we do not have a row of zeros. The fact of having rows of zeros implies that detM = 0. Thus, the matrix M is singular and there exist vectors x such that Ax = 0x. As a result, M has the eigenvalue λ = 0 with multiplicity given by the number of zero rows.
Suppose we have a fixed point with two gauge couplings equal to zero. Then the stability matrix will have two zero rows, that we can assume to be the last two.
This implies that the n − 2 eigenvectors corresponding to λ i = 0 have the form . . , V i n−2 , 0, 0]. The eigenvectors for λ = 0 lie in a 2-dimensional plane. There is a freedom in choosing these vectors, and we can take them to have the form  This implies that a n−2,1 a n−2,2 . . . a n−2,n−2 a n−2,n−1 a n−2,n 0 0 . . .
where we have labelled a i,j the non-zero entries and we have called b = 1/V n−1 n−1 , c = 1/V n n . Now, when we compute the form of the new variables z i = S −1 ij y j = S −1 ij (α j − α * j ), we observe that two of the new coordinates are just proportional to the asymptotically free variables, namely z n−1 = b·y n−1 = b·α n−1 , z n = c·y n = c·α n . This result has an important effect in the analysis. For the gauge β-functions, which in the case of the AF couplings reduces to We conclude that in order to know if a marginal coupling is relevant or irrelevant we need only check the sign of P iii . If P iii < 0, the coupling is marginally relevant.
If P iii > 0, the coupling is marginally irrelevant.
so-called collider bounds [64], namely In perturbation theory, central charges are expanded in series a =ã f ree +ã (1) (4π) 2 +ã There is a correlation between critical exponents and the change in central charges, which for the a-function can be explained as follows. At the fixed point we have,ã * = a * = a f ree + 1 4 where i runs over simple gauge groups, b 1 = B 1 , b 2 = −B 2 , b 3 = −B 3 are the oneloop coefficients of the gauge beta functions, and χ g i g i and A i are components of the Zamolodchikov metric, see [56]. One-loop critical exponent follows from β i = ±B i α 2 i (+ for the group U (1), − otherwise), and reads θ 1L = 2b i α * i . Then, which explains the correlation. α *