Discovering and quantifying nontrivial fixed points in multifield models
Abstract
We use the functional renormalization group and the \(\epsilon \)expansion concertedly to explore multicritical universality classes for coupled \(\bigoplus _i O(N_i)\) vectorfield models in three Euclidean dimensions. Exploiting the complementary strengths of these two methods we show how to make progress in theories with large numbers of interactions, and a large number of possible symmetrybreaking patterns. For the three and fourfield models we find a new fixed point that arises from the mutual interaction between different field sectors, and we establish the absence of infraredstable fixedpoint solutions for the regime of small \(N_i\). Moreover, we explore these systems as toy models for theories that are both asymptotically safe and infrared complete. In particular, we show that these models exhibit complete renormalization group trajectories that begin and end at nontrivial fixed points.
Keywords
Renormalization Group Universality Class Asymptotic Safety Functional Renormalization Group Symmetry Enhancement1 Introduction
The O(N) Wilson–Fisher fixed point appears in a large variety of systems where it controls the universal critical behavior in the infrared (IR) scaling regime [1, 2, 3, 4, 5]. Generalizations of this universality class appear in the context of coupledfield models, e.g., for the \(O(N_1) \oplus O(N_2)\) twofield model [6, 7, 8, 9, 10, 11]. Depending on the number of field components \(N_i\) and dimension d, one finds that different fixed points (FP) govern the IR behavior of the model. Two of these, the decoupled (DFP) and isotropic fixed point (IFP), can be deduced from the existence of the Wilson–Fisher fixed point. While the DFP is characterized by a complete decoupling of the fields and therefore can effectively be regarded as a model for two independent vector fields, the IFP displays a complete symmetry enhancement to an \(O(N_1 + N_2)\) rotational symmetry. However, the twofield model features another socalled biconical fixed point (BFP), which emerges due to the nontrivial interactions between the two field sectors. The BFP is fully coupled, i.e., mixed interactions are nonvanishing, but it does not show an enhanced symmetry, as the IFP does; see, e.g., Refs. [10, 11, 12]. Interestingly, it turns out that the \(O(N_1)\oplus O(N_2)\) model in \(d = 3\) dimensions exhibits exactly one IRstable fixed point (with no more than two relevant directions) for any pair of values \(N_1\) and \(N_2\). In this work, we address the question whether generalizations of the twofield model to the case of three and four fields allow for further unprecedented fixedpoint solutions that are relevant for the IR scaling behavior of the respective model. Moreover, we look for additional confirmation of our previous study of \(n = 3\) fields in three dimensions [13], where no stable fixed point (with no more than three relevant directions) was found for small values of \(N_i\) – in contrast to the twofield model.^{1} In our previous study of this system [13], we searched for fixed points using the nonperturbative functional renormalization group (RG) [14, 15, 16, 17, 18, 19]. Within this scheme, the \(\beta \)functions are nonpolynomial functions of the couplings and it is therefore challenging to make sure that numerical fixedpoint searches do indeed uncover all stable fixed points of the system. To address this problem, we match the solutions of the renormalization group \(\beta \)functions derived within the framework of the functional RG to those obtained with the Wilsonian momentumshell RG by employing an expansion in \(\epsilon = 4d\). The \(\epsilon \)expansion features \(\beta \)functions that are polynomials of the couplings. A comprehensive study of all fixed points that are continuously connected to the Gaussian FP at \(d = 4\), is therefore straightforward. On the other hand, the full nonlinear \(\beta \)functions of the functional RG yield reasonable estimates of the stability of fixed points already at low orders in the approximation. Indeed, comparing fixed points in both RG schemes, we can convincingly identify stable fixed points and determine their stability regions in the space spanned by the values of the field components \(N_i\) in arbitrary dimensions.
More recently, interacting fixed points have become an active field of research in fourdimensional models, which are explored in the context of an ultraviolet (UV) completion for gravity [20, 21] as well as QFTs including matter fields [22, 23, 24]. In this setting, an interacting fixed point provides a welldefined microscopic starting point from which a fundamental quantum field theory (QFT) valid on all scales, can be defined. Here, we add another example to the collection of toy models for asymptotic safety that apply to QFTs in low dimensions (see, e.g., Refs. [25, 26]). In our example, we focus on the question how both the UV and the IR limit of the RG trajectory are determined by interacting fixed points with different degrees of symmetry.
2 Effective action functional for multifield models
The \(\beta \)functions to oneloop order in the \(\epsilon \)expansion may also be obtained from the nonperturbative RG flow equation. This is achieved by employing an expansion around the upper critical dimension and restricting the functional space to those operators that appear in the bare action.^{3} In the case of two fields these \(\beta \)functions agree with those given in Ref. [11], as expected by oneloop universality. In the following, we consider a model in \(d = 3\) dimensions with three different field degrees of freedom, \(\phi _1, \phi _2\), and \(\phi _3\), with \(N_1\), \(N_2\), and \(N_3\) field components, respectively, and compare our results [13] explicitly with the Wilsonian momentumshell RG to oneloop order in the \(\epsilon \)expansion. As outlined in Sect. 1, our main goal is to complement the functional RG with the \(\epsilon \)expansion to identify and characterize all possible multicritical scaling solutions relevant in the IR scaling regime. A similar strategy was also chosen in Ref. [27], where the Polchinski version of the nonperturbative RG [28] was contrasted to the \(\epsilon \)expansion to investigate multicritical points for a scalar theory with a single order parameter.
3 RG fixed points to \(\mathcal {O}(\epsilon )\) in the threefield model
Let us briefly highlight the difference of the above \(\beta \)functions to those derived within the functional RG approach [13]: In the latter case higherorder interactions (generated by the RG flow toward the IR) are explicitly taken into account. Therefore, the \(\beta \)functions for the quartic couplings receive contributions from higherorder couplings (their scaledependence being characterized by additional \(\beta \)functions that are determined within this approach). Moreover, the functional RG represents a massive renormalization scheme, and accordingly, mass parameters explicitly enter all \(\beta \)functions, resulting in their nonpolynomial structure. Details can be found in Refs. [12, 13].

Isotropic fixed point (IFP), featuring a symmetry enhancement to an \(O(N_1+N_2+N_3)\) symmetry and consequently has coordinates \(\lambda _{200} = \lambda _{020} = \lambda _{002} = \lambda _{110} = \lambda _{101} = \lambda _{011}\);

Decoupled fixed point (DFP) is characterized by a complete decoupling of all sectors, i.e., \(\lambda _{110} = \lambda _{101} = \lambda _{011} = 0\);

Decoupled isotropic fixed points (DIFP) are characterized by a partial decoupling of the sectors as well as partial symmetry enhancement. One representative in this class is given by \(\lambda _{002} = \lambda _{020} = \lambda _{011}\), while \(\lambda _{101} = \lambda _{110} = 0\), and features an enhanced \(O(N_2+N_3)\) symmetry;

Decoupled biconical fixed points (DBFP) are partially decoupled but feature no symmetry enhancement, e.g., we might have \(\lambda _{101} = \lambda _{110} = 0\) and \(\lambda _{011} \ne \lambda _{020}\).
As we decrease the number of field components \(N_3\) and pass through the point \(N_1 = 1\), \(N_2 = N_3 = 8\) (\(\epsilon = 1\)), the fully coupled FP ceases to be IR stable – the scaling exponent \(\theta _4\) becomes nonnegative as the value of \(N_3\) is lowered
\(N_1\)  \(N_2\)  \(N_3\)  \(\lambda _{200}\)  \(\lambda _{020}\)  \(\lambda _{002}\)  \(\lambda _{110}\)  \(\lambda _{101}\)  \(\lambda _{011}\) 

1  8  8  0.105  0.062  0.062  0.020  0.020  \(\)0.002 
1  8  9  0.109  0.062  0.059  0.013  0.008  \(4\times 10^{4}\) 
1  8  10  0.110  0.063  0.056  0.012  0.002  \(7 \times 10^{5}\) 
\(N_1\)  \(N_2\)  \(N_3\)  \(\theta _4\)  \(\theta _5\)  \(\theta _6\)  \(\theta _7\)  \(\theta _8\)  \(\theta _9\) 
1  8  8  0  \(\)0.096  \(\)0.255  \(\)0.996  \(\)0.996  \(\)1.000 
1  8  9  \(\)0.019  \(\)0.046  \(\)0.274  \(\)0.977  \(\)0.999  \(\)1.000 
1  8  10  \(\)0.007  \(\)0.037  \(\)0.294  \(\)0.985  \(\)1.000  \(\)1.000 
Within the \(\epsilon \)expansion, we confirm our previous finding that the threefield models in \(d = 3\) dimensions exhibit regions in the space of field components \(N_i\) where no IRstable FP exists; cf. Fig. 1. Specifically, this implies that particular threefield models with a given set of \((N_1, N_2, N_3)\) do not feature multicritical behavior without additional fine tuning. A similar absence of IRstable multicritical FPs was observed in Ref. [29] where the effect of competing order was investigated on fermionic quantum criticality (see also Ref. [30]).
3.1 Fully coupled FPs
To uncover additional IRstable FPs, we inspect the scaling solutions as a function of the parameters \(N_i\); see Fig. 1.
3.1.1 Asymmetrically coupled FP
Our main result is the discovery of a new FP, which is completely coupled, i.e., \(\lambda _{101} \ne 0\), \(\lambda _{110}\ne 0\), and \(\lambda _{011} \ne 0\), but which does not feature any symmetry enhancement. In the following, we will refer to this scaling solution as the asymmetrically coupled fixed point (ACFP). It defines a genuine new universality class that cannot be obtained as a generalization of the Wilson–Fisher FP, and occurs for the first time in the threefield model. This new universality class relies crucially on the presence of three competing orders, and cannot occur in systems with a smaller number of order parameters.
To illustrate its properties, we give the corresponding values of the dimensionless, renormalized couplings \(\lambda _{m_1 m_2 m_3}\) and the critical exponents \(\theta _4 , \theta _5, \ldots \) at selected points in the \((N_2, N_3)\)plane, for \(N_1 = 1\); see Table 1. Mass parameters do not appear in the \(\beta \)functions to the given order of the \(\epsilon \)expansion. Therefore, the three relevant scaling exponents \(\theta _1\), \(\theta _2\), and \(\theta _3\) are not provided in the following.
We follow the fully coupled asymmetric FP along the \(N\equiv N_2 = N_3\) direction, where we expect that it should collide with the DFP at some critical value of N; cf. Fig. 2. In fact, we find that the two FPs exchange their stability properties at (\(N = 10\), \(N_1 = 1\)). That is, at the collision point the exponent that decides about the stability properties of the scaling solution, \(\theta _4\), changes its sign for each of the two solutions. If we attempt to continue the asymmetric FP to smaller values of N, we observe that it disappears into the complex plane at \(N = 8\), \(N_1 = 1\), together with another fully coupled FP which is always unstable – both FPs become inaccessible for small values of \(N_i\). From these results one might conclude that the fully coupled asymmetric FP will not be of any significance experimentally: The oneloop \(\epsilon \)expansion seems to suggest that there is a threshold value \(N_{i} \simeq 5\), for all \(i = 1,2,3\), below which the ACFP disappears completely (cf. Fig. 1). We show in Sect. 5 that the functional RG provides a quantitatively more reliable estimate for the critical values of \(N_{i}\).
3.1.2 Generalized BFP and regions without IR stable FP
In general, the space of renormalized couplings \(\lambda _{m_{_1} \,\ldots \, m_n}\) features closed subspaces that are characterized by enhanced symmetries and the decoupling phenomenon: Whenever one of the sectors decouples, and the couplings between sectors vanish, fluctuations cannot regenerate the mixed couplings, and therefore the RG flow stays within that space, making it an RGinvariant subspace. With the discovery of the new asymmetrically coupled FP we may complete this picture in the following way: We may state that each of these subspaces (excluding its symmetryenhanced or decoupled subspaces) contains at least one FP. In fact, from our analysis we find that almost all of these subspaces will feature an IRstable FP for a particular set of values \(N_i\), with one notable exception, the BIFP, cf. Appendix 1. This scaling solution is associated to the partially symmetryenhanced subspace and is nowhere stable.
While in principle such a universality class exists, it would require a higher degree of fine tuning to reach it. Thus, the associated pattern of symmetrybreaking is not expected to be relevant experimentally. The BIFP would be a natural candidate to take over stability from the IFP as soon as it becomes unstable, just as the BFP takes over stability from the IFP in the twofield model. The additional relevant directions of the BIFP prevent this scenario from being realized, and imply that the threefield case features a region in the space of the \(N_i\) that is devoid of stable FPs.
3.2 Multifield theories as toy models for asymptotic safety and IRcompleteness
In order for a QFT to provide a viable description of a set of degrees of freedom and their interactions on all scales, i.e., in order for the theory to be fundamental, it must reach a renormalization group FP in the UV and IR, respectively.^{6} Here, we provide a set of models that feature a large number of complete trajectories that run into nontrivial FPs both in the UV and IR.
In this context, it is important to realize that in principle a given FP can be reached asymptotically in either one of the two limits, if it features at least one critical exponent that differs in sign from the others. If a FP should be reached in the UV, all irrelevant couplings need to be tuned in such a way that the RG trajectory lies within the UVcritical hypersurface of the FP. In the context of highenergy physics, this implies that the values of all irrelevant couplings correspond to predictions of the model, i.e., for the model to be asymptotically safe, there is exactly one possible value for each irrelevant coupling. On the other hand, if the FP is reached in the IR, the renormalization group flow is automatically drawn toward it along the irrelevant directions, and it is the relevant directions that require tuning.
Stable, fully coupled FP in the fourfield case. There are four relevant directions; the corresponding exponents \(\theta _1\), ..., \(\theta _4\) are not provided
\(N_1\)  \(N_2\)  \(N_3\)  \(N_4\)  \(\lambda _{2000}\)  \(\lambda _{0200}\)  \(\lambda _{0020}\)  \(\lambda _{0002}\)  \(\lambda _{1100}\)  \(\lambda _{1010}\)  \(\lambda _{0110}\)  \(\lambda _{1001}\)  \(\lambda _{0101}\)  \(\lambda _{0011}\) 

1  9  9  9  0.110  0.059  0.059  0.059  0.007  0.007  \(1\times 10^{4}\)  0.007  \(1\times 10^{4}\)  \(1\times 10^{4}\) 
1  10  9  9  0.110  0.056  0.059  0.059  0.001  0.006  \(2\times 10^{5}\)  0.006  \(2\times 10^{5}\)  \(1\times 10^{4}\) 
1  10  9  8  0.109  0.055  0.059  0.062  0.003  0.008  \(8\times 10^{5}\)  0.013  \(1\times 10^{5}\)  \(3\times 10^{4}\) 
\(N_1\)  \(N_2\)  \(N_3\)  \(N_4\)  \(\theta _5\)  \(\theta _6\)  \(\theta _7\)  \(\theta _8\)  \(\theta _9\)  \(\theta _{10}\)  \(\theta _{11}\)  \(\theta _{12}\)  \(\theta _{13}\)  \(\theta _{14}\) 
1  9  9  9  \(\)0.012  \(\)0.028  \(\)0.028  \(\)0.295  \(\)0.295  \(\)0.295  \(\)0.986  \(\)1.000  \(\)1.000  \(\)1.000 
1  10  9  9  \(\)0.004  \(\)0.016  \(\)0.024  \(\)0.295  \(\)0.314  \(\)0.314  \(\)0.993  \(\)1.000  \(\)1.000  \(\)1.000 
1  10  9  8  \(\)0.009  \(\)0.022  \(\)0.046  \(\)0.294  \(\)0.306  \(\)0.314  \(\)0.975  \(\)0.999  \(\)1.000  \(\)1.000 
In particular, we will focus on two examples: The first involving the ACFP as a UV fixed point, thus defining a toy model for an asymptotically safe model. Here we pick \(N_1=1,N_2=N_3=11\), where the ACFP has one IRrelevant direction; this triggers a flow to the DFP in the IR. As a second example, we consider a region of \(N_i\) where, e.g., the IFP is stable (with three relevant directions) it is a natural candidate FP for RG trajectories in the IR; see Fig. 3. Thus, a model that has been rendered asymptotically safe, e.g., by defining it at the DFP, can only be infrared complete when at least three directions are tuned and it is the symmetryenhanced IFP which provides the lowest number of relevant directions. In this case, IR FPs with a lower degree of symmetry will typically require a higher degree of fine tuning.
4 Fourfield model to oneloop order in the \(\epsilon \)expansion
We proceed in an analogous manner for the fourfield model. Our main goal here is to confirm that the two novel features of the class of \(O(N_1)\oplus O(N_2)\oplus O(N_3)\)field models – the possible existence of theories without an IRstable FP and the existence of a new fully coupled FP – carry over to the case of larger numbers of fields. The \(\beta \)functions are given by the obvious generalization of Eqs. (5 6 7 8 9)–(10) to the case where one additional field degree of freedom with \(O(N_4)\) symmetry is added. Determining the zeros of the beta functions, we find that a new FP which is fully coupled and does not feature any symmetry enhancement, exists and is stable at selected points in the space of the \(N_i\), i.e., a FP that appears for the first time in the fourfield case similar to the role of the ACFP in the threefield case, cf. Table 2. In this context, stability is of course defined as the existence of no more than four relevant directions.
The new FP collides with the DFP at \(N_i=10\) and becomes unstable. Moreover, our results indicate that no FP is stable, e.g., at the point \(N_1=1\), \(N_2=N_3=N_4=8\). Together with the results in Table 2, this suggests that a structurally similar picture to the threefield case carries over to the fourfield case: The IFP will be stable for very small values of the \(N_i\), before it is destabilized. Keeping \(N_1=1\) fixed and increasing \(N_2=N_3=N_4\), we pass through a regime without a stable FP, i.e., with nonuniversal behavior only. At \(N_i=9\), the new FP then appears from the complex plane, and is stable until it collides with the DFP, that takes over stability for all larger values of the \(N_i\). Based on our findings in the threefield model, we expect that our \(\mathcal {O}(\epsilon )\) estimates for the \(N_i\), at which FPs are stable, are considerably larger than the correct values. As can be tested within, e.g., the LPA 4, the functional RG is more reliable when it comes to quantitative estimates.
Based on our findings in the three and fourfield case, we therefore conjecture that models with larger numbers of competing orders will not feature multicritical behavior without additional fine tuning.
5 Results from the functional RG
Employing an LPA truncation to fourth order in the fields and including the scaledependence of the renormalization factors, i.e., \(\eta _i \ne 0\) (which we will refer to as LPA \(4 + \eta \) in the following), we confirm the qualitative behavior of the \(\epsilon \)expansion: Fixing \(N_1 = 1\) and increasing \(N \equiv N_2=N_3\), the IFP becomes unstable around \(N \simeq 1.85\). For larger values of N there is no IRstable FP, until the new ACFP appears and becomes stable; cf. Fig. 4. In contrast to the \(\epsilon \)expansion, this already happens at \(N \simeq 2.8\). Finally, at \(N \simeq 3\) the asymmetrically coupled FP exchanges its stability with the DFP, which remains IR stable for all \(N \gtrsim 3\). The mechanism by which the new ACFP appears is completely analogous to the situation observed in the oneloop \(\epsilon \)expansion: It appears from the complex plane together with another FP and immediately takes over stability. Note, however, that the region of values N where the ACFP is stable is shifted to significantly smaller values of N bringing it into the reach of physically interesting models. Thus, the asymmetrically coupled FP might actually be of interest for efforts to establish the phase diagram of strongly correlated manybody systems either experimentally or via lattice Monte Carlo techniques, for an overview, see, e.g., Ref. [34]. Using scaling relations to estimate the stability regime for the DFP, we find that the LPA \(4 + \eta \) slightly overestimates the width of the region where the ACFP is stable. We expect that the region where the ACFP is stable becomes even smaller at higher orders of the LPA. In fact, this might account for the fact that it was not discovered in our previous analysis [13] based on a LPA to eighth order in the fields. We generically expect that a truncation of eighth will be sufficient to provide quantitatively reasonable estimates for the critical exponents.
Our present results clearly highlight the strength of a combination of the functional RG with the \(\epsilon \)expansion: The latter allows us to compile a complete list of all FPs that can be continuously connected to the Gaussian FP at \(d = 4\), whereas the former provides us with a quantitatively more reliable estimate of the stability regions of the different FPs. Combined, these methods allow us to arrive at a complete picture of stable FPs in the space of the \(N_i\) while minimizing the computational effort.
6 Conclusions
With this study we identify a new fully coupled FP in the \(d = 3\) dimensional three and fourfield models. While we find that this FP is indeed IR stable for some values of \(N_i\), it does not lie at real fixedpoint values for the couplings at other values of the \(N_i\). This is in stark contrast to the O(N) theory or the class of \(O(N_1)\oplus O(N_2)\) models [6, 7, 8, 9, 10, 11] where the relevant scaling solution(s) are either IR stable or can be reached via additional fine tuning. In addition to identifying a new FP, we confirm our previous finding [13] that for certain multifield models there is no IRstable multicritical scaling solution. This behavior is directly tied to the properties of the new fully coupled fixed points, the ACFP and the BIFP. Thus, this study has further clarified the reason for the absence of multicritical scaling solutions: While in the twofield model [6, 7, 8, 9, 10, 11] different FPs exchange stability only through a collision of two fixed points at real values of \(N_i\), the three and higherfield models feature the additional possibility that FPs emerge from the complex plane.
Our study plays out the strengths of two methods: The \(\epsilon \)expansion allows for a straightforward identification of all FPs that can be continuously connected to the Gaussian FP in \(d = 4\) dimensions, as the \(\beta \)functions are polynomial in the couplings. In contrast, the fixedpoint search is more involved with the functional RG due to the nonpolynomial nature of the \(\beta \)functions. However, the functional RG provides better quantitative results already at low orders of the LPA. This can be seen clearly for the example of the DFP. To estimate its stability regime, we may apply an exact scaling relation [8, 9, 35, 36, 37] to determine the exponent \(\theta _4\) from critical exponents of the O(N) Wilson–Fisher FP [1, 2, 3, 4, 5]. By doing so, we find that the result from the LPA at fourth order in the fields provides a quantitatively more reliable estimate for the scaling exponent than the \(\epsilon \)expansion at oneloop order. Taken together, the two methods thus allow for an efficient identification of all existing FPs, using the \(\epsilon \)expansion at low orders, followed by a leading order determination of the stability regimes and critical exponents with the functional RG.
The identification of distinct interacting FPs in three and fourfield models also allows us to explore RG trajectories that define both UV and IRcomplete QFTs in \(2 < d < 4\) dimensions. In general, multifield theories provide a large number of such trajectories and typically feature two distinct regimes when it comes to the question of symmetry enhancement in the IR: For values of \(N_i\) where a symmetryenhanced FP is IR stable, all other FPs require a higher degree of fine tuning to reach them in the IR. Thus, IR symmetry enhancement appears as a “natural” possibility that requires the least amount of fine tuning. In contrast, for other values of \(N_i\), the same symmetryenhanced FP will feature additional relevant directions, giving rise to the familiar notion that an enhancement of symmetry typically requires additional fine tuning.
Our findings might have implications for possible UV completions of coupled scalar models in \(d = 4\) dimensions. We observe that the asymmetrically coupled FP can be found in the \(\epsilon \)expansion, i.e., it emerges from the Gaussian FP at \(d < 4\). Thus we conclude that no nontrivial FP exists for these models in \(d=4\), unless it lies within a strongly nonperturbative regime at very large values of the couplings. This implies that, e.g., inflationary models with several scalar fields are not UV complete, but instead they most probably feature Landau poles at finite scales. Interestingly, a coupling to gravity could facilitate a UV completion in the context of asymptotically safe models. Studies suggest that a gravitational FP persists when the effects of several minimally coupled scalars are taken into account [38]. It is of course interesting to understand whether a similar statement applies to interacting matter models. In particular, the new universality classes that we discuss in this paper and which are inherent to n field models (\(n\ge 2\)) could potentially survive an extension to 4 dimensions, when gravitational effects are added, as these generically seem to shift Gaussian FPs to interacting FPs [39]. Thus gravity might extend the upper critical dimension for this interacting FP to \(d>4\). Following the methods discussed in [40, 41, 42], an assessment of this scenario could be possible. In the context of scalar darkmatter models, where the coupling to other matter fields is less relevant, the existence of such scalargravity FPs could provide a predictive UV completion.
Footnotes
 1.
 2.
Here, we include terms to \(\mathcal {O}(\partial ^2)\) and neglect a possible field dependence of the scaledependent renormalization factors \(Z_{i}\) (that are evaluated at the minimum of the effective potential U). Terms of the type \(\sim \big (\partial \phi _i^2\big )^2\), which in principle contribute at the same order in the derivative expansion, are not taken into account.
 3.
By virtue of oneloop universality, the \(\mathcal {O}(\epsilon )\)expanded functional RG \(\beta \)functions are exact and independent of the chosen (nonperturbative) regulator.
 4.We introduce the following notation for the \(\beta \)functions:which are expressed in terms of the rescaled couplings$$\begin{aligned} \beta _{m_1 \cdots \, m_n} \equiv k \frac{\partial \lambda _{m_1 \cdots \, m_n}}{\partial k} , \end{aligned}$$(3)where \(K_d = \left[ (4\pi )^{d/2} \Gamma (d/2 + 1) / 2 \right] ^{1}\) and the RG scale is given by \(k = e^{s} \Lambda \), \(\infty < s \le 0\).$$\begin{aligned} \lambda _{m_1 \cdots m_n} \rightarrow K_d k^{d} \big (\prod _{i=1}^n Z_i^{m_i} k^{(d2)m_i}\big ) \lambda _{m_1 \cdots m_n}, \end{aligned}$$(4)
 5.
To comply with the notation introduced in [13], we will assume that the eigenvalues are labeled in descending order, i.e., \(\theta _1 \ge \theta _2 \ge \ldots \,\), while \(\theta _{\mu } > 0\), for \(1\le \mu \le n\), corresponding to masslike perturbations (where \(n = 3\) for the threefield model).
 6.
In principle, more exotic scenarios as, e.g., limit cycles, might also be viable.
Notes
Acknowledgments
A.E. acknowledges support by the Perimeter Institute for Theoretical Physics through an Emmy Noether fellowship during the initial stages of this project. The work of A.E. is supported by an Imperial College Junior Research Fellowship. D.M. is supported by the European Research Council under the European Unions Seventh Framework Programme (FP7/20072013) / ERC Grant Agreement 339220. M.M.S. is supported by the Grant ERCAdG290623.
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