Asymptotic safety guaranteed for strongly coupled gauge theories

We demonstrate that interacting ultraviolet fixed points in four dimensions exist at strong coupling, and away from large-$N$ Veneziano limits. This is established exemplarily for semi-simple supersymmetric gauge theories with chiral matter and superpotential interactions by using the renormalisation group and exact methods from supersymmetry. We determine the entire superconformal window of ultraviolet fixed points as a function of field multiplicities. Results are in accord with the $a$-theorem, bounds on conformal charges, Seiberg duality, and unitary. We also find manifolds of Leigh-Strassler models exhibiting lines of infrared fixed points. At weak coupling, findings are confirmed using perturbation theory up to three loop. Benchmark models with low field multiplicities are provided including examples with Standard~Model-like gauge sectors. Implications for particle physics, model building, and conformal field theory are indicated.


I. INTRODUCTION
Ultraviolet fixed points are key for the renormalizability and predictive power of quantum field theories. The classic example is given by asymptotic freedom of the strong nuclear force, where the fixed point is noninteracting [1,2]. The possibility of interacting ultraviolet (UV) fixed points, often denoted as asymptotic safety [3], has been conjectured early on [4]. Recently, this field has taken up some speed due to the discovery of UV conformal fixed points in models of particle physics [5][6][7][8][9][10]. Conditions under which asymptotic safety arises in weakly coupled four-dimensional quantum field theories (without gravity) are by now well understood: Non-Abelian gauge fields are central [6], alongside Yukawa and scalar interactions and subject to a stable vacuum [7,8]. Templates with strict perturbative control have been found for unitary [5], orthogonal and symplectic [9], or product gauge groups [10], and supersymmetry [11]. This has also triggered new ideas for model building [12,13], asymptotically safe extensions of the Standard Model explaining the electron and muon g − 2 anomalies [14][15][16] whose new type of flavor phenomenology can be tested at colliders [17], and explanations of flavor anomalies as evidenced in rare B-meson decays [18]. Further results cover vacuum stability [19] including the Higgs [14][15][16]18], Abelian factors [13], global fixed points [20], aspects of radiative symmetry breaking [21], UV conformal windows [22], and fixed point mergers [23].
Despite the vast body of weakly coupled fixed points at hand, it has remained an open challenge to understand asymptotic safety of strongly coupled 4D quantum field theories from first principles. It would be desirable to have rigorous and explicit examples at hand, if only as a proof of principle, and to clarify whether large matter field anomalous dimensions or new phenomena may become an obstacle for safety in the UV. Moreover, it is well known that weakly coupled fixed points often require a large number of gauge and matter fields, such as in a Veneziano large-N limit, while decreasing the number of matter fields turns fixed point interactions stronger. In the context of asymptotically safe model building where only finitely many new matter fields are added to the Standard Model [12][13][14][15][16][17][18], it becomes paramount to control the size of UV conformal windows nonperturbatively and to understand how few matter fields can sustain an underlying fixed point [22,23]. It would be equally important to understand whether or not qualitatively new types of fixed points arise at strong coupling, beyond those discovered at weak coupling.
In this spirit, we put forward a nonperturbative search for interacting UV fixed points in conventional 4D quantum field theories, without gravity. With asymptotically safe fixed points being presently out of reach for lattice simulations or the conformal bootstrap, we turn, instead, to N ¼ 1 global supersymmetry as the nonperturbative tool of choice: Nonrenormalization theorems ensure that superpotential (Yukawa) couplings are renormalized nonperturbatively via the chiral superfield anomalous dimensions, and exact infinite-order perturbative renormalization group (RG) equations are available for gauge couplings [24,25]. Further, at superconformal fixed points, anomalous dimensions of all chiral superfields can be determined unambiguously using the method of a-maximization [26]. Finally, independent quartics do not arise and vacuum stability is automatically guaranteed provided gauge and Yukawa couplings take viable fixed points. Taken together, these ingredients prove sufficient to determine interacting fixed points reliably, including at strong coupling.
Here, we apply this methodology exemplarily to an asymptotically nonfree SUðN 1 Þ × SUðN 2 Þ supersymmetric gauge theory, with massless chiral matter and a superpotential, and determine the entire conformal window of interacting UV fixed points. Our basic setup is illustrated in Fig. 1 which shows the phase diagram of a semisimple gauge theory with superpotential interactions. Notice that asymptotic freedom is absent because one of the gauge sectors (α 2 ) is infrared-free while the other is not (α 1 ). The theory may nevertheless develop an asymptotically safe fixed point (UV) which allows a well-defined high energy limit and outgoing RG trajectories toward the IR and which is the central topic of this study. Consistency with Seiberg duality, and constraints from the a-theorem, global charges, and unitarity, are also observed [27]. Findings are confirmed independently using perturbation theory up to three-loop order. Implications of our findings for the asymptotic safety conjecture, model building, and conformal field theory are indicated.

II. SEMISIMPLE GAUGE THEORIES WITH MATTER
We consider semisimple Yang-Mills theories with product gauge group SUðN 1 Þ × SUðN 2 Þ coupled to chiral superfields ðψ; χ; Ψ; QÞ with flavor multiplicities ðN F ; N F ; 1; N Q Þ and gauge charges as in Table I. We require that the model has a global N ¼ 1 supersymmetry. It is then characterized by two gauge couplings g 1 and g 2 and a Yukawa coupling y via the superpotential where the trace sums over flavor and gauge indices. The superfields Q are not furnished with Yukawa interactions. The theory has a global SUðN F Þ L ×SUðN F Þ R ×SUðN Q Þ L × SUðN Q Þ R flavor and a Uð1Þ R symmetry and is characterized by the field multiplicities Asymptotic freedom and interacting infrared fixed points arise for suitable matter field multiplicities. For this study, the regime of interest is where the SUðN 1 Þ gauge sector is asymptotically free while the SUðN 2 Þ gauge sector is infrared-free. Accordingly, the free theory corresponds to a saddle and asymptotic freedom cannot be achieved ( Fig. 1), very much like in the non-Abelian gauge sectors of the minimal supersymmetric standard model (MSSM). In this light, the gauge coupling α 2 can be viewed as "dangerously irrelevant," in that it may become relevant due to the gauge-Yukawa fixed point in the other gauge sector. In the large-N Veneziano limit, the theory has previously been studied in perturbation theory, where it was found that interacting UV fixed points can arise at weak coupling with perturbatively small anomalous dimensions [11]. The main point of this study is to investigate the conditions under which theories with (6) may develop strongly interacting ultraviolet fixed points where anomalous dimensions become large, of order unity. Ordinarily, this would require strong coupling methods such as lattice simulations to establish the claim. In supersymmetry, however, powerful continuum methods beyond perturbation theory are available, to which we turn next.

III. RENORMALIZATION GROUP
To achieve our claim, we must find the renormalization group equations for all couplings and identify their fixed points and nonperturbative expressions for the chiral superfield anomalous dimensions. Here, we exploit key features of N ¼ 1 supersymmetric gauge theories that make this task feasible.
For supersymmetric gauge theories, closed all-order expressions for perturbative β-functions have been achieved by Novikov et al. (NSVZ) [24,25] (see also [28,29]). We introduce the gauge and Yukawa couplings as  Matter and denote the chiral superfield anomalous dimensions as γ a ða ¼ ψ; Ψ; χ; QÞ, with β-functions β i ≡ dα i =d ln μði ¼ 1; 2; yÞ defined as usual. The NSVZ beta functions for the gauge couplings of our models are then given by The scheme-dependent function FðαÞ is normalized to unity for vanishing coupling [30] and reads FðαÞ ¼ 1-2C G 2 α in the NSVZ scheme [24,25] with C G 2 the quadratic Casimir in the adjoint. Close to the Gaussian fixed point, the anomalous dimensions vanish and we can then read off the condition for a saddle in terms of the field multiplicities (2), This also covers the case where the one-loop coefficient of β 2 vanishes identically. For the Yukawa coupling, we exploit that supersymmetry dictates strict nonrenormalization theorems, which ensure that superpotential couplings are only renormalized through field anomalous dimensions. Hence, the RG running is given by valid to all orders in perturbation theory. Renormalization group fixed points ðβ i ¼ 0Þ correspond to superconformal field theories. All perturbative fixed points ðα Ã i ≪ 1Þ for this theory have previously been found in [11]. In general, these are either Banks-Zaks fixed points, where some of the gauge couplings are nonzero but α Ã y ¼ 0, or gauge-Yukawa (GY) fixed points with one (GY 1 ), the other (GY 2 ), or both gauge couplings nonzero (GY 12 ) and α Ã y > 0 [6,7]. It has also been shown that if GY 1 is UV, its outgoing trajectory is connected with the IR fixed point (GY 12 ), and similarly for GY 2 [11].
In this work, we focus on the regime (6) and search for nonperturbative fixed points with the property Our ansatz states that the SUðN 2 Þ gauge sector is infraredfree, meaning β 2 > 0 in the vicinity of the Gaussian. If β 2 < 0 in the vicinity of the interacting fixed point, the SUðN 2 Þ gauge sector suddenly becomes asymptotically free, and the fixed point is ultraviolet. In its vicinity, α 2 is the only relevant coupling in the UV, which runs out of the fixed point as with δα 2 ðΛÞ a small deviation at the high scale Λ, B 2;eff ¼ B 2 þ N F γ χ þ 4N 1 γ Ψ > 0 the interaction-induced oneloop coefficient, and B 2 < 0 given by (minus) the oneloop coefficient of (5) at the Gaussian. The couplings α 1 and α y are irrelevant interactions in the UV and their running is fully determined by the one of α 2 [11]. This scenario is schematically depicted in Fig. 1. The necessary and sufficient conditions for the partially interacting fixed point to turn the dangerously irrelevant coupling α 2 into a relevant one are given by where the two equations determine the fixed point, while the inequality ensures the sign flip from β 2 ≥ 0 close to the Gaussian to β 2 < 0 close to the fixed point (8).

IV. a-MAXIMIZATION
The beta functions (4), (5), and (7), and the conditions (10), still depend on the superfield anomalous dimensions. These can be determined either perturbatively, as we will do in Sec. VIII below, or exactly, as we do here. To that end, we exploit that fixed points of the renormalization group correspond to superconformal field theories. Hence, superfields must transform in representations of the superconformal algebra. Focusing on the bosonic part, we observe an extra global Uð1Þ R symmetry in addition to the ordinary conformal algebra. The global and anomaly free Uð1Þ R symmetry prescribes global charges R i for all chiral superfields at any superconformal fixed point and thereby determines anomalous dimensions γ i of chiral superfields via The task then reduces to the determination of R-charges at superconformal fixed points for which we use the method of a-maximization [26]. Specifically, at the fixed point GY 1 with (8) and characterized by (10), we find and Δ > 0 the positive root of The ψ Q fermions do not interact at the fixed point (8), hence γ Q ¼ 0 and none of the R-charges depend on N Q . The results for the R-charges uniquely determine the remaining anomalous dimensions nonperturbatively and provide closure for the beta functions (4), (5), and (7). At weak coupling, anomalous dimensions are small and R-charges are close to their classical values ðR i ≈ 2 3 Þ. Moreover, unitarity mandates that scaling dimensions D of spinless operators must satisfy D ≥ 1 [27], additionally implying The conditions for unitary quantum field theories with interacting UV fixed points (10) only depend on ratios of field multiplicities. Therefore, we can reduce the fourdimensional parameter space of field multiplicities (2) to a three-dimensional one. Following [11], we do so by scaling-out one of the field multiplicities, say N 1 , and by introducing three suitable ratios of field multiplicities instead, The color ratio R (not to be confused with the R-charges R i ) is sensitive to the relative size of gauge groups. The parameter P < 0 is proportional to the ratio of one-loop gauge coefficients. The parameter ϵ, which in the region of interest is negative ϵ < 0, can be made arbitrarily small in a Veneziano limit where it controls perturbative fixed points provided jϵj ≪ 1. Since field multiplicities are semipositive numbers, we further observe The parameters (14) take discrete values for integer field multiplicities except in a Veneziano large-N limit where they become continuous.

V. FIXED POINTS AND CONFORMAL WINDOWS
We are now in a position to investigate the range of field multiplicities for which the theory displays an interacting ultraviolet fixed point. Exploiting all constraints, we find that the parameter P is bounded from above and from below P min ðR; ϵÞ < P ≤ P max ðR; ϵÞ: The upper boundary P max relates to (6) and P ≤ 0, and to the physicality of field multiplicities (15), whichever is stronger. The lower boundary P min relates to the sign flip for induced asymptotic freedom of SUðN 2 Þ. Explicitly, In the above, the R-charges R Ψ and R χ are understood as functions of ðR; ϵÞ via (12) and (14). The parameters R and ϵ are constrained globally by the physicality of couplings (α ≥ 0) and unitarity, Figure 2 shows a contour plot of the superconformal window (16) and (19) in the ðϵ; RÞ plane. The left panel shows the lower boundary P min , while the right panel shows the accessible range of P values between P min and P max . The lower boundary in both graphs is given by Roughly speaking, the width in P is largest around the full white line. The upper boundary R max ðϵÞ arises as the solution of a cubic, approximately given by We notice that the width P max − P min vanishes at R max , while it remains small but non-zero at R min , except at the end points. Weakly coupled fixed points correspond to parameters ðP; R; ϵÞ close to the boundary where jϵj ≪ 1.
The conformal window is further illustrated in Fig. 3 showing its projection onto the ðP; RÞ plane. Within the yellow-shaded area, weakly coupled fixed points arise toward the right of the dashed line, while strongly coupled fixed points can arise anywhere. Outside the yellow-shaded area, asymptotic safety is not available and the corresponding quantum field theories must be viewed as effective rather than fundamental.
It is interesting to discuss the N F dependence of fixed points within the conformal window while keeping ðN 1 ; N 2 ; PÞ fixed. In Fig. 2, this effectively corresponds to varying ϵ along horizontal cuts. We notice that the conformal window splits into two distinct types of models, separated by dashed lines in Figs. 2 and 3, to which we refer as "mostly weakly" and "mostly strongly" coupled. Specifically, the first subset of models are those above the dashed line in Fig. 2 and on the right of the dashed line in Fig. 3. Their conformal window is characterized by Then, for any admissible value of R within (20), there is a range of viable parameters P within ð−1; 0 such that the UV conformal window in ϵ covers the range The lower bound ϵ max ¼ 3 2 ðR − 2Þ may become as low as − 3 2 . The significance of this result is as follows. Perturbative fixed points are controlled by the parameter jϵj ≪ 1 in a large-N Veneziano limit, meaning that this part of the UV conformal window contains all perturbatively controlled superconformal UV fixed points found previously in [11]. Hence, we observe that all perturbative fixed points extend into a nonperturbative conformal window for ϵ given precisely by the range (21). The same considerations apply for finite N, away from a Veneziano limit, the only difference being that ðR; P; ϵÞ take discrete rather than continuous values. Since all theses fixed points are linked to weakly coupled ones, and for want of terminology, we refer to the models within (20) as mostly weakly coupled.
Next, we turn to the part of the conformal window below the dashed line in Fig. 2 and to the left of the dashed line in Fig. 3, characterized by For any admissible value of R within (22), there is a range of viable parameters P within ð−4; 0 such that the UV conformal window in ϵ covers the range P min Here, the lower bound ϵ min ¼ 2ðR − 1Þ < 0 ensures induced asymptotic freedom for the coupling α 2 . Most importantly, we observe the existence of a finite gap ½ϵ min ; 0 in ϵ within which no asymptotically safe fixed points can be found. This implies that none of the fixed points within (22) can be achieved with strict perturbative control ðϵ → 0 − Þ, not even in a Veneziano limit. It is in this sense that the fixed points within (22) are nonperturbative and parametrically disconnected from the free theory, quite different from those in (20). For these reasons, we refer to these superconformal fixed points as mostly strongly coupled. Another important aspect of the conformal window relates to the saddle close to the Gaussian which needs to be overcome by the interacting fixed point. Recall that the asymptotically free (infrared-free) direction is characterized by the one-loop gauge coefficient , with (minus) their ratio given by the imbalance parameter for which we have I ≥ 0. Quantum effects at the interacting fixed point overcome the positive or vanishing one-loop coefficient B 2 and turn it, effectively, into a negative one (see Fig. 1). It is then important to understand the largest imbalance that can be achieved without spoiling asymptotic safety. We find that the imbalance is bounded from above, where we recall that field multiplicities obey (6). In other words, as soon as one gauge sector is as or more infraredfree at the Gaussian fixed point than the other gauge sector is ultraviolet-free, asymptotic safety at an interacting fixed point cannot arise. In Figs. 2 and 3, the small imbalance region 0 ≤ I ≪ 1 is realized close to the upper boundaries where P ≈ 0. Here, many ultraviolet fixed points can be found in large parts of the parameter space including perturbative and nonperturbative ones. With growing imbalance I → 1, the set of perturbatively controlled fixed points shrinks to the vicinity of a single point ðP; RÞ ¼ ð−1; 1Þ. 1 Nonperturbatively, however, many more fixed points realize a near-maximal imbalance, corresponding to the lower boundary in Fig. 2 with parameters given by the line R ¼ R min ðϵÞ, P ¼ −1=R min , and ϵ ∈ ½− 3 2 ; 0Þ. In Fig. 3, the near-maximal imbalance is realized along the lower boundary to the left of the dashed line. We conclude that all models that may afford the near-maximal imbalance are contained in the mostly strongly coupled part of the conformal window (22).
We emphasize that the boundary at P min in (18) is not part of the conformal window. The reason for this is that whenever P → P min the inequality in (10) becomes an equality, meaning that β 2 → 0 nonperturbatively at the fixed point (8). At this point the UV fixed point GY 1 merges with yet another fixed point (a nonperturbative IR fixed point GY 12 [11]) in the limit P → P min . In consequence, the fixed point (8) degenerates into a line of fixed points for any α 2 , illustrated in Fig. 4. As such, this limit offers a manifold of Leigh-Strassler-type models [31], each of which is characterized by a line of interacting superconformal field theories, disconnected from the free theory and generated by an exactly marginal operator. Further, each of these lines of fixed points corresponds to an infrared sink because the previously relevant perturbation, given by α 2 , has become strictly marginal. This includes all settings with maximal imbalance I ¼ 1. Concrete examples for strongly coupled Leigh-Strassler models are delegated to Sec. X below. 2 Figure 5 shows the superfield anomalous dimensions within the entire conformal window. Since the spectator fermions ψ Q are free at the fixed point (8), the chiral superfield anomalous dimensions are only functions of ðR; ϵÞ and independent of P. Overall, we find that anomalous dimensions grow with growing jϵj. For models within (20) or (22), the chiral anomalous dimensions cover the range FIG. 4. Schematic phase diagram of Leigh-Strassler-type models in the plane of gauge couplings in the limit where the UV fixed point degenerates into a line of IR fixed points (see Fig. 1). 1 In a Veneziano limit, this are the models with N 1 ; N 2 ; N F → ∞, The fate of an IR sink can be evaded by adding mass term perturbations for the N F superfields ψ, which may lift the degeneracy and allow RG trajectories to emanate from the line of fixed points. However, this mechanism defies the original setup (6) in that the removal of the ψ degrees of freedom would make the theory asymptotically free from the outset.
with extremals reached at the ϵ ¼ − 3 2 and ϵ ¼ 0 boundaries of the conformal window. For models within (22) anomalous dimensions tend to take larger values than for those in (20). A comparison with perturbation theory is given in Sec. VIII. Incidentally, the monotonicity of γ i with growing ϵ establishes that anomalous dimensions do not vanish unless the fixed point is free.

VI. CENTRAL CHARGES AND THE a-THEOREM
Superconformal fixed points can also be characterized by central charges a, b, and c, the anomaly coefficients [32,33]. Their values have to satisfy certain conditions and can be used to constrain viable fixed points. The global charges a and c can be expressed in terms of the R-charges of chiral superfields, Given that the positivity conditions a, b, c > 0 are satisfied nonmarginally for free theories, they are guaranteed to be satisfied for perturbative theories. Similarly, the conformal collider bound [34] cannot be violated for weakly interacting theories. Here, we find that the positivity conditions and the conformal collider bound are satisfied nonperturbatively, in the entire conformal window, as they must. We now turn to the a-theorem. It states that the central charge a must be a decreasing function along RG trajectories in any 4D quantum field theory [35,36]. We find that a UV − a Gauss < 0, which confirms that none of the UV fixed points is connected by an RG trajectory with the free Gaussian fixed point, in accord with Fig. 1. Further, we noted earlier that any interacting UV fixed point in the conformal window comes bundled with a fully interacting nonperturbative IR fixed point (GY 12 ) [11]. We find a UV − a IR > 0, meaning that both conformal fixed points are connected by RG trajectories flowing from the former to the latter as in Fig. 1. We conclude that our results are consistent with the a-theorem, as they must.

VII. SEIBERG DUALITY
Next, we comment on how our results relate to Seiberg's electric-magnetic duality [37]. At the partially interacting fixed point (8) where the SUðN 2 Þ gauge sector is free, the flavor symmetry of the theory is enhanced. This manifests itself as an exchange symmetry under N 2 ↔ N F , whereby the quarks ψ and Ψ interchange their roles while the fields χ remain unchanged, see (12). Hence, we observe an interacting "magnetic" SUðN 1 Þ gauge theory which has N F þ N 2 flavors of "magnetic quarks" ψ and Ψ, alongside ðN F þ N 2 Þ 2 singlet mesons ðψ L ψ R Þ, χ L , χ R , and ðΨ L Ψ R Þ [38]. In this light, the gauge-Yukawa fixed point (8) corresponds to a free non-Abelian magnetic phase whose superconformal window is known to cover the range 3 2 N 1 < N F þ N 2 < 3N 1 [37], which corresponds to the parameter range − 3 2 < ϵ < 0 observed in (19). Notice, however, that the conformal window in Fig. 2 has additional constraints on ðR; PÞ that ensure that the SUðN 2 Þ gauge sector becomes a relevant perturbation at the fixed point.
Seiberg duality predicts the existence of a dual "electric" theory which must have gauge group SUðN F þ N 2 − N 1 Þ coupled to N F þ N 2 flavors of electric quarks. The conformal window of the non-Abelian Coulomb phase is given This corresponds to the exact same parameter band in ϵ as the one in (19). In the special case where N F ¼ N 2 , given by the upper boundary in Fig. 2, the R-charges simplify and become One recognizes the R-charges for the quarks and singlet mesons of supersymmetric magnetic QCD [37], whose Seiberg dual is characterized by electric quarksψ andΨ with We refer to [38] for further aspects of Seiberg duality in theories with product gauge groups.

VIII. PERTURBATION THEORY
In this section, we contrast the nonperturbative determination of anomalous dimensions with perturbation theory, using general expressions for perturbative beta functions up to three loop, obtained in the dimensional reduction scheme [39,40]. The main additions are perturbative expressions for anomalous dimensions which at superconformal fixed points can be used to cross-check the nonperturbative results from a-maximization.
For the sake of this comparison, it is convenient to perform a Veneziano limit and rescale gauge couplings as α i → N i α i and Yukawa couplings as α y → N 1 α y . We also use the parametrization (14). The parameter 0 < jϵj ≪ 1 then serves as a small expansion parameter to ensure rigorous control of fixed points in perturbation theory. To find fixed points, scaling exponents, and anomalous dimensions to first (second) order in ϵ, we must retain terms up to two (three) loop in the gauge coupling and up to one (two) loop in the Yukawa beta functions [7,22]. We refer to these approximations as next-to-leading order (NLO) and next-tonext-to-leading order (NNLO), respectively. Using the general results of [39,40] we find the gauge beta functions up to three loop for our models as and β ð1Þ 2 ¼ 2α 2 2 Pϵ; The perturbative Yukawa beta function is given by (7) for any loop order. The anomalous dimensions of the superfields are required up to two-loop accuracy. They read and respectively. Then, interacting fixed points with (8) are determined via a Taylor expansion of couplings in ϵ around the zeros of (33) and (7), also using (35) and (36). Notice that (34) is only used to determine whether the sign change from β 2 > 0 at the Gaussian to β 2 < 0 at (8) has taken place. Inserting the perturbative results for the fixed point into (35) and (36) provides explicit expressions for the anomalous dimensions of the form The six coefficients A ðnÞ i for i ¼ ψ; Ψ; χ which for n ¼ 1, 2 arise from the perturbative fixed point solutions at NLO and NNLO accuracy [10,22] still depend on the parameter R (but not on P). Their explicit expressions are not given because they do not offer further insights for the present purposes.
The expressions (37) must be compared with the exact R-charges from a-maximization (12). Writing the exact anomalous dimensions in terms of the R-charges, and expanding the expressions to second order in ϵ with the help of (14), we find the identical result for the coefficients A ðnÞ i in (37). This establishes consistency of findings between a-maximization and perturbation theory, as it must.

IX. ANOMALOUS DIMENSIONS
In Fig. 5 we have shown the exact anomalous dimensions across the entire UV conformal window. Here, we have a closer look into anomalous dimensions and compare with the NLO and NNLO predictions from perturbation theory. More specifically, in Figs. 6 and 7, we compare results at fixed color ratio R ¼ 3 2 , 1, 2 3 , and 1 3 , and for any ϵ within ½− 3 2 ; 0, which corresponds to horizontal cuts across Figs. 2 and 5.
We begin with Fig. 6 where our results for R ¼ 3 2 (left panel) and R ¼ 1 (right panel) are shown. For small jϵj ≪ 1, perturbation theory matches the exact results as it must. With growing jϵj, all anomalous dimensions grow in magnitude. For R ¼ 3 2 , the gray-shaded area indicates that fixed points are no longer in the UV conformal window, leading to a lower bound for ϵ, see (21). Also, the NLO results for γ ψ and γ Ψ coincide accidentally. At NNLO, perturbative results for all three anomalous dimensions are quite close to the exact results in the admissible range for ϵ. For R ¼ 1, the extension of the conformal window is maximal. Here, anomalous dimensions correspond to the fixed points along the dashed lines in Figs. 2, 3, and 5, which marks the boundary between the mostly weakly and mostly strongly coupled quantum field theories, see (20) vs (22). Maximal values γ ψ;Ψ → − 1 2 and γ χ → 1 are reached for ϵ → − 3 2 . For γ ψ , we observe that the NLO and NNLO results are close to the exact one over the entire range for ϵ. For γ Ψ , the NLO correction vanishes accidentally. Similarly, for γ χ , the NLO result strongly underestimates the exact value with growing jϵj. However, at NNLO, perturbative results for all three anomalous dimensions are close to the exact findings in the entire range of ϵ. We conclude that differences between NNLO and exact results are moderate over the entire interval. Note that this does not hold true in general, but when it does, one may use this near coincidence to extract estimates for fixed points at strong coupling which otherwise are not easily accessible.
The same analysis is repeated in Fig. 7 for the parameter choices R ¼ 2 3 (left panel) and R ¼ 1 3 (right panel). These horizontal cuts through Fig. 2 project onto the more strongly coupled fixed points in (22). For either of these, the small ϵ regions are excluded (gray-shaded areas) because the corresponding fixed points are not ultraviolet. In the increasingly narrow regions of interest (23), which are ϵ ∈ ½− 3 2 ; − 2 3 and ϵ ∈ ½− 3 2 ; − 4 3 , respectively, we observe that differences between NLO, NNLO, and exact results become rather large for γ χ , γ ψ , and γ Ψ . Outside the UV conformal window, we also notice that γ Ψ is no longer monotonous with ϵ, but instead changes sign for nonvanishing ϵ and taking values outside the range (26). Perhaps unsurprisingly, this confirms that the leading orders of perturbation theory cease to offer a good approximation for anomalous dimensions at strong coupling, once jϵj is large.

X. BENCHMARKS FOR MODELS OF PARTICLE PHYSICS
At weak coupling, ultraviolet fixed points are often characterized by a large number of field multiplicities [11]. At strong coupling, fixed points can arise with a much lower number of matter fields. Therefore, we benchmark models with minimal numbers of gauge and matter fields and explore whether variants could serve as templates for asymptotically safe Standard Model extensions.
Our benchmarks of superconformal fixed points with low numbers of fields are summarized in Table II. They represent strongly coupled models in that they have large jϵj and anomalous dimensions. The benchmarks also cover the full range of imbalance parameters I. Note that because the ψ Q fermions are free at the superconformal fixed point, anomalous dimensions of chiral superfields agree between different models as long as they only differ in the N Q multiplicity, see models 2-4, models 5 and 6, or models 7 and 8.
We begin with model 1, which is the sole example with SUð2Þ × SUð2Þ gauge symmetry, the simple reason being that the only other integer solution ðN F ; N Q Þ ¼ ð3; 2Þ to the constraint (6) does not lead to asymptotic safety. Here, anomalous dimensions are still small and well approximated by the perturbative three-loop result (see Fig. 6, right panel). Moreover, in models 1 and 2, the SUð2Þ gauge sector has a vanishing one-loop coefficient. In both cases a viable interacting UV fixed point is found, located at a boundary of the conformal window (Fig. 2) where I ¼ 0.
For the gauge group SUð3Þ × SUð2Þ, we find a total of five solutions (models 2-6). For all of these SUð2Þ is infrared-free at the Gaussian. Settings where SUð3Þ is infrared-free do not lead to asymptotic safety. Also, all models are within the strongly coupled domain where three-loop perturbation theory does not offer an accurate approximation (Fig. 7, left panel). Model 3 has N F ¼ 3 and , the main change is that jϵj, and hence the anomalous dimensions come out slightly smaller than in models 2-4. Overall, increasing the number of matter fields charged under SUð2Þ also increases the imbalance parameter. Models 3-6 have some similarities with the minimal supersymmetric standard model. Unlike in the Standard Model, the SUð2Þ sector of the MSSM becomes infraredfree due to extra gauge charges from supersymmetry partners, while the SUð3Þ sector remains asymptotically free. Hence, the Gaussian corresponds to a saddle with imbalance I MSSM ¼ 1 3 , just as in model 5. The imbalance could even be made larger such as in model 4 or model 6 and still yield an asymptotically safe fixed point. Hence, we see that supersymmetric gauge theories with the SM gauge groups (neglecting hypercharge) and imbalances similar to the MSSM may very well become asymptotically safe [41].
The constraints (6) imply that maximal values for anomalous dimensions (26) cannot arise if gauge group factors are as small as SUð2Þ or SUð3Þ. For larger gauge groups, however, we can find models realizing maximal anomalous dimensions. An example is given by an SUð6Þ × SUð2Þ gauge theory with N F ¼ 7 (model 7) which has a near-maximal imbalance parameter ðI ¼ 7 9 Þ. The model is located at the ϵ ¼ − 3 2 boundary of the conformal window and leads to an asymptotically safe fixed point with nonperturbatively large chiral field anomalous dimensions, FIG. 7. Same as Fig. 6, except that the projections are along R ¼ 2 3 (left) and R ¼ 1 3 (right). Decreasing R considerably narrows the UV conformal window. The excluded (gray-shaded) areas refer to regions where some matter field field multiplicities would be unphysical (negative).
We conclude that the benchmark models are lowfield-multiplicity realizations of asymptotic safety with phase diagrams as in Fig. 1. Most of them are nonperturbative with large jϵj and large anomalous dimensions, whose features are not captured reliably by a few leading orders of perturbation theory (Fig. 7). With increasing sizes of gauge group factors ðN 1 ; N 2 Þ, many more solutions ðN F ; N Q Þ for the constraint (6) arise, and a fair fraction of these lead to an asymptotically safe fixed point within the UV conformal window (Fig. 2). Finally, we discuss two benchmark models with ϵ ¼ − 3 2 and maximal anomalous dimensions. These strongly coupled models are of the Leigh-Strassler type and narrowly outside the UV conformal window. The first one is model 8 with SUð6Þ × SUð2Þ gauge symmetry and N F ¼ 7, which differs from model 7 only in that N Q ¼ 1 instead of N Q ¼ 0, thus leading to a larger imbalance (I ¼ 8 9 ). In consequence, the ultraviolet fixed point of model 7 degenerates into a line of fixed points due to β 2 j Ã vanishing identically at the gauge-Yukawa fixed point. The exactly marginal coupling α 2 becomes a free parameter and extends the fixed point into an infrared attractive line. The same phenomenon occurs for model 9, which has an SUð8Þ × SUð2Þ gauge symmetry with N F ¼ 10 and imbalance I ¼ 1, situated at the lowerleft corner of Fig. 2. Although interesting in their own right, models 8 and 9 no longer serve the purpose of ultraviolet fixed points. However, we stress that the method of amaximization has been key in identifying fixed points with exactly marginal directions.

XI. CONCLUSIONS
We have shown, as a proof of principle, that interacting UV fixed points exist in strongly coupled quantum field theories including away from Veneziano (large-N) limits. Fixed point anomalous dimensions of matter fields can grow large, taking values in the entire range dictated by unitarity (26). Thereby, previously found perturbative fixed points [11] extend naturally into a nonperturbative conformal window (Fig. 6). Interestingly, a novel range of fixed points has become available, with no perturbative counterparts in a Veneziano limit (Fig. 7). We thus conclude that the nonperturbative "phase space" of fixed points is large and substantially larger than the one accessible at weak coupling. Further, all conformal fixed points (Fig. 2) are in accord with the a-theorem, bounds on central charges, Seiberg duality, and unitarity. We also found a manifold of Leigh-Strassler models arising at the sign-flip-controlling boundary of the conformal window, where theories display a line of IR fixed points generated by an exactly marginal gauge interaction (Fig. 4). It is understood that similar UV conformal windows, bounded by Leigh-Strassler manifolds, arise naturally in other semisimple gauge theories coupled to matter. From the viewpoint of particle physics and model building, it is quite promising that fixed points persist at low field multiplicities (Table II), including with Standard-Model-like gauge groups. The next natural questions to ask are whether fixed points also arise in MSSM extensions [41] or in nonsupersymmetric settings at strong coupling. From the viewpoint of conformal field theory (CFT), it will be interesting to classify asymptotically safe supersymmetric theories more systematically and to extract CFT data or structure coefficients directly from the RG fixed points.  II. Parameter and anomalous dimensions for a selection of benchmark models with superconformal fixed points and a low number of field multiplicities, including Leigh-Strassler-type models (8 and 9).

Model
Gauge group N F N Q I R P ϵ γ ψ γ Ψ γ χ