PHYSICAL REVIEW D 97, 085008 (2018)
More asymptotic safety guaranteed
† Andrew D. Bond* and Daniel F. Litim Department of Physics and Astronomy, U Sussex, Brighton BN1 9QH, United Kingdom
(Received 2 February 2018; published 10 April 2018)
We study interacting fixed points and phase diagrams of simple and semisimple quantum field theories in four dimensions involving non-Abelian gauge fields, fermions and scalars in the Veneziano limit. Particular emphasis is put on new phenomena which arise due to the semisimple nature of the theory. Using matter field multiplicities as free parameters, we find a large variety of interacting conformal fixed points with stable vacua and crossovers inbetween. Highlights include semisimple gauge theories with exact asymptotic safety, theories with one or several interacting fixed points in the IR, theories where one of the gauge sectors is both UV free and IR free, and theories with weakly interacting fixed points in the UV and the IR limits. The phase diagrams for various simple and semisimple settings are also given. Further aspects such as perturbativity beyond the Veneziano limit, conformal windows, and implications for model building are discussed.
DOI: 10.1103/PhysRevD.97.085008
I. INTRODUCTION An important new development in the understanding of Asymptotic freedom is a key feature of non-Abelian asymptotic safety has been initiated in [32] where it was gauge theories [1,2]. It predicts that interactions weaken shown that certain four-dimensional quantum field theories with growing energy due to quantum effects, thereby involving SUðNÞ gluons, quarks, and scalars can develop reaching a free ultraviolet (UV) fixed point under the weakly coupled UV fixed points. Results have been renormalization group. Asymptotic safety, on the other extended beyond classically marginal interactions [33]. hand, stipulates that running couplings may very well Structural insights into the renormalization of general asymptote into an interacting UV fixed point at highest gauge theories have led to necessary and sufficient con- energies [3,4]. The most striking difference between ditions for asymptotic safety, alongside strict no go asymptotically free and asymptotically safe theories relates theorems [34,35]. Asymptotic safety invariably arises as to residual interactions in the UV. Canonical power count- a quantum critical phenomenon through cancellations at loop level for which all three types of elementary degrees ing is modified, whence establishing asymptotic safety in a — — reliable manner becomes a challenging task [5]. of freedom scalars, fermions, and gauge fields are Rigorous results for asymptotic safety at weak coupling required. Findings have also been extended to cover have been known since long for models including either supersymmetry [36] and UV conformal windows [37]. scalars, fermions, gauge fields or gravitons, and away Throughout, it is found that suitable Yukawa interactions from their respective critical dimensionality [4,6–16].In are pivotal [34,35]. In this paper, we are interested in fixed points of these toy models asymptotic safety arises through the semisimple gauge theories. Our primary motivation is cancellation of tree level and leading order quantum the semisimple nature of the standard model, and the terms. Progress has also been made to substantiate the prospect for asymptotically safe extensions thereof [38]. asymptotic safety conjecture beyond weak coupling [5]. We are particularly interested in semisimple theories where This is of particular relevance for quantum gravity interacting fixed points and asymptotic safety can be where good evidence has arisen in a variety of different established rigorously [34]. More generally, we also wish settings [17–31]. to understand how low- and high-energy fixed points are generated dynamically, what their features are, and whether *[email protected] novel phenomena arise owing to the semisimple nature of † [email protected] the underlying gauge symmetry. Understanding the stabil- ity of a Higgs-like ground state at interacting fixed points Published by the American Physical Society under the terms of is also of interest in view of the “near-criticality” of the the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to standard model vacuum [39,40]. the author(s) and the published article’s title, journal citation, We investigate these questions for quantum field theories 3 and DOI. Funded by SCOAP . with SUðNCÞ × SUðNcÞ local gauge symmetry coupled to
2470-0010=2018=97(8)=085008(35) 085008-1 Published by the American Physical Society ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) massless fermionic and singlet scalar matter. Our models A. Fixed points in perturbation theory alsohaveaglobalUðNFÞL × UðNFÞR × UðNf ÞL × UðNfÞR We are interested in the renormalization of general flavor symmetry, and are characterized by up to nine gauge theories coupled to matter fields, with or without independent couplings. Matter field multiplicities serve as Yukawa couplings. The running of the gauge couplings free parameters. We obtain rigorous results from the leading α 2 4π 2 μ i ¼ gi =ð Þ with the renormalization group scale is orders in perturbation theory by adopting a Veneziano limit. determined by the beta functions of the theory. Expanding We then provide a comprehensive classification of quantum them perturbatively up to two loop we have field theories according to their UV and IR limits, their fixed points, and eigenvalue spectra. Amongst these, we find 2 4 μ∂μα ≡β ¼ α ð−B þC α −2Y4 ÞþOðα Þ; ð1Þ semisimple gauge theories with exact asymptotic safety in i i i i ij j ;i the UV. We also find a large variety of theories with where a sum over gauge group factors j is implied. The crossover- and low-energy fixed points. Further novelties one- and two-loop gauge contributions B and C and the include theories with inequivalent yet fully attractive IR i ij two-loop Yukawa contributions Y4 are known for general conformal fixed points, theories with weakly interacting ;i gauge theories, see [34,41–44] for explicit expressions. fixed points in both the UVand the IR, and massless theories While B and C may take either sign, depending on the with a nontrivial gauge sector which is UV free and IR free. i ii matter content, the Yukawa contribution Y4;i and the off- We illustrate our results by providing general phase diagrams ≠ for simple and semisimple gauge theories with and without diagonal gauge contributions Cij (i j) are strictly positive Yukawa interactions. in any quantum field theory. Scalar couplings do not play The paper is organized as follows. General aspects of any role at this order in perturbation theory. The effect of weakly interacting fixed points in 4d gauge theories are Yukawa couplings is incorporated by projecting the gauge β 0 laid out in Sec. II, together with first results and expressions beta functions (1) onto the Yukawa nullclines ð Y ¼ Þ, for universal exponents. In Sec. III we introduce concrete leading to explicit expressions for Y4;i in terms of the families of semisimple gauge theories coupled to elemen- gauge couplings gj. Moreover, for many theories the tary singlet “mesons” and suitably charged massless Yukawa contribution along nullclines can be written as α ≥ 0 fermions. Perturbative RG equations for all gauge, Y4;i ¼ Dij j with Dij [34]. We can then go one Yukawa and scalar couplings and masses in a Veneziano step further and express the net effect of Yukawa couplings → 0 limit are provided to the leading non-trivial orders in as a shift of the two loop gauge contribution, Cij Cij ¼ perturbation theory. Section IV presents our results for Cij − 2Dij ≤ Cij. Notice that the shift will always be by all interacting perturbative fixed points and their universal some negative amount provided at least one of the Yukawa scaling exponents. Particular attention is paid to new effects couplings is nonvanishing. It leads to the reduced gauge which arise due to the semisimple nature of the models. beta functions Section V provides the corresponding fixed points in the scalar sector. It also establishes stability of the quantum β α2 − 0 α α4 i ¼ i ð Bi þ Cij jÞþOð Þ: ð2Þ vacuum whenever a physical fixed point arises in the gauge sector. Using field multiplicities as free parameters, Sec. VI Fixed points solutions of (2) are either free or interacting provides a complete classification of distinct models with and α ¼ 0 for some or all gauge factors is always a self- asymptotic freedom or asymptotic safety in the UV, or consistent solution. Consequently, interacting fixed points without UV completions, together with their scaling in the are solutions to deep IR. In Sec. VII, the generic phase diagrams for simple and semisimple gauge theories with and without Yukawas 0 α α 0 Bi ¼ Cij j ; subject to i > ; ð3Þ are discussed. The phase diagrams, UV—IR transitions, and aspects of IR conformality are analysed in more where only those rows and columns are retained where depth for sample theories with asymptotic freedom and gauge couplings are interacting. asymptotic safety. Further reaching topics such as exact Next we discuss the role of Yukawa couplings for the perturbativity, extensions beyond the Veneziano limit, and fixed point structure. In the absence of Yukawa couplings, conformal windows are discussed in Sec. VIII. Section IX 0 the two-loop coefficients remain unshifted Cij ¼ Cij.An closes with a brief summary. immediate consequence of this is that any interacting fixed point must necessarily be IR. The reason is as follows: for II. FIXED POINTS OF GAUGE THEORIES an interacting fixed point to be UV, asymptotic freedom In this section, we discuss general aspects of interacting cannot be maintained for all gauge factors, meaning that fixed points in semisimple gauge theories which are weakly some Bi < 0. However, as has been established in [34], coupled to matter, with or without Yukawa interactions, Bi ≤ 0 necessarily entails Cij ≥ 0 in any 4d quantum gauge following [34,35]. We also introduce some notation and theory. If the left-hand side of (3) is negative, if only for a α conventions. single row, positivity of Cij requires that some j must take
085008-2 MORE ASYMPTOTIC SAFETY GUARANTEED PHYS. REV. D 97, 085008 (2018) negative values for a fixed point solution to arise. This, TABLE I. Conventions to denote the basic fixed points (Gaus- however, is unphysical [45] and we are left with Bi > 0 for sian, Banks-Zaks, or gauge-Yukawa) of simple gauge theories each i, implying that asymptotic freedom remains intact in weakly coupled to matter. all gauge sectors. Besides the Gaussian, the theory may Fixed point α α have weakly interacting infrared Banks-Zaks fixed points in gauge Yukawa each gauge sector, as well as products thereof, which arise Gauss G ¼ 0 ¼ 0 Banks-Zaks BZ ≠ 0 ¼ 0 as solutions to (3) with the unshifted coefficients. ≠ 0 ≠ 0 In the presence of Yukawa couplings, the coefficients Gauge-Yukawa GY 0 Cij can in general take either sign. This has far reaching implications. First, the theory can additionally display interactions take Gaussian values, the numbers C and G are gauge-Yukawa fixed points where both the gauge and i i the two-loop coefficients which arise owing to the gauge the Yukawa couplings take interacting values. Most impor- loops and owing to the mixing between gauge groups, tantly, solutions to (3) are then no longer limited to theories meaning C ≡ C (no sum), and G1 ≡ C12, G2 ≡C12, with asymptotic freedom. Instead, interacting fixed points i ii see (1). In this case, we also have that C , G ≥ 0 as soon can be infrared, ultraviolet, or of the crossover type. In i i as B < 0.1 For theories where Yukawa couplings take general we may expect gauge-Yukawa fixed points for each i C G independent Yukawa nullcline. In summary, perturbative interacting fixed points the numbers i and i receive C ≡ C0 fixed points are either (i) free and given by the Gaussian, or corrections due to the Yukawas, i ii (no sum), and ≡ 0 ≡ 0 (ii) free in the Yukawa but interacting in the gauge sector G1 C12, G2 C12,see(2). Most notably, strict positivity (Banks-Zaks fixed points), or (iii) simultaneously interact- of Ci and Gi is then no longer guaranteed [34]. ing in the gauge and the Yukawa sector (gauge-Yukawa In either case, the fixed points of the combined system fixed points), or (iv) combinations and products of (i), (ii), are determined by the vanishing of (4). For a general and (iii). Banks-Zaks fixed points are always IR, while the semisimple gauge theory with two gauge factors, one finds Gaussian and gauge-Yukawa fixed points can be either UV four different types of fixed points. The Gaussian fixed or IR. Depending on the details of the theory and its point Yukawa structure, either the Gaussian or one of the α α 0 0 interacting gauge-Yukawa fixed points will arise as the ð 1; 2Þ¼ð ; Þð5Þ “ultimate” UV fixed point of the theory and may serve to define the theory fundamentally [35]. always exists (see Table I for our conventions). It is the UV fixed point of the theory as long as the one-loop The effect of scalar quartic self-couplings on the fixed 0 point is strictly subleading in terms of the values of the coefficients obey Bi > . The theory may also develop fixed points, as they do not affect the running of gauge partially interacting fixed points, couplings at this order of perturbation theory. However, B2 as to have a true fixed point we must acquire one in all ðα1; α2Þ¼ 0; ; ð6Þ couplings, they provide additional constraints on the C2 physicality of candidate gauge-Yukawa fixed points, as B1 we additionally require that the quartic couplings take fixed ðα1; α2Þ¼ ; 0 : ð7Þ points which are both real-valued, and lead to a bounded C1 potential which leads to a stable vacuum state. Here, one of the gauge coupling is taking Gaussian values whereas the other one is interacting. The interacting fixed B. Gauge couplings point is of the Banks-Zaks type [46,47], provided Yukawa Let us now consider a semisimple gauge-Yukawa theory interactions are absent. This then also implies that the with non-Abelian gauge fields under the semisimple gauge coupling is asymptotically free. Alternatively, the gauge group G1 ⊗ G2 coupled to fermions and scalars. interacting fixed point can be of the gauge-Yukawa type, We have two non-Abelian gauge couplings α1 and α2, provided that Yukawa couplings take an interacting fixed which are related to the fundamental gauge couplings point themselves. In this case, and depending on the details α 2 4π 2 via i ¼ gi =ð Þ . The running of gauge couplings within of the Yukawa sector, the fixed point can be either IR or perturbation theory is given by UV. Finally, we also observe fully interacting fixed points β − α2 α3 α2α − − 1 ¼ B1 1 þ C1 1 þ G1 1 2; C2B1 B2G1 C1B2 B1G2 ðα1; α2Þ¼ ; : ð8Þ 2 3 2 C1C2 − G1G2 C1C2 − G1G2 β2 ¼ −B2α2 þ C2α2 þ G2α2α1: ð4Þ
1 Here, Bi are the well known one-loop coefficients. In General formal expressions of loop coefficients in the con- theories without Yukawa interactions, or where Yukawa ventions used here are given in [34].
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As such, fully interacting fixed points (8) can be either reduces to (4) whereby the two-loop coefficients Ci of the UV or IR, depending on the specific field content of the gauge beta functions are shifted according to theory. In all cases we will additionally require that the F1 couplings obey α ≠ 0∶ → 0 − ≤ Y C1 C1 ¼ C1 D1 C1; E1 α1 ≥ 0; α2 ≥ 0: ð9Þ 0 F2 αy ≠ 0∶ C2 → C2 ¼ C2 − D2 ≤ C2: ð12Þ to ensure they reside in the physical regime of the E2 theory [45]. Notice also that in this model the values for the mixing terms Gi do not depend on whether the corresponding C. Yukawa couplings Yukawa couplings vanish, or not, due to the fact that no In order to proceed, we must specify the Yukawa sector. fermions charged under both groups are involved in We assume three types of nontrivially charged fermions Yukawa interactions. Owing to the fixed point structure with charges under G1 and G2. Some or all of the fermions of the Yukawa sector (11), the formal fixed points (5), (6), which are only charged under G1 (G2) also couple to scalar (7), and (8) have the multiplicity 1,2,2, and 4, respectively. fields via Yukawa couplings αY (αy), respectively. The In total, we end up with nine qualitatively different fixed scalars may or may not be charged under the gauge points FP1–FP9, summarized in Table III:FP1 denotes the symmetries. They will have quartic self couplings which unique Gaussian fixed point. FP2 and FP3 correspond to a play no primary role for the fixed point analysis at weak Banks-Zaks fixed point in one of the gauge couplings, and coupling [34]. Within perturbation theory, the beta func- a Gaussian in the other. They can therefore be interpreted tions for the gauge and Yukawa couplings are of the form effectively as a “product” of a Banks-Zaks with a Gaussian fixed point. Similarly, at FP4 and FP5, one of the Yukawa 2 3 2 2 β1 ¼ −B1α1 þ C1α1 − D1α1αY þ G1α1α2; couplings remains interacting, and they can therefore 2 effectively be viewed as the product of a gauge-Yukawa β E1α − F1α α1; Y ¼ Y Y (GY) type fixed point in one gauge coupling with a 2 3 2 2 β2 ¼ −B2α2 þ C2α2 − D2α2αy þ G2α2α1; Gaussian fixed point in the other. The remaining fixed 2 points FP6–FP9 are interacting in both gauge couplings. β ¼ E2α − F2α α2: ð10Þ y y y These fixed points are the only ones which are sensitive to the two-loop mixing coefficients G1 and G2.AtFP6, both The renormalization group (RG) flow is given up to two-loop Yukawa couplings vanish meaning that it is effectively a in the gauge couplings, and up to one-loop in the Yukawa product of two Banks-Zaks type fixed points. At FP7 and couplings. We refer to this as the next-to-leading-order FP8, only one of the Yukawa couplings vanish, implying (NLO) approximation, see Table II for the terminology. We are interested in the fixed points of the theory, defined implicitly via the vanishing of the beta functions for TABLE III. The various types of fixed points in gauge-Yukawa all couplings. The Yukawa couplings can display either a theories with semisimple gauge group G1 ⊗ G2 and (10), (12). Gaussian or an interacting fixed point We also indicate how the nine qualitatively different fixed points can be interpreted as products of the Gaussian (G), Banks-Zaks F1 (BZ) and gauge-Yukawa (GY) fixed points as seen from the α 0 α α Y ¼ ; Y ¼ 1; individual gauge group factors (see main text). E1
F2 Gauge couplings Yukawa couplings αy ¼ 0; αy ¼ α2: ð11Þ Fixed Fixed point E2 α α α α point 1 2 Y y type FP G G Depending on whether none, one, or both of the Yukawa 1 0000 · B1 FP2 000BZ · G couplings take an interacting fixed point, the system (10) C1 B2 FP3 0 00G · BZ C2 TABLE II. Relation between approximation level and the loop B1 F1 FP4 0 0 α1 0 GY · G order up to which couplings are retained in perturbation theory, C1 E1 B2 F2 following the terminology of [32,48]. FP5 0 0 0 α2 G · GY C2 E2 C2B1−B2G1 C1B2−B1G2 FP6 00BZ · BZ Coupling Order in perturbation theory C1C2−G1G2 C1C2−G1G2 0 − β 1 FP C2B1−B2G1 C1B2 B1G2 F1 α GY BZ gauge 12 2n þ 7 0 − 0 − 1 0 · C1C2 G1G2 C1C2 G1G2 E1 β 0 − Yukawa 01 1 n C2B1 B2G1 C1B2−B1G2 F2 FP8 0 0 0 α2 BZ · GY β C1C −G1G2 C1C −G1G2 E2 scalar 00 1 n 2 2 0 − 0 − FP C2B1 B2G1 C1B2 B1G2 F1 α F2 α GY GY 0 0 9 0 0 − 0 0 − 1 2 · Approximation LO NLO NLO nNLO C1C2 G1G2 C1C2 G1G2 E1 E2
085008-4 MORE ASYMPTOTIC SAFETY GUARANTEED PHYS. REV. D 97, 085008 (2018) that these are products of a gauge-Yukawa with a Banks- from now on denote relevant and marginally relevant ones Zaks fixed point. Finally, at FP9, both Yukawa couplings as ϑ ≤ 0, and vice versa for irrelevant ones. are nonvanishing meaning that this is effectively the Given that the scalar couplings do not feed back to the product of two gauge-Yukawa fixed points. gauge-Yukawa sector at the leading non-trivial order in In theories where none of the fermions carries gauge perturbation theory, we may neglect them for a discussion charges under both gauge groups, we have that of the eigenvalue spectrum G1 ¼ 0 ¼ G2. In this limit, and at the present level of approximation, the gauge sectors do not communicate with fϑi;i¼ 1; 4g; ð15Þ each other and the “direct product” interpretation of the fixed points as detailed above becomes exact. For the related to the two gauge and Yukawa couplings. The fixed purpose of this work we will find it useful to refer to point FP1 is Gaussian in all couplings, and the scaling of “ ” the effective product structure of interacting fixed points couplings are either marginally relevant or marginally even in settings with G1, G2 ≠ 0. Whether any of the fixed irrelevant. Only if Bi > 0 trajectories can emanate from points is factually realized in a given theory crucially the Gaussian, meaning that it is a UV fixed point iff the depends on the explicit values of the various loop coef- theory is asymptotically free in both couplings. Furthermore, ficients. We defer an explicit investigation for certain asymptotic freedom in the gauge couplings entails asymp- “ ” minimal models to Sec. III. totic freedom in the Yukawa couplings leading to four marginally relevant couplings with eigenvalues D. Scalar couplings In [34], it has been established that scalar self-inter- ϑ1; ϑ2; ϑ3; ϑ4 ≤ 0 ð16Þ actions play no role for the primary occurrence of weakly interacting fixed points in the gauge- or gauge-Yukawa The fixed points FP2 and FP3 are products of a Banks-Zaks sector. On the other hand, for consistency, scalar couplings in one gauge sector with a Gaussian fixed point in the other. must nevertheless take free or interacting fixed points on Scaling exponents are then of the form their own. The necessary and sufficent conditions for this to arise have been given in [34]. First, scalar couplings must ϑ1; ϑ2; ϑ3 ≤ 0 < ϑ4 ð17Þ take physical (real) fixed points. Second, the theory must display a stable ground state at the fixed point in the scalar provided the gauge sector with Gaussian fixed point is sector. Below, we will analyse concrete models and show asymptotically free. For IR free gauge coupling, we instead that both of these conditions are nonempty. have the pattern
E. Universal scaling exponents ϑ1 < 0 ≤ ϑ2; ϑ3; ϑ4: ð18Þ We briefly comment on the universal behavior and scaling exponents at the interacting fixed points of At the fixed points FP4 and FP5, the theory is the product of a Table III. Scaling exponents arise as the eigenvalues ϑi Gaussian and a gauge-Yukawa fixed point. Consequently, of the stability matrix four possibilities arise: Provided that the theory is asymp- totically safe at the gauge-Yukawa fixed point and asymp- ∂β ∂α totically or infrared free at the Gaussian, scaling exponents Mij ¼ i= jj ð13Þ are of the form (17) or (18), respectively. Conversely, if the at fixed points. Negative or positive eigenvalues correspond gauge Yukawa fixed point is IR, the eigenvalue spectrum to relevant or irrelevant couplings respectively. They imply reads that couplings approach the fixed point following a power- law behavior in RG momentum scale, ϑ1; ϑ2 ≤ 0 ≤ ϑ3; ϑ4 ð19Þ X ϑ μ n if the Gaussian is asymptotically free. Finally, if the Gaussian α μ − α c Vn subleading: 14 ið Þ i ¼ n i Λ þ ð Þ is IR free and the gauge-Yukawa fixed point IR, all couplings n are UV irrelevant and Classically, we have that ϑ ≡ 0. Quantum-mechanically, and at a Gaussian fixed point, eigenvalues continue to 0 ≤ ϑ1; ϑ2; ϑ3; ϑ4: ð20Þ vanish and the behavior of couplings is determined by higher order effects. Then couplings are either exactly More work is required to determine the scaling exponents at − marginal ϑ ≡ 0 or marginally relevant ϑ → 0 or margin- the fully interacting fixed points FP6–FP9. To that end, we ally irrelevant ϑ → 0þ. In a slight abuse of language we will write the characteristic polynomial of the stability matrix as
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X4 TABLE IV. Representation under the semisimple gauge sym- n Tnϑ ¼ 0: ð21Þ metry (24) together with flavor multiplicities of all fields. Gauge n¼0 (fermion) fields are either in the adjoint (fundamental) or trivial representation. The coefficients Tn are functions of the loop coefficients. Introducing B ¼jB1j and B2 ¼ PB1, with P some free Fermions Scalars Gauge fields parameter, we can make a scaling analysis in the limit Representation Qqψ Hh Aμ aμ ≪ 1 1 B . Normalizing the coefficient T4 to unity, T4 ¼ ,it 2 − 1 Under SUðNCÞ NC 1 NC 11NC 1 then follows from the structure of the beta functions that 2 − 1 6 4 2 Under SUðNcÞ 1 Nc Nc 11 1 Nc T0 ¼ OðB Þ;T1 ¼ OðB Þ;T2 ¼ OðB Þ and T3 ¼ OðBÞ to 2 2 Multiplicity NF Nf Nψ NF Nf 11 leading order in B. In the limit where B ≪ 1 we can deduce exact closed expressions for the leading order behaviour of the eigenvalues from solutions to two quadratic equations, with field strength Fμν, and SUðNcÞ gauge fields aμ with 2 field strength fμν. The gauge fields are coupled to N 0 ¼ ϑ þ T3ϑ þ T2 F 2 flavors of fermions Qi, Nf flavors of fermions qi, and Nψ 0 ¼ T2ϑ þ T1ϑ þ T0: ð22Þ flavors of fermions ψ i. The fermions ðQ; q; ψÞ transform in the fundamental representation of the first, the second, and The general expressions are quite lengthy and shall not be both gauge group(s) (24), respectively, as summarized in given here explicitly. We note that the four eigenvalues of the Table IV. The Dirac fermions ψ are responsible for the four couplings at the four fully interacting fixed points – semisimple character of the theory and serve as messengers FP6 FP9 are the four solutions to (22). Irrespective of their to communicate between gauge sectors. All fermions are signs, and barring exceptional numerical cancellations, we Dirac to guarantee anomaly cancellation. The fermions conclude that two scaling exponents are quadratic and two ðQ; qÞ additionally couple via Yukawa interactions to an are linear in B, NF × NF matrix scalar field H and an Nf × Nf matrix scalar qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi field h, respectively. The scalars H and h are invariant 1 2 2 ϑ1 2 − T3 T − 4T2 O B ; ¼ 2 3 ¼ ð Þ under UðNFÞL × UðNFÞR and UðNf ÞL × UðNf ÞR global qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 flavor rotations, respectively, and singlets under the gauge ϑ − 2 − 4 O symmetry. They can be viewed as elementary mesons in 3;4 ¼ 2 T1 T1 T0T2 ¼ ðBÞ: ð23Þ T2 that they carry the same global quantum numbers as the singlet scalar bound states ∼hQQ¯ i and ∼hqq¯i. The fer- This is reminiscent of fixed points in gauge-Yukawa theories mions ψ are not furnished with Yukawa interactions. with a simple gauge group. The main reason for the The fundamental action is taken to be the sum of the appearance of two eigenvalues of order O B2 relates to ð Þ individual Yang-Mills actions, the fermion kinetic terms, the gauge sector, where the interacting fixed point arises the Yukawa interactions, and the scalar kinetic and self- through the cancellation at two-loop level. Conversely, two interaction Lagrangeans L ¼ L þL þL þL þL , eigenvalues of order OðBÞ relate to the Yukawa couplings, YM F Y S pot with as they arise from a cancellation at one-loop level. This completes the discussion of fixed points in general weakly 1 1 − μν − μν coupled semisimple gauge theories. LYM ¼ 2TrF Fμν 2Trf fμν L ¼ TrðQi¯ =DQÞþTrðqi¯ =DqÞþTrðψ¯ i=DψÞ III. MINIMAL MODELS F L ¼ YTrðQ¯ HQ þ Q¯ H†Q ÞþyTrðq¯ hq þ q¯ h†q Þ In this section we introduce in concrete terms a family of Y L R R L L R R L ∂ †∂μ ∂ †∂μ semisimple gauge theories whose interacting fixed points LS ¼ Trð μH HÞþTrð μh hÞ will be analyzed exactly within perturbation theory in the L ¼ −UTrðH†HÞ2 − VðTrH†HÞ2 Veneziano limit. pot − uTrðh†hÞ2 −vðTrh†hÞ2 − wTrH†HTrh†h: ð25Þ A. Semisimple gauge theory We consider families of massless four-dimensional The trace Tr denotes the trace over both color and quantum field theories with a semisimple gauge group flavor indices, and the decomposition Q ¼ QL þ QR with 1 QL=R ¼ 2 ð1 γ5ÞQ is understood for all fermions Q and q. SUðNCÞ × SUðNcÞð24Þ Mass terms are neglected at the present stage as their effect is subleading to the main features developed below. In four ≥ 2 ≥ 2 for general non-Abelian factors with NC and Nc . dimensions, the theory is renormalizable in perturbation Specifically, our models contains SUðNCÞ gauge fields Aμ theory.
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The theory has nine classically marginal coupling con- Nψ ¼ 0 ð30Þ stants given by the two gauge couplings, the two Yukawa couplings, and five quartic scalar couplings. We write the interaction between gauge sectors reduces to effects them as mediated by the portal coupling αw ≠ 0, which are strongly 2 2 2 2 loop-suppressed. In this limit, results for fixed points and g1NC g2Nc Y NC y Nc α1 ¼ ; α2 ¼ ; α ¼ ; α ¼ ; running couplings fall back to those for the individual ð4πÞ2 ð4πÞ2 Y ð4πÞ2 y ð4πÞ2 gauge sectors [32]. Results for fixed points for general 2 2 uNF vN uNf vN Nψ are deferred to Appendix A. Here, we will set Nψ to a α ¼ ; α ¼ F ; α ¼ ; α ¼ f ; U ð4πÞ2 V ð4πÞ2 u ð4πÞ2 v ð4πÞ2 finite value, ð26Þ Nψ ¼ 1: ð31Þ where we have normalized the couplings with the appro- This leaves us with four free parameters. In order to achieve priate loop factor and powers of NC, Nc, NF and Nf in view of the Veneziano limit to be adopted below. Notice the exact perturbativity, we perform a Veneziano limit [49] by sending the number of colors and the number of flavors additional power of NF and Nf in the definitions of the scalar double-trace couplings. We normalize the quartic ðNC;Nc;NF;Nf Þ to infinity but keeping their ratios fixed. “portal” coupling as This reduces the set of free parameters of the model down to three, which we chose to be wN N α ¼ F f : ð27Þ N N N w ð4πÞ2 R ¼ c ;S¼ F ;T¼ f : ð32Þ NC NC Nc It is responsible for a mixing amongst the scalar sectors The ratio starting at tree level. Below, we use the shorthand notation β ≡∂α 1 2 i t i with i ¼ð ; ;Y;y;U;u;V;v;wÞ to indicate the N β-functions for the couplings (26). To obtain explicit F ¼ f ð33Þ N expressions for these, we exploit the formal results sum- F – marized in [41 43]. The semisimple character of the theory is then no longer a free parameter, but fixed as F ¼ RT=S is switched off if the Nψ messenger fermions (which carry from (32). By their very definiton, the parameters (32) charges under both gauge groups) are replaced by N1 and are positive semidefinite and can take values 0 ≤ F;R; N2 Yukawa-less fermions in the fundamental of SUðNCÞ S;T ≤∞. However, we will see below that their values are and SUðNcÞ, respectively, with further constrained if we impose perturbativity for all couplings. N1 ¼ NcNψ ;N2 ¼ NCNψ : ð28Þ C. Perturbativity to leading order If in addition αw ¼ 0, the theories (25) reduce to a “direct product” of simple gauge Yukawa theories with The RG evolution of couplings is analyzed within the (28). Also, in the limit where one of the gauge groups is perturbative loop expansion. To leading order (LO), the α ≡ 0 α ≡ 0 β − α2 switched off, 1 (or 2 ), one gauge sector and the running of the gauge couplings reads i ¼ Bi i (no sum), scalars decouples straightaway, and we are left with a with Bi the one-loop gauge coefficients for the gauge 0 simple gauge theory. Finally, if N1 ¼ ¼ N2, we recover coupling αi. In the Veneziano limit, the one-loop coeffi- the models of [32] in each gauge sector (displaying cients take the form asymptotic safety for certain field multiplicities). Below, we will find it useful to contrast results with those from the 4 B ¼ − ϵ : ð34Þ “direct product” limit. i 3 i
In terms of (32) and in the Veneziano limit, the parameters B. Free parameters and Veneziano limit ϵi are given by We now discuss the set of fundamentally free parameters of our models. On the level of the Lagrangian, the free 11 1 11 ϵ1 ¼ S þ R − ; ϵ2 ¼ T þ − : ð35Þ parameters of the theory are the matter field multiplicities 2 R 2 We can therefore trade the free parameters ðS; TÞ defined NC;Nc;NF;Nf ;Nψ : ð29Þ in (32) for ðϵ1; ϵ2Þ and consider the set Notice that the Nψ fermions ψ are centrally responsible for ϵ ϵ interactions between the gauge sectors. In the limit ð 1; 2;RÞð36Þ
085008-7 ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) as free parameters which characterize the matter content of the additional rescaling with R does not affect the relative the theory. Under the exchange of gauge groups we have sign between ϵ1 and ϵ. In this manner we have traded the free parameters ðϵ1; ϵ2;RÞ for −1 ðϵ1; ϵ2;RÞ → ðϵ2; ϵ1;R Þ: ð37Þ ðR; P; ϵÞ: ð42Þ 11 For fixed R, we observe that R − 2 ≤ ϵ1 < ∞ and 1 − 11 ≤ ϵ ∞ =R 2 2 < . Perturbativity in either of the gauge Notice that the parameter P can be expressed as couplings requires that both one-loop coefficients Bi are parametrically small compared to unity. Therefore we 1 þðN − 11 N Þ=N P f 2 c C 43 impose ¼ 1 − 11 ð Þ þðNF 2 NCÞ=Nc 0 < jϵ j ≪ 1: ð38Þ i in terms of the field multiplicities (29). It thus may take any −∞ ∞ This requirement of exact perturbativity in both gauge real value of either sign with 2 11 N − 11 N ϵ 1 F 2 C 11 Outside of this range, no physical values for S and T can be In this parametrization, the ratio of fermion flavor multi- found such that (38) holds true. Inside this range, physical plicities (33) becomes values are constrained within 0 ≤ S; T ≤ 11 − 2 . The 2 11 parameters (36) have a simple interpretation. The small 11R − 2 2R P 11R − 2 2 paramaters ϵ control the perturbativity within each of F ¼ þ − ϵ þ Oðϵ Þ: ð45Þ i 11 − 2R 11 − 2R R 11 − 2R the gauge sectors, whereas the parameter R controls the “interactions” between the two gauge sectors. It is the We also observe that the substitution presence of R which makes these theories intrinsically semisimple, rather than being the direct product of two ϵ → −1 −1 ϵ simple gauge theories. Perturbativity is no longer required ðR; P; Þ ðR ;P ;P Þð46Þ in the limit where one of the gauge sectors is switched off, and the constraint (39) is relaxed into relates to the exchange of gauge groups. The parametriza- tion (42) is most convenient for analysing the various 11 interacting fixed points and their scaling exponents (see 0 ≤ R< if α ≡ 0; 2 2 below). This completes the definition of our models. 2 085008-8 MORE ASYMPTOTIC SAFETY GUARANTEED PHYS. REV. D 97, 085008 (2018) 11 γ ϵ − α ξ α O α2 E. Running couplings beyond the leading order Q ¼ 2 þ 1 R Y þ 1 1 þ ð Þ; We now go beyond the leading order in perturbation 11 1 2 theory and provide the complete, minimal set of RG γ ¼ þ ϵ2 − α þ ξ2α2 þ Oðα Þ; q 2 R Y equations which display exact and weakly interacting fixed points. To that end, we must retain terms up to two loop γ ξ α ξ α O α2 ψ ¼ 1 1 þ 2 2 þ ð Þ; ð48Þ order in the gauge coupling, or else an interacting fixed point cannot arise. At the same time, in order to explore the where ξ1 and ξ2 denote the gauge fixing parameters for the feasibility of asymptotically safe UV fixed points we must first and second gauge group respectively. retain the Yukawa couplings [34], minimally at the leading The anomalous dimension for the scalar mass terms can nontrivial order which is one loop. Following [32] we refer be derived from the composite operator ∼M2TrH†H and to this level of approximation in the gauge-Yukawa sector ∼m2Trh†h. Introducing the mass anomalous dimension as next-to-leading order (NLO). In the presence of scalar 2 γM ¼ d ln M =d ln μ, and similarly for m, one finds fields, we also must retain the quartic scalar couplings at their leading nontrivial order. We refer to this approxi- 0 γ ¼ 8α þ 4α þ 2α þ Oðα2Þ mation of the gauge-Yukawa-scalar sector as NLO [48], M U V Y see Table II. This is the minimal order in perturbation γ 8α 4α 2α O α2 m ¼ u þ v þ y þ ð Þ; ð49Þ theory at which a fully interacting fixed point can be determined in all couplings with canonically vanishing to one-loop order. We also compute the running of the mass mass dimension. terms for the scalars In general, the RG flow for the gauge and Yukawa couplings at NLO0 is strictly independent of the scalar 2 2 2 2 β 2 γ 2 α O α α couplings owing to the fact that scalar loops only arise M ¼ MM þ Fm w þ ð ; mFÞ; 2 −1 2 2 2 starting from the two loop order in the Yukawa sector, and β 2 γ 2 α O α α m ¼ mm þ F M w þ ð ; mFÞ; ð50Þ at three (four) loop order in the gauge sector, if the scalars are charged (uncharged). Furthermore, the scalar sector ≡ 0 where the parameter F Nf =NF solely depends on R to at NLO depends on the Yukawa couplings, but not on the leading order in ϵ, see (45). Notice that the coupling αw gauge couplings owing to the fact that the scalars are induces a mixing amongst the different scalar masses uncharged. Consequently, we observe a partial decoupling already at one-loop level. of the gauge-Yukawa sector ðα1; α2; αY; αyÞ and the scalar Analogously, the anomalous dimension for the sector ðαU; αV; αu; αv; αwÞ. This structure will be exploited Δ 3 γ fermion mass operator is defined as F ¼ þ MF with systematically below to identify all interacting fixed points. γ ≡ μ MF d ln MF=d ln , and MF stands for one of the We begin with the gauge-Yukawa sector where we find fermion masses with F ¼ Q, q or ψ. Within perturbation the coupled beta functions (10) which are characterized by theory, the one loop contributions read ten loop coefficients Ci, Di, Ei, Fi and Gi (i ¼ 1, 2), together with the coefficients Bi given in (34) or, equiv- 13 2 alently, the perturbative control parameters (35). The one- γ ¼ α þ ϵ1 − R − 3α1 − þOðα Þ; MQ Y 2 loop coefficients arise in the Yukawa sector and take the values 13 1 2 γ ¼ α þ ϵ2 − − 3α2 þ Oðα Þ Mq y 2 R E1 ¼ 13 þ 2ðϵ1 − RÞ;F1 ¼ 6; γ −3 α α O α2 Mψ ¼ ð 1 þ 2Þþ ð Þ: ð51Þ 1 E2 ¼ 13 þ 2 ϵ2 − ;F2 ¼ 6: ð53Þ R For the fermion masses we have the running At the two-loop level we have six coefficients related to the β ¼ γ m ; β ¼ γ m ; β ¼ γ mψ : mQ MQ Q mq Mq q mψ Mψ gauge, Yukawa, and mixing contribution, which are found ð52Þ to be γ γ γ 26 11 2 We note that Mψ is manifestly negative. For MQ and Mq 25 ϵ 2 ϵ − 2 C1 ¼ þ 3 1;D1 ¼ 1 R þ 2 ;G1 ¼ R we observe that the gauge and Yukawa contributions arise with manifestly opposite signs in the parameter regime 26 1 11 2 2 C2 ¼ 25 þ ϵ2;D2 ¼ 2 ϵ2 − þ ;G2 ¼ (38), (39). Hence either of these may take either sign, 3 R 2 R depending on whether the gauge or Yukawa contributions dominate. ð54Þ 085008-9 ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) β − 11 2 ϵ − α2 4α α 2α A few comments are in order. First, the loop coefficients U ¼ ½ þ ð 1 RÞ Y þ Uð Y þ UÞ; 0 Di, Ei, Fi, Gi > as they must for any quantum field β 12α2 4α α 4α α α2 V ¼ U þ Vð V þ U þ YÞþ w; theory. Additionally we confirm that Ci > 0 [34], provided 1 the parameters ϵi are in the perturbative regime (38). 2 βu ¼ − 11 þ 2 ϵ2 − αy þ 4αuðαy þ 2αuÞ; Second, provided that R ¼ 0 in the expressions for ϵ1, R E1 and G1, and 1=R ¼ 0 in those for ϵ2, E2, and G2, the 2 2 β ¼ 12α þ 4α ðα þ 4α þ α Þþα ; system (10) falls back onto a direct product of simple v u v v u y w gauge-Yukawa theories, each of the type discussed in [32]. βw ¼ αw½8ðαU þ αuÞþ4ðαV þ αvÞþ2ðαY þ αyÞ : ð57Þ Notice that this limit cannot be achieved parametrically in R. The reason for this is the presence of Nψ fermions Their structure is worth a few remarks: First, in the β which are charged under both gauge groups. They con- Veneziano limit, w contains no term quadratic in the α O −1 −1 tribute with reciprocal multiplicity R ↔ 1=R to the coupling w as the coefficient is of the order ðNF Nf Þ Yukawa-induced loop terms Di and Ei as well as to the and suppressed by inverse powers in flavour multiplicities. β α mixing terms Gi. Exact decoupling of the gauge sectors Second, we notice that w comes out proportional to w. then becomes visible only in the parametric limit where Consequently, αw is a technically natural coupling accord- Nψ → 0 whereby all terms involving R or 1=R drop out. ing to the rationale of [50], unlike all the other quartic Finally, we note that the exchange of gauge groups interactions, implying that G1 ↔ G2 corresponds to R ↔ 1=R and S ↔ T, implying α 0 ϵ1 ↔ ϵ2 and P ↔ 1=P, respectively. Evidently, at the w ¼ ð58Þ symmetric point R ¼ 1 and ϵ1 ¼ ϵ2 (or P ¼ 1)wehave exact exchange symmetry between gauge group factors. constitutes an exact fixed point of the theory. Comparison Inserting (53), (54) and (34) into the general expression with (49) shows that the proportionality factor is the sum of β α γ γ (10), we obtain the perturbative RG flow for the gauge- the scalar mass anomalous dimensions, w ¼ wð M þ mÞ. α Yukawa system at NLO accuracy The quartic coupling w would be promoted to a free parameter characterizing a line of fixed points with exactly 4 26 11 2 marginal scaling provided that its beta function vanishes β ϵ α2 25 ϵ α3 − 2 ϵ − α2α identically at one loop. This would require the vanishing 1 ¼ 3 1 1 þ þ 3 1 1 1 R þ 2 1 Y of the sum of scalar anomalous mass dimensions at the 2 þ 2Rα α2; fixed point, 1 4 26 1 11 2 2 3 2 β2 ¼ ϵ2α þ 25 þ ϵ2 α − 2 ϵ2 − þ α α γ þ γ ¼ 0: ð59Þ 3 2 3 2 R 2 2 y M m 2 2 Below, however, we will establish that such scenarios are þ α α1; R 2 incompatible with vacuum stability (see Sec. V). Moreover, 2 β ¼½13 þ 2ðϵ1 − RÞ α − 6α α1; at interacting fixed points we invariably find that Y Y Y 1 2 γ γ 0 β ¼ 13 þ 2 ϵ2 − α − 6α α2: ð55Þ M þ m > ð60Þ y R y y as a consequence of vacuum stability. This implies that αw We observe that the running of Yukawa couplings at one constitutes an infrared free coupling at any interacting fixed loop is determined by the fermion mass anomalous point with a stable ground state. For the purpose of the dimension (51), present study, we therefore limit ourselves to fixed points with (58). We then observe that the running of the remaining scalar couplings is solely fuelled by the β ¼ 2γ α ; β ¼ 2γ α : ð56Þ Y MQ Y y Mq y Yukawa couplings. Furthermore, the scalar subsectors associated to the different gauge groups are disentangled 3 The result for the mass anomalous dimensions (51) can also in our approximation. Interestingly, the beta functions α α α α be derived diagrammatically from the flow of the Yukawa for ð U; VÞ and ð u; vÞ are related by the substitution vertices (55), thus offering an independent confirmation for R ↔ 1=R and ϵ1 ↔ ϵ2. Moreover, the double trace scalar the link (56). couplings do not couple back into any of the other Next, we turn to the scalar sector and the running of couplings and their fixed points are entirely dictated by quartic couplings to leading order in perturbation theory, the corresponding single trace scalar and the Yukawa which is one loop. At NLO0 accuracy, we have (55) together with the beta functions for the quartic scalar 3The degeneracy is lifted as soon as the quartic coupling couplings which are found to be αw ≠ 0, see (25), (26). 085008-10 MORE ASYMPTOTIC SAFETY GUARANTEED PHYS. REV. D 97, 085008 (2018) TABLE V. Gauge and Yukawa couplings at interacting fixed points following Table III to the leading order in ϵ and in terms of ðR; P; ϵÞ. Valid domains for ðϵ;P;RÞ in (61) are detailed in Tables VII, VIII. Fixed point Gauge couplings Yukawa couplings Type FP α 0 α 0 α 0 α 0 G G 1 1 ¼ , 2 ¼ , Y ¼ , y ¼ , · FP α − 4 ϵ α 0 α 0 α 0 BZ G 2 1 ¼ 75 R , 2 ¼ , Y ¼ , y ¼ , · FP α 0 α − 4 Pϵ α 0 α 0 G BZ 3 1 ¼ , 2 ¼ 75 R , Y ¼ , y ¼ , · FP α 2 ð13−2RÞRϵ α 0 α 4Rϵ α 0 GY G 4 1 ¼ 3 ð2R−1Þð3R−19Þ, 2 ¼ , Y ¼ ð2R−1Þð3R−19Þ, y ¼ , · FP α 0 α 2 ð13−2=RÞ Pϵ α 0 α 4Pϵ=R G GY 5 1 ¼ , 2 ¼ 3 ð2=R−1Þð3=R−19Þ R , Y ¼ , y ¼ ð2=R−1Þð3=R−19Þ, · FP α −4ð25−2P=RÞ ϵ α −4ð25−2R=PÞ Pϵ α 0 α 0 BZ BZ 6 1 ¼ 1863 R , 2 ¼ 1863 R Y ¼ , y ¼ , · 2 ð13−2RÞð25−2P=RÞ 4 25−2P=R FP7 α 2 Rϵα2 Rϵ GY BZ 1 ¼ 9 50R −343Rþ167 Y ¼ 3 50R −343Rþ167 · α 4 ð13−2RÞR=Pþð2R−1Þð19−3RÞ Pϵ α 0 2 ¼ 9 50R2−343Rþ167 R y ¼ 4 ð13−2=RÞP=Rþð2=R−1Þð19−3=RÞ FP8 α 2 Rϵα0 BZ · GY 1 ¼ 9 50=R −343=Rþ167 Y ¼ α 2 ð13−2=RÞð25−2R=PÞ Pϵ α 4 25−2R=P Pϵ 2 ¼ 9 50=R2−343=Rþ167 R y ¼ 3 50=R2−343=Rþ167 R 13−2 13−2 2−1 3 −19 6α 2 ð RÞ½ð =RÞP=RþðR Þð =R Þ 1 FP9 α 2 2 Rϵα GY GY 1 ¼ 9 ð19R −43Rþ19Þð2=R −13=Rþ2Þ Y ¼ 13−2R · 13−2 13−2 2 −1 3 −19 6α α 2 ð =RÞ½ð RÞR=Pþð R Þð R Þ Pϵ α 2 2 ¼ 9 ð19=R2−43=Rþ19Þð2R2−13Rþ2Þ R y ¼ 13−2=R 2 11 coupling [32]. This structure allows for a straightforward characteristic values 11 1 þðN − 11 N Þ=N B. Partially and fully interacting fixed points P f 2 c C ¼ 1 − 11 þðNF 2 NCÞ=Nc Gauge theories with (55), (57) can have two types of interacting fixed points: partially interacting ones where one Nc R ¼ gauge coupling takes the Gaussian fixed point (FP2,FP3, N C FP4,FP5), and fully interacting ones where both gauge sectors 11 remain interacting (FP6,FP7,FP8,FP9), see Table III.At sgnϵ ¼ sgn N þ N − N ; ð61Þ c F 2 C partially interacting fixed points, one gauge sector decouples and the semisimple theory with (55), (57) effectively reduces which only depend on the matter and gauge field multi- to a simple gauge theory. Simple gauge theories have three plicities (29), and Nψ ¼ 1. Results for general Nψ are possible types of perturbative fixed points: the Gaussian (G), given in Appendix A. We also observe (39), unless stated the Banks-Zaks (BZ), and gauge-Yukawa (GY) fixed points otherwise. Constraints on the parameters ðR; P; ϵÞ and for each independent linear combination of the Yukawa other information is summarized Figs. 1–5 and in couplings [34]. In our setting, at FP2 and FP4 we have that Tables VI–VIII for the various fixed points. Below, certain α2 ≡ 0, and the theory reduces to a simple gauge theory with 085008-11 ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) 4 26 11 2 1 2 3 2 For ϵ1 > 0 and R<2 asymptotic safety is guaranteed with β1 ¼ ϵ1α1 þ 25 þ ϵ1 α1 − 2 ϵ1 − R þ α1αY 3 3 2 ϑg < 0 < ϑy; ϑu; ϑv, showing that the UV fixed point has one β 13 2 ϵ − α2 − 6α α relevant direction. The scaling exponents reduce to those in Y ¼½ þ ð 1 RÞ Y Y 1 1 [32] for R ¼ 0. Conversely, for ϵ1 < 0 and R> the theory is β − 11 2 ϵ − α2 4α α 2α 2 U ¼ ½ þ ð 1 RÞ Y þ Uð Y þ UÞ; asymptotically free and the interacting fixed point is fully IR β 12α2 4α α 4α α 0 ϑ ϑ ϑ ϑ V ¼ U þ Vð V þ U þ YÞ; ð63Þ attractive with < g; y; u; v. Results straightforwardly translate to the partially interacting fixed points FP3 and FP5 0 at NLO accuracy, where the parameter R with where α1 ≡ 0.Theexplicitβ-functions in the other gauge 11 – ϵ ↔ ϵ 0 ≤ N1 sector are found from (63) (66) via the replacements 1 2 R ¼ < 2 ð64Þ ↔ 1 NC and R =R, leading to measures the number of Yukawa-less Dirac fermions N1 in 2 4 2 26 3 1 11 2 the fundamental representation in units of NC. Notice that N1 β2 ¼ ϵ2α þ 25 þ ϵ2 α − 2 ϵ2 − þ α α 3 2 3 2 R 2 2 y is related to Nψ via (28) in the theories (25).Ontheotherhand, N1 can be viewed as an independent parameter (counting the 1 2 β ¼ 13 þ 2 ϵ2 − α − 6α α2 Yukawa-less fermions in the fundamental representation of y R y y the gauge group) if one were to switch off the semisimple 1 0 2 character of the theory. For R ¼ the theory (63) reduces to βu ¼ − 11 þ 2 ϵ2 − αy þ 4αuðαy þ 2αuÞ; the one investigated in [32]. The lower bound on R (39) is R 2 relaxed in (64), because the requirement of perturbativity for βv ¼ 12αu þ 4αvðαv þ 4αu þ αyÞ: ð67Þ an interacting fixed point in the other gauge sector has become redundant. We observe the R-independent Banks-Zaks (BZ) Evidently, the coordinates of the fully interacting gauge- 4 fixed point α1 ¼ 75 ϵ1 which is, invariably, IR. To leading Yukawa fixed point and the corresponding universal scaling order in ϵ1 we also find a gauge-Yukawa (GY) fixed point exponents of (67) are given by (65), (66) after obvious replacements. Moreover, in (67) the parameter R with 26−4R ϵ1 α ¼ 1 11 g 57−9R1−2R 0 ≤ N2 ¼ < 2 ð68Þ 4 ϵ1 R Nc α ¼ Y 19−3R1−2R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi measures the number of Yukawa-less Dirac fermions N2 in 23−4R−1 ϵ1 the fundamental representation in units of N ,see(28).The α ¼ c U 19−3R 1−2R only direct communication between the different gauge ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p sectors in (25) is through the off-diagonal two-loop gauge −2 23−4Rþ 20−4Rþ6 23−4R ϵ1 α ¼ : ð65Þ contributions G . Were it not for the fermions ψ which are V 19−3R 1−2R i charged under both gauge groups, the theory (25) with (55), “ ” For ϵ1 > 0, the GY fixed point is UV and physical as long as (57) would be the direct product of the simple model (63), 0≤R<1. It can be interpreted as a “deformation” of the UV (64) with its counterpart (67), (68). In this limit we will find 2 “ ” fixed point analyzed in [32] owing to the presence of charged nine direct product fixed points with scaling exponents 1 from each pairing of the possibilities (G, BZ, GY) in each Yukawa-less fermions. Once R> , however, the fixed point is 2 sector. physical iff ϵ1 <0 where it becomes an IR fixed point. This new Below, we contrast findings for the full semisimple regime is entirely due to the Yukawa-less fermions and does setting (55), (57) with those from the “direct product” not arise in the model of [32]. This pattern can also be read off limit in order to pin-point effects which uniquely arise from from the scaling exponents, which, at the gauge Yukawa fixed the semisimple character of the theories (25). point and to the leading nontrivial order in ϵ1,aregivenby At any of the partially interacting fixed points, the 104 1 − 2 R ϵ2 semisimple character of the theory becomes visible in ϑ − 13 1 the noninteracting sector. In fact, contributions from the ψ g ¼ 171 1 − 3 1 − 2 19 R R → 0 fermions modify the effective one-loop coefficient Bi Bi 4 1 ϵ ϑ 1 according to y ¼ 19 3 1 − 2 1 − 19 R R 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α1 ¼ 0∶ B1 → B1 ¼ B1 þ G1α2 16 23 − 4R ϵ1 ϑ ¼ α 0∶ → 0 α u 19 − 3R 1 − 2R 2 ¼ B2 B2 ¼ B2 þ G2 1: ð69Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 20 þ 6 23 − 4R − 4R ϵ1 “ ” ϑ ¼ : ð66Þ No such effects can materialize in a direct product limit. v 19 − 3R 1 − 2R Moreover, these contributions always arise with a positive 085008-12 MORE ASYMPTOTIC SAFETY GUARANTEED PHYS. REV. D 97, 085008 (2018) 0 coefficient (B >B) and are absent if Nψ ¼ 0 (where Gi ¼ 0). For Nψ ≠ 0, asymptotic freedom can thereby be changed into infrared freedom, but not the other way around. This result is due to the fact that the Yukawa 1 couplings are tied to individual gauge groups, and so by 1 11 this structure we cannot have any Yukawa contributions 25 to B0. In principle, the opposite effect can equally arise: it 2 would require Yukawa couplings which contribute to both P 0 gauge coupling β-functions, and would therefore have to 1 0, P 0, B2' 0 3 involve at least one field which is charged under both 2 0, P 0, B2' 0 gauge groups [34]. Table VI shows the effective one loop 3 0, P 0, B2' 0 1 coefficients at partially interacting fixed points as a function of field multiplicities. FP2 C. Gauss with Banks-Zaks 2 1 1 2 11 11 2 2 Next, we discuss all fixed points one-by-one, and deter- R mine the valid parameter regimes ðR; P; ϵÞ for each of them. We recall that Nψ ¼ 1 in our models. Whenever appropriate, we also compare results with the “direct product” limit, 2 whereby the diagonal contributions from the Yukawa-less 25 11 ψ-fermions are retained but their off-diagonal contributions 1 to the other gauge sectors suppressed (see Sec. IV B). This 1 comparison allows us to quantify the effect related to the semisimple nature of the models (25). For convenience and better visibility, we scale the axes in P 0 Figs. 1–5 as 3 1 0, P 0, B1' 0 2 0, P 0, B ' 0 X 1 X → where X ¼ P or R; ð70Þ 3 0, P 0, B1' 0 1 þjXj 1 2 11 and within their respective domains of validity R ∈ ð11 ; 2 Þ and P ∈ ð−∞; ∞Þ. The rescaling permits easy display of FP3 the entire range of parameters. 2 1 1 2 11 11 2 2 Figure 1 shows the results for FP2 (BZ · G, upper) and R FP3 (G · BZ, lower panel), and parameter ranges are given in Table VII. We observe that the Banks-Zaks fixed point FIG. 1. The phase space of parameters (61) for the partially always requires an asymptotically free gauge sector. Hence, interacting fixed points FP2 (upper panel) and FP3 (lower panel) FP2 exists for any R as long as ϵ < 0. Provided that Pϵ < 0, where one of the two gauge sectors remains interacting and all the other gauge sector either remain asymptotically free Yukawa couplings vanish. The inset indicates the different (region 1) or becomes infrared free (region 2). On the other parameter regions and conditions for existence, including whether 0 0 hand, if Pϵ > 0, the other gauge sector is invariably the non-interacting gauge sector is asymptotically free ðB > Þ or infrared free B0 < 0 , see Tables VI, VII. infrared free. This is a consequence of (69) which states ð Þ that the interacting gauge sector can turn asymptotic freedom of the non-interacting gauge sector into infrared Under the exchange of gauge groups we have freedom (region 2), but not the other way around. The ðR; P; ϵÞ ↔ ðR−1;P−1;PϵÞ, see (46). On the level of existence of the parameter region 2 is thus entirely due to Fig. 1 this corresponds to a simple rotation by 180 degree the semisimple character of the theory which cannot arise around the symmetric points ðR; PÞ¼ð1; 1Þ (for P>0) from a “direct product.” and ðR; PÞ¼ð1; −1Þ (for P<0), owing to the rescaling of The Banks-Zaks fixed point is invariably attractive in the parameters. Consequently, the results for FP3 can be gauge coupling, and repulsive in the Yukawa coupling. The deduced from those at FP2 by a simple rotation, see eigenvalue spectrum in the gauge-Yukawa sector is there- Fig. 1. More generally, this exchange symmetry relates fore of the form (17) or (18), depending on whether the free the partially interacting fixed points FP2 ↔ FP3 (Fig. 1), gauge sector is asymptotically free or infrared free, see FP4 ↔ FP5 (Fig. 2), and the fully interacting fixed points Table VII. FP7 ↔ FP8 (Fig. 4). The exchange symmetry is manifest at 085008-13 ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) the fully interacting fixed points FP6 (Fig. 3) and sector either remains asymptotically free (region 2) or FP9 (Fig. 5). becomes infrared free (region 1), whereas for Pϵ > 0 the other gauge sector invariably remains infrared free D. Gauss with Gauge-Yukawa (region 3). Conversely, if the interacting gauge coupling is infrared free ðϵ > 0Þ and provided that Pϵ < 0, the other In Fig. 2 we show the domains of existence for FP4 gauge sector either remains asymptotically free (region 5) (GY · G, upper) and FP5 (G · GY, lower panel). We observe or becomes infrared free (region 4), whereas for Pϵ > 0 that the fixed point exists for any parameter choice though the other gauge sector invariably remains infrared free its features vary with matter multiplicities. Specifically, (region 6). Moreover, as explained in Table VI, the interact- for FP4, six qualitatively different parameter regions are ing gauge sector can turn asymptotic freedom of the non- found. If the interacting gauge coupling is asymptotically interacting gauge sector into infrared freedom (region 1 free ðϵ < 0Þ and provided that Pϵ < 0, the other gauge and 4). The eigenvalue spectrum in the gauge-Yukawa sector has therefore no relevant eigendirection (20) in region 1 FP4 and 3, one relevant eigendirection (18) in region 4 and 6, two relevant eigendirections (19) in region 2, and three relevant eigendirections (17) in region 5, see Table VII. 1 We make the following observations. First, we note that 2 11 FP4 in region 1 and 3 corresponds to a fully attractive IR 25 fixed point with all RG trajectories terminating in it. The 6 1 fixed point then acts as an infrared “sink” for massless 0 P trajectories and all canonically marginal couplings of the 278 4 3 1421 1 0, P 0, B2' 0 theory. Once scalar masses are switched on, RG flows may 5 2 0, P 0, B2' 0 run away from the hypercritical surface of exactly massless 1 3 0, P 0, B2' 0 theories, leading to massive phases with or without 4 0, P 0, B2' 0 spontaneous breaking of symmetry. The quantum phase 5 0, P 0, B2' 0 transition at FP4 in region 1 and 3 is of the second order. 6 0, P 0, B2' 0 Notice that in the “direct product” limit only models with Pϵ > 0 > ϵ and R>1 (analogous to region 3) would lead 2 1 1 2 11 2 11 2 2 to a fully infrared attractive “sink.” Hence, the availability R of region 1 is an entirely new effect, solely due to the ψ fermions and the semisimple nature of our models. We conclude that the semisimple structure opens up new types 1 FP5 25 of fixed points which cannot be achieved through a product 11 structure. In region 2, we find that FP4 has two relevant 1 2 eigendirections as it would in direct product settings. Second, in regions 4 and 6, FP4 shows a single relevant eigendirection. In the direct product limit, only models with 6 P; ϵ > 0 and R<1 (analogous to region 6) would lead to a P 2 0 single relevant direction. Again, the availability of region 4 3 5 1 0, P 0, B1' 0 is a novel feature, and solely due to the ψ fermions and thus 2 0, P 0, B1' 0 a consequence of the semisimple nature of the model. 1 3 0, P 0, B1' 0 In the parameter region 5 the fixed point shows the 4 0, P 0, B1' 0 largest number of UV relevant directions as it would 5 0, P 0, B1' 0 without the ψ fermions. Moreover, in this parameter regime 1421 6 0, P 0, B ' 0 the Gaussian fixed point has only two relevant directions 1 4 278 (ϵ > 0;Pϵ < 0). Therefore FP4 in region 5 qualifies as an 2 1 1 2 11 11 2 2 asymptotically safe UV fixed point. On the other hand, in R region 2,4 and 6, it takes the role of a cross-over fixed point. Results for FP5 (Fig. 2, lower panel) follow from those for FIG. 2. The phase space of parameters for the partially interact- FP4 via (46), and the distinct regions stated for FP5 relate to ing fixed points FP4 and FP5, where the gauge and Yukawa the same physics as those for FP4. coupling in one gauge sector take interacting fixed points while those of the other sector remain trivial. The insets indicate the E. Banks-Zaks with Banks-Zaks different parameter regions and conditions for existence, and whether the non-interacting gauge sector is asymptotically free Next, we turn to fully interacting fixed points where ðB0 > 0Þ or infrared free ðB0 < 0Þ, see Tables VI, VII. both gauge couplings are nonvanishing, see Table VIII.In 085008-14 MORE ASYMPTOTIC SAFETY GUARANTEED PHYS. REV. D 97, 085008 (2018) general, the eigenvalue spectrum is determined through of certain parameter regions (white regions). We conclude (22) with solutions (23), with ϵ taking the role of the that the semisimple nature of the theory leads to parameter parameter B. In the direct product limit, fully interacting restrictions without otherwise changing the overall appear- fixed points reduce to direct products from each pairing of ance of the fixed point. the possibilities (BZ, GY) in each of the simple gauge ≠ 0 ψ sectors. For Nψ , the fermions introduce a direct F. Banks-Zaks with Gauge-Yukawa mixing between the gauge groups and we may then expect to find something close to a product structure, potentially At the interacting fixed points FP7 (BZ · GY, upper modified by new effects parametrized via R in fixed points panel), and FP8 (GY · BZ, lower panel), we have that both not involving Gaussian factors. gauge and one of the Yukawa couplings are nontrivial. Our results for the condition of existence and the eigenvalue The first such fixed point is FP6 (BZ · BZ), where each gauge sector achieves a Banks-Zaks fixed point. spectra are displayed in Fig. 4. By definition, this type of Yukawa couplings play no role, see Fig. 3. The fixed point invariably requires ϵ < 0 and Pϵ < 0 and entails an eigenvalue spectrum with two relevant directions of order 1 2 Oðϵ2Þ, and two irrelevant directions of order OðϵÞ asso- 25 11 ciated to the Yukawas, 1 ϑ ϑ 0 ϑ ϑ 11 1; 2 < < 3; 4: ð71Þ 25 The quartics are marginally irrelevant. The Gaussian is P 0 necessarily the UV fixed point in these settings which 278 1421 makes FP6 a cross-over fixed point. The accessible param- 1 0, P 0, 2 0 eter region, shown in Fig. 3, is invariant under the exchange 3 2 0, P 0, 2 0 1 of gauge groups (46). The direct product limit has quali- 3 0, P 0, 2 0 tatively the same spectrum (71). The main effect due to the semisimple character of the theory relates to the exclusion FP7 2 1 1 2 11 11 2 2 FP6 R FP 25 25 8 11 11 1 11 P 1 0, P 0, 1,2 0 3,4 25 2 1 P 0 11 3 25 1 0, P 0, 2 0 1 2 0, P 0, 2 0 3 0, P 0, 2 0 1421 0 278 2 1 1 2 11 11 2 2 2 1 1 2 11 R 11 2 2 R FIG. 3. The phase space of parameters for the interacting fixed point FP6 (red) where both gauge sectors take interacting and FIG. 4. The phase space of parameters for the fixed points FP7 physical fixed points while all Yukawa couplings vanish. The and FP8 where two gauge and one of the Yukawa couplings take eigenvalue spectrum at the fixed point always displays exactly two interacting and physical fixed points, while the other Yukawa relevant eigenvalues of OðϵÞ and two irrelevant eigenvalues of coupling remains trivial. The inset indicates the signs for ϵ and 2 order Oðϵ Þ,seeTableVIII. Note that this fixed point invariably Pϵ, together with the sign for the eigenvalue ϑ2, Table VIII (see requires asymptotic freedom for both gauge sectors (see main text). main text). 085008-15 ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) fixed point requires that either ϵ < 0 or Pϵ < 0, or both, 1 FP9 meaning that at least one of the gauge sectors is asymp- 25 totically free. In Fig. 4, this relates to three different 11 parameter regions (see inset for the color coding). In region 1 2 1 and 2, the theory is asymptotically free in both gauge 11 sectors, whereas in region 3 the theory is asymptotically 25 free in only one gauge sector. We observe that large regions 1 of parameter space are excluded. Valid parameter regions P 0 278 are further distinguished by their eigenvalue spectrum 4 which either takes the form (19) or (18), meaning that 1421 1 0, P 0, 1 0 minimally one and maximally two eigenoperators con- 2 0, P 0, 0 structed out of the gauge kinetic terms and the Yukawa 1 1 3 0, P 0, 1 0 interactions are UV relevant, ϑ1 < 0 < ϑ3; ϑ4. The sign of 4 0, P 0, 1 0 1421 ϑ2 depends on the matter field multiplicities. In region 1 3 278 and 3, and for either of FP7 and FP8, we find that 2 1 1 2 11 ϑ ϑ 0 ϑ ϑ 11 2 2 1; 2 < < 3; 4: ð72Þ R In region 2, conversely, we have FIG. 5. The phase space of parameters for the fully interacting fixed point FP9 where all gauge and all Yukawa couplings are ϑ1 < 0 < ϑ2; ϑ3; ϑ4: ð73Þ nontrivial. The coloured regions relate to the portions of parameter space where the fully interacting fixed point is physical. The Hence, at FP7 and in the regime where both gauge sectors inset provides additional information including the sign for the ϑ are asymptotically free ðP>0 > ϵÞ, two types of valid eigenvalue 4 (see main text). fixed points are found. For sufficiently low R 085008-16 MORE ASYMPTOTIC SAFETY GUARANTEED PHYS. REV. D 97, 085008 (2018) TABLE VI. Shown are the effective one-loop coefficients B0 for Valid fixed points then correspond to region 3 or region 4, the noninteracting gauge coupling at FP2,FP3,FP4, and FP5, and respectively. In either of these cases, the eigenvalue their dependence on model parameters. B0 > 0 corresponds to spectrum shows a single relevant direction, (75). This 0 asymptotic freedom. Notice that B changes sign across the agrees qualitatively with the eigenvalue spectrum in the 2 25 25 2 ˜ boundaries P ¼ R= ; R= ;XðRÞ, and XðRÞ, respectively, direct product limit. We conclude, once more, that the main with X and X˜ given in (B1). impact of the ψ fields relates to the boundaries in parameter Fixed point B0 Coefficient space which restrict the fixed point’s domain of availability. ϵ 0 0 4 2 Pϵ Finally, for ;P> , the theory is infrared free in FP2 B ¼ − ð1 − R=PÞ 2 3 25 R both gauge sectors. We observe that no such interacting FP 0 − 4 1 − 2 ϵ 3 B1 ¼ 3 ð 25 P=RÞR fixed point arises, irrespective of matter multiplicities. FP 0 − 4 1 − Pϵ 4 B2 ¼ 3 ð XðRÞ=PÞ R Interestingly though, such fixed points do exist in the 0 4 ˜ FP5 B1 ¼ − 3 ð1 − P=XðRÞÞRϵ direct product limit with spectrum ϑ1; ϑ2 < 0 < ϑ3; ϑ4: ð76Þ and is therefore a novel feature, entirely due to the semi- simple nature of the theory. In this regime, FP9 corresponds to a crossover fixed point (the Gaussian is the UV fixed The reason for their nonexistence in our models is the point) with a single unstable direction where trajectories presence of the ψ fermions. The requirement of perturba- escape either towards a weakly coupled IR fixed point, or tivity in both gauge couplings then leads to limitations on towards a regime of strong coupling with (chiral) symmetry the parameter R which cannot be satisfied at FP9 with breaking, confinement, or infrared conformality. eigenvalue spectrum (76). This result provides us with For ϵ > 0 >Por Pϵ > 0 > ϵ, the theory is asymptoti- an example where the semisimple nature of the theory cally free in one and infrared free in the other gauge sector. “disables” a fixed point. This completes the overview of TABLE VII. Parameter regions where the partially interacting fixed points FP1–FP5 exist, along with regions of relevancy for eigenvalues and effective one-loop terms, where applicable. The boundary functions XðRÞ and X˜ ðRÞ are given in (B1). The coefficient B0 for the gauge coupling at the Gaussian fixed point is given in Table VI. Parameter range Fixed point sign ϵ RPEigenvalue spectrum Info 2 11 FP1 ð11 ; 2 Þð−∞; þ∞Þ (16), (19),or(20) Gaussian FP2 Fig. 1 (upper panel) 2 11 2 − ð11 ; 2 Þð25 R; þ∞Þ ϑ1;2;3 ≤ 0 < ϑ4 region 1 2 11 2 − ð11 ; 2 Þð0; 25 RÞ ϑ1 < 0 ≤ ϑ2;3;4 region 2 2 11 − ð11 ; 2 Þð−∞; 0Þ ϑ1 < 0 ≤ ϑ2;3;4 region 3 FP3 Fig. 1 (lower panel) 2 11 25 − ð11 ; 2 Þð0; 2 RÞ ϑ1;2;3 ≤ 0 < ϑ4 region 1 2 11 25 − ð11 ; 2 Þð2 R; þ∞Þ ϑ1 < 0 ≤ ϑ2;3;4 region 2 2 11 þð11 ; 2 Þð−∞; 0Þ ϑ1 < 0 ≤ ϑ2;3;4 region 3 FP4 Fig. 2 (upper panel) 1 11 − ð2 ; 2 Þð−∞;XðRÞÞ 0 ≤ ϑ1;2;3;4 region 1 & 3 1 11 − ð2 ; 2 ÞðXðRÞ; þ∞Þ ϑ1;2 ≤ 0 < ϑ3;4 region 2 2 1 þð11 ; 2ÞðXðRÞ; þ∞Þ ϑ1 < 0 ≤ ϑ2;3;4 region 4 & 6 2 1 þð11 ; 2Þð−∞;XðRÞÞ ϑ1;2;3 ≤ 0 < ϑ4 region 5 FP5 Fig. 2 (lower panel) 2 ˜ − ð11 ; 2ÞðXðRÞ; þ∞Þ 0 ≤ ϑ1;2;3;4 region 1 2 ˜ − ð11 ; 2Þð0; XðRÞÞ ϑ1;2 ≤ 0 < ϑ3;4 region 2 2 þð11 ; 2Þð−∞; 0Þ 0 ≤ ϑ1;2;3;4 region 3 11 ˜ − ð2; 2 Þð−∞; XðRÞÞ ϑ1 < 0 ≤ ϑ2;3;4 region 4 11 ˜ − ð2; 2 ÞðXðRÞ; 0Þ ϑ1;2;3 ≤ 0 < ϑ4 region 5 11 þð2; 2 Þð0; þ∞Þ ϑ1 < 0 ≤ ϑ2;3;4 region 6 085008-17 ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) TABLE VIII. Parameter regions where the fully interacting fixed points FP6–FP9 exist, along with the eigenvalue spectrum for the various parameter regions. Parameter range Fixed point Sign ϵ RPEigenvalue spectrum Info 2 11 2 25 FP6 − ð11 ; 2 Þð25 R; 2 RÞ ϑ1;2 < 0 < ϑ3;4 Fig. 3 FP7 Fig. 4 (upper panel) 2 1 25 − ð11 ; 2Þð2 R; þ∞Þ ϑ1;2 < 0 < ϑ3;4 region 1 1 25 − ð2 ;R1Þð2 R; XðRÞÞ ϑ1;2 < 0 < ϑ3;4 region 1 11 25 − ðR1; 2 ÞðXðRÞ; 2 RÞ ϑ1 < 0 < ϑ2;3;4 region 2 2 1 þð11 ; 2Þð−∞;XðRÞÞ ϑ1;2 < 0 < ϑ3;4 region 3 FP8 Fig. 4 (lower panel) 11 ˜ 2 − ðR4; 2 ÞðXðRÞ; 25 RÞ ϑ1;2 < 0 < ϑ3;4 region 1 & 3 2 2 ˜ − ð11 ;R4Þð25 R; XðRÞÞ ϑ1 < 0 < ϑ2;3;4 region 2 FP9 Fig. 5 2 1 ˜ − ð11 ; 2ÞðXðRÞ; þ∞Þ ϑ1 < 0 < ϑ2;3;4 region 1 1 ˜ − ð2 ;R2ÞðXðRÞ;XðRÞÞ ϑ1 < 0 < ϑ2;3;4 region 1 11 ˜ − ðR3; ÞðXðRÞ;XðRÞÞ ϑ1 < 0 < ϑ2 3 4 region 1 & 4 2 ˜ ; ; − ðR2;R3ÞðXðRÞ; XðRÞÞ 0 < ϑ1;2;3;4 region 2 2 1 þð11 ; 2Þð−∞;XðRÞÞ ϑ1 < 0 < ϑ2;3;4 region 3 interacting fixed points in the gauge-Yukawa sector and fixed points are. Note also that the slope of the nullcline their key properties. remains positive and finite for all R within the domain (39). Hence strict perturbativity in the Yukawa couplings follows V. SCALAR FIXED POINTS AND from strict perturbativity in the gauge couplings, in accord VACUUM STABILITY with the general discussion in [34] based on dimensional analysis. In this section, we analyse the scalar sector and establish Next we turn to the scalar nullclines. Since the beta conditions for stability of the quantum vacuum. We also functions for the two scalar sectors decouple at this order, provide results for all scalar couplings at all interacting we may analyse their nullclines individually.4 All results for fixed points, Table IX. the subsystem ðαU; αVÞ can straightforwardly be translated to the subsystem ðα ; α Þ by substituting R ↔ 1=R, also A. Yukawa and scalar nullclines u v using (38). Furthermore, since the scalars are uncharged, Following [34], we begin by exploiting the results (53) to their one loop beta functions are independent of the gauge express the Yukawa nullclines in terms of the gauge coupling. Dimensional analysis then shows that all non- couplings and the parameter R. To leading order in the trivial scalar nullclines are proportional to the correspond- small expansion parameters (38), and using (53) together ing Yukawa coupling [34]. The scalar nullclines represent β 0 with (10), the nontrivial Yukawa nullclines Y ¼ and exact fixed points of the theory provided the Yukawa β 0 y ¼ take the explicit form couplings take interacting fixed points. Perturbativity of scalar couplings at an interacting fixed point then follows α 6 α 6 Y ¼ ; y ¼ : ð77Þ from the perturbativity of Yukawa couplings which, in turn, α1 13 − 2R α2 13 − 2=R follows from perturbativity in the gauge couplings. Specifically, the nullclines for the single trace scalar For fixed gauge couplings, we observe that the Yukawa couplings are found from (57) by resolving βU ¼ 0 for αU. couplings are enhanced over their values in the absence of We find two solutions the fermions ψ. The relevance of nullcline solutions (77) is as follows. By their very definition, the Yukawa couplings α 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi no longer run with the RG scale when taking the values U −1 23 − 4 α ¼ 4 R : ð78Þ (77). If at the same time the gauge couplings take fixed Y points on their own, the nullcline relations then provide us with the correct fixed point values for the Yukawa cou- 4This simplification solely arises provided the mixing coupling plings. Evidently, (77) together with (39) guarantees that αw takes its exact Gaussian fixed point (58). For nontrivial αw the the Yukawa fixed points are physical as long as the gauge nullclines take more general forms. 085008-18 MORE ASYMPTOTIC SAFETY GUARANTEED PHYS. REV. D 97, 085008 (2018) TABLE IX. Quartic scalar couplings at all weakly interacting fixed points to leading order in ϵ following Table III using the auxiliary functions (92). Same conventions as in Table V. Within the admissible parameter ranges (Tables VII, VIII) we observe vacuum stability. Fixed point Quartic scalar couplings FP α 0 α 0 α 0 α 0 1−3 U ¼ , V ¼ , u ¼ , v ¼ , FP α 4F1ðRÞRϵ α 4F2ðRÞRϵ α 0 α 0 4 U ¼ ð2R−1Þð3R−19Þ, V ¼ ð2R−1Þð3R−19Þ, u ¼ , v ¼ , FP α 0 α 0 α 4F1ð1=RÞPϵ=R α 4F2ð1=RÞPϵ=R 5 U ¼ , V ¼ , u ¼ ð2=R−1Þð3=R−19Þ, v ¼ ð2=R−1Þð3=R−19Þ, FP α 0 α 0 α 0 α 0 6 U ¼ , V ¼ , u ¼ , v ¼ , 4 ð25−2P=RÞF1ðRÞ 4 ð25−2P=RÞF2ðRÞ FP7 α 2 Rϵ α 2 Rϵ α 0 α 0 U ¼ 3 50R −343Rþ167 , V ¼ 3 50R −343Rþ167 , u ¼ , v ¼ , 4 ð25−2R=PÞF1ð1=RÞ Pϵ 4 ð25−2R=PÞF2ð1=RÞ Pϵ FP8 α 0 α 0 α 2 α 2 U ¼ , V ¼ , u ¼ 3 50=R −343=Rþ167 R , v ¼ 3 50=R −343=Rþ167 R , 4 ½ð13−2=RÞP=Rþð2=R−1Þð3=R−19Þ F1ðRÞ 4 ½ð13−2RÞR=Pþð2R−1Þð3R−19Þ F1ð1=RÞ Pϵ FP9 α 2 2 Rϵ α 2 2 U ¼ 3 ð19R −43Rþ19Þð2=R −13=Rþ2Þ , u ¼ 3 ð19=R −43=Rþ19Þð2R −13Rþ2Þ R , 4 ½ð13−2=RÞP=Rþð2=R−1Þð3=R−19Þ F2ðRÞ 4 ½ð13−2RÞR=Pþð2R−1Þð3R−19Þ F2ð1=RÞ Pϵ α 2 2 Rϵ α 2 2 V ¼ 3 ð19R −43Rþ19Þð2=R −13=Rþ2Þ , v ¼ 3 ð19=R −43=Rþ19Þð2R −13Rþ2Þ R . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Note that the double trace coupling does not couple back v −2 23 − 4 20 − 4 6 23 − 4 into the running of the single trace coupling. Within the α ¼ 4 =R =R þ =R ; α 0 α y parameter range (39) we observe that Uþ > > U−. Next, we consider the nullclines for the double-trace quartic ð82Þ α α β 0 coupling V. Inserting Uþ into V ¼ , we find a pair of nullclines given by α ≥ 0 0 α α and we end up with uþ together with > vþ > v−. α qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi On the other hand, the solution u− does not lead to real α 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi solutions for α . This completes the overview of Yukawa V −2 23 − 4 20 − 4 6 23 − 4 v ¼ R R þ R : and scalar nullcline solutions. αY 4 ð79Þ B. Stability of the vacuum We are now in a position to reach firm conclusions Both nullclines take real values for all R within the concerning the stability of the ground state at interacting α ≥ 0 range (39), and we end up with Uþ together with fixed points. The reason for this is that this information is 0 α α α β 0 > Vþ > V−. Analogously, inserting U− into V ¼ , encoded in the scalar nullclines. The explicit form of the we find a second pair of nullclines given by fixed point in the gauge-Yukawa sector is not needed. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi To that end, we recall the stability analysis for potentials of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α 2 1 p p the form V ¼ 2 23 − 4R 20 − 4R − 6 23 − 4R : αY 4 ∝ α † 2 α † 2 ð80Þ W UTrðH HÞ þ V=NFðTrH HÞ ; ð83Þ In this case, however, the result (80) comes out complex In the limit where αw ¼ 0 the scalar field potential in our α within the parameter range (39), meaning that even if Y models (25) are given by (83) together with its counterpart takes a real positive fixed point the corresponding scalar ðH; αU; αVÞ ↔ ðh; αu; αvÞ. For potentials of the form (83), fixed point is invariably unphysical. the general conditions for vacuum stability read [48,51] The replacement R → 1=R in (78) and (79), (80) allows us to obtain explicit expressions for the nullclines for α α α α α 0 α α 0 u = y and v = y. The real solutions are given by aÞ U > and U þ V > α 0 α α 0 bÞ U < and U þ V=NF > ð84Þ α 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ¼ −1 23 − 4=R ð81Þ αy 4 and similarly for ðαU; αVÞ ↔ ðαu; αvÞ. In the Veneziano limit, case (b) effectively becomes void and cannot be α ≥ 0 α α α α with uþ > u−. The solution uþ leads to real satisfied for any U, irrespective of the sign of V . Inserting nullclines for the double-trace coupling αv given by the fixed points into (84) we find 085008-19 ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C. Portal coupling α α Y 20 − 4R 6 23 − 4R Uþ þ Vþ ¼ 4 þ þ Now we clarify whether the stability of the vacuum is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi affected by the presence of the “portal” coupling αw ≠ 0 − 23 − 4R − 1 ≥ 0; which induces a mixing between the scalar sectors. In this qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi case the scalar potential is given by W ¼ −L in (25), α pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pot α α Y − 20 − 4R 6 23 − 4R † 2 † 2 † 2 Uþ þ V− ¼ 4 þ W ¼ UTrðH HÞ þ VðTrH HÞ þ uTrðh hÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ vðTrh†hÞ2 þ wTrðH†HÞTrðh†hÞ; ð88Þ − 23 − 4 − 1 ≤−α R Y; ð85Þ where H and h are NF × NF and Nf × Nf matrices, respectively. Following the reasoning of [48,51],weobserve ⊗ ⊗ Stability of the quantum vacuum is evidently achieved at that the potential has a global UðNFÞL UðNFÞR α α UðN Þ ⊗ UðN Þ symmetry which allows us to bring the fixed point ð Uþ; VþÞ following case (a) and irre- f L f R spective of the value for the Yukawa fixed point as long as each of the fields into a real diagonal configuration, H ¼ α 0 diagðΦ1;Φ2;…Þ and h ¼ diagðϕ1;ϕ2;…Þ.Asthepotential Y > . The potential (83) becomes exactly flat at the fixed 4 point iff R ¼ 11. In this case, higher order or radiative is homogeneous in either field, WðcH; chÞ¼c WðPH; hÞ,it 2 Φ2 corrections must be taken into consideration to guarantee sufficestoguaranteepositivityonafixedsurfaceP i i ¼ 1 ϕ2 stability in the presence of flat directions. Stability is not ¼ j j which is implemented using Lagrange multi- α α Λ λ achieved at the fixed point ð Uþ; V−Þ, for any R. Turning pliers and .From to the second scalar sector, we find ∂ W 2Φ 2 Φ2 2 − 2Λ ∂Φ ¼ ið U i þ V þ w Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αy α α 20 − 4 6 23 − 4 ∂W 2 uþ þ vþ ¼ þ =R þ =R 2ϕ 2 ϕ 2 − 2λ 4 ∂ϕ ¼ ið u i þ v þ w Þ; ð89Þ i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 23 − 4=R − 1 ≥ 0; it follows that extremal field configurations are those qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where all non-zero fields take equal values. If we have M pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Φ ϕ αy nonzero fields and m nonzero fields, the extremal field α α − 20 − 4 6 23 − 4 1 uþ þ v− ¼ =R þ =R Φ2 0 Φ2 ϕ2 0 4 values are i ¼ or i ¼ M alongside with i ¼ or ϕ2 1 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i ¼ m. Three nontrivial cases arise. If m ¼ the extremal − 23 − 4 − 1 ≤−α =R y; ð86Þ potential is We ¼ U=M þ V. Likewise if M ¼ 0 we have We ¼ u=m þ v. Lastly, if both m; M ≠ 0,wehave We ¼ U=M þ V þ u=m þ v þ w.ThevaluesofM, m for where the bounds refer to R varying within the range (39). which these extremal potentials are minima depend on the This part of the potential becomes exactly flat at the fixed signs of the couplings U, u, leaving us with the four possible 2 point iff R ¼ 11. The result establishes vacuum stability at cases U; u > 0, u>0 >U, U>0 >u,and0 >U;u.We α α the fixed point ð uþ; vþÞ. We also confirm that the fixed thus obtain four distinct sets of conditions for vacuum stability α α point ð uþ; v−Þ does not lead to a stable ground state. which we summarize as follows: We conclude that vacuum stability is guaranteed at the ðaÞ α ; α ≥ 0; α þ α ≥ 0; α þ α ≥ 0; interacting fixed points ðα ; α Þ and ðα ; α Þ, u U U V u v Uþ Vþ uþ vþ α α together with α ¼ 0, irrespective of the fixed points in u þ v w FðαU þ αVÞþ þ αw ≥ 0; the gauge Yukawa sector, as long as the later is physical. F α Out of the a priori 23 different fixed point candidates in the V ðbÞ αu > 0 > αU; αU þ ≥ 0; αu þ αv ≥ 0; scalar sector at one loop (half of which lead to real fixed NF points) the additional requirement of vacuum stability has αV αu þ αv αw αU þ þ þ ≥ 0; identified a unique viable solution. In this light, vacuum NF FNf Nf stability dictates that the anomalous dimensions (49) are α α 0 α α α ≥ 0 α v ≥ 0 strictly positive at interacting fixed points, (60), with ðcÞ U > > u; U þ V ; u þ ; Nf α α α α qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α v U þ V w ≥ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u þ þ F þ ; γ α 20 − 4 6 23 − 4 0 Nf NF NF M ¼ Y R þ R > ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αV αv pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdÞ 0 ≥ αu; αU; αU þ ≥ 0; αu þ ≥ 0; γ α 20 − 4 6 23 − 4 0 N N m ¼ y =R þ =R > ; ð87Þ F f αv αV αw αu þ þ F αU þ þ ≥ 0: ð90Þ and provided that (39) is observed. Nf NF NF 085008-20 MORE ASYMPTOTIC SAFETY GUARANTEED PHYS. REV. D 97, 085008 (2018) Notice that we have rescaled the couplings as in (26) and (27) to make contact with the notation used in this paper. The ≡ 0 25 parameter F Nf =NF > can be expressed in terms of the parameter R to leading order in ϵ ≪ 1,see(45). 11 We make the following observations. In all four cases, the additional condition owing to the mixing coupling (27) 11 25 takes the form of a lower bound for αw. Furthermore, αw is allowed to be negative without destroying the stability of P 0 the potential, provided it does not become too negative. We 278 also note that none of the three cases (b), (c), or (d) in (90) 1421 can have consistent solutions in the Veneziano limit where → ∞ NF;Nf . This uniquely leaves the case (a) as the only possibility for vacuum stability in the parameter regions considered here. These solutions neatly fall back onto the 1421 solutions discussed previously in the limit αw → 0. As long 278 as the auxiliary condition 2 1 1 2 11 11 2 2 −1 R αw ≥−½FðαU þ αVÞþF ðαu þ αvÞ ð91Þ FIG. 6. The “phase space” of quantum field theories with is satisfied, we can safely conclude that a nonvanishing fundamental action (25) expressed as a function of field multi- αw ≠ 0 does not spoil vacuum stability, not even for plicities and written in terms of ðP; RÞ, see (61). The 22 different negative portal coupling αw. parameter regions are indicated by roman letters. Theories with parameters in region X are dual to those in region Xb under the D. Unique scalar fixed points exchange of gauge groups following the map (46). Further details on fixed points and their eigenvalue spectra per parameter region In Table IX, we summarize our results for the quartic are summarized in Figs. 7, 8, and 9. scalar couplings at all weakly interacting fixed points to leading order in ϵ following Table III, using (61). We also introduce the auxiliary functions (25) in view of their fixed point structure at weak coupling, together with their behaviour in the deep UV and IR. 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Theories differ primarily through their matter multiplicities 23 − 4 − 1 F1ðxÞ¼4 x ; (32), which translate to the parameters ðP; RÞ and the sign qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ “ ” 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of , (61). As such, the phase space shown in Fig. 6 arises 20 − 4 6 23 − 4 − 2 23 − 4 as the overlay of Figs. 1–5. Distinctive parameter regions F2ðxÞ¼4 x þ x x ð92Þ are separated from each other by the seven characteristic 0 ˜ ˜ 1 which originate from the scalar nullclines. The main result curves P ¼ ;X;X; Y or Y and R ¼ 2 or 2. The functions ˜ ˜ is that vacuum stability together with a physical fixed point XðRÞ; XðRÞ;YðRÞ, and YðRÞ are given explicitly in (B1). in the gauge-Yukawa sector singles out a unique fixed point Overall, this leads to the 22 distinct regions shown in Fig. 6 in the scalar sector. The scalar fixed points do not offer and denoted by capital letters. Together with the sign of ϵ further parameter constraints other than those already stated this leaves us with 44 different cases. Some of these are in Tables VII and VIII. Within the admissible parameter redundant and related under the exchange of gauge groups, ranges we invariably find that the scalar couplings are either see (46). In fact, for P>0 and for either sign of ϵ, we find strictly irrelevant (at interacting fixed points) or marginally nine fundamentally independent cases corresponding to the irrelevant (at the Gaussian fixed point). parameter regions VI. ULTRAVIOLET COMPLETIONS A; B; C; D; E; F; G; H; I ð93Þ In this section, we discuss interacting fixed points and given in Fig. 6. Theories with parameters in the regime the weak coupling phase structure of minimal models (25) in dependence on matter field multiplicities. Differences Ab; Bb; Cb; Db; Eb; Fb; Gb; Hb; ð94Þ from the viewpoint of their high- and low-energy behavior are highlighted. are “dual” to those in (93) under the exchange of gauge groups (X ↔ Xb) and for the same sign of ϵ, except for the A. Classification theories within (I, ϵ), which are “self-dual” and mapped In Figs. 6–9 we summarize results for the qualitatively onto themselves under (46).ForP<0 we find five different types of quantum field theories with Lagrangian parameter regions for either sign of ϵ, 085008-21 ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) J; K; L; Kb; Jb: ð95Þ nontrivial at the fixed point; a hyphen indicates that the IR regime is strongly coupled. For these, the manifest “duality” under exchange of B. Asymptotic freedom gauge groups involves a change of sign for ϵ with (X, We discuss main features of the different quantum field ϵ < 0) being dual to (Xb, −ϵ > 0) except for the parameter theories (25) starting with those where each gauge sector region L which is self-dual. In total, we end up with is asymptotically free from the outset ðP>0 > ϵÞ, corre- 2 × 9 þ 5 ¼ 23 fundamentally distinct scenarios underneath sponding to the cases 1–17 in Fig. 7. The Gaussian fixed 2 9 8 5 44 the × ð þ þ Þ¼ cases tabulated in Figs. 7, 8,and9 point FP1 is always the UV fixed point. Any other weakly and discussed more extensively below. interacting fixed point displays a lower number of relevant A comment on the nomenclature: in each row of Figs. 7, directions. All weakly interacting fixed points can be 8, and 9, we indicate the parameter region ðP; RÞ as in reached from the Gaussian. Another point in common is Fig. 6 together with the sign of ϵ (if required), followed by that all theories are completely asymptotically free meaning the set of fixed points. For each of these, the (marginally) that—besides the gauge and the Yukawa couplings—all relevant and irrelevant eigenvalues in the gauge-Yukawa quartics reach the Gaussian UV fixed point. sector are indicated by a − and þ sign. For the Gaussian Differences arise as to the set of interacting fixed points, fixed point FP1, the signs relate pairwise to the SUðNCÞ and summarized in Fig. 7. Overall, theories display between SUðNcÞ gauge sector, respectively; for all other fixed points three and eight distinct weakly interacting fixed points. The eigenvalues are sorted by magnitude. Red shaded slots partial Banks-Zaks fixed points (FP2,FP3) are invariably indicate eigenvalue spectra which uniquely arise due to the present in all 17 cases. This is a consequence of a general semisimple character of the theory. The column “UV” theorem established in [34], which states that the two loop states the UV fixed point, differentiating between complete gauge coefficient is strictly positive for any gauge theory in asymptotic freedom (AF), asymptotic safety (AS), asymp- the limit where the one-loop coefficient vanishes. This totic freedom in one sector without asymptotic safety in guarantees the existence of a partial Banks-Zaks fixed point the other (pAF), asymptotic safety in one sector without in either gauge sector. At least one of the partial gauge- asymptotic freedom in the other (pAS), or none of the Yukawa fixed points (FP4,FP5) also arises in all cases. above. The column “IR” states the fully attractive IR fixed Moreover, the fully interacting Banks-Zaks (FP6) as well as point (provided it exists), distinguishing the cases where the fully interacting gauge-Yukawa fixed points (FP7,FP8, none (0), one (Y) or (y), or both (Yy) Yukawa couplings are FP9) are present in many, though not all, cases. All nine FIG. 7. Shown are the fixed points and eigenvalue spectra of quantum field theories with Lagrangean (25) for the 17 parameter regions with ϵ < 0 and P>0 in Fig. 6. Scalar selfinteractions are irrelevant at fixed points. All cases display complete asymptotic freedom in the UV. Red shaded slots indicate eigenvalue spectra which arise due to the semisimple character of the theory. In the deep IR, various types of interacting conformal fixed points are achieved depending on whether both, one, or none of the Yukawa couplings Y and y vanish (from left to right). Regimes with “strong coupling only” in the IR are indicated by a hyphen. 085008-22 MORE ASYMPTOTIC SAFETY GUARANTEED PHYS. REV. D 97, 085008 (2018) distinct fixed points are available in the “most symmetric” We note that FP6,FP7 and FP8 are natural IR sinks, with or parameter region I (case 9). without ψ fermions, provided that all Yukawa couplings of It is noteworthy that many theories display a fully IR those fermions which interact with the Banks-Zaks fixed attractive “sink”, invariably given by an IR gauge-Yukawa point(s) vanish. On the other hand, the result that FP2 and fixed point in one (FP4,FP5) or both gauge sectors (FP9). In FP3 may become IR sinks is a strict consequence of the ψ Fig. 6, this happens for matter field multiplicities in the fermions and would not arise otherwise. Once more, one regions A, B, C, E, F, G, I and their duals (cases 1, 2, 3, 5, 6, of the gauge sectors becomes IR free owing to the BZ fixed 7, 9, 11, 12, 13, 15, 16 and 17 of Fig. 7). point in the other, an effect which is mediated via the ψ At FP9, the fully IR attractive fixed point is largely a fermions. In the presence of nontrivial Yukawa couplings, consequence of IR attractive fixed points in each gauge no fully IR stable fixed point arises for theories with field sector individually. This is not altered qualitatively by the multiplicities in the parameter regions D and H (case 4, 8, semisimple nature of the model. As such, a fully IR 10, and 14). Generically, trajectories will then run towards attractive fixed point FP9 also arises in the “direct product” strong coupling with e:g.. confinement or strongly- limit where the ψ fermions are removed. coupled IR conformality. Analogous conclusions hold At FP4 and FP5, in contrast, the IR sink is a direct true in settings with fully attractive IR fixed points consequence of the semisimple nature of the theory in that provided their basins of attraction do not include the it would be strictly absent as soon as the messenger Gaussian. fermions ψ are removed. Most importantly, the IR gauge Finally, another interesting feature which is entirely due Yukawa fixed point in one gauge sector changes the sign of to the semisimple nature of the theory are models where the effective one loop coefficient in the other, mediated via FP9 has a single relevant direction (cases 1, 2, 5, 6, 12, 13, the ψ fermions. This secondary effect means that one gauge 16, and 17). Whenever this arises, the theory also always sector becomes IR free dynamically, rather than remaining displays a fully IR attractive fixed point (FP4,FP5, or both). UV free. Overall, the fixed point becomes IR attractive in all canonically marginal couplings (including the quartic C. Asymptotic safety couplings). In most cases the IR sink is unique except in parameter regions B and F (case 2, 6, 12, and 16) where We now turn to quantum field theories with (25) where we find two competing and inequivalent IR sinks asymptotic safety is realized. Asymptotic safety relates to (FP4 versus FP5). settings where some or all couplings take non-zero values Provided that one or both Yukawa couplings take in the UV [34]. A prerequisite for this is the absence of Gaussian values, other fixed points may take over the asymptotic freedom in at least one of the gauge sectors. We role of IR “sinks.” In these settings, one or both of the find two such examples provided P<0 (cases 22 and 23 in elementary “meson” fields remain free for all scales and Fig. 8), corresponding precisely to settings where one decouple from the outset. Specifically, the IR sink is given gauge sector is QCD-like whereas the other is QED-like. by FP6 provided that y ¼ 0 ¼ Y (cases 5–13); by FP2 or For these theories, we furthermore find that all other FP7 provided that y ¼ 0 (cases 14 or 7–9, respectively); interacting fixed points are also present, except those of and by FP3 or FP8 provided that Y ¼ 0 (cases 4 or 9–11). the Banks-Zaks type involving the QED-like gauge sector. FIG. 8. Same as Fig. 7, covering the 10 parameter regions with P<0 of Fig. 6. Notice that FP6 is absent throughout. Exact asymptotic safety (AS) is realized in the cases 22 and 23. Red shaded slots indicate eigenvalue spectra which arise due to the semisimple character of the theory. For the cases 18–21 and 24–27, partial asymptotic freedom (pAF) or partial asymptotic safety (pAS) is observed whereby one gauge sector decouples entirely at all scales. The latter theories are only UV complete in one of the two gauge sectors and must be viewed as effective rather than fundamental. 085008-23 ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) More specifically, in case 22 the role of the asymptotically safe UV fixed point is now taken by FP5. The UV critical surface is three-dimensional, in distinction to asymptoti- cally free settings where it is four-dimensional. This reduction, ultimately a consequence of an interacting fixed point in one of the Yukawa couplings, leads to enhanced predictivity of the theory. The Gaussian necessarily becomes a cross-over fixed point with both attractive and repulsive directions, similar to the interacting FP8. Also, FP2 and FP9 are realized with a one-dimensional critical surface. The fully IR attractive FP4—the counter- part of the UV fixed point FP5—takes the role of an IR sink. In the low energy limit, the theory displays free SUðNcÞ gluons in one gauge sector and weakly interacting SUðNCÞ gluons in the other. Moreover, the spectrum includes both free and weakly interacting mesons related to the former and the latter sectors, as well as free and weakly interacting fermions. Qualitatively, a similar result arises in the direct product limit, showing that the semi- simple nature of (25) is not crucial for this scenario. A noteworthy feature of semisimple theories with asymptotic safety is that they connect an interacting UV FIG. 9. Same as Figs. 7 and 8, covering the 17 parameter fixed point with an interacting IR fixed point. Hence, our regions where ϵ > 0 and P>0 in Fig. 6. Asymptotic freedom is models offer examples of quantum field theories with exact absent in both gauge sectors implying that FP2,FP3,FP6,FP7, UV and IR conformality, strictly controlled by perturbation and FP8 cannot arise. Partial asymptotic safety (in one gauge theory for all scales. In the massless limit, the phase sector) is observed in case 28,31,32,35,37,40,41, and 44, whereby the other gauge sector remains free at all scales diagram has trajectories connecting the interacting UV (pAS). All models must be viewed as effective rather than fixed point with the interacting IR fixed point. Some fundamental theories. All theories become trivial in the IR. trajectories may escape towards the regime of strong coupling where the theory is expected to display confine- can be interacting when viewed as an effective theory, ment, possibly infrared conformality. The same picture 1 very much like the Uð ÞY sector of the Standard Model. arises in case 23 after exchange of gauge groups. This setting requires P<0 and is realized in cases 18–21 No asymptotically safe fixed point arises if both gauge and 23–27. ϵ 0 sectors are IR free ðP; > Þ. This result is in marked Second, we find models with partial asymptotic safety contrast to findings in the direct product limit where models (pAS), where one gauge sector becomes asymptotically — with an interacting UV fixed points exist simply because safe whereas the other remains free at all scales. All such it exists for the simple gauge factors (63) and (67),given models display a UV gauge-Yukawa fixed point. When suitable matter field multiplicities. We conclude that it is viewed as a fundamental theory, these semisimple gauge precisely the semisimple nature of the specific set of theories in fact reduce to a simple asymptotically safe theories (25) which disallows asymptotic safety for settings ϵ 0 gauge theory (which is UV complete). The IR-free sector with P; > , see (61). can be interacting when viewed as a non-UV complete effective theory. This setting mostly requires P; ϵ > 0 and D. Effective field theories is realized in cases 28, 31, 32, 35, 37, 40, 41, and 44. We now turn to quantum field theories with (25) which Curiously, pAS is also realized in cases 21 and 24 where are not UV complete semisimple gauge theories and, as P<0 alongside pAF in the other gauge sector—such such, must be seen as effective field theories. We find models have two disconnected UV scenarios, where we can three different types of these. First, we find models with choose to have either asymptotic freedom in one sector, or partial asymptotic freedom (pAF), where one gauge sector asymptotic safety in the other, in each case with the remains asymptotically free whereas the other stays infra- remaining sector decoupling at all scales. Once more, if red free. These models always realize a Banks-Zaks fixed both gauge sectors are interacting these models must be point (as they must), and some also realize an IR gauge- viewed as (non-UV complete) effective theories. Yukawa fixed point. When viewed as a fundamental theory, Finally, we find models with none of the above. In these the IR free sector decouples exactly, for all RG scales, settings (cases 29, 30, 33, 34, 36, 38, 39, 42 and 43), and the theory becomes a simple asymptotically free both gauge sectors are IR free and no other weakly coupled gauge theory (which is UV complete). The IR-free sector fixed points are realized, leaving us with no perturbative 085008-24 MORE ASYMPTOTIC SAFETY GUARANTEED PHYS. REV. D 97, 085008 (2018) UV completion. In the cases 28–44, the Gaussian acts [34,35]. Qualitatively different cases realized amongst the as in IR sink for RG trajectories. Along these, the long- theories (25) are summarized in Fig. 10 for semisimple distance behavior is trivial, characterized by free massless gauge theories with two gauge groups G1 × G2. Results non-Abelian gauge fields, quarks, and elementary mesons. generalize to more gauge groups in an obvious manner. In summary, the semisimple gauge Yukawa theories (25) Specifically, Fig. 10(a) shows theories with asymptotic have a well-defined UV limit with either asymptotic free- freedom but without any BZ fixed points. UV free dom or asymptotic safety in 9 þ 1 ¼ 10 cases out of the 23 trajectories emanate out of the Gaussian fixed point and fundamentally distinct parameter settings covered in Fig. 6. invariably escape towards strong coupling where the theory The remaining 4 þ 9 ¼ 13 parameter settings do not offer a is expected to display confinement, or IR conformality. well-defined UV limit at weak coupling. This completes Similarily, Fig. 10(b) shows theories with asymptotic the classification of the models with (25). freedom and a BZ fixed point in one of the gauge sectors. The other gauge coupling remains an IR relevant pertur- VII. PHASE DIAGRAMS OF GAUGE THEORIES bation even at the BZ. Therefore UV free trajectories will again escape towards strong coupling in the IR. In this section, we discuss the phase diagrams of UV Figure 10(c) shows asymptotic freedom with a BZ fixed complete theories of the type (25), particularly in view of point in both gauge sectors individually. Here, and much theories with asymptotic freedom or asymptotic safety. unlike Fig. 10(b), one of the BZ fixed points has turned into an exact IR sink, and both BZ fixed points are connected by A. Semisimple gauge theories without Yukawas a separatrix. As we have already noticed in Sec. VI B, the We begin with settings where Yukawa couplings are presence of an interacting fixed point in one gauge sector switched off. In these cases, interacting fixed points can can turn the other gauge sector from UV free to IR free. only arise for asymptotically free gauge sectors, and fixed This new type of phenomenon has become possible owing points are of the Banks-Zaks type or products thereof to the ψ fermions and is once again due to the semisimple (a) (b) (c) (d) FIG. 10. Phase diagrams of asymptotically free semisimple gauge theories (two gauge groups) coupled to matter without Yukawas, covering (a) asymptotic freedom and the Gaussian (G) without interacting fixed points and trajectories running towards strong coupling and confinement, (b) the same, with an additional Banks-Zaks (BZ) fixed point, (c) two BZ fixed points, one of which turned into an IR sink for all trajectories, or (d) three BZ fixed points, the fully interacting one now becoming the IR sink. Axes show the running gauge couplings, fixed points (black) are connected by separatrices (red), and red-shaded areas cover all UV free trajectories with arrows pointing from the UV to the IR. 085008-25 ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) nature of the theory. Therefore, all UV free trajectories (25). Explicit examples are given for models without invariably are attracted into the IR sink. In the deep IR, the Yukawa couplings ðY ¼ 0 ¼ yÞ and for field multiplicities theory approaches a conformal fixed point with massless in the parameter regions (a) ϵ1, ϵ2 < −75=26,(b)ϵ1 < and unconfined free and weakly coupled gluons and −75=26 and −75=26 < ϵ2 < 0,orðϵ1 ↔ ϵ2Þ, (c) the cases – – – quarks. Regimes of strong coupling cannot be reached. 1 4and14 17 of Fig. 7, and (d) the cases 5 13 of Fig. 7. Figure 10(d) shows asymptotic freedom with a (partial) BZ fixed point in either gauge sector individually, as well as B. Simple gauge theories with Yukawas a fully interacting BZ fixed point. Most notably, all UV free We continue the discussion of phase diagrams with trajectories are attracted by the later, which acts as an simple gauge theories with gauge group G and a single IR sink. No trajectories can escape towards strong cou- Yukawa coupling. Four distinct cases can arise [34,35], pling. The long distance physics is characterized by an summarized in Fig. 11. For asymptotically free settings, the interacting conformal field theory with massless weakly theory either shows (a) only the Gaussian UV fixed point, coupled gauge fields and fermions. Here, and unlike in (b) the Gaussian together with the Banks-Zaks, or (c) the Fig. 10(c), all fields remain weakly coupled in the IR. Gaussian together with the Banks-Zaks and an IR gauge- In the scenarios of Figs. 10(a) and 10(b) UV free Yukawa fixed point. Simple gauge theories can also trajectoires run towards strong coupling and confinement become asymptotically safe, in which case (d) a UV in the IR, in one or both gauge sectors. In contrast, the gauge-Yukawa fixed point arises. Trajectories are directed scenarios in Figs. 10(c) and 10(d) show that all UV free towards the IR. The red-shaded areas indicate the set of UV trajectories are attracted by an IR-stable conformal fixed complete trajectories emanating out of the UV fixed point. point. These theories remain unconfined and perturbative at We genuinely observe a two-dimensional area of trajectories all scales. All four scenarios in Fig. 10 are realized for our for asymptotically free settings, which is reduced to a one- template of semisimple gauge theories with Lagrangean dimensional set in the asymptotically safe scenario. The IR (a) (b) (c) (d) FIG. 11. Phase diagrams of UV complete and weakly interacting simple gauge theories coupled to matter with a single Yukawa coupling, covering (a) asymptotic freedom with the Gaussian UV fixed point and no other weakly interacting fixed point, (b) asymptotic freedom with a Banks-Zaks (BZ) fixed point, (c) asymptotic freedom with a Banks-Zaks and an IR gauge-Yukawa (GY) fixed point, and (d) asymptotic safety with an UV gauge-Yukawa fixed point. Axes display the running gauge and Yukawa couplings, fixed points (black) are connected by separatrices (red), and red-shaded areas cover all UV free trajectories with arrows pointing from the UV to the IR [34,35]. Examples are given by (63), (67) (see main text). 085008-26 MORE ASYMPTOTIC SAFETY GUARANTEED PHYS. REV. D 97, 085008 (2018) An economic way to display phase diagrams for semi- simple theories with or without Yukawas is achieved by introducing a schematic diagrammatic language, see Fig. 12. Each of the four basic phase diagrams in Fig. 11 are (a) (b) (c) (d) represented by a “primitive” diagram, Fig. 12, where full dots indicate (free or interacting) fixed points, red arrows indicate the outgoing trajectories, and RG flows schemati- cally run “top-down” from the UV to the IR. Also, at each fixed point the number of outgoing arrows indicates the FIG. 12. “Primitives” for phase diagrams of simple gauge- dimensionality of the fixed point’s “UV critical surface.” Yukawa theories with asymptotic freedom (AF) or asymptotic Fixed points are connected by separatrices. We use straight safety (AS), corresponding to the different setting shown in lines to indicate separatrices involving the BZ fixed point, Fig. 11. Arrows point from the UV to the IR and connect the curved lines to indicate separatrices connecting GY fixed different fixed points. Open arrows point towards strong coupling points with the Gaussian, and open-ended lines to denote RG in the IR. The number of outgoing red arrows gives the trajectories running towards strong coupling without reach- dimensionality of the UV critical surface. The separate UV safe ing any weakly coupled fixed points. trajectory towards strong coupling in case (d) is not indicated. Specifically, in case (a), a two-dimensional array of RG Yukawa-induced IR unstable directions in (a,b) or gauge Yukawa fixed points in (c,d) are absent as soon as Yukawa interactions are flows are running out of the Gaussian UV fixed point switched off from the outset. towards strong coupling, with no weakly interacting fixed points. In case (b), we additionally observe a Banks-Zaks regime is characterized by either strong interactions and fixed point. It is connected with the Gaussian by a separatrix confinement such as in Figs. 11(a,b,d), or by an interacting shown in red. Arrows invariably point towards the IR. conformal field theory with weakly coupled gluons and Yukawa couplings act as an unstable direction at both fixed fermions alongside free or interacting scalar mesons— points. In case (c), we additionally observe a gauge-Yukawa corresponding to the BZ fixed points in Figs. 11(b) and fixed point besides the Gaussian and the BZ. All three fixed 11(c), or the IR GY fixed point in Fig. 11(c), respectively— points are connected by separatrices. Note that two lines or by Gaussian scaling, Fig. 11(d). emanate from the Gaussian, reflecting that the UV critical All four scenarios in Fig. 11 are realized for simple surface is two dimensional. The GY fixed point arises as an gauge theories with (63) corresponding to the parameter IR sink, which attracts all UV-free trajectories emanating out 1 of the Gaussian. In case (d), the model is asymptotically safe regions (a) ϵ1 < −75=26, (b) −75=26 < ϵ1 < 0 and R>2, −75 26 ϵ 0 1 ϵ 0 1 and the GY fixed point has become the interacting UV fixed (c) = < 1 < and R<2, or (d) 1 > and R<2, point. A Banks-Zaks fixed point can no longer arise [34]. respectively, with R additionally bounded by (64). The theory has a one-dimensional UV critical surface connecting the GY fixed point with the IR Gaussian fixed point via a separatrix. A second UV safe trajectory which (a) (b) (c) leaves the GY fixed point towards strong coupling is not depicted. Finally, we note that the Yukawa-induced IR unstable directions in (a) and (b) or gauge Yukawa fixed points in (c) and (d) are absent as soon as Yukawa interactions are switched off from the outset. C. Semisimple gauge theories with asymptotic freedom We consider phase diagrams for semisimple theories (25) FIG. 13. Schematic phase diagram for asymptotically free with complete asymptotic freedom, exemplified by all semisimple gauge theories (25) with Banks-Zaks type fixed models in Fig. 7. When Yukawa couplings are absent, points without Yukawas and exact IR conformality. Field the mesonlike scalar degrees of freedom remain free at all multiplicities correspond to the cases (a) 1–4, (b) 5–13, and scales and decouple from the theory. In the regime with – (c) 14 17 of Fig. 7, respectively, with scalars decoupled. RG asymptotic freedom solely Banks-Zaks fixed points can flows point from the UV to the IR (top to bottom). At each fixed arise in the IR. Figures 10(d) and 13(b) shows settings point, the dimensionality of the UV critical surface is given by the where all Banks-Zaks fixed points are present, correspond- number of outgoing red arrows. All UV free trajectories terminate ing to the cases 5–13 of Fig. 7. RG flows point from the UV at FP2,FP6, and FP3, respectively, which act as fully attractive IR sinks. The topology of the phase diagram (b) is the “square” of to the IR (top to bottom) and connect the Gaussian UV Fig. 12(b), representing Fig. 10(d). The phase diagrams (a) and fixed point (FP1) with either of the partially (FP2 and FP3) (c), representing Fig. 10(c), cannot be constructed from the and the fully interacting (FP6) Banks-Zaks fixed points. primitives in Fig. 12. The latter is fully attractive and acts as an IR sink. 085008-27 ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) FIG. 14. Asymptotic freedom and schematic phase diagram for FIG. 15. Asymptotic freedom and schematic phase diagrams semisimple gauge-Yukawa theories with field multiplicities as in for semisimple gauge-Yukawa theories with field multiplicities as case 9 of Fig. 7. RG flows point from the UV to the IR (top to in case 8 of Fig. 7. Flows point from the UV to the IR (top to bottom). Besides the Gaussian UV fixed point (FP1), the theory bottom). The theories display five weakly interacting fixed points displays all eight weakly interacting fixed points, see Table III. besides the Gaussian UV fixed point (FP1). The unavailability of At each fixed point, the dimensionality of the UV critical surface FP5,FP8, and FP9 implies that some trajectories escape towards is given by the number of outgoing red arrows. FP9 is fully strong coupling (short arrows), and none of the fixed points acts attractive and acts as an IR sink. The topology of the phase as a complete IR attractor. The topology of the phase diagram is diagram is the square of Figs. 11, 12(c); see main text. the “direct product” of Figs. 11, 12(c) with Figs. 11, 12(b); see main text. The IR unstable direction is removed provided that the ≡ 0 The topology of the phase diagram is the “square” of Yukawa coupling y , in which case the singlet mesons h decouple. Figs. 11, 12(b). In the deep IR the theory is unconfined yet weakly interacting, and the elementary gauge fields A, a and fermions Q, q and ψ appear as massless particles at Yukawa fixed point individually, and all nine fixed points the IR conformal fixed point. The phase diagrams in are realized in the full theory. Figs. 13(a) and 13(c) cannot be constructed out of the Since the sign pattern of the eigenvalue spectra at all simple primitives, Fig. 12. The reason for this is that the fixed points is equivalent to the direct product limit, the eigenvalue spectrum at one of the fixed points deviates topology of the semisimple phase diagram is the square of from the direct product spectrum due to interactions. Fig. 12(c)—shown in Fig. 14. Fixed points are connected Next we include Yukawa interactions. We have already by separatrices (red lines), and arrows always point towards concluded from Fig. 7 that the eigenvalue spectrum in the the IR. From top to bottom, the fixed points FP1 (FP2;3) cases 8, 9, and 10 agrees qualitatively, for all fixed points, [FP4;5;6] (FP7;8) and FP9 have a 4 (3) [2] (1) and with the eigenvalue spectrum in the corresponding direct 0-dimensional UV critical surface, respectively, corre- product limit. In these settings, we may then use the sponding to the number of outgoing red arrows. FP9 acts primitives in Fig. 12 to find the semisimple phase diagrams. as an IR attractor for all trajectories within its basin of We consider the case where the parameters (61) take values attraction. Consequently, the elementary quarks and gluons within the range I of Fig. 6 and for ϵ < 0, corresponding to are not confined and the theory corresponds to a conformal case 9 of Fig. 7. This family of theories includes the field theory of weakly interacting massless gluons, fer- “symmetric” setup ðR; PÞ¼ð1; 1Þ where symmetry under mions, and mesons in the deep IR. For certain fine-tuned the exchange of gauge groups is manifest. The UV fixed settings, the IR limit would, instead, correspond to one of point is given by the Gaussian (FP1), and the UV critical the other interacting fixed points FP2–FP8, relating to surface at the Gaussian is four-dimensional, owing to the different conformal field theories. Also, while all other marginal UV relevancy of the two gauge and the two fixed points can be reached from the Gaussian FP1 (whose Yukawa couplings. All scalar couplings are irrelevant in the UV critical surface has the largest dimensionality), it is not UV and can be expressed in terms of the gauge and the true in general that a fixed point with a smaller UV critical Yukawa couplings along UV-free trajectories. Moreover, dimension can be reached from a fixed point with a larger each gauge sector displays the Banks-Zaks and a gauge- one. Fixed points are also not connected “horizontally.” 085008-28 MORE ASYMPTOTIC SAFETY GUARANTEED PHYS. REV. D 97, 085008 (2018) As a further example we consider a less symmetrical setting given by models with (61) in the parameter range H (or Hb) of Fig. 6, and for ϵ < 0. In these theories, only one of the two gauge sectors can achieve a gauge-Yukawa fixed point. Consequently, six different types of fixed points are realized. The sign pattern of the eigenvalue spectrum (cases 8 or 10, Fig. 7) ensures that the topology of the semisimple phase diagram obtains as the direct product of Fig. 12(b) with Fig. 12(c), shown in Fig. 15. From top to bottom, the fixed points FP1 (FP2;3) [FP4;6], and FP7 have a 4 (3) [2] and 1-dimensional UV critical surface, respectively. Fixed points are connected by separatrices. The absence of FP5, FP8, and FP9 implies that some trajectories escape towards strong coupling, indicated by short arrows, from each of the fixed points. The unstable direction relates to the Yukawa coupling y in (25). Provided it is switched off, FP7 would become the fully attractive IR sink. In this case, the elementary mesons h are spectators and remain free at all scales. Also, the elementary quarks and gluons remain FIG. 16. Asymptotic safety and schematic phase diagram of unconfined. In the deep IR, the theory corresponds to a semisimple gauge-Yukawa theories with field multiplicities as in conformal field theory of weakly interacting massless case 22 of Fig. 8. Besides the partially interacting UV fixed gluons A, fermions Q, ψ and mesons H, together with point (FP5), the theory displays five weakly interacting fixed free and massless gluons a, fermions q and mesons h, see points. The Gaussian (FP1) takes the role of a crossover Table IV. For certain fine-tuned settings, the IR limit would, fixed point and FP4 takes the role of an IR sink. The topology instead, correspond to one of the other interacting fixed of the phase diagram is the direct product of Figs. 11, 12(c) with points FP2–FP8, relating to different conformal field Figs. 11, 12(d); see main text. theories. The phase diagrams of asymptotically free theories in the cases 1–7 and 11–17 of Fig. 7 cannot be constructed out of interacting UV fixed point. At each fixed point, the number the simple primitives, Fig. 12. The reason for this is that of outgoing directions indicate the dimensionality of the their eigenvalue spectrum at some of the interacting fixed fixed point’s critical hypersurface. From top to bottom, the points deviates from the direct product spectrum. Once fixed points FP5 (FP1;8) [FP2;9] and FP4 have a 3 (2) [1] and again this effect is due to the semisimple nature of the 0-dimensional UV critical surface. UV finite trajectories theory. A more detailed study of these cases is left for connect FP5 via intermediate cross-over fixed points future work. with the fully IR attractive fixed point FP4, which is of the GY · G type. At weak coupling, all UV-IR connecting D. Semisimple gauge theories with asymptotic safety trajectories proceed either via the Gaussian FP1 (G · G) and We finally turn to the phase diagram of semisimple FP2 (BZ · G), or via FP8 (BZ · GY) and FP2 or FP9 gauge theories with exact asymptotic safety. From Figs. 8 (GY · GY). The Gaussian fixed point is IR free in one and 9 we conclude that asymptotic safety arises through a of the gauge couplings meaning that is necessarily arises as partially interacting UV fixed point where one gauge sector a cross-over fixed point. There are no trajectories connect- is interacting whereas the other gauge sector is free. This is ing the fixed points FP1 with FP9 because the sole relevant achieved for matter field multiplicities (61) taking values direction at the latter is an irrelevant direction at the former. within the range J or Jb of Fig. 6, corresponding to cases 22 FP4 acts as an IR “sink” for RG trajectories. While all other or 23 of Fig. 8. Once more, the eigenvalue spectra at all fixed points can be reached from the interacting UV fixed fixed points are equivalent to the ones in the direct product point FP5 (whose UV critical surface has the largest limit, implying that the phase diagram arises as the direct dimensionality), it is not true in general that a fixed product of the corresponding simple factors Figs. 11, 12(c) point with a smaller UV critical dimension can be reached and Figs. 11, 12(d). from a fixed point with a larger one (e.g. FP9 cannot be Figure 16 shows the schematic phase diagram for reached from FP1). Fixed points are also not connected case 22, where the asymptotically safe UV fixed point “horizontally”. FP5 is of the G · GY type (see Tables III and V). Unlike the An intriguing novelty of our models with asymptotic cases with asymptotic freedom, here, the UV hypercritical safety is that both the deep UV and the deep IR limits are surface is three rather than four dimensional. The reason for characterized by weakly interacting conformal field theo- this is that one of the Yukawa couplings is taking an ries. For example, in the deep UV the theories of case 22 085008-29 ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) correspond to conformal field theories of weakly interact- A. Gap, universality, and operator ordering ψ ing massless gluons a, fermions q, and mesons h, At partially or fully interacting fixed points, the degen- together with free and massless gluons A, fermions Q eracy of the nine classically marginal couplings (26), (27) is — and mesons H. Along the UV IR transition, the fields partly or fully lifted. We have computed scaling exponents “ ” ðA; Q; HÞ and ða; q; hÞ effectively interchange their roles, to the leading nontrivial order in ϵ. Interacting fixed points ultimately approaching conformal field theories of weakly have nontrivial exponents of order ∼ϵ, except if a gauge ψ interacting massless gluons A, fermions Q, , and mesons coupling is involved in which case one of the exponents is H, together with free and massless gluons a, fermions q parametrically smaller ∼ϵ2. Hence, the eigenvalue spectrum and mesons h in the IR. Hence, one may say that IR opens up ∼ϵ because eigenvalues of order ϵ are invariably conformality in the SUðNcÞ gauge sector arises from UV present at any of the interacting fixed points. It is “ conformality in the SUðNCÞ gauge sector through a see- convenient to denote the difference between the smallest ” ψ saw mechanism transmitted via the fermions, i.e., the negative eigenvalue and the smallest positive eigenvalue as only fields which are interacting at all scales including the the “gap” in the eigenvalue spectrum, which serves as an UVand the IR limits. For certain fine-tuned settings, the IR indicator for interaction strength [5,24]. Simple SUðNÞ limit would, instead, correspond to one of the other gauge theories in the Veneziano limit such as (63) display a interacting fixed points FP1,FP2,FP8,orFP9, relating gap of order ∼ϵðϵ2Þ at the Banks-Zaks or the UV gauge to different conformal field theories. Also, for certain UV Yukawa (IR gauge Yukawa) fixed point, respectively [32]. parameters, theories may escape towards strong coupling in In semisimple theories, and depending on the specifics the IR. of the fixed point, we again find that the gap is either of order ϵ or of order ϵ2. (The gap trivially vanishes if one of E. Mass deformations and phase transitions the gauge sectors is asymptotically free and takes Gaussian In the vicinity of fixed points phase transitions between values.) The gap still depends on the remaining free different phases arise once mass terms are switched on. At parameters ðP; RÞ. weak coupling mass anomalous dimensions are perturba- Also, all results for fixed points and scaling exponents tively small (Sec. III D). The running of scalar or fermion are universal and independent of the RG scheme, although mass terms, once switched on, will then be dominated by we have used a specific scheme (MS bar) throughout. This their canonical mass dimensions—modulo small quantum is obviously correct for dimensionless couplings at one corrections. Consequently, mass terms add additional loop where divergences are logarithmic. We have checked relevant directions at all fixed points (e.g., Figs. 14–16). that it also holds at two loop level both for the gauge Each of the eight interacting UV fixed points relates to a sectors, and for the Yukawa contributions to the running of quantum phase transition between phases with and without the gauge coupling(s) [32]. The field strengths and the Yukawa couplings are marginally relevant operators at spontaneous breaking of symmetry where the vacuum – expectation value of the scalar fields serves as an order asymptotically free Gaussian UV fixed points (case 1 17 parameter. In particular, fixed points which act as IR sinks of Fig. 7). At asymptotically safe UV fixed points, one of the field strengths becomes relevant and the corresponding for the canonically marginal interactions (such as FP9 in Yukawa coupling irrelevant (case 22, 23 of Fig. 8). There Fig. 14 and FP4 in Fig. 16) develop new unstable directions is no UV fixed point where both gauge sectors remain driven by the mass. Scalar fields may or may not develop interacting. The scalar self-interactions are (marginally) vacuum expectation values leading to symmetric and irrelevant at any fixed point. symmetry broken phases, respectively. Also, fermions may acquire masses spontaneously. Thereby a variety of different phases may arise, connected by first and higher B. Elementary gauge fields and scalars order quantum phase transitions. Close to interacting fixed Triviality bounds relate to perturbative UV Landau poles points, phase transitions are continuous and, in some cases, of infrared free interactions. They limit the predictivity of of the Wilson-Fisher type with a single relevant parameter. theories to a maximal UV extension [52]. For theories We leave a more detailed investigation of phase transitions with action (25), perturbative UV Landau poles can arise for a future study. for gauge couplings in the absence of asymptotic freedom or asymptotic safety. Examples for this are given in cases 18–21 and 24–27 of Fig. 8 where one gauge sector is IR VIII. DISCUSSION free, as well as in cases 28–44 of Fig. 9 where both In this section, we address further aspects of interacting gauge sectors are IR free. In these cases the theories can at fixed points covering universality and operator ordering, best be treated as effective rather than fundamental (see triviality bounds, perturbativity in and beyond the Veneziano Sec. VI D). Conversely, triviality in gauge sectors is limit, conformal symmetry, and conformal windows. trivially avoided in settings with asymptotic freedom 085008-30 MORE ASYMPTOTIC SAFETY GUARANTEED PHYS. REV. D 97, 085008 (2018) – α α O ϵnþ1 (such as in cases 1 17), and nontrivially in settings with i ¼ i jnNLO0 þ ð Þð96Þ asymptotic safety (case 22 and 23). In the latter cases, the loss of asymptotic freedom is compensated through an for all couplings (26), (27) and all fixed points, with interacting fixed point in the Yukawa and scalar couplings, corrections from the ðn þ 1ÞNLO0 level being at least which enabled a fixed point for the gauge coupling [32]. one power in ϵ smaller than those from the preceding We stress that scalar fields and Yukawa interactions play a level. We conclude that the expressions for the interacting key role. Without them, triviality of any QED-like gauge α ϵ fixed points i jnNLO0 are accurate polynomials in up to theories cannot be avoided [34,35]. including terms of order ϵn, for all n. Triviality also relates to the difficulty of defining Beyond the Veneziano limit, the parametrically small elementary self-interacting scalar quantum fields in four control parameter ϵ is no longer available. Instead, ϵ will dimensions [53–55]. It is interesting to notice that the take finite, possibly large, values dictated by the (finite) quartic scalar couplings always take a unique physical fixed field multiplicities. Still, for sufficiently large matter field points as soon as the gauge and Yukawa coupling take multiplicities, ϵ remains sufficiently small and perturba- weakly coupled fixed points. Hence, in theories with (25) tivity remains in reach [38]. It is then conceivable that the scalar fields can be viewed as elementary and triviality is fixed points found in the Veneziano limit persist even for evaded in all settings with asymptotic freedom and asymp- finite N.5 At finite N, however, we stress that the nNLO0 totic safety. In either case gauge fields play an important approximations and (96) are no longer exact order-by- role, albeit for different reasons [32]. For gauge interactions order. It then becomes important to check numerical with asymptotic freedom, the running of gauge couplings convergence of higher loop approximations, including dictates the running for Yukawa and scalar couplings, and nonperturbative resummations. In this context it would conditions for complete asymptotic freedom have been be particularly useful to know the radius of convergence of derived [56] which ensure that gauge theories coupled to beta functions (in ϵ) in the Veneziano limit. A finite radius matter reach the free UV fixed point [57]. For theories with of convergence has been established rigorously in certain asymptotic safety, scalars are required to help generate a large-NF limits of gauge theories without Yukawa inter- combined fixed point in the gauge, Yukawa, and quartic actions [58,59] which makes it conceivable that the radius 6 scalar couplings. This leads invariably to a “reduction of of convergence might be finite here as well. If so, this couplings” and enhanced predictivity over models with would offer additional indications for the existence of asymptotic freedom through a reduced UV critical surface. interacting fixed points beyond the Veneziano limit. C. Veneziano limit and beyond D. Conformal symmetry and conformal windows Our findings, throughout, rely on the existence of exact By their very definition, the gauge-Yukawa theories small parameters ϵ1 ≪ 1 and ϵ2 ≪ 1 (35) [or ϵ ≪ 1 see investigated here are scale-invariant at (interacting) fixed (41)] in the Veneziano limit, which relate to the gauge one points. Conditions under which scale invariance entails exact loop coefficients. Consequently, an iterative solution of conformal invariance have been discussed by Polchinski [60] perturbative beta functions becomes exact and interacting (see also [61]). Applied to the theories (25) at weak coupling, fixed points arise as exact power series in the small it implies that exact conformal invariance is realized at all parameters. More specifically, the leading non-trivial interacting fixed points discovered here. It would then be approximation which is NLO0 (Table II) retains the gauge interesting to find the full conformally invariant effective beta functions up to two loop, and the Yukawa and scalar action beyond the classically marginal invariants retained in (25). First steps into these directions have been reported in beta functions up to one loop. The parametric smallness of [33]. Moreover, for a quantum theory to be compatible with the gauge one-loop coefficients allows an exact cancella- unitarity, scaling dimension of (primary) scalar fields must tion of one and two loop terms implying that interacting be larger than unity. This is confirmed for all fixed points by fixed points for the gauge couplings must be of the order of using the results of Sec. III D for the anomalous dimensions the one loop coefficient ∼ϵ. The Yukawa nullclines at one of fields and composite scalar operators, together with the loop imply that Yukawa couplings are necessarily propor- results for fixed points at NLO0 accuracy (Tables VII and tional to the gauge couplings, and the scalar nullcline VIII). We conclude that the residual interactions are impose that scalar couplings are proportional to the compatible with unitarity. Yukawas (see Sec. VA); hence either of these come out Away from the Veneziano limit, findings for the various ∼ϵ 0 . Higher order loop approximations nNLO starting with interacting conformal fixed points persist once ϵ is finite. n ¼ 2 then correspond to retaining n þ 1 loops in the gauge, and n loops in the Yukawa and scalar beta functions 5An example for a conformal window with asymptotic safety is respectively, see Table II. Hence, solving the beta functions given in [37] for the model introduced in [32]. for interacting fixed points order-by-order in perturbation 6Results for resummed beta functions of large-N gauge theory ðn → n þ 1Þ we have that theories with Yukawa couplings are presently not available. 085008-31 ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) One may then think of keeping the parameters in the gauge supersymmetric counterparts in [36], make it conceivable sectors ðNC;Nc) fixed and finite while varying the matter that semisimple theories display interacting UV fixed field content ðNF;Nf ;Nψ Þ. Then, the domain of existence points even beyond the Veneziano limit, thus further paving for each of the interacting fixed points (Tables VII, VIII) the way for asymptotic safety beyond the Standard Model turns into a “conformal window” as a function of the matter [38]. The stability of the vacuum (Sec. V) in all models field multiplicities. The fixed point ceases to exist outside studied here suggests that the near-criticality of the standard the conformal window. The conformal window for asymp- model Higgs [39,40] can very well expand into full totic safety with a simple SUðNÞ gauge factor has been criticality at an interacting UV fixed point [38]. determined in [37]. Boundaries of conformal windows In addition, we have investigated phase diagrams for can be estimated within perturbation theory though more simple and semisimple gauge theories with and without accurate results invariably require nonperturbative tools.7 Yukawa interactions, continuing an analysis initiated in [34,35]. We find that transitions from the UV to the IR can IX. SUMMARY proceed from free or interacting fixed points to confinement and strong coupling. We also find transitions from free We have used perturbation theory and large-N tech- to interacting (Figs. 10–15) or from interacting to other niques for a rigorous and comprehensive investigation of interacting conformal fixed points (Fig. 16). In the latter weakly interacting fixed points of gauge theories coupled “ ” to fermionic and scalar matter. For concrete families of cases, theories display a variety of exact IR sinks, simple and semisimple gauge theories with action (25) and meaning free or interacting IR conformal fixed points following the classification of fixed points put forward in which are fully attractive in all classically marginal inter- [34,35], we have discovered a large variety of exact high- actions. Once more, many new features have come to light and low-energy fixed points (Tables III, VII, VIII). These beyond those observed in simple gauge theories [34,35]. include partially interacting ones (Table VII) where one Our study used minimal models with a low number of gauge sector remains free, and fully interacting ones Yukawa and gauge couplings. Already at this basic level, an (Table VIII) where both gauge sectors are interacting. intriguing diversity of fixed points and scaling regimes has We have determined the domains of existence for all of emerged, with many novel characteristics both at high them (Figs. 1–5). Interestingly, we also find that the and low energies. We believe that these findings warrant requirement of vacuum stability always singles out a more extensive studies in view of rigorous results [34,36], unique viable fixed point in the scalar sector. extensions towards strong coupling [33], and its exciting As a function of field multiplicities, the phase space of potential for physics beyond the standard model [38]. distinct quantum field theories (Fig. 6) includes models with asymptotic safety and asymptotic freedom, and ACKNOWLEDGMENTS effective theories without UV completion (Figs. 7, 8, and 9). In the IR, theories display either strong coupling Some of the results have been presented at the workshops and confinement, or weakly coupled fixed points where the The Exact Renormalization Group, Trieste (Sept 2016), and elementary gauge fields and fermions are unconfined and Understanding the LHC, Bad Honnef (Feb 2017). This work appear as massless particles. Many features are a conse- is supported by the Science and Technology Research quence of the semisimple nature and would not arise in Council (STFC) under the Consolidated Grant [ST/ simple (or direct products of simple) gauge theories. L000504/1] and by an STFC studentship. Highlights include massless semisimple gauge-matter the- ories where one gauge sector can be both UV free and IR APPENDIX A: GENERAL EXPRESSIONS free owing to a fixed point in the other, Fig. 10(c), and FOR FIXED POINTS theories with inequivalent scaling limits in the IR. Semisimple effects are particularly pronounced for asymp- Most results in the main text relate to the choice Nψ ¼ 1. totically free theories where they enhance the diversity of For completeness, we summarize fixed point results for different IR scaling regimes (Fig. 7). general Nψ species of fermions in the fundamental of both Another central outcome of our study is the first explicit gauge groups SUðN Þ and SUðN Þ. We observe that Nψ is “ ” C c proof of existence for asymptotic safety in semisimple restricted within the range quantum field theories with elementary gauge fields, scalars and fermions. It establishes the important result 11 0 ≤ ≤ that asymptotic safety is not limited to simple gauge Nψ 2 : ðA1Þ factors [32], fully in line with general theorems and structural results [34]. Our findings, together with their Outside of this range, exact perturbativity is lost. Substituting Nψ into the RG coefficients and solving for 7See [62] for lattice studies of conformal windows in QCD fixed points, we find the following expressions at the with fermionic matter (Banks-Zaks fixed points). partially interacting Banks-Zaks fixed points FP2 and FP3, 085008-32 MORE ASYMPTOTIC SAFETY GUARANTEED PHYS. REV. D 97, 085008 (2018) 4 FP ∶ α − ϵ For the interacting fixed points FP7 and FP8 we find 2 1 ¼ 75 R ðA2Þ 4 ϵ 8 P 13−2 25−2 FP3∶ α2 ¼ − ðA3Þ > 2 ð Nψ RÞð Nψ P=RÞ > α1 ¼ 2 2 2 2 Rϵ 75 R > 3 150Nψ R −ð4Nψ þ1025ÞNψ Rþ26Nψ þ475 <> FP ∶ 4 ð13−2Nψ RÞNψ R=Pþð2Nψ R−1Þð3Nψ R−19Þ Pϵ At the partially interacting fixed points FP4 and FP5 we 7 α2 ¼ − 3 2 2 2 2 > 150Nψ R −ð4Nψ þ1025ÞNψ Rþ26Nψ þ475 R have > :> 4ð25−2Nψ P=RÞ α ¼ 2 2 2 2 Rϵ 8 Y 150Nψ R −ð4Nψ þ1025ÞNψ Rþ26Nψ þ475 < 2 13−2Nψ R α1 ¼ Rϵ 3 ð2Nψ R−1Þð3Nψ R−19Þ ðA7Þ FP4∶ A4 : 4 ð Þ α ¼ Rϵ Y ð2Nψ R−1Þð3Nψ R−19Þ 8 < 2 13−2Nψ =R Pϵ 8 α2 ¼ 3 2 −1 3 −19 ð Nψ =R Þð Nψ =R Þ R > 4 ð13−2Nψ =RÞNψ P=Rþð2Nψ =R−1Þð3Nψ =R−19Þ FP5∶ ðA5Þ > α1 ¼ − 2 2 2 2 Rϵ : 4 ϵ > 3 150Nψ =R −ð4Nψ þ1025ÞNψ =Rþ26Nψ þ475 α P <> y ¼ ð2Nψ =R−1Þð3Nψ =R−19Þ R FP ∶ 2 ð13−2Nψ =RÞð25−2Nψ R=PÞ Pϵ 8 α2 ¼ 3 2 2 2 2 > 150Nψ =R −ð4Nψ þ1025ÞNψ =Rþ26Nψ þ475 R For the Banks-Zaks times Banks-Zaks-type fixed point FP6 > :> 4ð25−2Nψ R=PÞ Pϵ α ¼ 2 2 2 2 we find y 150Nψ =R −ð4Nψ þ1025ÞNψ =Rþ26Nψ þ475 R 8 4 25−2Nψ P=R ðA8Þ < α1 ¼ − 2 Rϵ 3 625−4Nψ FP6∶ ðA6Þ : 25−2 α − 4 Nψ R=P Pϵ 2 ¼ 3 625−4 2 R Nψ Finally, at the fully interacting fixed point FP9 we have 8 > 2 ð13−2Nψ RÞ½ð13−2Nψ RÞNψ P=Rþð2Nψ =R−1Þð3Nψ =R−19Þ Rϵ > α1 ¼ 3 2 2 2 4 2 2 > 114Nψ ðR þ1=R Þþð32Nψ þ1512Nψ þ361Þ−ð220Nψ þ779ÞðRþ1=RÞ > > 2 ð13−2Nψ =RÞ½ð13−2Nψ =RÞNψ R=Pþð2Nψ R−1Þð3Nψ R−19Þ Pϵ=R < α2 ¼ 2 2 2 4 2 2 3 114Nψ ðR þ1=R Þþð32Nψ þ1512Nψ þ361Þ−ð220Nψ þ779ÞðRþ1=RÞ FP9∶ ðA9Þ > 4½ð13−2Nψ RÞNψ P=Rþð2Nψ =R−1Þð3Nψ =R−19Þ Rϵ > αY ¼ 2 2 2 4 2 2 > 114Nψ ðR þ1=R Þþð32Nψ þ1512Nψ þ361Þ−ð220Nψ þ779ÞðRþ1=RÞ > :> 4½ð13−2Nψ =RÞNψ R=Pþð2Nψ R−1Þð3Nψ R−19Þ Pϵ=R α ¼ 2 2 2 4 2 2 : y 114Nψ ðR þ1=R Þþð32Nψ þ1512Nψ þ361Þ−ð220Nψ þ779ÞðRþ1=RÞ All expressions reduce to those given in the main body in the ð2R − 13ÞR 25 XðRÞ¼ ;YðRÞ¼ R; limit Nψ ¼ 1. We note that the parameter range in which ð2R − 1Þð3R − 19Þ 2 fixed points exist changes both qualitatively and quantita- ð2=R − 1Þð3=R − 19Þ 2 tively when varying Nψ within the range (A1).Moreover,we ˜ ˜ XðRÞ¼ 2 − 13 ; YðRÞ¼25 R: ðB1Þ also observe that the characteristic boundaries in paramater ð =R Þ=R space depend on Nψ , indicating that domains of existence These appear as boundaries of the “phase space” of and eigenvalue spectra depend on Nψ . It is straightforward, if tedious, to investigate regions of validity and scaling parameters ðR; PÞ characterizing valid fixed points. Note ˜ ˜ “ ” exponents for the general case, and to find the analogues that the functions ðX; XÞ and ðY;YÞ in (B1) are dual to each other, of Figs. 1–5 and of Tables VI–VIII for general Nψ . XðRÞ · X˜ ðR−1Þ¼1 ¼ YðRÞ · Y˜ ðR−1Þ: ðB2Þ APPENDIX B: BOUNDARIES A further set of boundaries is given by the straight lines We find that the existence and relevancy of fixed points R ¼ Rlow or Rhigh, with in the parameter space ðP; RÞ, see (61), is controlled by characteristic curves P ¼ XðRÞ;YðRÞ; X˜ ðRÞ or Y˜ ðRÞ with 1 R ¼ R ¼ 2: ðB3Þ the functions low 2 high 085008-33 ANDREW D. BOND and DANIEL F. LITIM PHYS. REV. D 97, 085008 (2018) The boundaries P ¼ XðRÞ;YðRÞ; X˜ ðRÞ,orY˜ ðRÞ with (B1) with R1 and R2 arising from together with (B3) delimit the qualitatively different quan- tum field theories in the “phase space” shown in Fig. 6. XðR1Þ¼YðR1Þ; Certain characteristic values for the parameter R arise in ˜ 2 11 XðR2Þ¼XðR2ÞðB5Þ its domain of vailidity 11 [1] D. J. Gross and F. Wilczek, Ultraviolet Behavior of [17] M. Reuter, Nonperturbative evolution equation for quantum Nonabelian Gauge Theories, Phys. Rev. Lett. 30, 1343 gravity, Phys. Rev. D 57, 971 (1998). (1973). [18] D. F. 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