AAsymptotic safety guaranteed
Daniel F. Litim ∗ and Francesco Sannino † Department of Physics and Astronomy, U Sussex, Brighton, BN1 9QH, U.K. CP -Origins & the Danish Institute for Advanced Study Danish IAS,Univ. of Southern Denmark, Campusvej 55, DK-5230 Odense We study the ultraviolet behaviour of four-dimensional quantum field theoriesinvolving non-abelian gauge fields, fermions and scalars in the Veneziano limit. Ina regime where asymptotic freedom is lost, we explain how the three types of fieldscooperate to develop fully interacting ultraviolet fixed points, strictly controlled byperturbation theory. Extensions towards strong coupling and beyond the large- N limit are discussed. Preprint: CP3-Origins-2014-24, DNRF90 & DIAS-2014-24 ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ h e p - t h ] D ec Contents
I. Introduction II. Origin of interacting UV fixed points
III. From asymptotic freedom to asymptotic safety
IV. Consistency
V. Towards asymptotic safety at strong coupling
VI. Conclusion Acknowledgments References I. INTRODUCTION
It is widely acknowledged that ultraviolet (UV) fixed points are central for quantumfield theories to be fundamental and predictive up to highest energies [1, 2]. A well-knownexample is asymptotic freedom of quantum chromodynamics where the UV fixed point isnon-interacting [3, 4]. In turn, neither the U (1) nor the scalar sector of the standard modelare asymptotically free. This is known as the triviality problem, which limits the predictivityto a scale of maximal UV extension [5]. High-energy fixed points may also be interacting, ascenario referred to as asymptotic safety [6]. It is then tempting to think that theories whichare not asymptotically free, or not even renormalisable by power-counting, may well turn outto be fundamental in their own right, provided they develop an interacting UV fixed point[7]. In recent years, asymptotic safety has become a popular scenario to address quantumaspects of gravity [7–13]. In a similar vein, UV conformal extensions of the standard modelwith and without gravity have received some attention in view of interacting fixed points[14–29] and scale invariance in particle physics and cosmology [30–53].The most notable difference between asymptotically free and asymptotically safe theoriesrelates to residual interactions at high energies. Canonical power counting becomes modifiedand the relevant or marginal invariants which dominate high energy physics are no longerknown a priori. Couplings may become large and small expansion parameters are often notavailable. Establishing or refuting asymptotic safety in a reliable manner then becomes achallenging non-perturbative task [54]. A few rigorous results for asymptotically safe UVfixed points have been obtained for certain power-counting non-renormalisable models bytaking the space-time dimensionality as a continuous parameter [6, 16, 55–59] in the spiritof the (cid:15) -expansion [60], or by using large- N techniques [19, 61–68]. Asymptotic safety thenarises in the vicinity of the Gaussian fixed point where perturbation theory is applicable.The success of well-controlled model studies provides valuable starting points to search forasymptotic safety at strong coupling.In this paper, we are interested in the UV behaviour of interacting gauge fields, fermionsand scalars in four dimensions. In the regime where asymptotic freedom is lost, we ask thequestion whether the theory is able, dynamically, to develop an interacting UV fixed point.Our main tool to answer this question is a suitably chosen large- N limit [69] (where N refersto the number of fields), whereby the theory is brought under strict perturbative control.Banks and Zaks [70] have used a similar idea to investigate the presence of interactinginfrared (IR) fixed points in gauge theories with fermionic matter. Here, we will discoverthat all three types of fields are required for an asymptotically safe UV fixed point to emerge.Given that exactly solvable models in four dimensions are hard to come by, our findings area useful starting point to construct UV safe models of particle physics.We illustrate the main outcome for gauge-matter theories in the absence of asymptoticfreedom in terms of a running non-Abelian coupling ( α g ) and a running Yukawa coupling( α y ). Fig. 1 shows the phase diagram close to the Gaussian fixed point to leading order inperturbation theory. Renormalization group trajectories are directed towards the IR. Weobserve that both the Yukawa and the gauge coupling behave QED-like close to the Gaussian.Consequently, not a single trajectory emanates from the Gaussian, meaning that it is an UVG
A BC D w e a k s t r o n g Α g Α y Figure 1: The phase diagram of certain 4D gauge-Yukawa theories in the absence of asymptoticfreedom and supersymmetry, in the vicinity of an asymptotically safe fixed point (UV). The renor-malization group flow for the gauge coupling α g and the Yukawa coupling α y is pointing towardsthe IR. The two renormalization group trajectories emanating out of the asymptotically safe fixedpoint define UV finite quantum field theories which at low energies correspond to a weakly (G)and a strongly coupled theory. Other trajectories exist as well, but they do not lead to UV finitetheories (within perturbation theory). Parameter choices and further details are given in Sec. III G. IR fixed point. The main novelty is the occurrence of an interacting fixed point inducedby fluctuations of the gauge, fermion, and scalar fields. The fixed point is located close tothe Gaussian and controlled by perturbation theory. The UV nature of the fixed point isevidenced by the (two) UV finite renormalization group trajectories running out of it. Theylead to sensible theories at all scales: in the UV, they are finite due to the interacting fixedpoint. In the IR, they correspond to weakly coupled theories with Gaussian scaling, andto strongly coupled theories with confinement and chiral symmetry breaking or conformalbehaviour, respectively. We also find UV unstable trajectories which do not emanate fromthe UV fixed point. They equally approach sensible IR theories, yet their UV predictivity islimited (at least in perturbation theory) by a scale of maximal UV extension. In this sense,asymptotic safety guarantees the existence of UV finite matter-gauge theories even in theabsence of asymptotic freedom or supersymmetry.In the rest of the paper we provide the details of our study. We recall the perturbativeorigin of interacting UV fixed points and asymptotic safety for sample theories of self-interacting gravitons, fermions, gluons, and scalars (Sec. II). We then explain why and howasymptotic safety can arise for certain gauge-Yukawa theories in strictly four dimensions.Fully interacting UV fixed points are found and analysed together with their universal scalingexponents, the UV critical surface, and the phase diagram of Fig. 1 (Sec. III). Aspects suchas stability, Weyl consistency, unitarity, and triviality are discussed (Sec. IV), together withfurther directions for asymptotic safety within perturbation theory and beyond (Sec. V).We close with some conclusions (Sec.VI).
II. ORIGIN OF INTERACTING UV FIXED POINTS
In this section, we recall the perturbative origin of asymptotic safety for certain quantumfield theories. We discuss key examples and introduce some notation.
A. Asymptotic safety
Asymptotic safety is the scenario which generalises the notion of a free, Gaussian, ultravi-olet fixed point to an interacting, non-Gaussian one. An asymptotivcally safe UV fixed pointthen acts as an anchor for the renormalisation group evolution of couplings, allowing themto approach the high-energy limit along well-defined RG trajectories without encounteringdivergences such as Landau poles.Within perturbation theory, the origin for asymptotic safety is best illustrated in terms ofa running dimensionless coupling α = α ( t ) of a hypothetical theory (to be specified below)with its renormalisation group (RG) β -function given by ∂ t α = A α − B α . (1)Here, t = ln( µ/ Λ) denotes the logarithmic RG ‘time’, µ the RG momentum scale and Λa characteristic reference scale of the theory. A and B are numbers. We assume that (1)arises from a perturbative expansion of the full β -function ( β ≡ ∂ t α ), with α reasonablysmall for perturbation theory to be applicable. The linear term relates to the tree levelcontribution, reflecting that the underlying coupling is dimensionful with mass dimension − A . The quadratic term stands for the one loop contribution. Evidently, the flow displaystwo types of fixed points, a trivial one at α ∗ = 0, and a non-trivial one at α ∗ = A/B . (2)In the spirit of perturbation theory, the non-trivial fixed point (2) is accessible in the domainof validity of the RG flow (1) as long as α ∗ (cid:28)
1. This can be achieved in two manners,either by having A (cid:28) B , or by making 1 /B (cid:28) A . (Below, we discussexamples where both of these options are realised.) Integrating (1) in the vicinity of thefixed point, one finds that small deviations from it scale as δα = ( α − α ∗ ) ∝ (cid:18) µ Λ c (cid:19) ϑ , (3)thereby relating the characteristic energy scale Λ c of the theory to the deviation from thefixed point δα at the RG scale µ and a universal number ϑ . The role of Λ c here is similarto that of Λ QCD in QCD as it describes the cross-over between two different scaling regimesof the theory. The universal scaling index ϑ arises as the eigenvalue of the one-dimensional‘stability matrix’ ϑ = ∂β∂α (cid:12)(cid:12)(cid:12)(cid:12) ∗ . (4)It is given by ϑ = − A at the non-trivial fixed point (2), and by ϑ = A at the Gaussian fixedpoint. We have that ϑ <
A >
0. Consequently,small deviations from the fixed point (3) decrease with increasing RG momentum scalemeaning that (2) is an UV fixed point. If, additionally,
B >
0, the fixed point obeys α ∗ > A = 0 the model (1) displays a doubly-degenerate Gaussian fixed point. In this light, as soon as the canonical mass dimension of theunderlying coupling becomes negative, A >
0, such as in theories which are power-countingnon-renormalisable, the degeneracy of the perturbative β -function is lifted leading to a pairof non-degenerate fixed points. Provided that α is the sole relevant coupling of the modelunder consideration, the existence of an interacting UV fixed point can be used to definethe theory fundamentally. This are the bare bones of asymptotic safety [1, 2, 6]. B. Gravitons
We now recall specific examples for asymptotically safe quantum field theories wherethe mechanism just described is at work. We start with Einstein gravity with action(16 π G N ) − (cid:82) √ det gR in D dimensions. Newton’s coupling G N has canonical mass dimen-sion [ G N ] = 2 − D and the theory is power-counting non-renormalisable above its criticaldimension D c = 2. In units of the RG scale, the dimensionless gravitational coupling of themodel reads α = G N ( µ ) µ D − . (5)In D = D c + (cid:15) dimensions, one finds the RG flow (1) with A = (cid:15) (cid:28) B = 50 /
3, andan UV fixed point (2) in the perturbative regime by analytical continuation of space-timedimensionality [6, 55, 56, 71]. Results have been extended to two-loop order, also including acosmological constant [72]. It is the sign of
B > The leading exponent ϑ is related to the exponent ν in the statistical physics literature as ν = − /ϑ . C. Fermions
Next we consider a purely fermionic theory of N F selfcoupled massless Dirac fermionswith, examplarily, Gross-Neveu-type selfinteraction g GN ( ¯ ψψ ) , [80]. The quartic fermionicselfcoupling g GN has canonical mass dimension [ g GN ] = 2 − D and the model is perturbativelynon-renormalisable above its critical dimension D c = 2. The dimensionless coupling reads α = g GN ( µ )2 πN F µ − D . (6)Close to two dimensions the β -function (1) for (6) can be computed within the (cid:15) -expansionby setting D = D c + (cid:15) . The coefficient A , given by minus the canonical mass dimension,becomes A = (cid:15) (cid:28)
1, while the coefficient
B >
0, to leading order in (cid:15) , is of order one andgiven by the 1-loop coefficient in the two-dimensional theory. Hence, the model has a reliableUV fixed point (2) in the perturbative regime. Its renormalisability has been establishedmore rigorously in [58, 59] with the help of the non-perturbative renormalisation group.Similarly, in the large- N F limit and at fixed dimension D = 3, one finds that A ∝ /N F (cid:28) B >
D. Gluons
Next, we consider pure SU ( N C ) Yang-Mills theory with N C colors in D = 4 + (cid:15) dimen-sions, with action ∼ / ( g ) (cid:82) Tr F µν F µν . The canonical mass dimension of the couplingreads [ g ] = 4 − D and the theory is perturbatively non-renormalisable above its criticaldimension D c = 4. Introducing the running dimensionless strong coupling as α = g N C (4 π ) µ D − , (7)one may derive its β -function (1) to leading order in the (cid:15) -expansion, (cid:15) = D − D c . Again,one finds that A = (cid:15) (cid:28) B > (cid:15) [16] suggesting that this fixedpoint persists in D = 5, in accord with a prediction using functional renormalisation [85]. E. Scalars
Finally we turn to self-interacting scalar fields. For scalar field theory with linearlyrealised O ( N ) symmetry, the dimensionless version of its quartic self-coupling is given by α = λ N (4 π ) µ D − . (8)The critical dimension of these models is D c = 4. Within the (cid:15) -expansion away from the crit-ical dimension, using D = 4 − (cid:15) , the one loop β -function is of the form (1) with A = − (cid:15) < B <
0. Therefore, the fixed point (2) is physical below the critical dimension, andit is an infrared one, i.e. the seminal Wilson-Fisher fixed point. For the physically rele-vant dimension D = 3, its existence has been confirmed even beyond perturbation theory[86, 87]. In this light, the Wilson-Fisher fixed point can be viewed as the infrared analogueof asymptotic safety. The search for interacting UV fixed points in D > σ -models, i.e. scalar theories with non-linearly realised internal symmetry,the critical dimension is reduced to D c = 2. The relevant coupling then displays an UV fixedpoint within perturbation theory [90, 91]. Fixed points of non-linear σ -models have also beeninvestigated for their similarity with gravity [92]. Lattice results for an interacting UV fixedpoint in D = 3 in accord with functional renormalisation have recently been reported in[93]. III. FROM ASYMPTOTIC FREEDOM TO ASYMPTOTIC SAFETY
In this section, we explain the perturbative origin for asymptotic safety in a class ofgauge-Yukawa theories in strictly four dimensions.
A. Gauge-Yukawa theory
We consider a theory with SU ( N C ) gauge fields A aµ and field strength F aµν ( a =1 , · · · , N C − N F flavors of fermions Q i ( i = 1 , · · · , N F ) in the fundamental represen-tation, and a N F × N F complex matrix scalar field H uncharged under the gauge group.The fundamental action is taken to be the sum of the Yang-Mills action, the fermion ki-netic terms, the Yukawa coupling, and the scalar kinetic and self-interaction Lagrangean L = L YM + L F + L Y + L H + L U + L V , with L YM = −
12 Tr F µν F µν (9) L F = Tr (cid:0) Q i /D Q (cid:1) (10) L Y = y Tr (cid:0) Q L HQ R + Q R H † Q L (cid:1) (11) L H = Tr ( ∂ µ H † ∂ µ H ) (12) L U = − u Tr ( H † H ) (13) L V = − v (Tr H † H ) . (14)Tr is the trace over both color and flavor indices, and the decomposition Q = Q L + Q R with Q L/R = (1 ± γ ) Q is understood. This theory has been investigated for its interestingproperties in [65–68]. We will motivate it for our purposes while we progress. In fourdimensions, the model has four classically marginal coupling constants given by the gaugecoupling, the Yukawa coupling y , the quartic scalar couplings u and the ‘double-trace’ scalarcoupling v , which we write as α g = g N C (4 π ) , α y = y N C (4 π ) , α h = u N F (4 π ) , α v = v N F (4 π ) . (15)We have normalized the couplings with the appropriate powers of N C and N F preparingfor the Veneziano limit to be considered below. Notice the additional power of N F in thedefinition of the scalar double-trace coupling, meaning that v/u becomes α v / ( α h N F ). Wealso use the shorthand notation β i ≡ ∂ t α i with i = ( g, y, h, v ) to indicate the β -functions forthe couplings (15). To obtain explicit expressions for these, we use the results [94–96]. B. Leading order
We begin our reasoning with the RG flow for the gauge coupling to one-loop order usingthe SU ( N C ) Yang-Mills Lagrangean (9) coupled to N F fermions (10) in the fundamentalrepresentation, β g = ∂ t α g = − B α g . (16)Note that a linear term Aα g is absent, unlike in (1), as we strictly stick to four dimensions.To this order the gauge β -function (16) displays a doubly-degenerated fixed point at α ∗ g = 0 (17)which is an UV fixed point for positive B . Provided that the coefficient B is numericallyvery small, | B | (cid:28)
1, however, we also have that ∂ t α g (cid:28) < α ∗ g (cid:28) B >
0, (17) corresponds to asymptotic freedom,in which case (18) would then correspond to a conformal infrared fixed point. In turn,for
B < B depends on both N C and N F . Explicitly, B = − (cid:15) , where (cid:15) = N F N C − . (19)For (cid:15) >
0, asymptotic freedom of the gauge sector is lost. The prerequisition for an asymp-totically safe fixed point within the perturbative regime thus translates into0 ≤ (cid:15) (cid:28) . (20) There is a vast body of work dealing with the availability of this Banks-Zaks type IR fixed point [70, 97]in the continuum and on the lattice (for recent overviews see [98, 99] and references therein). N C and N F to infinity, butkeeping their ratio N F /N C fixed [69]. The parameter (19) thereby becomes continuous andcan take any real value including (20). In most of the paper, the parameter (cid:15) will be ourprimary perturbative control parameter in the regime (20), except in Sec. V where we alsodiscuss the regime where (cid:15) becomes large. C. Next-to-leading order
We now must check whether this scenario can be realized upon the inclusion of higher loopcorrections. At the next-to-leading (NLO) order in perturbation theory, which is two-loop,the RG flow for the gauge coupling takes the form ∂ t α g = − B α g + C α g . (21)As such, the gauge β -function (21) may display three fixed points, a doubly-degenerated oneat α ∗ g = 0, and a non-trivial one at α ∗ g = B/C . (22)The non-trivial fixed point is perturbative as long as 0 ≤ α ∗ g (cid:28) | B | (cid:28) C of order unity, and B/C >
0. For this fixed point to be an asymptotically safe one wemust have
B <
C <
0. However, in the absence of Yukawa interactions one finds C = 25 [97] to leading order in (cid:15) . Consequently, the would-be fixed point (22) resides in theunphysical domain α ∗ g < β -function depends on the Yukawa coupling starting from the two-loop order. Forthis reason, to progress, we must first evaluate the impact of non-trivial Yukawa couplings.At the same time, since the fixed point for the gauge coupling depends on the Yukawacoupling, we must retain its RG flow to its first non-trivial order, which is one-loop. Havingintroduced Yukawa couplings means that we also have dynamical scalars, (11). The simplestchoice here is to assume that the scalars are uncharged under the gauge group, whence (12).Then neither the gauge nor the Yukawa RG flows depend on the scalar couplings at this orderand we can neglect their contribution for now. This ordering of perturbation theory for thedifferent couplings is also favored by considerations related to Weyl consistency conditions(see Sec. IV B). Hence, following this reasoning and in the presence of the Yukawa term (11),we end up with β g = α g (cid:40) (cid:15) + (cid:18)
25 + 263 (cid:15) (cid:19) α g − (cid:18)
112 + (cid:15) (cid:19) α y (cid:41) ,β y = α y (cid:110) (13 + 2 (cid:15) ) α y − α g (cid:111) . (23)1 Α g Α y Α h Ε Figure 2: The coordinates of the UV fixed point as a function of (cid:15) at NLO (dashed) and NNLO(full and short-dashed lines). Gauge, Yukawa and scalar couplings are additionally shown in red,magenta and blue, respectively. NNLO corrections lead to a mild enhancement of the gauge andYukawa couplings over their NLO values. The scalar and Yukawa fixed point couplings are nearlydegenerate at NNLO.
The coupled system (23) admits three types of fixed points within perturbation theory (20).The system still displays a Gaussian fixed point( α ∗ g , α ∗ h ) = (0 ,
0) (24)irrespective of the sign of (cid:15) . For (cid:15) >
0, neither the gauge coupling nor the Yukawa couplingare asymptotically free at this fixed point. Ultimately, this is related to the sign of either β -function being positive arbitrarily close to (24), making it an IR fixed point. A second,non-trivial fixed point is found as well, ( α ∗ g , α ∗ h ) (cid:54) = (0 ,
0) which is an UV fixed point withcoordinates α ∗ g = 26 (cid:15) + 4 (cid:15) − (cid:15) − (cid:15) = 2657 (cid:15) + 14243249 (cid:15) + 77360185193 (cid:15) + O ( (cid:15) ) α ∗ y = 12 (cid:15) − (cid:15) − (cid:15) = 419 (cid:15) + 1841083 (cid:15) + 1028861731 (cid:15) + O ( (cid:15) ) . (25)Numerically, the series (25) reads α ∗ g = 0 . (cid:15) + 0 . (cid:15) + 0 . (cid:15) + O ( (cid:15) ) α ∗ y = 0 . (cid:15) + 0 . (cid:15) + 0 . (cid:15) + O ( (cid:15) ) (26)This UV fixed point is physically acceptable in the sense that ( α ∗ g , α ∗ y ) > (0 ,
0) for (cid:15) >
0. Itarises because the gauge and Yukawa couplings contribute with opposite signs to β g at thetwo-loop level, allowing for an asymptotically safe fixed point in the physical domain.As an aside, we also notice the existence of a second interacting fixed point within per-turbation theory located at ( α ∗ g , α ∗ y ) = ( − (cid:15)
75 + 26 (cid:15) , . (27)2 e v e h e g e y Ε Figure 3: Components of the eigenvectors (31) and (45) (absolute values) corresponding to therelevant eigenvalue ϑ at the UV fixed point at NLO (dashed) and NNLO (full lines), respectively,as functions of (cid:15) . From top to bottom, the gauge, Yukawa and scalar, and double-trace scalarcomponents are shown. Notice that the relevant eigendirection is largely dominated by the gaugecoupling. Each component varies only mildly with (cid:15) . For (cid:15) > α g > (cid:15) <
0, (27) takes the role of an interacting infrared fixedpoint `a la
Caswell, Banks and Zaks [70, 97]. Interacting IR fixed points play an importantrole in extensions of the standard model with a strongly interacting gauge sector and modelswith a composite Higgs (see [98, 99] and references therein). The IR fixed point arises evenin the absence of scalar fields, whereas the UV fixed point necessitates scalar matter withnon-vanishing Yukawa interactions. We conclude that the IR fixed point (27) is profoundlydifferent from the UV fixed point (25).Returning to our main line of reasoning, we linearize the RG flow in the vicinity of itsUV fixed point (25), β i = (cid:88) j M ij ( α j − α ∗ j ) + subleading (28)where i = ( g, y ) and M ij = ∂β i /∂α j | ∗ denotes the stability matrix. The eigenvalues of M are universal numbers and characterise the scaling of couplings in the vicinity of the fixedpoint. They can be found analytically. The first few orders in (19) are ϑ = − (cid:15) + 22963249 (cid:15) + 13877681666737 (cid:15) + O ( (cid:15) ) ϑ = 5219 (cid:15) + 91401083 (cid:15) + 2518432185193 (cid:15) + O ( (cid:15) ) . (29)Numerically, the eigenvalues (29) read ϑ = − . (cid:15) + 0 . (cid:15) − . (cid:15) + O ( (cid:15) ) ϑ = 2 . (cid:15) + 8 . (cid:15) + 13 . (cid:15) + O ( (cid:15) ) (30)3A few comments are in order. Firstly, the gauge-Yukawa system at NLO has developed arelevant and an irrelevant eigendirection with eigenvalues ϑ < ϑ >
0, respectively. Secondly, the relevant eigenvalue ϑ is found to be of order (cid:15) , whereas the irrelevant oneis of order (cid:15) and thus parametrically larger. This is not a coincidence, and its origin can beunderstood as follows: All couplings settle at values of order (cid:15) at the fixed point. Hence, β g ∼ (cid:15) and β y ∼ (cid:15) in the vicinity of the fixed point. The relevant eigenvalue originatesprimarily from the gauge sector, because asymptotic freedom is destabilised due to (20).Consequently, at the fixed point, the relevant eigenvalue must scale as ϑ ∼ ∂β g /∂(cid:15) | ∗ ∼ (cid:15) .Conversely, the irrelevant eigenvalue scales as ϑ ∼ ∂β y /∂(cid:15) | ∗ ∼ (cid:15) . We conclude that theparametric dependence ∼ (cid:15) of the relevant eigenvalue arises because the fixed point in thegauge sector stems from a cancellation at two-loop level. Conversely, the behaviour ∼ (cid:15) of the irrelevant eigenvalue stems from cancellations at the one-loop level. This feature isa direct consequence of the vanishing mass dimension of the couplings. Asymptotic safetythen follows as a pure quantum effect rather than through the cancelation of tree-level andone-loop terms.Finally, introducing a basis in coupling parameter space as a = ( α g , α y ) T , we denote therelevant eigendirection at the UV fixed point as e = ( e g , e y ) T . (31)The absolute values of its entries are shown in Fig. 3 (dashed lines). We find that therelevant eigenvalue is largely dominated by the gauge coupling for all (cid:15) . Furthermore, therelevant eigendirection is largely independent of (cid:15) , up to (cid:15) < . D. Next-to-next-to-leading order
We now move on to the next-to-next-to-leading order (NNLO) in the perturbative expan-sion where the scalar sector is no longer treated as exactly marginal. Identifying a combinedUV fixed point in all couplings becomes a consistency check for asymptotic safety in the fulltheory. In practice, this amounts to adding the quartic selfinteraction terms (13) and (14).The one-loop running of the quartic couplings given by β h = − (11 + 2 (cid:15) ) α y + 4 α h ( α y + 2 α h ) , (32) β v = 12 α h + 4 α v ( α v + 4 α h + α y ) . (33) Strictly speaking, there are two further marginal eigenvalues ϑ , = 0 related to the scalar selfinteractionswhich we have taken as classical. β (2) y = α y (cid:26) (cid:15) − α g + (49 + 8 (cid:15) ) α g α y − (cid:18) (cid:15) + (cid:15) (cid:19) α y − (44 + 8 (cid:15) ) α y α h + 4 α h (cid:27) ∆ β (3) g = α g (cid:26)(cid:18) (cid:15) − (cid:15) (cid:19) α g −
278 (11 + 2 (cid:15) ) α g α y + 14 (11 + 2 (cid:15) ) (20 + 3 (cid:15) ) α y (cid:27) . (34)Several comments are in order. Firstly, both scalar β functions are quadratic polynomialsin the couplings and display a Gaussian fixed point. The full system at NNLO then displaysa Gaussian fixed point ( α ∗ g , α ∗ y , α ∗ h , α ∗ v ) = (0 , , ,
0) as it must.Secondly, and unlike all other β -functions, the RG flow (33) for the double-trace couplingshows no explicit dependence on (cid:15) . Also, to leading order in 1 /N C and 1 /N F , (33) staysquadratic in its coupling to all loop orders [101].Finally, and most notably, the β -functions of the gauge, Yukawa and single-trace scalarcoupling remain independent of the double-trace scalar coupling α v . In consequence thedynamics of α v largely decouples from the system and it acts like a spectator withoutinfluencing the build-up of the asymptotically safe UV fixed point in the gauge-Yukawa-scalar subsector. Its own RG evolution is primarily fueled by the Yukawa and the singletrace coupling, and as such indirectly sensitive to the gauge-Yukawa fixed point. In turn,the scalar coupling α h couples back into the Yukawa coupling, though not into the gaugecoupling.Since the RG flows at NNLO partly factorize, we can start by first considering the cor-rections to (25) induced by the scalar coupling α h . This leads to UV fixed points in thegauge-Yukawa-scalar subsystem ( α ∗ g , α ∗ y , α ∗ h ) (cid:54) = (0 , , . (35)In the background of the gauge-Yukawa fixed point (25), the RG flow (32) for α h admitstwo fixed points α ∗ h < < α ∗ h , with α ∗ h ,h = ( ±√ − (cid:15)
19 + O ( (cid:15) ) , (36)irrespective of α v . Inserting α ∗ h together with (25) into (33) we then also find two solutionsfor the double trace coupling, with α ∗ v < α ∗ v . These are discussed in more detail below.Conversely, inserting α ∗ h together with (25) into (33) does not offer a fixed point for α v . Weconclude that the fixed point α ∗ h ≡ α ∗ h > α h which leadsto a UV fixed point in the scalar subsystem. We return to this in Sec. III E.Using (23), (32), and (34), the coordinates of the gauge, Yukawa and scalar coupling areobtained analytically and can be expressed as a power series in (cid:15) starting as α ∗ g = 2657 (cid:15) + 23(75245 − √ (cid:15) + O ( (cid:15) ) α ∗ y = 419 (cid:15) + (cid:32) − √ (cid:33) (cid:15) + O ( (cid:15) ) α ∗ h = √ − (cid:15) + 1168991 − √ √ (cid:15) + O ( (cid:15) ) . (37)5Numerically, the first few orders in the (cid:15) -expansion read α ∗ g = 0 . (cid:15) + 0 . (cid:15) + 3 . (cid:15) + O ( (cid:15) ) α ∗ y = 0 . (cid:15) + 0 . (cid:15) + 2 . (cid:15) + O ( (cid:15) ) α ∗ h = 0 . (cid:15) + 0 . (cid:15) + 2 . (cid:15) + O ( (cid:15) ) . (38)The addition of the scalar selfcouplings has led to a physical fixed point α ∗ h > (cid:15) .NNLO corrections to α ∗ g and α ∗ y arise only starting at order (cid:15) without altering the NLOfixed point (25). Performing the expansion (38) to high orders in (cid:15) one finds its radius ofconvergence as (cid:15) ≤ (cid:15) max = 0 . · · · (39)At (cid:15) max in (39), the NNLO equations display a bifurcation and the UV fixed point ceases toexist through a merger with a non-perturbative IR fixed point, and the relevant eigenvaluedisappears at (cid:15) max . The merger at (cid:15) max indicates that our working assumption (20) shouldbe superseeded by 0 < (cid:15) (cid:28) (cid:15) max .Since α v does not contribute to the RG flow of the subsystem ( α g , α y , α h ), the computationof scaling exponents equally factorizes. Linearizing the RG flow in the vicinity of the fixedpoint, we have (28) where now i = ( g, y, h ). The eigenvalues ϑ n are found analytically as apower series in (cid:15) , the first few orders of which are given by ϑ = − (cid:15) + 22963249 (cid:15) + 4531558295989 − √ (cid:15) + O ( (cid:15) ) ϑ = 5219 (cid:15) + 136601719 − √ (cid:15) + O ( (cid:15) ) ϑ = 16 √ (cid:15) + 4 217933589 √ − (cid:15) + O ( (cid:15) ) (40)For the relevant eigenvalue, we notice that the first two non-trivial orders have remainedunchanged. Numerically, the eigenvalues read ϑ = − . (cid:15) + 0 . (cid:15) + 2 . (cid:15) + · · · ϑ = 2 . (cid:15) + 6 . (cid:15) + · · · ϑ = 4 . (cid:15) + 14 . (cid:15) + · · · . (41)The cubic and quartic corrections to the relevant eigenvalue both arise with a sign oppositeto the leading term, which is responsible for the smallness of ϑ even for moderate values of (cid:15) . In turn, the irrelevant eigenvalues receive larger corrections and reach values of the orderof 0 . ÷ (cid:15) .We now discuss the role of the double-trace scalar coupling α v . Its fixed points are entirelyinduced by the fixed point in the gauge-Yukawa-scalar subsystem (37). Two solutions arefound, α ∗ v ,v = − (cid:18) √ ∓ (cid:113)
20 + 6 √ (cid:19) (cid:15) + O ( (cid:15) ) (42)6 (cid:74) (cid:72) Ε (cid:76) (cid:74) (cid:72) Ε (cid:76) (cid:45)(cid:74) (cid:72) Ε (cid:76) (cid:45) (cid:45) (cid:45) Ε Figure 4: Universal eigenvalues ϑ < < ϑ < ϑ at the UV fixed point at NLO (dashed lines) andNNLO (full lines) as function of (cid:15) (19). For all technical purposes, the eigenvalues are dominatedby the NLO values except for the close vicinity of (cid:15) ≈ (cid:15) max . Numerically α ∗ v = − . (cid:15) and α ∗ v = − . (cid:15) , up to quadratic corrections in (cid:15) . Inprinciple, either of these fixed points can be used in conjunction with (37) to define thecombined UV fixed point of the theory. We also note that α ∗ h + α ∗ v < < α ∗ h + α ∗ v , (43)showing that the scalar field potential is bounded (unbounded) from below for the latter(former) choice of couplings. At the fixed point α ∗ v , the invariant (14) becomes perturba-tively irrelevant and adds a positive scaling exponent to the spectrum. Conversely, at thefixed point α ∗ v , the invariant (14) has become perturbatively relevant. Since the RG flow(33) is quadratic in the coupling α v to all orders, the corresponding scaling exponents areequal in magnitude with opposite signs, ϑ = 8 (cid:15) (cid:113)
20 + 6 √
23 + O ( (cid:15) ) . (44)Numerically, ϑ = 2 . (cid:15) + O ( (cid:15) ). The occurence of an additional negative eigenvalue isinduced by the interacting UV fixed point (37).We may introduce a basis in coupling parameter space as a = ( α g , α y , α h , α v ) T to de-note the eigendirections at the UV fixed point as e i . They obey the eigenvalue equation M e i = ϑ i e i . The normalised relevant eigendirection e corresponding to the UV attractiveeigenvalue has the components e = ( e g , e y , e h , e v ) T (45) An inspection of radiative corrections confirms that the effective potential for the scalar fields is stableclassically and quantum-mechanically [102]. ϑ is independent of the gauge, Yukawa and scalar coupling, e = (0 , , , E. UV scaling and Landau pole
We summarize the main picture (see also Tab. I). The asymptotically safe UV fixedpoint (25) in the gauge-Yukawa system bifurcates into several fixed points once the scalarfluctuations are taken into account. In addition to the universal eigenvalues of the gauge-Yukawa system ϑ and ϑ , the scalar sector add the eigenvalues ± ϑ and ± ϑ . To the leadingnon-trivial order in (cid:15) , these are ϑ = − . (cid:15) + O ( (cid:15) ) ϑ = 2 . (cid:15) + O ( (cid:15) ) ϑ = 4 . (cid:15) + O ( (cid:15) ) ϑ = 2 . (cid:15) + O ( (cid:15) ) . (46)Complete asymptotic safety, e.g. asymptotically safe UV fixed points in all couplings, isachieved at two UV fixed points, FP and FP . At FP , both scalar invariants are pertur-batively irrelevant and the eigenvalue spectrum is ϑ < < ϑ < ϑ < ϑ . (47)At FP , the partly decoupled double-trace scalar interaction term with coupling α v becomesrelevant in its own right, and the eigenvalue spectrum, instead, reads − ϑ < ϑ < < ϑ < ϑ . (48)At either of these, the UV limit can be taken. We recall that the scalar field potential isstable (unstable) at FP (FP ), indicating that FP corresponds to a physically acceptabletheory at highest energies. Finally, a fixed point FP exists for the gauge-Yukawa-scalarsubsystem. Here, the scalar coupling α h becomes relevant and the eigenvalue spectrumreads − ϑ < ϑ < < ϑ . (49)Here, however, asymptotic safety is not complete. In fact, the β -function for the double-trace scalar coupling does not show a fixed point in perturbation theory as it remains strictlypositive, leading to Landau poles α v → ±∞ in the IR and in the UV. In the UV, this regimeresembles the U (1) or Higgs sector of the standard model. In either case it is no longer underperturbative control. Unless strong-coupling effects resolve this singularity in the UV, thisbehaviour implies a limit of maximal UV extension of the model close to FP .8 fixed point couplings eigenvaluesgauge Yukawa scalar double-trace relevant irrelevantFP α ∗ g α ∗ y α ∗ h α ∗ v ϑ ϑ , ϑ , ϑ FP α ∗ g α ∗ y α ∗ h α ∗ v − ϑ , ϑ ϑ , ϑ FP α ∗ g α ∗ y α ∗ h Landau pole − ϑ , ϑ ϑ Table I: Summary of UV fixed points of the gauge-Yukawa theory and the number of (ir-)relevanteigenvalues. To leading non-trivial order, the fixed point values are given by (25), (36), (42) andthe exponents by (46).
F. UV critical surface
The existence of relevant and irrelevant direction in the UV implies that the short-distancebehaviour of the theory is described by a lower-dimensional UV critical surface. We discussthe NLO case and FP at NNLO in detail. This is straightforwardly generalised to the casewith two relevant eigendirections. On the critical surface in coupling constant space, wemay express the RG running of the irrelevant coupling, say α i with ( i = y, h, v ), in terms ofthe relevat one, say α g , leading to relations α i = F i ( α g ) . (50)To see this more explicitly, we integrate the RG flow in the vicinity of the UV fixed pointto find the general solution α i ( µ ) = α ∗ i + (cid:88) n c n V ni (cid:18) µ Λ c (cid:19) ϑ n + subleading . (51)Here, Λ c is a reference energy scale, ϑ n are the eigenvalues of the stability matrix M , V n the corresponding eigenvectors, and c n free parameters. The eigenvectors generically mixall couplings. At NLO the eigenvalues obey ϑ < < ϑ . For all coupling α i to reach theUV fixed point with increasing RG scale 1 /µ → c = 0. Conversely, the parameter c remains undetermined and should be viewed as a freeparameter of the theory. Since this holds true for each coupling α i , we can eliminate c from(51) to express the irrelevant coupling in terms of the relevant one. At NLO, one finds F y ( α g ) = (cid:18) − (cid:15) (cid:19) α g + 8171 (cid:15) + O ( (cid:15) ) (52)to the first few orders in an expansion in (cid:15) . At NNLO, the hypercritical surface is extendedand receives corrections due to the scalar couplings. At FP , for example, one finds F y ( α g ) = (0 . . (cid:15) ) α g − . (cid:15) + O ( (cid:15) ) F h ( α g ) = (0 . . (cid:15) ) α g − . (cid:15) + O ( (cid:15) ) ,F v ( α g ) = − (0 . . (cid:15) ) α g + 0 . (cid:15) + 0 . (cid:15) + O ( (cid:15) ) (53)9 Β g sep Β y sep UV G Α g Figure 5: The gauge and Yukawa β -functions projected along the separatrix of the phase diagramgiven in Fig. 1 (NLO with (cid:15) = 0 . β → β/ ( α ∗ g ) , and α g → α g /α ∗ g ; see main text. to the first few orders in (cid:15) , and in agreement with (52) to order (cid:15) . The significance of theUV critical surface is that couplings can reach the UV fixed point only along the relevantdirection, dictated by the eigenperturbations. This imposes a relation between the relevantand the irrelevant coupling, both of which scale out of the fixed point with the same scalingexponent ϑ . On the critical surface and close to the fixed point, the gauge coupling evolvesas α g ( µ ) = α ∗ g + (cid:16) α g (Λ c ) − α ∗ g (cid:17) (cid:18) µ Λ c (cid:19) ϑ ( (cid:15) ) (54)and the irrelevant couplings follow suit via (50) with (52) and (53), and with α ∗ g and ϑ given by the corresponding expressions at NLO and NNLO, respectively. G. Phase diagram
In Fig. 1, we show the phase diagram of the asymptotically safe gauge-Yukawa theory forsmall couplings at NLO, where we have set (cid:15) = 0 .
05. The RG trajectories are obtained fromintegrating (23) with arrows pointing towards the IR. For small couplings, one observesthe Gaussian and the UV fixed points. At vanishing Yukawa (gauge) coupling, the RGequations (23) become infrared free in the gauge (Yukawa) coupling, corresponding to thethick red horizontal (vertical) trajectory in Fig. 1. This makes the Gaussian fixed point IRattractive in both couplings. The UV fixed point has a relevant and an irrelevant direction,corresponding to the two thick red trajectories one of which is flowing out of and the otherinto the UV fixed point. These trajectories are distinguished in that they also divide thephase diagram into four regions
A, B, C and D .0The trajectory connecting the UV with the Gaussian fixed point is a separatrix, whichdefines the boundary between the regions A and C , and B and D . Close to the UV fixedpoint, its coordinates are given by the UV critical surface (52). The RG flow along theseparatrix is given analytically to very good accuracy by (54) and (52), and for suitableinitial conditions α g ( µ = Λ c ) on the separatrix. Note that due to the smallness of therelevant UV eigenvalue compared to the irrelevant one, | ϑ /ϑ | = (cid:15) + O ( (cid:15) ), the RG flowruns very slowly along the separatrix. In turn, trajectories entering into the separatrix (withdecreasing RG scale µ ) run much faster, reflected by their near-perpendicular angle betweenthese trajectories and the separatrix; see Fig. 1.Trajectories in region A run towards the Gaussian FP in the IR, and towards strongYukawa coupling in the UV. Trajectories in B run towards a strong coupling regime inthe IR. In region C , trajectories approach the Gaussian FP in the IR limit. Finally, inregion D trajectories approach a strongly coupled regime in the IR, outside the domainof applicability of our equations. Notice that the Gaussian fixed point is attractive in alldirections. Hence, asymptotically safe trajectories emanating from the UV fixed point eitherrun towards a weakly coupled phase controlled by the Gaussian fixed point in the deep IR,or towards a strongly coupled QCD-like phase characterised by chiral symmetry breakingand confinement. More generally, in Fig. 1, the boundary between weakly ( A and C ) andstrongly ( B and D ) coupled phases at low energies is given by the UV irrelevant direction,i.e. the two full (red) trajectories running into the UV fixed point.In Fig. 5 , we show the gauge and Yukawa β -functions projected along the separatrix asfunctions of the gauge coupling, β sep g ( α g ) ≡ β g ( α g , α y = F y ( α g )) β sep y ( α g ) ≡ β y ( α g , α y = F y ( α g )) (55)also using (50) with (52). Both of them display the Gaussian and the UV fixed point. Wealso recover the UV relevant eigenvalue ϑ = dβ sep i dα g (cid:12)(cid:12)(cid:12) ∗ = ∂β i ∂α g (cid:12)(cid:12)(cid:12) ∗ + ∂F y ∂α g ∂β i ∂α y (cid:12)(cid:12)(cid:12) ∗ (56)from either of these ( i = g, y ). Close to the UV fixed point the RG running is power-like. Close to the IR fixed point, the running becomes logarithmic. Quantitatively, alongthe separatrix the crossover from UV scaling to IR scaling takes place once couplings arereduced to about ∼
65% of their UV fixed point values.
H. Mass terms and anomalous dimensions
If mass terms are present, their multiplicative renormalisation is induced through the RGflow of the gauge, Yukawa, and scalar couplings. We now discuss the scaling associated tothe scalar wave function renormalization, the scalar mass, and the addition of the fermionmass operator. The former is identified with the anomalous dimension γ H for the scalar1wave function renormalization Z H ,∆ H = 1 + γ H , γ H ≡ − d ln Z H d ln µ . (57)Within perturbation theory, the one and two loop contributions to (57) read (see e.g. [103]) γ (1) H = α y , (58) γ (2) H = − (cid:18)
112 + (cid:15) (cid:19) α y + 52 α y α g + 2 α h . (59)Inserting the UV fixed point FP and expanding the anomalous dimension in powers of (cid:15) ,we find γ H = 4 (cid:15)
19 + 14567 − √ (cid:15) + O ( (cid:15) ) . (60)Notice that the leading and subleading terms are both positive. The anomalous dimensionfor the scalar mass term m can be derived from the composite operator ∼ m Tr H † H .Introducing γ m = d ln m /d ln µ one finds the mass anomalous dimension γ (1) m = 2 α y + 4 α h + 2 α v (61)to one-loop order. We find that (61) becomes as large as γ (1) m ≈ .
09 for (cid:15) ≈ . .Evidently, the loop corrections remain small compared to the tree level term leaving m = 0as the sole fixed point within perturbation theory. Analogously, the anomalous dimensionfor the fermion mass operator is defined as∆ F = 3 − γ F , γ F ≡ d ln Md ln µ (62)where M stands for the fermion mass. Within perturbation theory, the one and two loopcontributions read γ (1) F = 3 α g − α y (cid:18)
112 + (cid:15) (cid:19) , (63) γ (2) F = (44 + 8 (cid:15) ) α g α y + (cid:18) − (cid:15) (cid:19) α g + 14 (cid:18)
112 + (cid:15) (cid:19) (cid:18)
232 + (cid:15) (cid:19) α y . (64)Inserting the UV fixed point FP and expanding in (cid:15) we find γ F = 4 (cid:15)
19 + 4048 √ − (cid:15) + O ( (cid:15) ) . (65)The leading and subleading terms are both positive. Interestingly, to one-loop order, thescalar anomalous dimension and the fermion mass anomalous dimension coincide in magni-tude. The quantum corrections are bounded, | γ (1) H | < /
40. We stress that the leading orderresults are entirely fixed by the NLO fixed point (25), and insensitive to the details of thescalar sector. The latter only enter starting at order (cid:15) . The bare and renormalized fields here are related via H B = Z H H . Also the wave function definition in[95] is the inverse of the one here. IV. CONSISTENCY
In this section, we discuss aspects of consistency and the validity of results.
A. Stability
In the LO, NLO and NNLO approximations, we have retained the β -functions of thegauge, Yukawa and quartic couplings at different loop levels within perturbation theory. Aswe have argued, the ordering as shown in Tab. II is dictated by the underlying dynamicstowards asymptotic safety, centrally controlled by the gauge coupling.The selfconsistency of our reasoning is confirmed a posteriori by the stability of the result.Firstly, the leading coefficients in (cid:15) of the NLO fixed point α ∗ g and α ∗ y remain numericallyunchanged at NNLO, see (25) and (37). We therefore expect that all coefficients up to (cid:15) of α ∗ g and α ∗ y and the (cid:15) coefficient of α ∗ h and α ∗ v in (37), (36) and (42) remain unchanged beyondNNLO. Secondly, the stability also extends to the universal eigenvalues. Interestingly, here,the first two non-trivial coefficients (up to order (cid:15) ) for the relevant eigenvalue ϑ at NLOagree with the NNLO coefficients, see (29) and (40). For the leading irrelevant eigenvalue ϑ , this agreement holds for the leading (order (cid:15) ) coefficient.All couplings of the theory have become fully dynamical at NNLO. At N LO in theexpansion, no new consistency conditions arise. Instead, higher loop corrections will lead tohigher order corrections in the results established thus far. Based the observations above,we expect that all coefficients up to (cid:15) ( (cid:15) ) [ (cid:15) ] of the universal eigenvalues ϑ ( ϑ ) [ ϑ , ] atNNLO in (40) are unaffected at N LO and beyond.
B. Weyl consistency
At a more fundamental level an argument known as Weyl consistency conditions [104, 105]lends a formal derivation of this hierarchical procedure of Tab. II. Replacing the couplings(15) by the set { g i } ≡ { g, y, u, v } with β functions β i = dg i /d ln µ , the Weyl consistencycondition ∂β j ∂g i = ∂β i ∂g j (66)relates partial derivatives of the various β functions to each other, and β i ≡ χ ij β j . Thefunctions χ ij plays the role of a metric in the space of couplings. The relations are expectedto hold in the full theory, and hence it is desirable to obey (66) even within finite approx-imations. The crucial point here is that the metric itself is a function of the couplings.Therefore, a consistent solution to (66) will generically relate different orders within a na¨ıve fixed-order perturbation theory. In [106] it was shown that these conditions hold for thestandard model. For the gauge-Yukawa theory studied here, the metric χ has been givenexplicitly in [68] showing that the ordering laid out in Tab. II is consistent with (66).3 coupling order in perturbation theory α g α y α h α v C. Universality
For our explicit computations we have used known RG equations in the MS-bar regu-larisation scheme. In general, the expansion coefficients of β -functions are non-universalnumbers and depend on the adopted scheme. On the other hand, it is well-known that one-loop RG coefficients for couplings with vanishing mass dimension are scheme-independentand universal. Furthermore, the two-loop gauge contribution to the gauge β -function is alsoknown to be universal, provided a mass-scale independent regularisation scheme is adopted.Coefficients at higher loop order are strictly non-universal. The main new effect in our workarises from the two-loop coefficients in the gauge sector, and from the interacting UV fixedpoint in the Yukawa RG flow at one-loop (23). Expressing the Yukawa fixed point in termsof the gauge coupling α ∗ y = α ∗ y ( α g ), one then shows that the fixed point in the gauge sec-tor is invariant to leading order in (cid:15) under perturbative (non-singular) reparametrisations α g → α (cid:48) g ( α g ), see (23). We therefore conclude that the interacting UV fixed point arisesuniversally, irrespective of the regularisation scheme. D. Operator ordering
Unlike in asymptotically free theories, at an interacting UV fixed point it is not knownbeforehand which invariants will become relevant since canonical power counting cannot betaken for granted [54]. For asymptotically safe theories with perturbatively small anomalousdimensions and corrections-to-scaling, however, canonical power counting can again be usedto conclude that invariants with canonical mass dimension larger than four will remainirrelevant at a perturbative UV fixed point. The reason for this is that corrections to scaling,in the regime (20), are too small to change canonical scaling dimensions by an integer, andhence cannot change irrelevant into relevant operators. If masses are switched-on, two suchoperators are the fermion and scalar mass terms, both of which receive only perturbativelysmall corrections at the fixed point. We conclude that the relevancy of operators continuesto be controlled by their canonical mass dimension [54].On the other hand, residual interactions, even if perturbatively weak, control the scalingof invariants which classically have a vanishing canonical mass dimension and can changethese into relevant or irrelevant ones, see Fig. 6. In our model, we find that the operator4 classical UV fourfolddegeneracy (cid:68) (cid:45) (cid:45) (cid:45) Ε Figure 6: The fourfold degeneracy of the classically marginal invariants (9), (11), (13) and (14) —schematically indicated by a thick grey line (left panel) — is lifted by residual interactions in theUV, 0 < (cid:15) (right panel). Also shown are the universal eigenvalues ϑ < < ϑ < ϑ < ϑ (bottomto top, respectively) of the fixed point FP , and the interaction-induced gaps ∆ in the eigenvaluespectrum at NNLO as functions of (cid:15) . ordering of the classically marginal invariants at the fixed point is reflected by our searchstrategy, see Tab. II. At LO, the SU ( N C ) Yang-Mills Lagrangean (9) coupled to N F fermions(10) is assumed to become a relevant operator in the regime (20) because asymptotic freedomis lost. This assumption is tested and confirmed at NLO against the inclusion of Yukawainteractions (11). The eigenvalue ϑ is dominated by the gauge and ϑ dominated by theYukawa coupling. This is consistent with the initial assumption inasmuch as the scaling ofthe Yukawa term provides a subleading correction to the scaling of the Yang-Mills term. AtNNLO, two quartic scalar selfinteractions are introduced whose non-trivial fixed points addtwo eigenvalues to the spectrum. At FP , both of these are irrelevant. At FP , the doubletrace scalar selfinteraction becomes relevant. The structure of the scalar sector is inducedby the fixed point in the gauge-Yukawa subsector. In general, for other values of the gaugeand Yukawa couplings, the scalar sector may not offer a fixed point at all. E. Gap
Residual interactions at the UV fixed point have lifted the fourfold degeneracy amongstthe classically marginal couplings. In Fig. 6, we show the eigenvalues to leading order in (cid:15) at the fixed point FP , except for ϑ which is shown at order (cid:15) . The difference betweenthe smallest negative and the smallest positive eigenvalue, which we denote as the gap ofthe eigenvalue spectrum ∆, is then a good quantitative measure for the strength of residualinteractions. At the UV fixed point we have ∆ = ϑ − ϑ . Quantitatively, the gap in the5eigenvalue spectrum read ∆ = 5219 (cid:15) + O ( (cid:15) ) , (67)where the (sub)leading term in (cid:15) arises from the (N)NLO approximation. Classically, wehave ∆ = 0. We notice that the leading and subleading term have the same sign, increasingthe gap with increasing (cid:15) . We stress that the gap in the eigenvalue spectrum is insensitiveto the details of the scalar sector and only determined by the gauge-Yukawa subsystem. F. Unitarity
An important constraint on quantum corrections relates to the scaling dimension of pri-mary fields such as scalar fields themselves. For a quantum theory to be compatible withunitarity, it is required that the scaling dimension must be larger than unity, ∆ H >
1. Thisbehaviour can be observed in the result. To leading order in (cid:15) , γ H is negative and hence∆ H >
1. At NNLO, we observe cancellations in (59) ensuring that γ H remains negative.Overall, fluctuation-induced corrections reach values of up to 5% for moderate (cid:15) .For the composite scalar operator δ ij ¯ Q i Q j , the leading order corrections in (cid:15) decrease itsscaling dimension ∆ F below its classical value ∆ F = 3, see (62). This is further decreasedat NNLO where all corrections to ∆ F have the same sign and no cancellations occur. TheNNLO corrections are thus stronger than those for ∆ H . Here, corrections push ∆ F downfrom its classical value by up to 10%, leaving ∆ F >
1. We conclude that the effects of residualinteractions are compatible with basic constraints on the scaling of scalar operators.
G. Triviality
Triviality bounds often arise when infrared free interactions display a perturbative Landaupole towards high energies, limiting the predictivity of the theory to the scale of maximalUV extension [5]. On a more fundamental level, triviality relates to the difficulty of defininga self-interacting scalar quantum field in four dimensions [60, 107–109], which also puts theexistence of elementary scalars into question. In the standard model, the scalar and the U (1) sectors are infrared free. At the UV fixed points detected here, triviality for all threetypes of fields is evaded through residual interactions. This also indicates that the scalardegrees of freedom may indeed be taken as elementary.Moreover, we also observe that the avoidance of triviality in the scalar sector is closelylinked to the presence of gauge fields, be they asymptotically free or asymptotically safe. Infact, an interacting fixed point in the scalar sector would not arise without an interactingfixed point for the Yukawa coupling, see (32), (33). Furthermore, without gauge fields, thefermion-boson subsystem does not generate an interacting UV fixed point, and couplingscannot reach the Gaussian fixed point in the UV. With asymptotically free gauge fields(say, for small (cid:15) < (cid:15) > N limit withand without asymptotic freedom in the gauge sector, although the specific details differ.Asymptotic freedom in the gauge sector had to be given up for the Yukawa and scalarsectors to develop interacting UV fixed points. V. TOWARDS ASYMPTOTIC SAFETY AT STRONG COUPLING
It would be useful to understand the existence or not of UV fixed points in non-Abeliangauge theories with matter and away from the regime where asymptotic safety is realisedperturbatively and (cid:15) is small. In this section, we indicate some directions towards larger (cid:15) ,with and without scalar matter.
A. Beyond the Veneziano limit
Presently, our study is bound to the second nontrivial order within perturbation theoryand to the leading order in 1 /N F , /N C (cid:28)
1, allowing for an accurate determination of theUV fixed point in the regime (20). The stability in the result makes it conceivable that theUV fixed point may persist even for finite values of (cid:15) . With increasing (cid:15) , the upper bound(39) which has arisen at NNLO comes into play. Solutions ( N C , N F ) to the constraint0 ≤ (cid:15) ( N C , N F ) < (cid:15) max , (68)where we take for (cid:15) max its value at NNLO given in (39), would then be likely candidatetheories where the fixed point may exist even for finite but small couplings. The first fewsuch solutions with the smallest numbers of fields are( N C , N F ) = (5 , , (7 , , (9 , , (10 , , (11 , , (12 , , · · · . (69)Once N C >
12, more than one solution for N F may exist. Extending our study to N LOshould improve the estimate for the window (68) for large N . For finite values of N F and N C , the existence of an asymptotically safe window can in principle be tested using non-perturbative tools such as functional renormalisation [111–114], or the lattice. B. Infinite order perturbation theory
Interestingly, an infinite order result is available for nonabelian gauge theories with afinite number of colors N C < ∞ , without scalars, but with N F → ∞ many Dirac fermionstransforming according to a given representation of the gauge group [115], see also [116, 117]and references therein. In the terminology of this work, this corresponds to the parameterregime 1 (cid:28) (cid:15) , (70)see (19). In this limit the parametric deviation from asymptotic freedom is large, and themodel becomes partly abelian [118]. Defining x = 4 N F T R α with α = g / (4 π ) and T F the7 Β (cid:72) x (cid:76) UV G x Figure 7: The fully resummed gauge β function (71) is shown to leading order in 1 /N F includingthe Gaussian and the UV fixed point ( N C = 5 and N F = 28). trace normalization, it is possible to sum exactly the infinite perturbative series for the gauge β -function for large numbers of flavors. The all-order result has the form [116, 118, 119]32 x β ( x ) x = 1 + H ( x ) N F + O (cid:0) N − F (cid:1) (71)and an integral representation of H ( x ) can be found in [116] (see Fig. 7 for an example).Adopting T F = , one can show that the function H ( x ) is finite for 0 ≤ x < x = 3 where H ( x ) = N C / × ln | − x | + const. + O (3 − x ). (Similarresults are found for other representations as well.) This structure implies the existence ofa nontrivial UV fixed point to leading order in 1 /N F . As a word of caution, however, weremind the reader that an infinite order perturbative result may be upset non-perturbatively,or by higher order terms in N − F (see [116] for a discussion of the latter in QED). Expandingabout the fixed point, the two leading terms read α ∗ = 32 N F − N F exp (cid:18) − a · N F N C + b ( N C ) (cid:19) (72)where a = 8, and b ( N C ) (cid:39) .
857 + 2 . /N C . The UV fixed point starts dominating theRG running once ( α ∗ − α ) /α ∗ ∼ 0. Eigenvalues which grow rapidly with the number of degrees of freedomhave been observed previously for quantum gravity in the large dimensional limit in thecontinuum [17, 18, 73] and from lattice considerations [120]. C. Finite order perturbation theory The origin of asymptotic safety in Yang-Mills theory with (72), (73) is different fromthe one observed in Sec. III, because the vanishing of the gauge β -function (71) arises asan infinite order effect due to gluon and fermion loops for large (cid:15) , rather than through anorder-by-order cancellation of fluctuations from gauge, fermion and scalar fields for small (cid:15) . It would be useful to understand whether the result (72) persists beyond the limit ofinfinite N F with fixed and finite N C . To that end, we test the continuity of the fixed pointin ( N F , N C ) by combining two observations. Firstly, we notice that a precursor of the fixedpoint (72) is already visible within perturbation theory at finite orders. To see this explicitly,we come back to our equations at NNLO and switch off the Yukawa and scalar coupling, α h = 0 , α y = 0 and α v = 0. In the parameter regime (70), we then find the UV fixed point α ∗ g = 3 / (2 √ (cid:15) ) + subleading, and the eigenvalue ϑ = − √ (cid:15)/ √ α ∗ = 32 √ (cid:112) N C N F ,ϑ = − √ (cid:114) N F N C , (74)to leading order in 1 /(cid:15) and 1 /N F . A few comments are in order. Comparing (74) with(72), (73) for fixed N C , we find that the 1 /N F decay of the fixed point is replaced by asofter square-root decay due to the finite order approximation in perturbation theory. Thenon-analytic dependence on N F and N C develops into the result (72) with increasing ordersin perturbation theory where the power law behaviour becomes α ∗ ∼ N (2 − p ) / ( p − F [115],provided the p -loop coefficient is negative [117]. We also find that the eigenvalue ϑ in(74) grows large in the regime (70), modulo subleading corrections. While the growth rate ϑ ∼ −√ N F in (74) is weaker than the one observed in (73), the correlation length exponent ν shows the same qualitative behaviour ν → /N C → N F /N C fixed can be accessed, and hence finite values for (cid:15) with (70), because the underlyingNNLO equations remain valid in this parameter regime. Note that this limit is not coveredby the rationale which has led to (71). The UV fixed point then reads α ∗ = 33 + 6 (cid:15) √ (cid:15) N F . (75)For fixed (cid:15) , the fixed point shows the same 1 /N F behaviour as the fixed point (72). Thecoefficient in front of 1 /N F in (75) is larger than the fixed point (72) for all finite (cid:15) . Unlike9(73), its eigenvalue (74) remains bounded since N F /N C is finite. It would thus seem thatthe inclusion of more gluons or less fermions maintains the UV fixed point, albeit witha softened UV scaling behaviour and at stronger coupling. The continuity of results in( N F , N C ) suggests that the UV fixed point (72) is not an artefact of the large- N F limit, butrather a fingerprint of a fixed point in the physical theory.In summary, the observations in this section indicate that the matter-gauge systems stud-ied here have a sufficiently rich structure to admit asymptotically safe UV fixed points alsofor finite ( N C , N F ), with and without scalar matter, in addition to the weakly coupled UVfixed point for small (cid:15) . More work is required to identify them reliably within perturbationtheory and beyond, and for generic values of (cid:15) . VI. CONCLUSION We have used large- N techniques to understand the ultraviolet behaviour of theoriesinvolving fundamental gauge fields, fermions, and scalars. In strictly four space-time di-mensions, and in the regime where the gauge sector is no longer asymptotically free, wehave identified a perturbative origin for asymptotic safety. We found that all three types offields are necessary for an interacting UV fixed point to arise. The primary driver towardsasymptotic safety are the Yukawa interactions, which source the interacting fixed point forboth the gauge fields and the scalars. In return, the gauge fields stabilise an interactingfixed point in the Yukawa sector. Fixed points are established in the perturbative domain,consistent with unitarity. Triviality bounds and Landau poles are evaded. Here the scalarfields can be considered as elementary.It would be worth extending this picture within perturbation theory and beyond, alsotaking subleading corrections into consideration, and for fields with more general gaugecharges, gauge groups, and Yukawa interactions. Once the number of fields is finite, asymp-totic safety can in principle be tested non-perturbatively using the powerful machinery offunctional renormalisation [111–114], or the lattice. In a different vein, one might wonderwhether the weakly coupled ultraviolet fixed point has a strongly coupled dual. First stepsto extend the ideas of Seiberg duality [121] to non-supersymmetric theories have been dis-cussed in [122, 123]. It has also been suggested that UV conformal matter could simplifythe quantisation of canonical gravity [39], or help to resolve outstanding puzzles in particlephysics and cosmology. Our study offers such candidates. Acknowledgments This work is supported by the Science Technology and Facilities Council (STFC) [grantnumber ST/J000477/1], by the National Science Foundation under Grant No. PHYS-1066293, and by the hospitality of the Aspen Center for Physics. The CP -Origins centre is0partially funded by the Danish National Research Foundation, grant number DNRF90. 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