Directions for model building from asymptotic safety
Andrew D. Bond, Gudrun Hiller, Kamila Kowalska, Daniel F. Litim
DDO-TH 17/02, QFET-2017-04
Directions for model building from asymptotic safety
Andrew D. Bond, Gudrun Hiller, Kamila Kowalska, and Daniel F. Litim Department of Physics and Astronomy, University of Sussex, Brighton, BN19QH, United Kingdom Institut f¨ur Physik, Technische Universit¨at Dortmund, D-44221 Dortmund, Germany
Building on recent advances in the understanding of gauge-Yukawa theories we explorepossibilities to UV-complete the Standard Model in an asymptotically safe manner. Minimalextensions are based on a large flavor sector of additional fermions coupled to a scalar singletmatrix field. We find that asymptotic safety requires fermions in higher representations of SU (3) C × SU (2) L . Possible signatures at colliders are worked out and include R -hadronsearches, diboson signatures and the evolution of the strong and weak coupling constants. I . Introduction II . Basics of asymptotic safety for gauge theories
III . Asymptotic safety beyond the Standard Model N F approximation 15H. Synopsis of UV fixed points 18 IV . Matching onto the Standard Model V . Phenomenology R -hadrons 38D. Diboson spectra and resonances 38 VI . Summary A . Technicalities References a r X i v : . [ h e p - ph ] S e p I. INTRODUCTION
Asymptotic freedom plays a central role in the construction of the Standard Model (SM) ofparticle physics and extensions thereof [1, 2]. It predicts that interactions are dynamically switchedoff at highest energies due to quantum fluctuations. In the language of the renormalisation group,asymptotic freedom corresponds to a free ultraviolet (UV) fixed point. Asymptotic freedom fa-mously requires the presence of non-abelian gauge fields [3], together with suitable matter inter-actions to ensure that Yukawa and scalar couplings reach the free fixed point in the UV alongsidethe non abelian gauge coupling [4]. Identifying viable theories beyond the Standard Model (BSM)with complete asymptotic freedom continues to be an active area of research [5–7].Asymptotic safety states that fundamental quantum fields may very well remain interacting athighest energies [8, 9], implying that running couplings reach an interacting (rather than a free)UV fixed point under the renormalisation group evolution. If so, theories remain well-behaved andpredictive up to highest energies in close analogy to theories with complete asymptotic freedom.Asymptotic safety has initially been put forward as a scenario for quantum gravity [9] where alarge amount of evidence has arisen from increasingly sophisticated studies in four dimensionsincluding signatures at colliders (see [10] for an overview). More recently, necessary and sufficientconditions for asymptotic safety in general weakly coupled gauge theories (without gravity) havebeen derived, alongside strict no go theorems [11]. Most importantly, it was found that Yukawainteractions together with elementary scalar fields such as the Higgs offer a unique key towardsasymptotic safety [11]. Moreover, an important proof of existence has been provided in [12], andfurther expanded in [13], showing that exact asymptotic safety with a stable ground state can arisein SU ( N ) gauge theories under strict perturbative control in the Veneziano limit. The feasibilityof asymptotic safety is thus well motivated theoretically and opens intriguing new directions formodel building beyond the SM.In this paper, we make a first step to investigate asymptotically safe extensions of the SMand phenomenological signatures thereof at colliders. Our motivation for doing so is twofold.Firstly, we want to understand whether and how minimal extensions of the SM can be found withweakly interacting UV fixed points. We are particularly interested in the “phase space” of suchextensions, and in the concrete conditions under which interacting UV fixed points are connectedthrough well-defined trajectories with the SM at low energies. Secondly, we wish to understand howphenomenological constraints may arise through existing data, and, more generally, the conditionsunder which asymptotic safety can be tested at colliders. Our investigation is “top-down” in thatwe begin by requiring conditions under which weakly coupled asymptotic safety can be achieved.Our central new input are BSM fermions and scalars, some of which are charged under the gaugesymmetries of the SM. Our approach will be minimal in that we add a single BSM Yukawacoupling whose sole task is to negotiate asymptotically safe UV completions for the SM with SU (3) C ⊗ SU (2) L ⊗ U (1) Y gauge symmetry.The paper has the following format. In Sec. II we discuss the basic perturbative mechanismfor asymptotic safety in gauge theories including general conditions for existence. In Sec. III weinvestigate minimal extensions of the Standard Model in view of weakly interacting high energyfixed points. In Sec. IV , we explain the conditions under which interacting UV fixed points areconnected with the SM at low energies. Phenomenological implications are worked out in Sec. V .We summarize in Sec. VI . Appendix A contains technicalities summarising the perturbative loopcoefficients and group theoretical information, and details of UV-IR connecting separatrices. II. BASICS OF ASYMPTOTIC SAFETY FOR GAUGE THEORIES
In this section, we recall the basic mechanism for asymptotic safety in four-dimensional gaugetheories with matter and recall general theorems for asymptotic safety in weakly coupled gaugetheories following [11, 12]. We also introduce some notation and conventions.
A. Weakly interacting UV fixed points
We begin with a discussion of asymptotic safety in gauge theories and the renormalisation grouprunning of couplings. In the absence of asymptotic freedom, it is well-known that perturbativecouplings would grow towards higher energies thereby limiting predictivity to a highest energyscale Λ. The main feature of asymptotic safety, however, is that the growth of couplings is tamed,dynamically, through a weakly interacting fixed point. An explicit mechanism which allows quan-tum fields to avoid the notorious Landau poles of QED-like theories has recently been discoveredin [12]. Strict theorems for asymptotic safety in general weakly coupled gauge theories have beenderived in [11].To illustrate the mechanism, and to prepare for our models below, we consider the renormal-ization group (RG) flow for a simple gauge theory with gauge coupling α g = g / (4 π ) interactingwith scalars and fermions, with Yukawa coupling α y = y / (4 π ) . Within perturbation theory, theRG flow in the gauge-Yukawa system to the leading non-trivial order is given by β g ≡ dα g d ln µ = ( − B + C α g − D α y ) α g ,β y ≡ dα y d ln µ = ( E α y − F α g ) α y . (1)Scalar selfcouplings do not impact on interacting fixed points to leading order at weak coupling andcan be neglected. The various loop coefficients B, C, D, E and F depend on the matter contentof the theory, which we leave unspecified at this stage. The gauge coupling is asymptoticallyfree (infrared free) provided that the one loop gauge coefficient obeys B >
B < C may take either sign depending on the matter content. Provided thatasymptotic freedom is absent, B <
0, it has also been shown that
C >
D, E, F > B .In general, theories with (1) may have various types of fixed points, depending on the mattercontent. Equating β i = 0 for both couplings, three types of fixed points are found. The Gaussianfixed point ( α ∗ g , α ∗ y ) = (0 ,
0) (2)always exists, and corresponds to the UV (IR) fixed point provided that
B >
B < α ∗ g , α ∗ y ) = (cid:18) BC , (cid:19) . (3)This is the well-known Caswell–Banks-Zaks fixed point [14, 15] which requires B · C >
B/C (cid:28) B · C <
B < β y = 0, (1) implies that the gauge and Yukawa coupling are proportionalto each other, α y = FE α g . This nullcline condition modifies the running of the gauge coupling andturns (1) into β g = ( − B + C (cid:48) α g ) α g , (4)where the two loop term is effectively shifted C → C (cid:48) owing to Yukawa interactions, with C (cid:48) = C − D FE < C . (5)This shift term has important implications: Firstly, the fixed point is now fully interacting, withthe gauge coupling taking the form (3) with C shifted as in (5), together with the interacting fixedpoint for the Yukawa coupling, ( α ∗ g , α ∗ y ) = (cid:18) BC (cid:48) , BC (cid:48) FE (cid:19) . (6)Secondly, for theories with asymptotic freedom ( B > C (cid:48) >
0, the gauge-Yukawa fixed point (6) corresponds to an IR fixed point. It can be reached by RG trajectoriesemanating out of the Gaussian UV fixed point. Finally, for theories without asymptotic freedom(
B <
0) the gauge coupling may now take a viable interacting fixed point α ∗ g = B/C (cid:48) > C (cid:48) <
0. This is the interacting UV fixed point of asymptotic safety (see Tab. for a summary).The result is in stark contrast to theories without Yukawa interactions, where (3) cannot possiblybecome an UV fixed point. We conclude that the Yukawa interactions are of crucial importance forasymptotic safety [11]. Moreover, the necessary condition for asymptotic safety at weak coupling B, C (cid:48) <
0, see Tab. , now translates into a simple condition relating the one and two loopcoefficients appearing in (1), C (cid:48) < ⇔ DF − CE > . (7)In the remaining part of the paper, we evaluate whether the condition (7) can be achieved forextensions of the SM. B. Scaling behaviour
In the vicinity of a free (UV or IR) fixed point the running of couplings is logarithmically slow.In the vicinity of interacting (UV or IR) fixed points, instead, the running of couplings is powerlaw like, characterised by universal scaling exponents { ϑ i } . Linearising the RG flow in the vicinityof a fixed point β i = (cid:88) j M ij ( α j − α ∗ j ) + subleading , (8) case parameter fixed point info typea) B > , C > b) B > , C (cid:48) > c) B < , C (cid:48) < Table 1 . All weakly interacting fixed points α ∗ of simple gauge theories with (1) and their dependence onthe matter content expressed through the parameters B, C and C (cid:48) , see [11]. the scaling exponents can be derived as the eigenvalues of the stability matrix M ij = ∂β i /∂α j | ∗ .Eigendirections are termed relevant (irrelevant) provided that ϑ < ϑ > ϑ = 0 − ) such as in QCD, or marginally irrelevant ( ϑ = 0 + ) such as in QED,is determined beyond leading order. In the vicinity of interacting fixed points couplings scaleaccording to α i ( µ ) = α ∗ i + (cid:88) n c n V ni (cid:18) µµ (cid:19) ϑ n + subleading , (9)where V n are the eigenvectors of the stability matrix with eigenvalue ϑ n , µ denotes the RG scale,and c n are free numbers. The significance of (9) is as follows [16, 17]. In order to achieve awell-defined UV limit, the parameters c n related to irrelevant eigenvalues must be set identicallyto zero, or else the UV fixed point cannot possibly be reached from (9) in the limit µ → ∞ . Onthe other side, the relevant eigendirections are unconstrained and the corresponding numbers c n are free parameters of the theory. Provided that the number of relevant directions is finite, thetheory is predictive with a finite number of free parameters whose values must be determined byexperiment.Returning to the models at hand, three different types of interacting fixed points arise. At aCaswell–Banks-Zaks fixed point (3), scaling exponents are given by ϑ = − BF/C , (10) ϑ = B /C , (11)to leading order in B/C (cid:28)
1, with ϑ < < ϑ . Consequently, the fixed point has a relevantdirection corresponding to the Yukawa interaction, and an irrelevant one, corresponding to thegauge coupling, see Tab. . At the gauge-Yukawa fixed point (6), the scaling exponents of thetheory are given by ϑ = B /C (cid:48) , (12) ϑ = BF /C (cid:48) , (13)to leading order in B/C (cid:48) (cid:28) For asymptotically free theories, we note that 0 < ϑ < ϑ ,meaning that both directions are IR attractive, see Tab. . In the remaining part of the paperwe are particularly interested in theories with asymptotic safety where B <
0. For these, theeigenvalues are of the form ϑ < < ϑ , see Tab. . It states that the fixed point has a one Notice that (12) and (13) do not follow from (10) and (11) by substituting C → C (cid:48) . dimensional UV critical surface characterised by the relevant direction given through ϑ [12]. C. Theorems for asymptotic safety
General theorems for asymptotic safety in weakly coupled gauge theories have recently beenderived in [11]. In particular, it has been established that Yukawa interactions offer a unique mechanism towards asymptotic safety. Neither gauge interactions nor scalar self interactions areable to negotiate an interacting UV fixed point at weak coupling. Stated differently, it is impossibleto find an asymptotically safe and weakly coupled gauge theory with simple or product gaugegroups but without Yukawa interactions. Hence, asymptotic safety in four dimensional gaugetheories invariably requires elementary scalars and fermions, besides the gauge fields. Furthermore,fermions must minimally be charged under some or all of the gauge group(s). For general gaugetheories with product gauge group G = G ⊗ G ⊗ · · · ⊗ G n , weakly interacting fixed points arise assolutions to the linear equations [11] B (cid:48) i = C ij α ∗ j , subject to α ∗ j ≥ , (14)where C ij denotes the matrix of two loop gauge contributions, and B (cid:48) i = B i + 2 Y ∗ ,i the one-loopcoefficient shifted by the Yukawa terms Y ∗ ,i = Tr[ C F i Y A ∗ ( Y A ∗ ) † ] /d ( G i ) ≥ C F i denotes the Casimir of the fermions, Y A the matrix of Yukawa couplings, and d ( G i ) the dimension of the group G i following the conventions of [18–21]. It has also been shownin [11] that for any infrared free gauge factor ( B i < B (cid:48) i > . (15)It states that Yukawa interactions must effectively change the sign of the one loop coefficientfor the infrared free gauge couplings, generalising the necessary condition (7) to general gaugetheories. Sufficiency conditions for asymptotic safety, in addition to the mandatory presence ofYukawa couplings, have also been detailed in [11]. These relate to the specifics of the Yukawasector as well as to the viability of the scalar sector including the stability of the vacuum. It thenremains to investigate whether the mandatory and sufficient conditions for asymptotic safety haveviable weakly coupled UV fixed points as their solutions. In the remaining part of the paper, weinvestigate in concrete terms the availability of asymptotically safe solutions of (14), (15) for BSMextensions of the SU (3) C × SU (2) L sector of the SM. III. ASYMPTOTIC SAFETY BEYOND THE STANDARD MODEL
In this section, we investigate minimal extensions of the SM and conditions under which asymp-totic safety becomes available in the deep UV [11]. While designing the structure of the BSM sector,we make use of the properties of the gauge-Yukawa theory where asymptotic safety can be achievedby an interplay between the gauge and Yukawa interactions of vector-like fermions and a scalarmatrix field [12].
A. Minimal BSM extensions
Asymptotic safety in BSM extensions minimally require the presence of new matter fields whichcarry charges under the SM gauge groups and thereby modify the RG running of couplings. Guidedby the findings of [11, 12], we consider the existence of N F flavors of BSM vector-like fermions ψ which minimally couple to the SM gauge bosons. In general, the BSM fermions may carry chargesunder SU (3) C , SU (2) L , or hypercharge Y , meaning ψ i ( R , R , Y ) , (16)where i = 1 , · · · , N F denotes the flavor index. Furthermore, the BSM fermions couple via Yukawainteractions to complex scalar fields S ij which we take to be a singlet under the SM. Since theBSM fermions are taken to be vector-like, anomalies are not an issue. The Yukawa interactionsare given by L BSM , Yukawa = − y Tr( ψ L S ψ R + ψ R S † ψ L ) . (17)Here, y denotes the BSM Yukawa coupling, the trace Tr sums over color and flavor indices, andthe decomposition ψ = ψ L + ψ R with ψ R/L = (1 ± γ ) ψ is understood. Yukawa interactions arecrucial for asymptotic safety to arise in weakly coupled gauge theories. The BSM sector is invariantunder global U ( N F ) × U ( N F ) flavor rotations. The full Lagrangean for the BSM extension of theSM is given by L = L SM + L BSM , kin . + L BSM , pot . + L BSM , Yukawa . (18)Here, L SM denotes the SM Lagrangean and L BSM , pot . the interaction Lagrangean of the BSMscalars. The BSM scalars S can mix with the SM Higgs boson through suitable portal couplingcontained in L BSM , pot . . The BSM kinetic terms are given by L BSM , kin . = Tr (cid:0) ψ i /D ψ (cid:1) + Tr ( ∂ µ S † ∂ µ S ) . (19)The BSM fermions communicate to the SM through the gauge interactions, provided they arecharged accordingly. The scalar fields are taken to be singlets under the SM gauge groups. Weassume that the BSM matter fields develop soft scalar M S and fermion M ψ masses for the modelto be compatible with data.For the sake of this paper we make a few further simplifying assumptions. Firstly, we limitourselves to BSM fermions which carry no hypercharge. This assumption can be relaxed withoutchanging the overall picture of results. Secondly, we neglect the role of quartic self interactions ofthe BSM scalars as well as portal couplings to the Higgs. At weak coupling, neither of these are rel-evant for the primary existence of the UV fixed point in the gauge-Yukawa sector. Consequently,the free fundamental parameters of the BSM matter sector are given by their group-theoreticalrepresentation under SU (2) L and SU (3) C , and their flavor multiplicity N F ,( R , R , N F ) . (20)A key goal of our study will be to identify viable, weakly coupled UV fixed points for the BSMtheory (18) within the parameter space (20). A detailed analysis of the SM Yukawa and scalar sector will be given elsewhere.
B. Renormalisation group
In order to identify interacting fixed points, we must analyse the RG equations for the theory(18). Within perturbation theory, weakly interacting fixed points arise for the first time at thetwo loop level in the gauge sector and at the one loop level in the Yukawa and scalar sectors [11].Also, interacting UV fixed points necessarily require the presence of a fixed point in the Yukawainteractions. For these reasons, we consider the RG equations for (18) up to second order in bothgauge couplings, and up to first order in the BSM Yukawa coupling. This is the lowest order atwhich a weakly coupled UV fixed point may arise.To be concrete, we normalise the gauge and Yukawa couplings with the perturbative loop factorand introduce α = g (4 π ) , α = g (4 π ) , α y = y (4 π ) , (21)to denote the weak, strong, and BSM Yukawa coupling, respectively. Our study will be confinedto the perturbative domain where all couplings remain sufficiently small. For now, we use α < IV E. In terms of(21), the RG equations within dimensional regularisation and to the leading non-trivial order aregiven by [18–21] β ≡ dα d ln µ = ( − B + C α + G α − D α y ) α ,β ≡ dα d ln µ = ( − B + C α + G α − D α y ) α , (22) β y ≡ dα y d ln µ = ( E α y − F α − F α ) α y . A few comments are in order. The one loop gauge coefficients B i can take either sign, dependingon the BSM matter content. The two loop gauge coefficient C is positive throughout. Thetwo loop gauge coefficients C may take either sign if B >
0, but is strictly positive as soon as B ≤ G i as well as the two loop Yukawa contribution D i and the one loop Yukawa terms E and F i are always positive in any quantum field theory. TheYukawa couplings always contribute with a negative sign to the running of gauge couplings. Thisis centrally important for interacting UV fixed points to arise at weak coupling.Explicit expressions for the various loop coefficients and further details are summarised in theappendix, see ( A
2) – ( A A C. UV fixed points at weak coupling
Gauge-Yukawa theories with (22) may display up to four different types of weakly coupled UVfixed points, depending on whether the gauge couplings take free or interacting values in the UV. Our definition for the gauge couplings relates to the more standard definition α s = g / (4 π ) as α s = 4 π α , andsimilarly for α w . gauge couplings Yukawa couplingcase α ∗ α ∗ α ∗ y type infoFP G · G non-interacting FP B C (cid:48) F E α ∗ G · GY partially interacting FP B C (cid:48) F E α ∗ GY · G partially interacting FP C (cid:48) B − B G (cid:48) C (cid:48) C (cid:48) − G (cid:48) G (cid:48) C (cid:48) B − B G (cid:48) C (cid:48) C (cid:48) − G (cid:48) G (cid:48) F E α ∗ + F E α ∗ GY · GY fully interacting Table 2 . The four different types of UV fixed points FP – FP in minimal BSM extensions of the SMwith (22). The primed and unprimed loop coefficients are defined in App. A . We also indicate how thefixed points can be interpreted as products of the Gaussian (G) and gauge-Yukawa (GY) fixed points whenviewed from the individual gauge group factors (see main text). We refer to the different cases as FP – FP , defined asFP : α ∗ = 0 , α ∗ = 0 , FP : α ∗ > , α ∗ = 0 , FP : α ∗ = 0 , α ∗ > , FP : α ∗ > , α ∗ > , (23)see Tab. . The Gaussian fixed point FP , where all couplings vanish, always exists. It qualifies asa candidate for an asymptotically free extension of the SM provided that each gauge sector remainsasymptotically free individually. Using the explicit expressions ( A
2) this condition translates intobounds SU (2) L : N F < / (8 S ( R ) d ( R )) ,SU (3) C : N F < / (4 S ( R ) d ( R )) . (24)In Tabs. and we show the maximal number of BSM vector-like fermions ψ ( R , R ) compatiblewith asymptotic freedom, N F ≤ N AF for SU (2) L singlets, doublets and triplets, and for differentdimensions of the SU (3) C representations. We observe a small window for low-dimensional rep-resentations where asymptotic freedom persists. Asymptotic freedom is lost as soon as the BSMfermions transform under higher-dimensional representations of the gauge group. See [5] for arecent analysis of BSM extensions with complete asymptotic freedom.Theories with (22) may also display weakly interacting fixed points with α ∗ ≤
1. These areeither partially or fully interacting. Conditions for existence of partially interacting UV fixed pointssuch as FP and FP then reduce to those given in Sec. II A for simple gauge theories. Analogousconditions of existence arise for the fully interacting fixed point FP . In either of theses cases, forFP , FP or FP to qualify as asymptotically safe UV fixed points, the Yukawa coupling must takean interacting fixed point by itself. To the leading non-trivial order in perturbation theory, using0 FP R = 1 R = 2 R = 3 R ( p, q ) C ( R ) S ( R ) N AF N AS N AF N AS N AF N AS (1,0)
43 12
10 – 6 – 3 – (2,0)
103 52 (1,1) 3 3 1 (62) 96 – (127) 198 – (192) 299 (3,0) 6 – (16) 18 – (32) 34 – (48) 51 (2,1)
10 – (28) 30 – (55) 60 – (82) 90 (cid:48) (4,0)
283 352 – (16) 18 – (32) 33 – (48) 50
Table 3 . Asymptotic freedom versus asymptotic safety at the partially interacting fixed point FP : shownare the maximal numbers of BSM fermion flavors compatible with asymptotic freedom, N AF , and thesmallest number of flavors required for an asymptotically safe fixed point FP to exist, N AS , both independence on the fermion representations R and R under SU (2) L and SU (3) C , respectively. N AS valuesin brackets relate to the absolute lower bound, those without to fixed points with 0 < α ∗ , α ∗ y <
1. Alsoindicated are the weights ( p, q ), quadratic Casimir, and Dynkin index under SU (3). (22), it follows that the Yukawa coupling at a fixed point is linearly related to the gauge couplings,FP : α ∗ y = F E α ∗ , FP : α ∗ y = F E α ∗ , FP : α ∗ y = F E α ∗ + F E α ∗ , (25)depending on whether α , or α , or both, take interacting fixed points by themselves. Combining(25) with the vanishing of the gauge beta functions provides explicit expressions for the differentfixed points. An overview of fixed points and their properties is given in Tab. . Next we analyseminimal conditions that need to be fulfilled in the BSM sector in order to generate partially orfully interacting UV fixed points in the system (22). D. Partially interacting fixed points
The partially interacting fixed points FP and FP are characterised by one of the gaugecouplings, say α AS , taking an asymptotically safe fixed point in the UV whereby the other gaugecoupling, say α AF , becomes asymptotically free. The Yukawa couplings must take interactingvalues, α ∗ y ∝ α ∗ AS , see (25). The beta functions (22) then take the simplified form β AS = ( − B AS + C AS α AS − D AS α y ) α ,β y = ( E α y − F AS α AS ) α y . (26)These expressions formally agree with (1) and therefore offer the same type of fixed point solutions.The non-trivial UV fixed point is then of the form (5), (6), after substituting the appropriate loop1 FP R = 1 R = 3 R = 6 R (cid:96) C ( R ) S ( R ) N AF N AS N AF N AS N AF N AS
12 34 12
32 154
52 354 352 – (6) 7 – (16) 17 – (31) 33
Table 4 . Asymptotic freedom versus asymptotic safety at the partially interacting fixed point FP : shownare the maximal numbers of BSM fermion flavors N F < N AF compatible with asymptotic freedom, andthe smallest number N F ≥ N AS required for a weakly-coupled asymptotically safe fixed point, both independence on the fermion representations R and R under SU (2) L and SU (3) C , respectively. Values for N AS in brackets relate to the absolute lower bound, those without brackets to settings with 0 < α ∗ , α ∗ y < (cid:96) , the quadratic Casimir, and the Dynkin index under SU (2) L . coefficients. A minimal requirement for partially interacting fixed points to be UV fixed points isthe loss of asymptotic freedom in the gauge sector B AS <
0, meaning eitherFP : N F > / (8 S ( R ) d ( R )) , or FP : N F > / (4 S ( R ) d ( R )) , (27)thus reverting the condition (24). Associating suitable charges to the BSM fermions, it is thenpossible to satisfy either of the conditions in (27). Furthermore, the physicality condition (7)translates into FP : D F − E C > , FP : D F − E C > . (28)It remains to evaluate solutions to the conditions (28) separately for FP and FP , to which weturn next. Strong strong and weak weak gauge coupling . In Fig. we analyse the condition (28)exemplarily for FP where the strong coupling remains interacting in the deep UV whereas theweak coupling vanishes asymptotically. We assume that the BSM fermions carry no SU (2) L charges( R = ), but different SU (3) C representations R = , , and . We observe the followingpattern. For fermions in the fundamental, a narrow window of weakly interacting fixed pointsexists for a low number of flavors N F . These low- N F solutions come out as IR fixed points inthat they relate to settings with asymptotic freedom in both gauge sectors (see the discussion inSec. IV E). With increasing N F , the fixed point takes negative values and becomes unphysical.Conversely, for fermions in higher-dimensional representations (anything but the fundamental),we find that a fixed point exists for sufficiently large N F . No fixed points exist for intermediatevalues of N F . Occasionally we find that fixed points can exist for exceptionally low values of N F ,in which case the fixed point is IR rather than UV. In Fig. (middle panel), we show the setof parameters ( R , R , N F ) for which FP exists as an interacting UV fixed point. In Tab. we2 a R = a y unphysical - - N F a y R = a physical - - N F a R = a y - - N F a y R = a - - N F N F N F R = R = R = R = Figure 1 . Partially interacting fixed point FP with α ∗ = 0, showing the strong coupling α ∗ (blue, solidline) and the BSM Yukawa coupling α ∗ y (red, dashed) versus the number N F of the BSM flavors for R = and different SU (3) C representations R = , , and (see main text). summarise the minimum number of BSM fermions N AS which lead to a weakly coupled UV fixedpoint with α ∗ ≤ N F is sufficiently large. The necessary conditionfor existence (28) of FP turns into a quadratic polynomial in N F after inserting the explicitexpressions for the loop coefficients, X N F + Y N F − Z < , (29)with coefficients X = Z C ( R ) [5 − C ( R )] / ,Y = Z [ C ( R ) + 5] / − C ( R ) ,Z = C ( R ) d ( R ) d ( R ) , (30)and with C ( R ) and d ( R ) defined in ( A N F , the sign of the coefficient X dictates whether the condition (29) provides an upper or a lower bound on N F . If X >
0, thecondition (29) provides an upper bound on the number of the BSM fermions. However, we observethat
X > SU (3) C (see Tab. for explicit values of the Casimir invariant for several R of the lowestdimension). In this case it is readily confirmed that a solution to (29) is incompatible with thelower bound from (24) for any choice of R , meaning that such a fixed point is necessarily anIR fixed point. We conclude that asymptotic safety via a partially interacting fixed point cannot3 R N F R = R = R = R N F R = R = R = R N F R = R = R = Figure 2 . Availability of weakly interacting UV fixed points FP (left panel), FP (middle panel), andFP (right panel) in dependence on the representation ( R , R ) and the flavor multiplicities N F of BSMfermions. The pattern of results continues to higher ( R , R ). Partially interacting fixed points FP areabsent for any N F as soon as R = or ; FP is absent whenever R = or ; fully interacting UV fixedpoints FP are absent for R = or ( R , R ) = ( , ) , ( , ), and ( , ). be achieved within the fundamental representation of SU (3) C . On the other hand, for higher-dimensional representations the coefficient X becomes negative. Consequently, (29) provides alower bound on the number of BSM fermions required to achieve asymptotic safety, N F ≥ N AS .The case X = 0 has no physical solutions. Exemplary values for the lower bound , N F ≥ N AS for different representations R and R are given in Tab. , where we additionally require weakcoupling α ∗ i < Strong weak and weak strong gauge coupling . Next we turn to FP where the weaksector remains interacting in the deep UV whereas the strong coupling becomes asymptoticallyweak. Qualitatively, our findings for FP are very similar to those discussed previously for FP .The absence of asymptotic freedom in the SU (2) L gauge sector, (27), requires a minimal number ofBSM fermion flavors N F ≥ N AF . In Fig. (left panel), we show the set of parameters ( R , R , N F )for which FP exists as an interacting UV fixed point. In Tab. , we provide N AF for SU (3) C singlets, triplets and sextets, and for different dimensions of the SU (2) L representations. Sincethe SM contribution to the one-loop gauge coefficient is larger for SU (2) L than for SU (3) C , lowervalues for N F and lower dimensions of representations are required to lose asymptotic freedom for SU (2) L . Similarly, from (28) we find that asymptotic safety cannot be achieved with fermions inthe fundamental representations of SU (2) L . The minimal number of BSM fermion flavors requiredfor a weakly-coupled asymptotically safe fixed point, N AS , are given in Tab. for various choicesof R and R .4 - - N F B ' / C R = R = R = R = R = N F B ' / C R = - - N F B ' / C R = C Figure 3 . Asymptotic freedom at partially interacting fixed points FP . Shown is the condition forasymptotic freedom (32) for the effective coefficient B (cid:48) in units of the two loop coefficient C > for, exemplarily, R = (left panel) and R = (right panel), and as a function of R (color coding givenin the legend). We observe that asymptotic freedom in the weak sector is regained as soon as R > and R > , for any N F . For R = , a lower bound on N F is found (left panel). Qualitatively andquantitatively similar results are obtained at FP (not displayed). E. Regaining asymptotic freedom
Next we discuss the fate of the gauge coupling which vanishes at partially interacting fixedpoints FP or FP , and which we denote for notational simplicity as α AF . The coupling α AF mustbe asymptotically free for a partially interacting fixed point to be viable, or else the UV fixed pointcannot be reached by any finite RG trajectory along the α AF direction. In general, we find that B AF becomes negative as soon as BSM fermions carry charges of both gauge groups. However, thesign of B AF plays no role, as it no longer dictates whether this sector remains asymptotically freeor not. Rather, to leading order in the asymptotically free gauge coupling, we have β AF = (cid:0) − B AF + G AF α ∗ AS − D AF α ∗ y (cid:1) α + O ( α ) , (31)showing that the one loop coefficient B AF is replaced by B (cid:48) AF = B AF − G AF α ∗ AS + D AF α ∗ y . Westress that this shift is a consequence of partially interacting fixed points. It arises from residualinteractions at the UV fixed point due to asymptotic safety of the gauge coupling α AS and theBSM Yukawa coupling. Their residual interactions modify the running of the asymptotically freecoupling owing to fermions which carry charges under both gauge groups. Provided that the shiftedone loop coefficients B (cid:48) take positive values, B (cid:48) AF > , (32)the non-interacting gauge sector becomes asymptotically free in the deep UV. We also stress thatthe BSM Yukawa interactions play a central role: only Yukawa couplings add negatively to thebeta function (31). Without them, (32) cannot be achieved starting from B AF <
0. Using (22),5we have the following expressions for the shifted one loop coefficientsFP : B → B (cid:48) = B − G α ∗ + D α ∗ y , FP : B → B (cid:48) = B − G α ∗ + D α ∗ y . (33)We conclude that (32), (33) are necessary conditions for the corresponding partially interactingfixed point to qualify as UV completions of the SM.In Fig. the condition for asymptotic freedom (33) at FP is shown for models with R = (left panel) and R = (right panel) and various R > (recall that there are no viable UV fixedpoints FP for R ≤ , Fig. ). If R = , we observe that B (cid:48) is positive for sufficiently large N F , and negative for sufficiently low N F , thus leading to a lower bound. Conversely, if R = (orlarger), the sign of B (cid:48) is always positive. In this case asymptotic freedom is guaranteed withoutany further constraints as soon as α is asymptotically safe. The same pattern of results holdstrue for FP . We conclude that as soon as the BSM fermions carry a non-trivial charge underthe asymptotically free coupling R AF (cid:54) = , for any R AS (cid:54) = , the condition (32) follows from thecondition for asymptotic safety for α AS (28). For BSM fermions with R AF = , (32) entails anadditional lower bound on N F . F. Fully interacting fixed points
Finally we consider the case FP . In the case where both gauge couplings and the BSM Yukawacoupling remain weakly interacting at the fixed point in the asymptotic UV the overall behaviourof the system (22) depends on the interplay between one- and two-loop coefficients. Using theresults of Tab. , the necessary condition for a fixed point can be stated as α ∗ = C (cid:48) B − B G (cid:48) C (cid:48) C (cid:48) − G (cid:48) G (cid:48) > , α ∗ = C (cid:48) B − B G (cid:48) C (cid:48) C (cid:48) − G (cid:48) G (cid:48) > , (34)with primed two-loop coefficient given in ( A (right panel), we show the set of parameters ( R , R , N F ) for which FP exists as an interacting UV fixed point. Our results for the lowest number of flavor multiplicities N F ≥ N AS are summarised in Tab. . G. Large- N F approximation Some analytical insights about interacting UV fixed points can be obtained in the limit of manyflavors of fermions N F (cid:29)
1, which we discuss separately for either type of fixed point.
Partially interacting fixed points.
For a partially interacting fixed point, and using theexplicit solution for FP as given in Tab. , the large- N F approximation leads to( α ∗ , α ∗ , α ∗ y ) (cid:12)(cid:12)(cid:12) N F (cid:29) = 1 X (cid:18) , , C ( R ) N F (cid:19) + subleading , (35)6 FP R = 1 R = 3 R = 6 R = 8 R = 10 R N AS N AS N AS N AS N AS – – (130) 130 – (21) 21 – – (29) 35 (45) 56 (27) 29 – (23) 28 (27) 30 (38) 43 (33) 35 – (17) 18 (26) 28 (36) 39 (37) 39 – (15) 16 (27) 28 (36) 38 (40) 42 Table 5 . The minimal number of the BSM fermions flavors N F ≥ N AS required for the fully interactingfixed point FP to exist, in dependence on the fermion representations R and R under SU (2) L and SU (3) C . The values for N AS in brackets relate to the absolute lower bound, and those without to settingswith 0 < α ∗ , α ∗ , α ∗ y < where X ( R ) = 2 C ( R ) −
5. The subleading terms are at least one power in N F smaller thanthe leading order terms. Several observations can now be made. First of all, positivity of the fixedpoint couplings requires X ( R ) > C ( R ) > . Hence, our result confirms that asymptoticsafety cannot be achieved within the fundamental representations of SU (3) C even at large- N F ,owing to X (fund . ) <
0. Secondly, we observe that the Yukawa coupling scales like 1 /N F and canalways be made arbitrarily small. Conversely, the size of the gauge coupling is solely determined bythe quadratic Casimir C ( R ), and independent of N F in the large- N F limit. We stress that (35) isparametrically close to the Gaussian fixed point, provided that X becomes parametrically large.In addition, the necessary condition for asymptotic freedom (32) for the weak coupling simplifiesto leading order at large- N F and reads C ( R ) > . Interestingly, the condition for asymptoticfreedom exactly coincides with the condition for asymptotic safety of (35) at large- N F , C ( R ) > , or R ≥ . (36)We conclude that higher dimensional representations under SU (3) C with (35) are favoured for thetheory to display a perturbative UV fixed point in the SU (3) C coupling, and for SU (2) L sector toregain asymptotic freedom at the partially interacting UV fixed point FP , see Fig. . Analogousresults are established for FP , where the large- N F expansion starts off with( α ∗ , α ∗ , α ∗ y ) (cid:12)(cid:12)(cid:12) N F (cid:29) = 1 X (cid:18) , , C ( R ) N F (cid:19) + subleading , (37)and X ( R ) = 3 C ( R ) −
5. Again, subleading terms are suppressed by at least one additionalpower in N F over the leading terms. A necessary condition for asymptotic safety is X ( R ) > X (fund . ) <
0. We also conclude that FP is parametrically close to the Gaussian fixed point in the limit of high-dimensional representations X . Furthermore, to leading order at large- N F the condition for asymptotic freedom (32) for thestrong coupling becomes C ( R ) > , or R ≥ . (38)7 FP FP FP R R N F Figure 4 . Summary of weakly interacting UV fixed points of (22) in dependence on the fermion represen-tation and flavor multiplicities ( R , R , N F ). The different symbols relate to FP (gray circle), FP (bluesquare) and FP (red diamond). Overlapping symbols indicate that either type of fixed point can exist,with lower-lying symbols relating to fixed points which arise at a higher number of fermion flavors N F , seealso Fig. . Once more, this secondary condition coincides with the condition for asymptotic safety of (37).
Fully interacting fixed points.
For the fully interacting fixed point, using the explicitsolution for FP as given in Tab. and performing a large- N F limit, we find( α ∗ , α ∗ , α ∗ y ) (cid:12)(cid:12)(cid:12) N F (cid:29) = 1 X (cid:18) , , C ( R ) + 3 C ( R ) N F (cid:19) + subleading , (39)with X = 2 C ( R ) + 3 C ( R ) −
5. The above expression holds true provided that R (cid:54) = and R (cid:54) = . The requirement of asymptotic safety results in an inequality X >
0. Furthermore,the result also shows that the fully interacting fixed point is parametrically close to the Gaussianprovided that X is large. The explicit result explains why a fully interacting fixed point withasymptotic safety can be achieved even with BSM fermions in the fundamental representation of SU (3) C , as long as they transform under SU (2) L in a representation of a dimension higher thanthe fundamental. Analogously, FP exists for BSM fermions in the fundamental representation of SU (2) L provided that R > . Note that these large- N F estimates are in very good agreementwith the numerical findings in Tab. . In the special case where R > and R = , and insteadof (39), one obtains( α ∗ , α ∗ , α ∗ y ) (cid:12)(cid:12)(cid:12) N F (cid:29) = (cid:18) X , − X , C ( R ) N F X (cid:19) + subleading , (40)8with X = 2 C ( R ) + 3 C ( R ) − SU (2) L sector of the SM. The condition for existenceis now given by X > which translates into C ( R ) > . Solutions are given by the R = and R ≥ representations under SU (3) C . Curiously, the adjoint representation R = with R = is not a solution of (40) owing to the “would-be” Banks-Zaks IR fixed point. Finally, for R = and R ≥ one readily confirms that α ∗ and α ∗ cannot simultaneously take positive valuesmeaning that an asymptotically safe fixed point does not arise at large N F . H. Synopsis of UV fixed points
We are now in a position to summarise the main results for weakly interacting UV fixed pointsin extensions of the SM of the form (18). We have observed that interacting UV fixed pointscan arise as partially or fully interacting ones. In either of these cases, necessary conditions fortheir existence have been found, providing us with constraints on the remaining BSM parameters( R , R , N F ). We have also observed that for fixed ( R , R ), UV fixed points typically exists for all N F down to limiting values specified in Tab. , and . Fig. shows a summary of our findings,in dependence on the fermion representation ( R , R ) under SU (3) C ⊗ SU (2) L with differentsymbols relating to the different fixed points FP , FP , and FP . Broadly speaking, results showthe existence of fixed points both with increasing dimensionality of the fermion representation,and with increasing flavor multiplicities. Similarly, fixed points come out less strongly coupled thelarger their dimensionality R , R and the flavor multiplicity N F . We also observe that differenttypes of fixed points might coexist for BSM fermions with the same set of representations ( R , R ),starting from a lowest value for N F where the fixed point arises for the first time. In Fig. thepossibility of coexistence is indicated by overlapping symbols: the lower lying symbol relates to afixed point which arises for larger N F .Hence, four distinct cases arise: ( i ) For high dimensional representations, starting from R = and R = onward, all three types of fixed points are realised starting from some lowest valuefor N F . For fixed ( R , R ) but with increasing N F results show that the fully interacting fixedpoint FP is achieved first, followed by the partially interacting FP for the strong coupling, andultimately followed by the partially interacting fixed point FP in the weak coupling. ( ii ) If thefermions are in the fundamental representation of one of the two gauge groups, we find that thecorresponding partially interacting fixed point is absent throughout. However, the other two fixedpoints still exist and the order in which they appear, with increasing N F , is exactly the sameas the order observed for the higher dimensional representations. ( iii ) If the BSM fermions areuncharged under SU (3) C , only the partially interacting fixed point FP can arise, starting from R = onwards. ( iv ) If the BSM fermions are uncharged under SU (2) L , we find that FP arisesfirst, followed by FP , while FP is absent throughout. This holds true for all R = or higher,except for R = where only FP appears.Finally, we discuss the status of interacting fixed points for low numbers of flavors N F . Thelow- N F partially interacting fixed points FP at R = with R = , and all have B , B > . Similarly, for FP andFP we find a handful of low- N F fixed points all of which occur where asymptotic freedom persistsin both gauge groups (24). Conversely, we also have low- N F fixed points FP with R = , · · · , which have asymptotic freedom only in the strong gauge coupling, while the weak sector hasbecome infrared free. Such fixed points are not of phenomenological interest because they cannot9 Α Α
100 10 Μ (cid:72) GeV (cid:76)
Figure 5 . SM running of the strong and weak gauge coupling α (blue) and α (purple) from the Z mass upto Planckian energies. The SM GUT scale reads approximately µ GUT ≈ × GeV. UV safe trajectorieshave to coincide with SM values at the matching scale µ = M where BSM matter decouples. be linked with any finite α (cid:54) = 0 in the IR and shall be dropped.This completes our investigation of weakly coupled UV fixed points of (18) to the leading non-trivial order in perturbation theory, (22). In the next section, we explain whether and how thesefixed points are connected with the SM at low energies under the RG evolution of couplings. IV. MATCHING ONTO THE STANDARD MODEL
In this section, we evaluate the conditions under which BSM trajectories emanating out ofinteracting UV fixed points are connected with the SM at low energies.
A. Matching conditions
Any RG trajectory emanating from free or interacting UV fixed points qualifies as a UV com-plete quantum field theory. The UV critical surface then determine the set of UV-safe trajectories.The relevant or marginally relevant couplings in the UV determines the dimensionality of the UVcritical surface (9). Conversely, the irrelevant couplings are uniquely fixed by the relevant cou-plings in the UV. Consequently, the number of fundamentally free parameters which characterisethe UV-safe trajectories is given by the dimensionality of the UV critical surface. At low energies,physically viable BSM trajectories must connect with those of the SM, see Fig. , as soon as theBSM matter fields have decoupled.It remains to check whether the UV fixed points discovered in the previous section are connected0through well-defined RG trajectories to the SM at low energies. Away from the fixed point, BSMmatter fields will develop scalar and fermion masses M S and M ψ which are independent parametersof the theory. Phenomenological constraints for M S and M ψ are worked out in Sec. V below. ForRG scales much larger than the masses, the BSM matter fields are effectively massless, and theRG flow is given by (22). Conversely, for RG scales much lower than the masses, the BSM fieldsare taken to be infinitely heavy and decouple. The RG flow (22) reduces to the one of the SM,also restoring confinement of QCD at low energies. Hence, the BSM contributions to the runninggauge couplings decouple as soon as µ is of the order of the BSM fermion mass. Furthermore,threshold effects are subleading to the overall picture and will be neglected. Consequently, the RGflow of the SM is matched onto the RG flow of the BSM extension at the matching scale M , µ = M ≈ M ψ (41)below which the BSM fermions decouple. This leads to matching conditions between the RG flowof the SM at scales below (41), and BSM flows (22) above the mass scale (41), α i ( µ = M ) (cid:12)(cid:12)(cid:12) SM = α i ( µ = M ) (cid:12)(cid:12)(cid:12) BSM , (42)for i = 2 , i =AF,AS in settings with partially interacting fixed points). There will be nomatching condition for the BSM Yukawa coupling since it is not part of the SM. Rather, after thedecoupling of the BSM fields, the Yukawa coupling will “freeze out” at its value at decoupling. Forthe quantitative studies below, we use PDG SM reference values at the scale of the Z pole mass[22], α ( µ = m Z ) = 2 . × − ,α ( µ = m Z ) = 9 . × − , (43)together with the two loop perturbative running of gauge couplings in the SM, using (22) with N F = 0. Fig. illustrates the SM running between the mass of the Z boson ( m Z = 91 .
19 GeV)and Planckian energies. Note that equality of gauge couplings α ( µ ) = α ( µ ) (44)arises in the SM at the GUT scale µ GUT ≈ × GeV. We emphasize that the matching of theBSM extension (22) onto the SM (42) takes place at perturbatively small couplings.
B. Partially interacting fixed points
As has been detailed in Sec.
III
D, at partially interacting fixed points FP and FP , one ofthe two gauge couplings becomes asymptotically free, while the other one becomes asymptoticallysafe. Moreover, the asymptotically (free) safe coupling is (marginally) relevant and, hence, the UVcritical surface is invariably two-dimensional. On the other hand, the BSM Yukawa coupling α y isirrelevant and fully specified by the asymptotically safe coupling in the UV.In this light, a convenient choice for the two fundamentally free dimensionless parameters whichcharacterise UV-safe trajectories running out of the fixed point are the deviations of the gauge1 parameter UV fixed points typemodel ( R , R , N F ) α ∗ α ∗ α ∗ y (Fig. ) infoA ( , ,
12) 0 0.2407 0.3385 FP (cid:32) Fig. , low scale ∗ FP (cid:4) Fig. , low scale ∗ B ( , ,
30) 0 . . . FP (cid:7) Fig. , no match0.3317 0 0.0995 FP (cid:4) Fig. , low scale ∗ C ( , ,
80) 0.0503 0.0752 0.0292 FP (cid:7) Fig. , high scale0 0.8002 0.1500 FP (cid:32) Fig. , high scale0 0.0895 0.0066 FP (cid:32) (no Fig.), low scale ∗ D ( , , FP (cid:7) Fig. , low scale E ( , ,
72) 0.1499 0.2181 0.0471 FP (cid:7) Fig. , low scale Table 6 . UV fixed points and matching characteristics for various benchmark scenarios. An asteriskindicates that a matching is permitted at any scale including low (TeV) energy scales. couplings from their UV fixed point values at some high scale µ = Λ, δα i (Λ) = α ∗ i − α i (Λ) , (45)with i = 2 , i =AF,AS). We take the practical view that the high scale is essentially givenby the Planck scale. Quantum gravity effects should be retained at scales close to and above Λ.The BSM Yukawa coupling is an irrelevant coupling and entirely dictated by the UV hypercriticalsurface relating it with α AS and α AF , α y = F y ( α AS , α AF ) . (46)The parameters (45) will be used to match trajectories onto the SM. Specifically, the parameter δα AS controls at which energy scale the asymptotically safe coupling is crossing over from the UVfixed point towards the Gaussian IR fixed point of (22). For 0 < δα AS (cid:28) α AS will start out ofthe UV fixed point along the separatrix which connects the UV fixed point with the Gaussian inthe IR. In the immediate vicinity of the UV fixed point the RG flow is of the power-law type andthus fast, controlled by the relevant scaling exponent. Further away from the fixed point, as soon as α AS ≈ α ∗ AS and below [13], we observe a cross-over whereby the running becomes logarithmicallyslow instead, dominated by the “would-be” Gaussian IR fixed point of (22). Hence, the parameter δα AS allows us to chose at which scale α AS ( M ) has reached the desired SM value. Notice that thisdiscussion is largely independent of α AF provided the latter remains small.The running of α AF out of the UV fixed point is controlled by the RG flow (31), which in turnis largely determined by the parameter δα AF , together with the coefficients B AF and B (cid:48) AF , (33).Integrating (31) close to the UV fixed point gives1 α AF ( µ ) = 1 δα AF (Λ) + B (cid:48) AF ln( µ/ Λ) . (47)2One might expect that the two free parameters (45) are sufficient to match the RG flow in theUV to two preset values at low energies. We stress, however, that a matching may fail if the“would-be” asymptotically free coupling α AF runs into Landau poles at intermediate energies.Thus, we must explain how Landau poles are avoided. In the deep UV, we have that B (cid:48) AF > B (cid:48) AF depends on the asymptotically safe gauge coupling α AS and onthe Yukawa coupling. Both of these run out of the UV fixed point and induce an effective runningof B (cid:48) AF → B (cid:48) AF ( µ ) owing to (31). For sufficiently small α AS such as close to the matching scale µ = M , the coefficient B (cid:48) AF falls back onto the BSM one loop coefficient B (cid:48) AF → B AF . Close tothe matching scale the one loop approximation is viable and we have1 α i ( µ ) = 1 α i ( M ) + B i ln( µ/M ) (48)for both i = AF , AS. If B AF >
0, meaning that the gauge sector remains asymptotically free, wehave that B (cid:48) AF ( µ ) > B i < µ i M = exp (cid:18) − B i α i ( M ) (cid:19) . (49)For α AS the Landau pole is avoided automatically owing to the two loop Yukawa terms: withgrowing energy, once the scale µ AS is reached, the two loop terms kick in and α AS settles into itsUV fixed point, see Sec. II . For α AF it is not guaranteed that B (cid:48) AF ( µ ) changes sign in time for α AF to avoid the Landau pole. We find that α AF avoids a Landau pole provided that B AF · α AF ( M ) < B AS · α AS ( M ) . (50)The condition (50) ensures that the “would-be” one loop Landau pole for α AF arises at higherscales µ AF > µ AS > M than the one for α AS . The crucial point about scales µ ≈ µ AS is that thetwo loop terms have become active. Two loop terms also contribute to the running of α AF andthereby ensure that the sign of B (cid:48) AF has become positive. We conclude that (50) is sufficient toprovide an upper bound for viable matchings to the SM, ensuring that neither of the couplingsescapes a successful matching through a Landau pole at intermediate energies. Within the confinesof (50), this enables us to match SM and BSM running onto each other essentially at any scalebetween TeV and Planckian energies. C. Fully interacting fixed points
All fully interacting UV fixed points are characterized by a stability matrix (8) with a singlerelevant eigenvalue. This important result states that a linear combination of the gauge groups’kinetic terms together with the BSM Yukawa interaction term in the fundamental Lagrangean (18)correspond to the sole UV relevant operator in the theory. This result has important implications.Unlike in asymptotically free theories (or in asymptotically safe theories at partially interactingfixed points FP or FP ) where every gauge coupling corresponds to a UV relevant direction, here,instead, the UV critical surface is of a lower dimensionality. This new effect is a consequence ofcompeting gauge interactions in the UV. Most notably, it entails that the number of fundamentallyindependent parameters is reduced, leading to an enhanced level of predictivity.In our models, the UV critical surface at fully interacting UV fixed points becomes one-3dimensional, parametrised by a single free parameter. Consequently, only one out of the threecouplings ( α , α , α y ) may be considered as an independent variable. For FP , and without loss ofgenerality we chose this to be α . The UV critical surface then uniquely determines the weak andthe Yukawa coupling as functions of the strong coupling, α i = F i ( α ) for i = 2 , y . (51)Most importantly, the UV critical surface imposes a relation between the two gauge couplingswhich arises as a strict consequence of asymptotic safety at a fully interacting fixed point FP . Wemay then use the dimensionless parameter δα (Λ) = α ∗ − α (Λ) (52)at the high scale Λ to parametrise all UV safe trajectories running out of the fully interacting UVfixed point FP . The UV-IR connecting separatrix( α , α , α y )( µ ) ≡ ( α , F ( α ) , F y ( α ))( µ ) (53)uniquely determines the relation between the strong and the weak gauge coupling for all scalesabove the matching scale. The role of the free parameter δα (Λ) is to determine at which scalethe curves (53) display a cross-over from UV dominated running towards IR dominated running.The task to identify trajectories which can be matched onto the SM at some matching scale µ = M reduces to analysing the separatrix (53). Given that the set of determining equationsis over-constrained, a successful matching cannot be guaranteed from the outset, meaning thatthe viability needs to be checked for each FP . On the other hand, if trajectories emanating outof FP can be matched, the BSM extension implies a fundamental relation between both gaugecouplings, which would not exist otherwise. In settings where FP exists alongside FP , or FP ,or both, either of the partially interacting fixed points is the more relevant UV fixed point. TheirUV critical dimensions are larger, UV-IR connecting trajectories can be found which link FP orFP with FP . An example for this is discussed below in Fig. .To understand more explicitly how a matching of FP onto the SM depends on the fermionrepresentations and flavor multiplicities, we evaluate the link between the gauge couplings asdictated by the UV critical surface, (9). Since in the general case the separatrix cannot be resolvedanalytically, we use the critical surface approximation of (9), see App. A for the technicalities,keeping in mind that far from the UV fixed point the critical surface may deviate from the onegiven by the separatrix. The relation (51) then takes the simple linear form α ( M ) = − X + Y α ( M ) ,Y = − V /V ,X = − α ∗ + Y α ∗ . (54)The parameters V i are related to the UV relevant eigendirection (9) which characterises the UVcritical surface. Using (22), we find the explicit expressions X = B D − B D C D − G D , Y = C D − G D C D − G D , (55)in terms of the perturbative loop parameters. Whether a matching of trajectories ( α , α )( µ ) with4 Α Α Α y matching scale cross (cid:45) over scale R = , R = , N F (cid:61) (cid:45) Μ (cid:72) GeV (cid:76)
Figure 6 . Low scale matching of the partially interacting fixed point FP onto the SM for the benchmarkscenario A with ( R , R , N F ) = ( , ,
12) and M = 2 TeV. BSM (SM) running is shown by full (dashed)lines. Around the cross-over scale the BSM running of α and α y slows down from power-law to logarithmic.Notice that the running of the strong coupling is not modified by BSM fermions. (54) onto the SM is possible or not depends on the signs and magnitude of X and Y . In thelarge- N F limit, we find X (cid:12)(cid:12)(cid:12) N F (cid:29) = 215 C ( R ) − C ( R ) C ( R ) C ( R ) d ( R ) d ( R ) 1 N F + subleading ,Y (cid:12)(cid:12)(cid:12) N F (cid:29) = 32 + subleading . (56)With increasing N F we observe that Y > N F limit Y = . Conversely, X may have either sign depending on the representation ( R , R ), though not on N F .Also, X becomes parametrically small for high dimensional representations and for large flavormultiplicities N F . If X < α ( M ) > α ( M ) indicating that a matching below GUT-typescales is impossible. Furthermore, (54) also implies that α ( M ) > − X , stating that a matchingbecomes impossible at any scale if − X becomes too large. On the other hand, if X > α ( M ) < α ( M ). For intermediateand large values of N F the condition X > C ( R ) − C ( R ) > . (57)A few comments are in order:( i ) If R < , the condition (57) is satisfied for any R . Non-trivial constraints arise from(57) once R = or higher (for example, R = necessitates R ≥ , and similarly for higherdimensional representations R > ). Note that this ratio is a direct consequence of the SU (3) C and SU (2) L gauge groups of the SM. ii ) An increasing flavor multiplicity N F is required for models with a low dimensional SU (3) C representation R or high dimensional SU (2) L representation R to ensure that the magnitudeof X stays within the limits compatible for a matching onto SM values. Also, increasing thedimension of the matter representations lowers α ( M ). For these cases it then follows that thematching can only take place at a high scale.( iii ) The linear approximation for the separatrix (54) becomes exact once R = . This canbe seen as follows. Using (55) we find X = − at FP for any viable ( R , N F ). The exact sameresult is found if (54) is evaluated at the Banks-Zaks IR fixed point of the weak gauge coupling inthe SM ( α ∗ , α ∗ ) = ( , A coincides with the critical surface at the Banks-Zaks IR fixed point. The latter therefore controlsthe running of the weak coupling away from the fully interacting UV fixed point by directing allUV safe trajectories straight into the Banks-Zaks fixed point. As is shown more explicitly below,it is for this reason that a matching of FP with R = onto the SM is impossible.( iv ) For all scenarios considered in this work, we find that the condition (57) is a good estimatorfor the availability of a matching with the SM.This completes the general discussion of matching conditions for UV safe trajectories onto theSM. D. Benchmark scenarios
Let us now illustrate how the matching works in practice for a selection of benchmark scenarios,summarised in Tab. , covering low scale and high scale matchings. Benchmark scenario A . For this setting we assume that the BSM fermions do not carry SU (3) C charges. Following on from our earlier discussion, FP is the sole UV fixed point whichmay arise and neither FP nor FP are available, see Fig. . We consider the parameters( R , R , N F ) = ( , ,
12) (58)with RG trajectories displayed in Fig. . The matching scale M may take any value between TeVand Planckian energies. In Fig. , for illustration, we have set it to the low value M = 2 TeV (ver-tical dashed line). Evidently, the running of the strong coupling is not modified by BSM matterand remains SM-like throughout. Once the matching scale is fixed, the model predicts the valueof the (otherwise unconstrained) Yukawa coupling. For M = 2 TeV one obtains α y ( M ) = 0 . µ cr ∼ × GeV and much larger thanthe matching scale.
Benchmark scenario B . For this case we assume that the BSM fermions do not carry SU (2) L charges. From Fig. it follows that solely FP , possibly in conjunction with FP can arise.Informed by the results of Tabs. and we chose the parameters( R , R , N F ) = ( , ,
30) (59)to ensure that the model has both types of UV fixed points, FP and FP . The partially interact-ing fixed point FP can always be matched onto the SM at any scale. In Fig. , this is illustratedfor a low matching scale M = 2 TeV. In this model, the running of the weak gauge coupling is6 Α Α y Α matching scale cross (cid:45) over scale R = , R = , N F (cid:61) (cid:180) (cid:180) (cid:45) Μ (cid:72) GeV (cid:76)
Figure 7 . Low scale matching from the partially interacting fixed point FP onto the SM for the benchmarkscenario B with ( R , R , N F ) = ( , ,
30) and M = 2 TeV. BSM (SM) running is shown by full (dashed)lines. Around the cross-over scale the BSM running of α and α y slows down from power-law to logarithmic.Notice that the running of the weak coupling is not modified by the BSM fermions. FP H model B L a a y a m ê m weak BZ Figure 8 . Shown is the running of the gauge and BSM Yukawa couplings along the UV-IR connectingseparatrix emanating from FP for the benchmark model B , (59). The scale µ may take any valuedetermined by the free parameter δα (Λ). The weak gauge coupling is attracted towards its “would-be”Banks-Zaks fixed point (weak BZ) , see ( A R = , R = , N F (cid:61) Α y Α Α matching scale cross (cid:45) over scale (cid:180) (cid:180) (cid:45) (cid:45) Μ (cid:72) GeV (cid:76)
Figure 9 . Low scale matching from the partially interacting fixed point FP onto the SM for the benchmarkscenario C with ( R , R , N F ) = ( , ,
80) and M = 2 TeV. BSM (SM) running is shown by full (dashed)lines. Around the cross-over scale the BSM running of α and α y slows down from power-law to logarithmic.Notice that the approach towards asymptotic freedom of the weak coupling is enhanced by BSM fermions. not modified by BSM matter to the leading orders in perturbation theory. The crossover scale µ cr ∼ . × GeV is an order of magnitude larger than the matching scale. Elsewise the samereasoning as in Fig. applies. In contrast, the impossibility for a matching at FP is illustrated inFig. . The scale µ is arbitrary and can take any value upon tuning the UV parameter δα (Λ).However, the UV safe trajectory emanating out of FP is attracted towards the strongly coupleddomain, owing to the Banks-Zaks IR fixed point at ( α ∗ , α ∗ ) = ( ,
0) in the weak sector, ( A SU (2) L . Thenon-availability of FP persists for all models with R = , in line with our discussion in Sec. IV C.We now turn to benchmark scenarios where the BSM fermions carry both SU (2) L and SU (3) C charges. In these cases we find realisations for either of the partially interacting fixed points FP and FP , as well as for the fully interacting fixed point FP . Benchmark scenario C . As soon as R ≥ and R ≥ , and for sufficiently large N F ,all three types of UV fixed points arise, see Fig. . To illustrate such settings, we consider thebenchmark scenario C with parameters( R , R , N F ) = ( , ,
80) (60)which displays FP , FP and FP within the perturbative domain, see Tab. . In Fig. , we beginwith FP where α ∗ = 0 in the deep UV. Once more we observe that the matching condition (50)on the one loop BSM parameters for the strong and the weak coupling can be satisfied at any scalebetween a few TeV and Planckian energies including the low matching scale M = 2 TeV chosen in8 R = , R = , N F (cid:61) Α y Α Α matching scale cross (cid:45) over scale (cid:180) (cid:180) (cid:180) (cid:180) (cid:180) Μ (cid:72) GeV (cid:76)
Figure 10 . High scale matching of a fully interacting fixed point FP onto the SM for the benchmarkscenario C with M = 2 × GeV. Once BSM matter fields are active, the weak coupling approachesasymptotic safety more rapidly than the strong coupling. See also Fig. and Fig. . Fig. . Furthermore, the weak coupling α continues to decrease even directly below the matchingscale, for any matching scale. For the weak gauge coupling the approach towards asymptoticfreedom is accelerated over the SM rate owing to the two loop BSM Yukawa contributions whichare winning over the contributions by the strong gauge coupling along the entire UV-IR connectingseparatrix into the fixed point. This pattern is consistent with the matching condition (50), whichis fullfilled for any intermediate scale. For the example shown in Fig. , the cross-over scale andthe matching scale are separated by an order of magnitude.In Fig. we turn to the matching of the fully interacting fixed point FP corresponding to thesame parameter set (60). Couplings run out of the UV fixed point FP along a unique separatrix(53) connecting FP with the Gaussian fixed point. The separatrix thereby imposes a link between α and α . On the separatrix, and close to the fully interacting fixed point, the weak couplingis genuinely stronger than the strong coupling. At crossover, it becomes rapidly weaker than thestrong coupling. The gauge couplings also weaken more rapidly than the BSM Yukawa coupling.For the parameters (60), the separatrix dictates that the unique matching scale onto SM valuescomes out comparatively high, with M ≈ × GeV.In Fig. we consider the matching with FP where α ∗ = 0 in the UV. In this model, we findthat a matching at FP is more strongly constrained compared to a generic partially interactingfixed point. The reason for this is the influence of FP on UV-IR connecting trajectories and thenecessity to avoid an early Landau pole in the strong sector. Specifically, starting at some highscale Λ we observe that the weak and the Yukawa couplings decrease with energy, while the strongcoupling increases towards the IR. For too small δα (Λ) (45) the strong coupling does not growfast enough. For too large δα the strong coupling runs into a Landau pole at intermediate scales.Within a narrow window for δα , however, the growth of α is tamed due to FP . Then, trajectoriesare close to the separatrix connecting FP with FP , with a cross-over scale µ ≈ × GeV, seeFig. . Below the cross-over scale, couplings are attracted towards FP (see Tab. for the fixed9 R = , R = , N F = a a y a matching scale c r oss - ove r scale ¥ ¥ ¥ ¥ ¥ ¥ - m H GeV L c r o ss - ov e r I FP (model C) c r o ss - ov e r II m a t c h i ng s ca l e FP Figure 11 . High-scale matching of a partially interacting fixed point FP onto the SM for the benchmarkscenario C with ( R , R , N F ) = ( , , and initiallycross over into the vicinity of FP (indicated by arrows) at around 1 . × GeV. Subsequently, trajectoriesdisplay a second cross over to match with the SM at about M = 5 × GeV. point values) which, however, is not reached exactly. Instead, at scales about µ ≈ . × GeVthe couplings are driven away from FP , now following the separatrix which connects FP with theGaussian. In consequence, we find that couplings can be matched onto the SM at a high matchingscale of about M = 5 × GeV, close to the matching scale found for FP .This result is consistent with the “Landau pole avoidance condition” (50) derived in Sec. IV B,which for the parameters (60) has no solutions for low matching scales. For example, at M = 2 TeVthe SM predicts α SM3 ( M ) ≈ . α ( M ) ≈ . M ≈ × GeV, the condition (50) eases up andallows a consistent matching without the strong coupling prematurely running into a perturbativeLandau pole at intermediate scales.We conclude that all three fixed points qualify as UV completions for the SM, although thespecifics of the UV completion differ due to finer details of the fixed point structure.
Benchmark scenario D . For models with R = and R ≥ , and for sufficiently large N F ,we have observed that the UV fixed points FP and FP coexist, see Fig. . To illustrate thematching procedure for these settings, we set the parameters as( R , R , N F ) = ( , , . (61)The partially interacting fixed point FP can be matched onto the SM, particularly at low match-ing scale M (not displayed). Results for FP are displayed in Fig. . Couplings run out of theUV fixed point along the separatrix (53). Unlike the previous example of benchmark C (60), inthis case couplings display a cross over at much lower energies. In particular, a matching to SMvalues is possible at a low scale of about M ≈ . R = , R = , N F (cid:61) Α y Α matching scale cross (cid:45) over scale Α Μ (cid:72) GeV (cid:76)
Figure 12 . Low scale matching from the fully interacting fixed point FP onto the SM for the benchmarkscenario D with ( R , R , N F ) = ( , , M = 2 . FP H model E L a a ¥ - m ê m matching scale M/µ SMBSM
Figure 13 . The matching procedure at FP for the benchmark model E with parameters (62). The thicklines show the BSM running along the UV-IR connecting separatrix as a function of the scale µ , whichmay take any value determined by the free parameter δα (Λ). The dashed lines show the SM running ofcouplings from Fig. provided that µ = 1 GeV. It is observed that SM and BSM values both coincide atthe matching scale M ≈ . N F Μ high scaleno match (cid:72) weak (cid:76) low scaleno match (cid:72) strong (cid:76) R R Figure 14 . Summary of matching conditions at fully interacting UV fixed points FP of the RG system(22) in dependence on the BSM parameters ( R , R , N F ). Blue diamonds indicate low-scale matchings inthe multi-TeV regime. Red dots stand for a high matching scale beyond the reach of present day colliders.Black triangles indicate scenarios where a matching onto SM values is not available despite of both gaugecouplings approaching the Gaussian. Gray triangles indicate the unavailability of a matching due to strongcoupling phenomena in the weak gauge sector ( R = ). Arrows additionally illustrate how the number ofBSM fermion flavors N F (blue arrow) and the matching scale µ (red arrow) vary with the representation toensure a successful matching. is the necessity for a large multiplicity of flavors, significantly larger than N AS = 18 as minimallyrequired for weakly coupled asymptotic safety to arise (see Tab. ). It is worth contrasting thesuccessful matching at FP with the failure for the benchmark scenario B (59): unlike the weaksector of the SM, the strong sector does not display a “would-be” Banks Zaks IR fixed point, see( A are attracted towards the Gaussian fixedpoint rather than being diverted by an interacting fixed point as in (59). We conclude that the(non)-availability of a matching with FP in the benchmark scenario D ( B ) is dictated by featuresof the SM rather than the specifics of the BSM extension. Benchmark scenario E . For all settings with R ≥ and R ≥ , the theory can display allthree types of interacting UV fixed points. In any of these cases, FP arises at the lowest possiblevalue for N F , see Fig. . It is then interesting to evaluate scenarios where FP is the sole UV fixedpoint. Using our results from Tab. we consider exemplarily the case( R , R , N F ) = ( , , . (62)The UV-IR connecting separatrix is displayed in Fig. . We observe that both gauge couplingsdecrease towards the IR. The scale µ is a free parameter and solely fixed by the free parameter δα (Λ) in the deep UV. We confirm once more that the hierarchy α > α in the deep UV invariablytransforms into α < α once the RG flow falls below the cross-over scale. Also shown is the SM2 ··ÛÛ ÛÛıı ııııÌÌÌÌ ÒÒ a * a * C CC AD E B U ( ) Y L a nd a u p o l e a r i s i n g b e l o w M P l B D Figure 15 . Shown is a conservative estimate for the exclusion area (gray) of fixed point values for thegauge couplings ( α , α ) to ensure the absence of a Landau pole for the U (1) Y hypercharge below Planckianenergies. Equally shown are the partially and fully interacting fixed point values for the benchmark models A, B, C, D and E given in Tab. , for comparison. All benchmark models are UV-safe. running of gauge couplings (dashed lines) taken from Fig. , with data points starting from µ = 1TeV (corresponding to the choice µ = 1 GeV on the lower axis). Tuning the value of δα (Λ) (or µ ) along the separatrix amounts to shifting the separatrix in its entirety parallel to the lower axis.In Fig. , values have been chosen to exemplify that the separatrix can match SM values, (42),at the matching scale M ≈ . E. Synopsis of matching conditions
To summarise, we have established that partially interacting fixed points FP and FP cancomfortably be connected with the SM at low energies provided there are no nearby competingfixed points in the phase diagram of the theory. Moreover, in these cases the matching scaleand thus the masses of BSM matter fields remain freely adjustable parameters. The underlyingreason for this is that both gauge couplings remain relevant couplings in the deep UV. Typicalexamples for this are shown in Fig. , , and for FP of benchmark A , and FP of benchmark B and C , respectively. On the other hand, a matching to the SM becomes more contrived, or evenimpossible, if nearby competing fixed points influence the running of couplings. An example forthis is shown in Fig. for FP of benchmark C , where the nearby fully interacting fixed pointFP impacts on the UV-safe trajectories emanating out of the partially interacting fixed pointFP , thereby enforcing a high matching scale.The matching of fully interacting fixed points FP to the SM is qualitatively different. Thereason for this is that only one of the gauge couplings remains a relevant coupling, which reduces3the number of freely adjustable parameters in the UV by one. Unlike partially interacting ones,fully interacting fixed points predict a relation between the gauge couplings. The availability of amatching to the SM is then encoded in the UV-IR connecting separatrix and must be checked on acase by case basis, see Fig. . In Fig. we summarise our results for the matching conditions atFP in dependence on the BSM paramaters ( R , R , N F ). Low-scale matchings in the multi-TeVregime are indicated by blue diamonds. Examples for this relate to benchmark D and E displayedin Fig. and , respectively. This is contrasted with matchings at a high scale, beyond thereach of present day colliders (red dots). An example for the latter is furnished by benchmark C asshown in Fig. . On the other hand, matchings fail if gauge couplings along the separatrix neverhit SM values, Fig. , despite both gauge couplings approaching the Gaussian. In Fig. suchscenarios are indicated by black triangles. Finally, nearby competing fixed points may distort theUV-safe separatrix and disallow a matching to the SM. In Fig. , the unavailability of matchingdue to strong coupling in the weak gauge sector ( R = ) is indicated by gray triangles. Anexample for this is benchmark B in Fig. where the competing fixed point is the “would-be”Banks-Zaks fixed point of the SM weak sector. Arrows have been added in Fig. to illustratehow the number of BSM fermion flavors N F (blue arrow) and the matching scale µ (red arrow)vary with the BSM matter representation to ensure a successful matching.We briefly come back to the perturbativity of interactions in the fixed point regime. We havefound that the gauge couplings for all benchmarks take small values α ≈ . − .
8, see Tab. .Moreover, in the large- N F and large representation limit, fixed point couplings are parametricallysmall, (35) – (40). In models which permit an asymptotic large- N Veneziano limit, it has alsobeen shown that perturbativity in N F · α (cid:28) N F , the products N F · α come out of order O (1 −
10) for all benchmarks, hinting towards the onset of strong coupling.Ultimately, this pattern of result reflects the unavailability of a Veneziano limit because fixed pointsnecessitate representations higher than the fundamental, Fig. . Future studies should thereforeinclude loop corrections beyond the leading orders, and non-perturbative effects.Finally, we comment on the role of the U (1) Y hypercharge. The SM predicts a Landau pole forthe hypercharge many orders of magnitude beyond the Planck scale M Pl ∼ GeV. In our setupthe BSM fields do not carry hypercharge. The interesting case where BSM fields carry hyperchargewill be detailed elsewhere. Nevertheless, the running of the hypercharge nevertheless differs fromSM running above the matching scale because the strong or the weak or both gauge coupling(s)will grow and eventually settle at interacting fixed points. Interacting fixed points accelerate therunning of the hypercharge due to contributions at two loop. To exclude that a Landau pole mayarise below Planckian energies, we assume a “worst case” scenario in which ( α , α ) take fixedpoint values already at a very low scale of 1 . wherethe shaded area indicates the forbidden region of values for the UV fixed point. We observe thatgauge sectors must become strongly coupled already at low energies to inflict a Landau pole belowPlanckian energies for the hypercharge. For comparison, we also indicate the location of UV fixedpoints for the benchmark models A, B, C, D and E as given in Tab. . Quantitatively, none of thebenchmark models reach a Landau pole for the hypercharge below 10 GeV. We conclude thatall benchmark models are deeply in the UV-safe region of parameter space.4
V. PHENOMENOLOGY
In this section we discuss experimental signatures of asymptotically safe SM extensions. Weassume that the BSM sector can at least partially be accessed at the LHC, which implies a lowmatching scale and masses of the BSM matter fields in the multi-TeV range. An order of magnitudeheavier states can be considered at future colliders [23]. Because of the flavor symmetry the BSMfermions are stable in the model (18). Allowing the flavor symmetry to be broken, the lightestBSM fermion is still stable as long as Yukawa interactions with SM fermions are absent. Thelatter holds except for a few low-dimensional representations with tuned hypercharge of the BSMfermions. As we assume that the BSM fields do not carry hypercharge, these exceptional casescannot be realized. Without mixing with SM fermions, flavor physics constraints are not relevantto our models. We further assume R (cid:54) = . If the new fermions would be colorless, their productionat hadron colliders would be of higher order and suppressed. Scenarios with R = are certainlysuitable for study at an e + e − -machine operating at high energies [23–25].An obvious search strategy is to look for asymptotically safe BSM physics by probing thestrong running coupling evolution and the weak interaction. We discuss various constraints andopportunities from the running gauge couplings, from the weak sector, from direct searches forlong-lived QCD-bound states composed out of BSM fermions and SM partons, and from LHCdiboson searches, offering further constraints on BSM matter including the ( M ψ , M S ) parameterspace. A. Strong coupling constant evolution
The presence of a large number of fermions charged under SU (3) C × SU (2) L changes therunning of the corresponding gauge couplings drastically, as illustrated in Figs. - . The deviationfrom the SM, shown in Fig. , kicks in rather quickly with an order one increase in slope of theasymptotically safe coupling and provides a smoking gun signature of BSM physics considered inthis work. Threshold corrections are not expected to change this picture qualitatively, althoughthe onset of BSM effects may be somewhat smoother.The CMS collaboration has extracted the value of the QCD running coupling up to the scale2 TeV [26] using the measurement of the inclusive jet cross section for proton-proton collisions ata centre-of-mass energy of 7 TeV with data corresponding to an integrated luminosity of 5.0 fb − [27]. This determination is consistent with the SM. Also other measurements, including the oneof the inclusive 3-jet production differential cross section [28], have not observed deviations fromthe two-loop running predicted in the SM up to 1 . M ψ (cid:38) . . (63)In the following subsections we work out further experimental constraints on M ψ , which in turnimply limits on the matching scale. We recall that scenarios with partially interacting fixed pointsFP and FP can generically be matched at any scale (except in specific circumstances, see Fig. ),whereas the matching scale is uniquely fixed for fully interacting fixed points FP . The latterscenarios are therefore subject to stronger experimental constraints.In Fig. we show the running of the strong coupling (black dashed line) and its uncertainty(green band) as determined by CMS [26]. In addition, we show the running of α in the asymp-5 M od e l E M od e l C M od e l B
500 1000 1500 2000 2500 30000.0060.0070.0080.0090.010 Q (cid:64) GeV (cid:68) Α Figure 16 . SM running of the QCD coupling constant (black dashed line) and its uncertainty (green band)as determined by CMS [26]. Colored solid lines indicate the running of α in asymptotically safe benchmarkscenarios B , C and E summarized in Tab. with a low matching scale around 1 . totically safe benchmarks B , C and E introduced in Tab. for low matching scales around 1 . E (blue curve) relates to a fully interacting UV fixed point whosematching scale is fixed at 1 . E is already being probed exper-imentally. Threshold corrections may allow to evade the CMS limit, as the data near 2 TeV arealso losing statistics. B. The weak sector
The experiments at the LEP collider have probed the SM’s electroweak sector with scrutinyand found no significant deviation up to ∼
209 GeV [29]. The LHC has extended related SM testsinto the several O (100) GeV regime [30], still allowing for weakly-interacting uncolored vector-likefermions below the TeV-scale. Within asymptotically safe models, this can happen, for instance,in benchmark A by noting that since the fixed point is partial only, the matching scale can bedifferent than the one shown in Fig. . Electroweak vector boson scattering at the LHC and atfuture lepton colliders [25] is sensitive to such BSM effects.For R (cid:54) = contributions to the ρ -parameter arise if the BSM fermions encounter SU (2) L breaking due to mass splitting δM (cid:28) M ψ in the fermion multiplet. This implies [22] N F d ( R ) S ( R ) δM (cid:46) (40 GeV) , (64)a splitting below percent level for TeV-ish fermion masses and higher.In Fig. we show the running of the weak coupling (black dashed line) and, schematically,the region with agreement with the SM’s weak theory denoted by the hatched green band. Thesolid colored lines correspond to the asymptotically safe benchmarks A , D and E summarized inTab. for a low matching scale around 1 . M o d e l A M od e l D M od e l E
500 1000 2000 50000.00200.00250.00300.00350.00400.00450.0050 Q (cid:64) GeV (cid:68) Α Figure 17 . SM running of the weak coupling constant (black dashed line) and schematically indicatedby the green hatched band the region where the weak sector of the SM has passed experimental tests, seetext for details. Colored solid lines indicate the running of α in asymptotically safe benchmark scenariosintroduced in Tab. that allow for a low matching scale around 1 . Constraints from rare decays can be evaded as long as the BSM fermions do not couple directlyto SM Higgs, quarks, or leptons – as is the case in our setup. For BSM fermions with R > acontribution to the anomalous magnetic moment of the muon arises at 2-loop in the electroweakinteractions. We estimate this as ∆ a µ ∼ d ( R ) S ( R ) N F [ α/ (4 π )( m µ /M ψ )] . Comparison withdata ∆ a exp µ ∼ (2 − · − [22] yields the constraint d ( R ) S ( R ) N F (cid:18) TeV M ψ (cid:19) (cid:46) , (65)which is satisfied for all our benchmarks in Tab. and for M ψ above a TeV.Below the BSM mass threshold the effects of the BSM fermions can be studied indirectlythrough electroweak precision tests. Charged and neutral current Drell-Yan (DY) processes offera promising way to test such corrections [31], as they are both experimentally clean and verywell understood theoretically. The oblique parameter W [32, 33] is of particular interests since itsimpact increases with energy allowing high precision studies at present and future colliders [34]. W is directly related to the BSM contribution that modifies the electroweak beta function, W = − α M W M ψ B BSM , (66)where B BSM denotes the BSM contribution to the 1-loop coefficient B of β , see A displays W versus the matching scale M ψ for the benchmark scenarios defined in Tab. .In the left panel the experimental 95% C.L. upper limits [34] from LEP (red dashed line) and LHC8 TeV (blue dashed line) are shown. Constraints on negative W exist but are not relevant here.For the fully-interacting fixed points FP of benchmark D and E (full dot), a low matching scale7 LEPLHC 8 TeV AE B FP C FP D FP D FP (cid:45) M Ψ (cid:64) TeV (cid:68) W (cid:137) LHC 13 TeV 3 ab (cid:45) ILC 500 GeV 3 ab (cid:45) D FP C FP A
100 TeV collider 10 ab (cid:45) M Ψ (cid:64) TeV (cid:68) W (cid:137) Figure 18 . Shown is the electroweak precision parameter W (66) as a function of the BSM fermion massfor the low-scale benchmark models given in Tab. . In the left panel dashed lines show 95% C.L. upperlimits obtained from LEP (red) and the LHC at 8 TeV (blue). In the right panel dashed lines indicatethe projected reach of the LHC at 13 TeV (blue), the ILC 500 GeV (red), and a 100 TeV collider (gray).Experimental limits are taken from [34]. at around 2 TeV dictates a high multiplicity of BSM fermions, and a large contribution to (66).These scenarios turn out to be excluded. For scenarios with partially interacting fixed points (FP or FP ) the matching scale is a free parameter. For benchmark models C (magenta line) and D (green line) the matching scale must be larger than about 8 TeV to satisfy LEP constraints.Benchmark A (red line) is not yet constrained by the data owing to the low number of BSMfermion species N F , while benchmark B (black line) is not probed at this order because its BSMfermions are SU (2) L singlets.The right panel of Fig. shows the projected sensitivities of the LHC at 13 TeV with 3 ab − integrated luminosity (dashed blue line), the ILC 500 with 3 ab − (dashed red line), and a 100 TeVcollider with 10 ab − (dashed gray line). The precision of the W determination is expected to in-crease by two orders of magnitude, requiring the matching scale in all allowed benchmark scenariosto be above around 10 TeV.While this analysis demonstrates the importance of DY measurements for the type of BSMscenarios outlined here, constraints based on (66) must be taken with a grain of salt. The reasonfor this is that the corrections to W are only known to one loop order. In our framework, competingtwo loop corrections in the gauge beta functions play an important role as they are responsible forthe fixed point. To estimate two loop effects, we replace B BSM in (66) by the effective coefficient B (cid:48) evaluated on the RG trajectory near the matching scale. For benchmark models C where α becomes asymptotically safe, we find that the bound on the matching scale softens, from about8 TeV to about 5 TeV. Also, benchmark B now contributes negatively to W and can be probed inthe future. Similar two loop effects are expected for the other benchmark models. A complete twoloop analysis of the oblique parameter W , although desirable, is beyond the scope of this work.8 C. R -hadrons We assume that at least some of the BSM fermions can be pair-produced,2 M ψ < √ s , (67)where √ s denotes the accessible center of mass energy at the collider. At least the lightest of thefermions has a long life, longer than a typical hadronization time scale, and forms colorless QCDbound states with ordinary partons (quarks and gluons), the so-called R -hadrons.Both the ATLAS and CMS collaborations searched for heavy long-lived charged R -hadronsusing a data sample corresponding to 3.2 fb − of proton-proton collisions at √ s = 13 TeV. Nosignificant deviations from the expected background have been observed which allowed to put amodel-independent 95% confidence level (C.L.) upper limits on the production cross section oflong-lived R -hadrons. In the framework of supersymmetry, those results have been translated intoa lower bound on the mass of the fermionic partner of the gluon (gluino), which read 1.5 TeV forCMS [35] and 1.6 TeV for ATLAS [36]. Recently CMS has updated their analysis to 12.9 fb − ofdata [37], with the corresponding limit on the gluino mass increased to 1.7 TeV.At the LHC any colored and hypercharge-neutral BSM fermion would be produced in the sameway. Therefore one can easily recast the experimental limits for gluino searches in the frameworkof asymptotically safe scenarios considered in this study. In the leading order the ψ ¯ ψ pairs can beproduced by gluon fusion or by quark-anti-quark annihilation, where the former mechanism is adominant one. We can additionally assume that the main contribution to pp → ψ ¯ ψ comes from a t -channel exchange of a BSM fermion. In this case the production cross section σ ψ ¯ ψ depends on( R , R , N F ) and scales proportionally to the factor C , σ ψ ¯ ψ ∼ N F C with C = [ C ( R )] d ( R ) d ( R ) . (68)We can then put lower limits on the BSM fermion mass using the experimental limits for gluinosprovided in [35] and [36], rescaling the gluino production cross section by C . Notice also that itis possible for real representations, such as those with ( p, p ) for R , that ψ is a Majorana fermion;in all other cases we ignore the differences with respect to the gluinos in our estimates.In Tab. we show the lower bounds on M ψ in dependence of R and R together with C for N F = 1. The lower bound increases with increasing d ( R ). For R = (cid:48) and d ( R ) > N F , the bounds get stronger. For example, for benchmarks B , D and E defined in Tab. we find M min ψ = 2 . , . . C isbeyond 2.8 TeV. For benchmarks C and D the constraints from DY processes obtained in Sec. V Bare stronger than the R -hadron ones. D. Diboson spectra and resonances
Consider the situation where the BSM scalars are lighter than twice the mass of the fermionsand can be resonantly produced, M S < M ψ , and M S < √ s . (69)9 ψ ( R , R ) R = 1 R = 2 R = 3 R C M min ψ (TeV) C M min ψ (TeV) C M min ψ (TeV) (1.3) 10 (1.4) 16 1.5
72 1.7 144 1.8 216 1.9
360 2.0 720 2.1 1080 2.2 (cid:48) Table 7 . Lower limits on the mass of the lightest BSM fermion, M min ψ , as derived from the searches for long-lived charged particle by CMS [35, 37] and ATLAS [36] for N F = 1. We make explicit a dependence on thefermion representations R and R under SU (2) L and SU (3) C , respectively. The dominant contribution tothe production cross section is proportional to C , (68), which is also given. Values in parentheses correspondto scenarios ( R , R ) with no weakly interacting UV fixed points, see Fig. . In such a case the scalars cannot decay on-shell to any of the BSM fermions and because its mixingwith the SM Higgs boson is negligible, the only possible decay channels are loop-mediated decaysinto pairs of gauge bosons GG = gg, γγ, ZZ, Zγ, or W W . (70)The cross sections for these, as well as their relative strengths, depend directly on transformationproperties of the BSM fermions under SU (3) C and SU (2) L . Since S does not couple directly tothe SM fermions its dominant production mechanism is gluon fusion which proceeds through theloops containing ψ i . This process is schematically depicted in Fig. . Due to the particular flavorstructure of the asymptotically safe BSM sector, one needs to consider a simultaneous productionof N F scalars S ij , each of them coupled to exactly one fermion pair ¯ ψ i ψ j . However, since flavoris conserved in the fermion-gauge boson interactions, only diagonal couplings are allowed in thisprocess and the number of simultaneously produced scalars is reduced to N F . Due to interferenceeffects between the N F diagrams, it is useful to investigate separately the limiting cases of maximaland no interference. Maximum interference.
Provided that all scalars S ii have the same mass M S and totaldecay width Γ S , the interference between them is maximal and the cross section for a dibosonsignal GG (70) is given by [38, 39] σ ( pp → S ,...,N F → GG ) = N F σ ( pp → S )BR( S → GG ) = N F π M S Γ S I pdf Γ GG Γ gg . (71)Notice that (71) scales as N F times the cross section for one individual flavor. In the above, Γ GG denotes the partial decay width into two gauge bosons with only one generation of BSM fermionsin the loop. Similarly, Γ gg stands for the corresponding partial width into two gluons, and I pdf is the integral of parton (gluon) distribution function in proton, evaluated at the energy scale µ = M S with the center of mass energy √ s .0 S ii ψ i ψ i GG gg Figure 19 . Production via gluon fusion and decay of the scalar resonance S in the asymptotically safeSM extension (22). Here, i denotes the BSM flavor index and GG stands for any combination of SM gaugebosons (70). No interference . Interference effects are absent provided that the masses of the scalars S ii arenarrowly spaced with mass differences ∆ M below the detector mass resolution Γ det , and providedthat individual widths do not overlap. Consequently, the production cross section becomes σ ( pp → S ,...,N F → GG ) = N F σ ( pp → S ) BR( S → GG ) = N F π M S Γ det I pdf Γ GG Γ gg . (72)Notice that the total cross section scales as N F times the cross section for one individual flavor. Assuch, the absence of interference effects reduces the total cross section parametrically by a factorof N F over (71). We expect that settings with partial interference effects are well covered withinthe limits (71) and (72).The relevant energy scale in the process is µ ∼ M S , that is, the diboson invariant mass. Thematching scale M is of the order M ψ . For M S below M , roughly M S (cid:46) M ψ (“low M S ”), the gaugecouplings assume SM evolution. For M S above M (“high M S ”), the gauge couplings follow theBSM fixed-point trajectory. The kinematical range for the high M S scenario is M ψ (cid:46) M S ≤ M ψ . (73)Since characteristic diboson signatures can arise in many BSM scenarios, they have been intensivelysearched for at the LHC.Recently both ATLAS and CMS updated their 95% C.L. limits on the fiducial cross sectiontimes branching ratio ( σ × BR × acceptance A for a dijet analysis) for a general scalar resonancedecaying into gg [40, 41] (updating [42–44]), Zγ [45, 46], ZZ [47, 48], W W [49, 50] and γγ [51, 52].The exact limit in each case depends on the mass of the resonance, as well as on its total width. Inthe following we choose M S = 1 . S ≤ Γ det . First we considerlimits on σ ( pp → GG ) provided by dijet searches. The partial width of S into gluons [53] readsΓ gg = α s M S π (cid:12)(cid:12)(cid:12)(cid:12) yS ( R ) d ( R ) M ψ A / ( x ) (cid:12)(cid:12)(cid:12)(cid:12) , (74)where α s = 4 πα and the loop function is defined as A / ( x ) = x [ x + ( x −
1) arcsin( √ x ) ] and x = M S / (4 M ψ ). In Fig. we show σ ( pp → S → gg ) versus the mass of the BSM fermions M ψ for M S = 1 . Model BModel D M Ψ (cid:64) TeV (cid:68) Σ (cid:72) pp (cid:174) S (cid:174) gg (cid:76)(cid:64) f b (cid:68) Figure 20 . Dijet cross section as a function of the BSM fermion mass M ψ for benchmark B (thick bluecurves), and benchmark D (thin green curves) for M S = 1 . N F scalars, while dashed ones to no interference. The upper and lower horizontaldashed line denotes the ATLAS 95% C.L. limit [40] on the dijet cross section assuming 50% and 100%acceptance, respectively. refer to no interference. In the latter case the total width is assumed to be Γ det = 0 . M S , whilein the former we calculate Γ S = (cid:80) GG Γ GG (cid:39) Γ gg , which is below Γ det . Moreover, benchmark B ( R = , R = , N F = 30, thick blue lines) is contrasted with benchmark D ( R = , R = , N F = 290, thin green lines) in Fig. . The upper and lower horizontal dashed line indicatesthe ATLAS 95% C.L. dijet limit for an acceptance A = 50% and 100%, respectively. We use theNNLO parton distribution functions MSTW2008NNLO [54] with I pdf = 0 . M ψ (cid:38)
87 (62) TeV for benchmark D and M ψ (cid:38)
125 (89) TeV for benchmark B using A = 100% (50%). The bounds gradually becomeweaker when the interference decreases. In the limit of no interference, the respective lower limitsdrop to M ψ (cid:38) . .
2) TeV for benchmark B . For benchmark D the bounds drop below the onesfrom R -hadron searches given in Tab. . Depending on N F , the two limiting cases may differ byseveral orders of magnitude. Moreover, since Γ gg (cid:28) Γ det , the cross section pp → S → gg withoutinterference is additionally reduced relative to the maximal one. We conclude that, wheneverapplicable (69), the dijet searches can provide significantly stronger limits on the BSM fermionmass M ψ than the DY and R -hadron limits worked out in Sec. V B and Sec. V C, respectively.In Fig. we present exclusion limits in the M S − M ψ plane from R -hadron searches (greenhorizontal stripe) and the dijet cross section limit from ATLAS 95% C.L. — exemplarily for thebenchmark B ( R = , R = , N F = 30) —, also comparing settings with maximal interference(red solid line) and no interference (red dashed line) and A = 100%. Moreover, solid and dashedblack lines represent the borders of (73) where M S = 2 M ψ and M S = M ψ , respectively. Theexcluded areas are below the solid black lines and to the left and below the red lines. The darkand light blue areas indicate, respectively, the searchable parameter space (69) at the LHC (with2 M y @ TeV D M S @ T e V D LHC m a x . i n t e r f e r e n c e f e r e n c e R h a d r on s e a r c h e s B S M r u n n i n g d i - j e t li m i t s ( , ,30) i n t e r n o M od e l B ( , , ) Figure 21 . Excluded regions in the M ψ − M S plane combining R -hadron searches (green horizontal stripe)with dijet cross section limits from ATLAS 95% C.L. [40] for the benchmark B ( R = , R = , N F = 30).Excluded regions are also given for the limiting cases of maximal interference (below red solid curve) and nointerference (left of red dashed curve), with A = 100%. The dark (light) blue area indicates the searchableparameter space (69) at the LHC (future colliders). The strip (73) with M ψ (cid:46) M S < M ψ where anenhancement of α s due to asymptotically safe running takes place corresponds to the region between thefull and dashed black lines. √ s = 14 TeV) and at future colliders [23]. For masses within the range (73) — corresponding tothe band between the full and dashed black lines —, the strong coupling α s is already of non-SMtype and enhanced, α s ( M S ) (cid:38) α s ( M ψ ). This regime allows to probe higher values of M S .If the BSM fermions transform non-trivially under SU (2) L , two additional effects arise. Firstly,the lower dijet bound on the fermion mass increases since the cross section σ ( pp → S → gg )scales with d ( R ). Secondly, decays into electroweak gauge bosons V V = γγ, W W, ZZ, Zγ becomepossible. In order to discuss decays into weak gauge bosons in more detail, it is convenient tointroduce the reduced decay widths¯Γ V V = 1 F Γ V V Γ gg , with F = (cid:18) C ( R ) C ( R ) (cid:19) , (75)which expresses the widths Γ V V in units of Γ gg together with a group theoretical factor F whichtakes into account the quadratic Casimirs of the BSM fermions. In terms of (75), we find¯Γ W W = α α , ¯Γ ZZ = α α + α ) α , ¯Γ Zγ = α α ( α + α ) α , ¯Γ γγ = α α α + α ) α , (76)where α = g Y / (4 π ) is the hypercharge coupling. The reduced decay widths depend, in general,on the three gauge couplings. We note that ¯Γ W W stands out in that it is independent of α , andonly sensitive to the ratio of the other two gauge couplings.For low M S below the matching scale the ratios ¯Γ V V are solely determined by the SM gauge3couplings. In this case F can be determined from any of the V V modes, providing informationabout R and R . Measuring more than one mode serves as a consistency check. In Fig. we showthe ratios Γ W W / Γ gg (blue), Γ ZZ / Γ gg (green), Γ Zγ / Γ gg (red), and Γ γγ / Γ gg (orange), depending on d ( R ) for R = (left panel) and R = (right panel) for low M S . The hierarchies among thedifferent modes are fixed in this regime. Deviations can arise from switching on hypercharges ofthe fermions, or from running above the matching scale, at high M S , where the gauge couplingsexperience BSM running. This modifies the running of α and α whereas the running of α remains SM-like. We may neglect its slow logarithmic running and can take α as constant for thefollowing considerations.To discuss further BSM effects, we simplify (76) by exploiting that ( α /α ) (cid:46) .
08 at TeVenergies below the matching scale. We find¯Γ
W W = α α , ¯Γ ZZ ≈ α α , ¯Γ Zγ ≈ α α α α , ¯Γ γγ ≈ α α , (77)where “ ≈ ” means equality up to relative corrections of order O ( α /α ). We observe that ¯Γ γγ isno longer sensitive to α as it has reduced to a ratio of the other two gauge couplings, similarly to¯Γ W W . It follows that ¯Γ ZZ ∝ ¯Γ W W and that ¯Γ Zγ ∝ (¯Γ ZZ · ¯Γ γγ ) / .For models with fully interacting fixed points FP we recall that the weak and strong gaugecouplings start growing with RG scale above the matching scale, dictated by the underlying sep-aratrix into the UV fixed point, see Fig. . Their ratio increases from α /α < α /α → / α ( µ ) /α ( µ ) > α ( M ) /α ( M ) for µ > M which implies thatboth ¯Γ W W and ¯Γ ZZ increase accordingly with increasing µ > M . On the other hand ¯Γ γγ becomessuppressed. For ¯Γ Zγ , the situation is ambiguous: the growth of ¯Γ ZZ competes with the suppressionof ¯Γ γγ and the outcome in the cross-over region will be model-dependent. Quantitatively, for thefully interacting fixed points of benchmark D (benchmark E ) we find that both ¯Γ W W and ¯Γ ZZ grow from µ = M to µ = 2 M by factors of about 12 (3), and that ¯Γ γγ is suppressed by factors ofabout 13 (2). ¯Γ Zγ is very mildly suppressed only.For models with partially interacting fixed points FP we generically observe α ( µ > M ) >α ( M ) and α ( µ > M ) < α ( M ), e.g. Fig. . This implies that all reduced decay widths increasewith increasing µ > M , albeit with different factors, see (77). Conversely, for models with FP we have α ( µ > M ) > α ( M ) and α ( µ > M ) < α ( M ). As can be deduced from the explicitexpressions in (76) and (77), all four reduced decay widths decrease relative to Fig. .We conclude that diboson searches involving pairs of electroweak gauge bosons can providestronger limits than the dijet ones if d ( R ) is sufficiently large. Due to the a priori unknownhierarchy between M S and M ψ , correlations of V V with dijet limits cannot be interpreted unam-bigously. On the other hand, an observation of a GG -resonance determines M S , while a breakdownof SM-running of α , perhaps together with a similar effect in the weak coupling, determines M ψ .In these cases, extracting F is feasible at low M S .Resonance-induced diboson signatures can arise as well from decays of ( ψ ¯ ψ )-bound states,which are expected to form somewhat below center-of mass energies of 2 M ψ for R (cid:54) = [55]. Inour model such ψ -onia can start at about 2 M ψ (cid:38) S resonance (76). Further analysis isbeyond the scope of this work.4 (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) WW ZZZ
ΓΓΓ (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) R = (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:45) d (cid:72) R (cid:76) (cid:71) VV (cid:144) (cid:71) gg R = WW ZZ Z
ΓΓΓ (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:45) (cid:45) (cid:45) d (cid:72) R (cid:76) (cid:71) VV (cid:144) (cid:71) gg Figure 22 . The ratios Γ
W W / Γ gg (blue), Γ ZZ / Γ gg (green), Γ Zγ / Γ gg (red), and Γ γγ / Γ gg (orange) versus d ( R ) for R = (left) and R = (right) for low M S (see main text). VI. SUMMARY
The concept of an interacting UV fixed point in quantum field theory is of high interest perse; for particle physics it opens up “theory space” for model building. Here, we have investigatedasymptotically safe extensions of the Standard Model by adding new fermions and scalar singletfields. The new matter fields also interact, minimally, via a single Yukawa coupling to help generateinteracting UV fixed points. A large variety of stable high energy fixed points emerges where eitherthe strong, or the weak, or both couplings assume finite values, see Figs. , . Those where oneof the gauge couplings remains asymptotically free can flow into the Standard Model at any scaleabove O (1 −
2) TeV, modulo nearby competing fixed points. Many of the fully interacting fixedpoints can also be matched onto the Standard Model including at TeV scales, Fig. . Specifically,with fermions charged under SU (3) C × SU (2) L , we found that they must carry representationshigher than the fundamental in at least one of the gauge sectors, Fig. . Also, fully interactingfixed points cannot arise if the fermions are charged under SU (2) L only. An intriguing featureof models with fully interacting UV fixed points is a relation between gauge couplings, dictatedby asymptotic safety. The number of fundamentally free parameters is thereby reduced offeringan enhanced degree of predictivity compared to the Standard Model, quite similar to the idea ofunification. Our results have been obtained at two loop accuracy where couplings remain small forall scales, though not parametrically small such as in the Veneziano limit [12]. Of course, furtherstudy is needed to explore the full potential of this new direction.There are several opportunities to look for asymptotically safe BSM physics at colliders. Thepresence of a large number of new fermionic degrees of freedom from higher representations of SU (3) C × SU (2) L with large multiplicities implies striking new physics at the corresponding massdespite being weakly coupled, e.g. Figs. - . Irrespective of the choice of benchmark models, thequalitative features from the model ansatz laid out in Sec. III are rather generic. For low scalematching BSM physics can be just around the corner, as close as O (1 −
2) TeV: R -hadron signalsarise and the strong coupling evolution itself is altered and further collider tests should be pur-sued, see Fig. . For SU (2) L -charged fermions the weak interaction is modified, schematicallyshown in Fig. . Corresponding shifts in electroweak observables, including W W -production5appear above threshold. Loop-induced diboson spectra involving the scalar resonance Fig. aresensitive to about an order of magnitude higher scales Fig. . While the actual limits are rathermodel-dependent, this demonstrates that the phenomenology of asymptotically safe BSM can beprobed at the LHC at Run 2 and beyond. Tests of the weak interaction are also encouraged athigh energy e + e − colliders [23–25]. Acknowledgements
We thank Joachim Brod, John Donoghue, Veronica Sanz, Martin Schmaltz, and Enrico Sessolofor useful discussions. AB is grateful to the Physics Department at Boston University for itshospitality and stimulating environment while this project was pursued. AB is supported by anSTFC studentship, GH and KK in part by the DFG Research Unit FOR 1873 “Quark FlavorPhysics and Effective Field Theories”, and DL by the Science and Technology Facilities Council(STFC) under grant number ST/L000504/1.
Appendix A: Technicalities
The appendix summarises group theoretical formulæ and loop coefficients, together with adiscussion of UV-IR connecting separatrices.
Loop coefficients and group theoretical factors
We summarise formulæ for perturbative loop coefficients. We have exploited general expressionsas given in [18–21]. We consider the SM matter fields, together with N F vector BSM fermionsin the R and R representation under SU (3) C and SU (2) L , respectively. The beta functions arestated in (22). We reproduce them here for completeness, β = ( − B + C α + G α − D α y ) α ,β = ( − B + C α + G α − D α y ) α , ( A β y = ( E α y − F α − F α ) α y . The gauge one loop coefficients read B = 14 − N F S ( R ) d ( R ) ,B = 193 − N F S ( R ) d ( R ) . ( A C = −
52 + 4 N F S ( R ) d ( R ) (2 C ( R ) + 10) ,C = 353 + 4 N F S ( R ) d ( R ) (cid:18) C ( R ) + 203 (cid:19) , ( A G = 9 + 8 N F S ( R ) C ( R ) d ( R ) ,G = 24 + 8 N F S ( R ) C ( R ) d ( R ) . ( A D = 4 N F S ( R ) d ( R ) ,D = 4 N F S ( R ) d ( R ) . ( A E = 2[ N F + d ( R ) d ( R )] ,F = 12 C ( R ) ,F = 12 C ( R ) . ( A C ( R ), S ( R ) and d ( R ) denote the quadratic Casimir invariant, theDynkin index and dimension of the representation R , respectively. They are related by S ( R ) = d ( R ) C ( R ) /d (Adj) , ( A d (Adj). It is also convenient to parametrize theloop coefficients through the weights ( p, q ) for irreducible SU (3) representations R , and, similarly,through the highest weight (cid:96) for SU (2) representations R , d ( R ) = ( p + 1)( q + 1)( p + q + 2) ,C ( R ) = p + q + ( p + q + pq ) , with p, q = 0 , · · · ,d ( R ) = 2 (cid:96) + 1 ,C ( R ) = (cid:96) ( (cid:96) + 1) , with (cid:96) = 0 , , · · · . ( A N F = 0), the perturbative loop coefficients reduceto their SM values B SM3 = 14 , B
SM2 = 19 / ,C SM3 = − , C SM2 = 35 / ,G SM3 = 9 , G
SM2 = 24 , ( A E SM = F SM2 = F SM3 = 0. In this limit and at two loop accuracy, we observe thatthe SU (2) L sector displays a “would-be” Banks-Zaks type IR fixed point at( α ∗ , α ∗ ) = (cid:18) , (cid:19) . ( A SU (3) C sector does not display signs of a Banks-Zaks type fixed point owing to B /C < A
10) become visible for scenarios with BSM matter uncharged under SU (2) L .At places, primed two loop coefficients arise. They relate to their unprimed counterparts as C (cid:48) = C − D F /E ,C (cid:48) = C − D F /E ,G (cid:48) = G − D F /E ,G (cid:48) = G − D F /E . ( A UV-IR connecting separatrices
Next, we summarise formulæ and results related to the running of couplings along UV safetrajectories emanating out of an interacting UV fixed point. We are particularly interested in therunning of the relevant gauge coupling along the UV-IR connecting hypercritical surface down toenergy scales µ close to the mass M of the new matter fields, µ ≈ M , where the model connectswith the SM. We approximate the beta functions ( A
1) as ∂ t α = α ( − B + C α − Dα y ) ,∂ t α y = α y ( Eα y − F α ) . ( A B, C, D and F then take the values corresponding to the asymptotically safe gauge coupling. To findthe exact UV-IR connecting separatrix, the system ( A
12) must be solved numerically. However,for most cases of interest, approximate estimates can be obtained as well. We discuss two strategies.
UV critical surface approximation . Firstly, we may approximate α y along the separatrixthrough its values along the UV critical surface. The virtue of this approximation is that it becomessufficiently exact close to the UV fixed point. Quantitatively, on the UV hypercritical surface, theBSM Yukawa coupling is determined via the gauge coupling as α y = C y ( α − α ∗ ) + α ∗ y . ( A C y is defined via the relevant eigendirection at the UV fixed point, with (1 , C y ) T denoting the eigenvector with negative eigenvalue of the stability matrix M at the fixed point inthe basis ( α, α y ) T . In terms of the perturbative loop coefficients, it reads C y = 2 FE (cid:32) (cid:115) − BF ( C E − CDEF + 2 D F ) − B C E F ( DF − CE ) + BCEF ( DF − CE ) (cid:33) − . ( A A
13) into ( A
12) to find ∂ t α = α ( − ˜ B + ˜ C α ) , ( A B = B + D ( α ∗ y − C y α ∗ ) , ˜ C = C − D C y . ( A A
15) analytically, we find α ( µ ) at any RG scale µ in terms of α ( M ) deter-mined at some reference mass scale M , (cid:16) µM (cid:17) − ϑ = α ∗ − α ( M ) α ∗ − α ( µ ) α ( µ ) α ( M ) exp (cid:18) α ∗ α ( M ) − α ∗ α ( µ ) (cid:19) . ( A ϑ is given by ϑ = ˜ B / ˜ C < A A A
17) can be resolved for α ( µ ) with the help of the Lambert function, α ( µ ) = α ∗ W ( µ, M, ϑ ) , ( A W ( µ, M, ϑ ) ≡ W L [ z ( µ, M, ϑ )] with W L [ z ] denoting the Lambert function, defined implicitlythrough z = W L exp W L . The variable z ( µ, M, ϑ ) is given explicitly by z ( µ, M, ϑ ) = (cid:16) µM (cid:17) ϑ (cid:18) α ∗ α M − (cid:19) exp (cid:18) α ∗ α M − (cid:19) , ( A ϑ given in ( A α M ≡ α ( µ = M ) > α ( µ ) inbetween α M and α ∗ . The expression( A
19) can now be used to approximately determine the mass scale M by matching it to values ofthe SM. Yukawa nullcline approximation . Alternatively, we may use the Yukawa nullcline to es-timate α y along the UV-IR connecting separatrix. In the system ( A α y = FE α . ( A A
21) to approximate the separatrix is twofold. Firstly, close to the Gaussianfixed point, the UV-IR connecting separatrix and the nullcline coincide, meaning that ( A
21) isa very good approximation if the gauge coupling is matched to the SM at scales where α (cid:28) α ∗ .Secondly, rewriting ( A
21) as α y = FE ( α − α ∗ ) + α ∗ y , ( A UV (cid:45) IR connecting separatrixUV hypercritical surfaceYukawa nullcline (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45)
10 00.0010.010.11 ln (cid:72) Μ (cid:144) M (cid:76) Α Α (cid:42) Figure 23 . UV-IR connecting trajectories at the example of a partially interacting fixed point FP andparameters R = , R = , N F = 30, showing the exact separatrix (red line) in comparison with theUV hypercritical surface approximation (magenta), ( A A we conclude that the nullcline also coincides with the hypercritical surface at the UV fixed point.Hence, ( A
21) can be viewed as a “global linear approximation” for the UV-IR connecting separa-trix. Comparing this approximation with the UV hypercritical surface( A
13) in the limit
B/C (cid:48) (cid:28) C (cid:48) = C − DF/E ), we observe that ( A
14) becomes C y = F/E + O ( B/C (cid:48) ), establishing thatthe hypercritical surface ( A
22) exactly coincides with the Yukawa nullcline. For finite
B/C (cid:48) < A
21) into therunning of the gauge coupling ( A
12) we find ∂ t α = α ( − B + C (cid:48) α ) . ( A A
23) with initial condition α ( µ = M ) = α M is given by ( A
19) with( A ϑ which now reads ϑ = B /C (cid:48) < A A at the example of a partially interactingfixed point FP with R = , R = , N F = 30. We compare the exact numerical solution for α ( µ ) (full red line) with the hypercritical surface approximation (magenta) and with the Yukawanullcline approximation (blue). We observe that the UV region (IR region) is well-approximated bythe UV critical surface (Yukawa nullcline), respectively. We also observe that the exact separatrixis globally well approximated by the Yukawa nullcline, corresponding to ( A
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