JHEP09(2019)100 Springer July 29, 2019 : September 5, 2019 September 12, 2019 : : d,b Received Accepted Published Published for SISSA by https://doi.org/10.1007/JHEP09(2019)100 [email protected] , and Antonio D. Pereira c,b [email protected] , Astrid Eichhorn . 3 a,b 1907.11173 The Authors. c Models of Quantum Gravity, Nonperturbative Effects, Renormalization

Constraining quantum gravity from observations is a challenge. We expand , [email protected] E-mail: Philosophenweg 16, 69120 Heidelberg, Germany CP3-Origins, University of SouthernCampusvej Denmark, 55, DK-5230 Odense M,Instituto Denmark de Universidade F´ısica, Federal Fluminense, Campus da Praia Vermelha, Av. s/n, Litorˆanea 24210-346, Niter´oi,RJ, Brazil Centro Brasileiro de Pesquisas (CBPF), F´ısicas Rua Dr Xavier Sigaud 150,Institute Urca, for Rio Theoretical de Physics, Janeiro, University RJ, of Brazil, Heidelberg, CEP 22290-180 b c d a Open Access Article funded by SCOAP Group ArXiv ePrint: a dimensionless gravitational coupling,tribution we to also beta investigate functions whether in theGroup the matter techniques gravitational sector is con- calculated from universal, functionalparameterization by Renormalization and studying choice the of dependence gauge. on the regulator, metricKeywords: field — unimodular asymptotic safety, Weyl-squared gravitywith and constraints asymptotically arising safe from gravity demandingically, — an we ultraviolet show complete that StandardLandau-pole within Model. problems approximations, Specif- demanding in thatstrains Abelian the quantum gauge viable gravity gravitational solves couplings parameter the and space. Yukawa In couplings the case strongly of con- Weyl-squared gravity with Abstract: on the idea that thechallenge. interplay of Thus, quantum we gravity set with out matter could to be confront key different to potential meeting candidates this for quantum gravity Gustavo P. de Brito, A link that matters:of towards unimodular phenomenological asymptotic tests safety JHEP09(2019)100 ], 6 1 being Pl M 18 35 13 9 6 34 39 23 25 – 1 – 28 15 18 being the energy scale relevant for experiments, E 3 32 , with 17 # ) Pl 1 33 E/M 14 with ( 1 In the presence of a second “meso”-scale, as hinted at by some quantum-gravity approaches, e.g., [ 5.1 Higgs-Yukawa and5.2 (non-)Abelian gauge sector The in Higgs unimodular potential gravity in unimodular gravity theories 4.1 (Non-)Abelian4.2 gauge couplings Yukawa terms Weyl-squared gravity 3.1 Functional Renormalization3.2 Group for quantum Functional gravity Renormalization3.3 Group for unimodular Functional gravity Renormalization Group for Weyl-squared gravity 1 Observational constraints on quantumdimensional gravity argument, are one hard typicallyeffects expects to a come power-law by. suppression of Based quantum-gravity on a simple this situation can change. 1 Introduction B Fermions and the exponential parameterization C Unimodular gravity-matter systems in generalD dimensions Explicit results for the Reuter fixed point 7 Results for Weyl-squared gravity 8 Concluding remarks A Conventions 5 Results for unimodular gravity 6 Comparison with the Reuter fixed point 4 Quantum-gravity induced ultraviolet completion of Standard-Model like 3 Setup Contents 1 Introduction 2 Introduction to (unimodular) asymptotically safe gravity and JHEP09(2019)100 2 (1) O ], rules the 17 ] indicate that the non-perturbative 9 – 2 ] carry over to the full Standard Model. 16 – 14 – 2 – (1) at the electroweak scale. This is in contrast O theory [ 4 φ ] for an overview of potential mechanisms for asymptotic 18 ] and 13 – 0. Nevertheless, mathematical and internal consistency are not 10 most prominently in the Abelian gauge coupling as well as the > 3 cally safe/free model, andmarginal the couplings. fact The that a free UV-IR parameters link of can an be asymptotically established free/safe based model on are microscopic dynamics. Scale-symmetry isthe a case consequence of of asymptotic freedom, vanishingof or interactions a asymptotic in balance safety, see between residual [ interactionssafety. in Theories the which case pass this first test can be subjected to the second. Yukawa sector, must bedescription, resolved i.e., by valid up quantum-gravity to arbitrarily fluctuations. shortin distances, the quantum of A field the theory fundamental building framework blocks requiresor of theory nature safe. to be Both either cases asymptotically providetheories, free a framework in for which an ultraviolet an completion enhanced of effective symmetry, field quantum scale symmetry [ Standard Model, The first is a microscopic consistency test: the perturbative Landau poles of the The second test exploits the finite number of free parameters in an asymptoti- Therefore, the marginal couplings of the Standard Model are a prime target to set up Such a UV-IR connection appears to contradict the well-established principle of sepa- A consistent microscopic description of all degrees of freedom of nature must account Analogous considerations hold inLandau-poles settings in beyond the Abelian the gauge Standard and the Model. Higgs-Yukawa sector [ i) 2 3 ii) triviality problems from QED [ observational consistency tests for quantum gravity. Specifically,der they the provide assumption two tests, of un- no new physics between the electroweak scale and the Planck scale: at the Planck scaleto lead canonically to irrelevant couplings, changes which of dimension: are a power-law suppressed large due interval ofby to UV their the values canonical RG is flow. mappedcouplings to is Accordingly, a “washed rather out” the small by information interval the on of RG IR flow, microscopic values and physics not encoded accessible in in the those IR. ration of scales in nature,decouples which from loosely microscopic speaking physics. statesscales that Yet, effective typically physics descriptions at feature macroscopic for finitelyIn scales physics many the at Standard macroscopic parameters Model thatcrophysics. of are particle They sensitive physics exhibit to the marginal a the couplings logarithmic microphysics. are scale-dependence. sensitive to Accordingly, the changes mi- of including a metric as well asobservational consistency matter tests fields is in that the thedetermine microscopic properties some description. of properties the The of microscopic key description the pointi.e., actually matter for in sector at the energy infrared scalesthat (IR). far the below This resulting the low-energy allows Planck behavior mass, to is restrict consistent the with microscopic observations. dynamics by demanding the only conditions that coulddirect allow to probes constrain of candidate Planck-scale quantum-gravity theoriestency physics while tests remain for (mostly) quantum out gravity of arise reach. from the Observational interplay consis- for of both quantum gravity gravity and with matter. matter. In a quantum-field theoretic setting this can be achieved by the Planck mass and # JHEP09(2019)100 ]. 19 ] of the asymptotically safe Reuter 46 – The Reuter fixed point is an interacting 20 4 ]. 56 – 3 – ] for recent reviews, with promising indications 51 – 49 , 18 ] in four dimensions [ 55 – ], see, e.g., [ 52 48 , 47 try in the UV.agree The with theoretically their determined measured valuesasymptotically of values. free/safe irrelevant fixed couplings Hence, point need given is not not a automatically set phenomenologically of viable. fields and symmetries, an underlying asymptotic safety changesautomatically the correspond scaling, to and power-counting relevantthat couplings relevant is ones. no irrelevant longer atthe Conversely, a IR, a since fixed coupling it point isrelevant must not couplings. automatically a free Intuitively assume parameter, speaking, onein but the the specific determined ultraviolet powerful in value (UV) terms symmetry leaves in of imprints of the in scale-invariance finitely the many IR akin to any other enhanced symme- trigger a departure fromto scale the invariance IR. incouplings, In the i.e., the Renormalization couplings case Group with positive of (RG)i.e., mass asymptotic flow couplings dimension, with freedom, and vanishing marginally they mass relevant dimension aretum ones, and contribution. the a leading-order The powercounting antiscreening presence relevant quan- of residual interactions at the interacting fixed point the (marginally) relevant couplings, corresponding to those interactions which can Before focusing on the interplay of these gravity-models with matter, we review the Here, we use these ideas to make the first steps to differentiate between quantum- Weyl-squared gravity By Reuter fixed point we mean the class of fixed points associated with a theory space defined in terms 4 are present in anthe asymptotically couplings safe that context, there have to are infinitely be many satisfied relations as between aof consequence the of full diffeomorphism scalegravitational symmetry. universality symmetry. class In as Accordingly, well this as sense, its the deformations Reuter that fixed arise when point one encompasses adds both matter. the purely fixed point [ for a quantum-gravity induced UVpredictive power completion [ for thefixed Standard point Model of with the ansafe. Renormalization enhanced Group (RG) The flow, enhancedpected rendering scale-symmetry to the be controls model present the asymptotically at infinitely the Planck many scale operators from effective that field are theory arguments. ex- While they 2 Introduction to (unimodular) asymptotically safe gravity and There are strong indications for the existence [ est to understand the resultingsion differences of in the relation the of interplay various with symmetry-restrictions matter. on gravity For at a the classical clear level,motivation discus- see for [ them asquantum gravity well below. as their status as potential candidates for a description of gravity-matter models with regardscompare Weyl-squared gravity, to asymptotically safe their gravity based phenomenologicaland on unimodular viability. the asymptotically Reuter safe gravity fixed and We point, sectors. concentrate set on We Standard-Model out focus like matter on to thesemetric degrees three, of as freedom, they but are differ all in the based symmetries on that a are formulation realized. purely Thus in it terms is of of inter- JHEP09(2019)100 ] ]), 102 , 105 – 101 103 ], as well as the ], background in- 70 66 , – 28 63 ]. Moreover, it brings a 96 , 95 ] can now be tackled. Most of these ] that singularity resolution in simple 72 , 100 , 71 63 ]. A further motivation for unimodular gravity – 4 – ], singularity-resolution [ 94 62 ]. These are derived from the full quantum effective 98 , ], providing a basis for systematic approximations of the 97 , 61 95 – ], the relation to a Lorentzian setting [ 57 69 , – ], namely the transverse diffeomorphisms (“TDiff”, the local-volume 45 , 67 , 91 , ]. 43 , 90 59 89 , – ]. Finally, it has been argued [ 32 46 , ] which states that under suitable global causality conditions, the conformal 99 73 , 26 = 0, which motivates the conjecture that the most natural choice for a constant 93 , 56 , k – 23 92 52 Unimodular gravity is attractive due to several reasons: it is actually based on the corrections [ asymptotic-safety inspired models of blackmicroscopic holes fixed-point value requires for a the unimodularWhile cosmological setting, constant the accidentally unless equivalence happens the to to(and General vanish. Relativity extensions at including the higher classical order level operators is undisputed also [ have been investigated [ of the equations ofaction, motion [ where all quantumlonger fluctuations a have “typical”, been largescale integrated energy-scale is out. present in Therefore, theof there setting. integration is in Instead, no this theit setting most should has “natural” be been close shown to explicitly zero in that units the of cosmological the constant Planck scale. is not Indeed, subject to quantum the Hamiltonian is nonvanishingdifferent in perspective the to unimodular theWhile case the fine-tuning-questions observed [ surrounding value of the thedard cosmological cosmological asymptotically constant constant. safe appears gravity, to itstrajectory. be inclusion compatible Instead, with requires in stan- a the unimodular selection setting, of the cosmological a constant very appears specific at RG the level to fluctuate, but explicitly removesconformal-factor the conformal problem factor. in At Euclideanwould the gravity same therefore time, based also this on constitute solvestions the the an of Einstein-Hilbert interesting the action starting gravitationalas and path point the integral for starting [ point semi-classical for considera- the quantization of gravity is present in a canonical setting, where Einstein gravity carries through the presencethe of conformal a non-dynamical factor. degree of Itorem freedom, follows namely [ the spirit ofgeometry the of a Hawking-King-McCarthy-Malament the- spacetime isistic encoded starting in point the for causal quantum relations. gravity which This does suggests not a allow more all minimal- components of the metric e.g., [ symmetry group that follows fromPoincar´egroup [ an analysis of thepreserving massless diffeomorphisms). spin-2-representation of Further, the it gets rid of unnecessary “baggage” that classical relation to other quantum-gravityrelate approaches [ to the internalbefore structure an asymptotically and safe consistencyanother description of crucial of the requirement nature model thatwith can observational a and constraints. be viable need Here, deemed it model toallow viable. appears to be has to rule be out Yet, to answered such possible there a satisfy, to model is make namely based progress on the that phenomenological could consistency consistency in the matter sector, actions. In particular,point there [ are indicationsRG for flow. a Building on near-perturbativepertaining strong nature to indications of for the the the unitaritydependence Reuter fixed of [ fixed theory point, [ challenging questions the model remains predictive despite the existence of infinitely many higher-order inter- JHEP09(2019)100 (2.2) (2.1) to gauge trans- ] for older work 5 ]. We point out that 122 , 107 121 ]. For further discussions ], but most importantly , . with 2 117 125 106 – R 1 3 µναβ 111 + , C , under a Weyl transformation. Thus, 17 2 µν R µναβ µ νκλ g C µν C C √ R x = 2 → to second order, a ghost mode, i.e., a mode Z 2 ] to differ from the Reuter universality class. − 2 C w 1 µ – 5 – C 2 νκλ C 109 ] for a review) is in some sense the exact opposite , = µναβ R 110 WG 108 , S ) to be gauge transformations. This enhanced symmetry µναβ R 103 µν g ], such that it could potentially be a candidate for a UV ) = x 2 ( 2 126 ]. C Ω 120 → – ]. Upon spontaneous breaking of the conformal symmetry, this mode ac- µν 118 g 128 , ] and references therein for more recent work. We will not focus on this ver- 127 124 , being a dimensionless coupling. w 123 Weyl-squared gravity is power-counting renormalizable [ Alternatively, conformal invariance can be achieved in any action by means of the in- It is curious to observe that promoting conformal transformations Weyl-squared gravity (see, e.g., [ Throughout the paper we refer to the local rescaling of the metric by a conformal factor as a conformal 5 spacetime [ quires a mass that might be high enough to render this modetransformation, phenomenologically and irrelevant. caution that thison should flat not spacetime. be confused with the global action of the conformal group as Weyl-squared gravity. also asymptotically freecomplete [ description of gravity.around flat Yet, spacetime, there upon iswith expansion a negative of kinetic major term, propagates, problem signalling that an requires inconsistency of a the solution: theory about flat formations does notinvariant require action. to Nevertheless, one introduce canWeyl of a photon, course new introduce as such gauge part aand of gauge field [ field, the in the gravitational so-called order connection,sion to of see, Weyl-gravity write or e.g., conformal a [ gravity here. gauge- For clarity we refer to the theory we study troduction of extra fields such ason the dilaton, the see, role e.g., [ ofgravity, see, scale e.g., symmetry [ as a key ingredient of a fundamental theory for quantum the most general, local gravitationalexpressed action just invariant in under terms Weyl transformations of which the is metric is given by with in the action is the Weyl-squared invariant The Weyl-tensor transforms according to Further, there are different variants of unimodular gravity. of unimodular gravity: whereas thethe latter former features declares a conformal/Weyl fixed, transformations non-dynamical (i.e.,conformal conformal rescaling factor, factor, of the metricstrongly by a constrains local the viable dynamics. In fact, the only invariant term that can appear diffeomorphism (“Diff”) invariance) is undera debate decisive see, comparison e.g., of [ thecompletion. quantum theories Specifically, actually theory requires spaces someformer knowledge of of does TDiff the not versus UV contain Diffunimodular invariant the asymptotic gravity cosmological differ, safety constant. as [ the Therefore one would actually expect the relation of unimodular quantum gravity and “standard” quantum gravity (with full JHEP09(2019)100 ], 48 (3.1) ] and . This ]. 135 µν – h 131 – 133 ], although it 129 -term, conformal 132 , R 62 2 φ , see, e.g., [ ... + R 2 Pl , ν M κ ]. ]. )] · · 80 → h 142 – R 2 φ 139 [exp( – 6 – µκ . We will mostly focus on the exponential split g µν is a fluctuation field (of arbitrary amplitude). In ] and first used in the context of the functional RG g = ¯ µν µν 137 h g dimensions in [ ] for recent reviews. The key point of the FRG is the introduction  ] which has the form 136 , 51 is the full metric and 138 , , , where we study the parameterization dependence of the results in Weyl-squared µν 49 , g 7 108 , 18 The quantization is performed by integrating over the fluctuation field The functional renormalization group (FRG) has been extensively employed in Invariance under conformal transformations — as expected — precludes the existence Vacuum solutions to the Einstein equations, where the Planck mass drops out since 36 where section gravity, we also explore more general splits [ background field approach allows us to define an IR-suppression term that is added to the should distinguish UV fromintroduction IR of modes a in backgroundintroduced metric a in ¯ 2 local + coarse-grainingin scheme, [ it requires the UV complete matter sector, as discussed, e.g., in [ the asymptotic-safety programsee for [ quantum gravity,of an based IR-cutoff on which the allows us seminal to work capture [ the scale dependence of the dynamics. As it predictive, UV-complete fashion. Tomatter that under end, we coarse-graining explore steps,the the and dynamics change for search of matter, the for providingpoints dynamics scale-invariant a for depend points UV in on completion. thework. the The space existence Accordingly, gravitational the and of couplings microscopic location gravitational which of dynamics these we is treat constrained by as demanding free a parameters in this 3 Setup 3.1 Functional RenormalizationIn Group this for paper quantum we gravity setting. study quantum-gravity matter We systems aim within a at quantum-field discovering theoretic whether the formal path integral can be defined in a symmetry can bevalue, broken thereby spontaneously generating once anreferences the Einstein therein. scalar term acquires Clearly, thefrom a the theory vacuum ones space expectation explored of for Weyl-squared “standard” gravity and differs unimodular significantly quantum gravity. is not clear whetherCMB further fluctuations, evidence can for actually dark be matter, reproduced. coming, e.g., fromof the dimensionful spectrum couplings. of problematic, At as a gravity first at glancetroducing IR this a scales scalar could clearly field appear contains with to a a be conformal mass phenomenologically coupling scale, to the gravity, i.e., Planck a mass. In- evade such unitarity problems in curvature-squared gravity, see, e.g., [ the energy-momentum tensor vanishes, actuallygravity, including turn the Schwarzschild out solution. to Moreover,squared it be has gravity solutions even been might of argued reproduce Weyl-squared that Weyl- the observed galactic rotation curves [ We will not focus on this question in this paper. For recent discussions how to potentially JHEP09(2019)100 ]. 143 (3.4) (3.5) (3.2) (3.3) → ∞ 2 k contains the , k ) has to satisfy µν 2 J ¯ D µν . − ¯ k gh ( . The key advantage k S √ 2 interpolates between a x k ∆ R R k + . − k , S  αβ in that limit, see, e.g., [ ]  ∆ h k − J [ S lower than the momentum scale R k t 2 Z ∂ ghost µναβ ¯ S 1 D ln )] − − 2 − ) higher than −  l ¯ is driven by quantum fluctuations with D k λ of ) comes from modes with momenta close fixing µν k − ). A popular choice is the so-called Litim R l , its scale derivative actually acts as an − ( h 2 2 λ 3.5 k + ¯ D µν R gauge [ − (2) k based on the spectrum of the covariant back- S ¯ ( gJ > k plays the role of an IR cutoff, Γ – 7 – Γ k − µν . Further, it should diverge in the limit ] l √ , with Λ being a UV cutoff, and the full quantum ) are the Faddeev-Popov ghosts. The suppression k  2 µν R λ x Λ  µν ¯ g h h g ] highlight that the use of a different shape function Z S [ √ S < k  ¯ C,C = l − x STr 148 Z λ J 1 2 – Λ Φ is the Hessian and STr denotes the supertrace which ), defined as follows δ 1 2 C e k → = k Φ D Z 146 = k ] = sup ¯ , C /δ Γ k k t µν ). Accordingly, the physical interpretation of the flow equation 20 D S ∂ Γ ¯ g h acts as an IR cutoff scale. The function 2 , (= ln ∆ obeys an exact flow equation of one-loop structure, the Wetterich δ 3.5 k D k µν k : modes with eigenvalue as a modified Legendre transform of the scale-dependent Schwinger W = Γ. As the scale 2 h = Z [ ¯ k 0 k D . This translates the Wilsonian idea of performing the path integral in Γ (2) k → k − ], formally written as ] = k J ,Γ [ ], but, e.g., [ k 151 k – , i.e., the change of the dynamics at Z , which we will define just below, approaches ) vanishes for modes with k∂ k k 2 145 cutoff in eq. ( is an external source and ( , 149 ¯ = D and take a finite value for t µν − 144 ( ∂ J 2 k R are suppressed, i.e., > k l 2 is the following:thought under of as a the changemain “resolution” of contribution scale to of the the right-hand-side the momentum ofto theory eq. scale the ( — scale — the effective intuitivelymomenta dynamics speaking close changes. to to The be where contains a negative signAs for Grassmann-valued fields andultraviolet a factor of 2 for complex fields. effect of modes withof generalized this momentum setting (i.e., is thatequation Γ [ Note that we slightlythe abuse the expectation notation values here, ofthat as are the the integrated fields, arguments over of in whichmicroscopic the the we (bare) path flowing UV denote integral. action action by The are Γ effective flowing the action action same Γ Γ variable as the fields does not qualitatively alterthe the flowing action results Γ forgenerating the functional Reuter fixed point. This allows to define several requirements: in orderλ for it to actsuch as that an Γ IR suppressionThis term, leaves it some has freedom to incutoff vanish the [ for form of It suppresses quantum fluctuationsground in Laplacian k where term takes the form action appearing in the Euclidean generating functional JHEP09(2019)100 ). ), we can extract the anomalous 3.5 3.5 ). Due to the one-loop structure of A ) can be expressed diagrammatically as in 3.5 – 8 – ]. 152 ]. For a recent demonstration of the quantitative reliability of 61 – ) is formally exact, in practice it requires an approximation. In other 57 , 3.5 45 , which should not be confused with Feynman diagrams. , ]. 2 43 , 154 , . Diagrams encoding the quantum-gravity contributions to the anomalous dimensions of and 32 , 1 153 26 Let us add two cautionary remarks: firstly, we stress that our results are obtained Plugging a given truncation into the flow equation ( While eq. ( figures in Euclidean gravity, andLorentzian setting, there as is thee.g., no Wick-rotation [ is straightforward in way general to ill-defined extract in implications quantum gravity, in see, a gravitational context, see [ dimensions and the running of dimensionlessapplying couplings appropriate by taking projection functional rules derivativesthe and (see flow appendix equation, the right-hand side of eq. ( dynamics are those withmensions. positive, vanishing or This only motivatesmost slightly truncations negative higher-order that canonical operators. scaling follow di- see canonical This [ power-counting assumption and is neglect the corroborated FRG by and several the recent fast results, convergence of truncations for interacting fixed points in the non- greatly simplifies practical calculations. words, a truncationder of the the assumption that dynamicsquantum an has asymptotically gravity to safe is fixed be a point used. in near-Gaussian four-dimensional one. Euclidean In this This work, implies we that will the work relevant un- terms in the scalar field, the wigglyfield. line The denotes crossed circle a denotes gauge the field regulator and insertion in the eq. single ( solid linea stands momentum-shell-wise for fashion a into fermionic an equation that is structurally one-loop, a fact that Figure 1 the matter fields. Double lines correspond to the metric propagator, the dashed line represents a JHEP09(2019)100 ). y y as a (3.6) (3.7) D ω = µν g provide indications k , in the unimodular case TDiff . ] is not a physical scale, although g [ k UQG iS e ω , = UQG ) g µν – 9 – g D ( This is not the same as imposing det Z det ], that this could provide an interesting vantage point from 6 = 108 UQG Z -dependence simply because a scaling regime by definition does k denotes a fixed scalar density. . Further diagrams contributing to the running of the Yukawa coupling (denoted as ω We further stress that we explore the dependence of couplings on the RG cutoff scale Let us note, as already hinted at in [ 6 , not on physical momenta. A priori, the cutoff scale which to develop a more background-independentcoarse-graining, flow one equation: only in requires thefixed implementation a background of density definition the provides RG of this, asto a as a derive local local it a is “patch” flow sufficient over equation to which is define to a an local average intriguing volume. fluctuations. open question. Whether The this is sufficient instead of the fullon diffeomorphism group. a definition Formally, unimodular ofover quantum the gravity the functional is measure space based in ofinterested such in metrics a computing satisfying way functional that the integrals the of unimodularity integration the condition. is form performed In this sense, we are where gauge-condition in a path integralthe over all symmetry components group, of which the is metric. transverse The diffeomorphisms, difference lies in 3.2 Functional Renormalization GroupUnimodular for quantum gravity unimodular (UQG) gravity isspace characterized such by that a the restriction metric on the determinant configuration is non-dynamical, more specifically k external momenta can ofcesses. course We act use similarly, that i.e., inis as a expected an scaling to regime, mimick IR the the cutoff,not dependence e.g. feature of in any couplings scattering characteristic onwhether physical pro- physical asymptotic momenta scale. safety Hence, is fixed realized in points a in physical sense. In the Weyl-squared gravity caseWithin the only the non-vanishing diagram standard isdepicted asymptotically the above tadpole safe may in lead quantum the to first gravity non-trivial column. (ASQG) contributions (depending framework on all the the gauge choice). diagrams Figure 2 In the unimodular gravity framework only the triangle diagrams in the last column are non-zero. JHEP09(2019)100 B ) = (3.9) (3.8) µν g -term for 4 ) F ¯ ψψ . The fermion φ i yφ + → − ]. ψ gh. k φ / ∇ ¯ ψ . 155 i , + Γ ] h ψ ; g Z g.f. k [¯ ( k 108 , S ω ∆ + Γ √ from the path integral. Therefore, − x
] 107 g Z , [ µν µν h + R 103
UQG µν ) µν S g 2 − φ e = ¯ ( b R ¯ h V + . 2 UQG – 10 – +2 ) αβ φ g a R ν F D ( ], is generically nonzero at an interacting fixed point + ¯ µν φ∂ µ F Z 80 R ∂ νβ − = ]. More specifically, derivative interactions are induced g µν g , k ω µα φ 159 Z √ , under which the scalar transforms as – 5 ω g x UQG γ Z 2 ω √ Z x 156 √ N Z , iπ/ x Z A ¯ 1 4 ψe π G 80 1 2 Z , 16 → + + 78 ¯ ψ , = , ). Since the exponential parameterization allows us to express det( ψ ). As a consequence, the same condition on the full metric can be achieved by 74 UG k , one can impose the unimodularity condition on the background metric (namely 5 , ω γ h Γ 2 3.1 e = 61 ) , iπ/ e µν µν 56 We therefore assume that at this leading order, induced higher-order matter interac- In setting up this truncation, we follow a canonical power-counting scheme, as we In this work we employ a truncation containing matter fields (scalar, vector and spinor g g → (1). Thus, the canonically least irrelevant operators might be shifted into relevance tions and non-minimal couplingsglobal can symmetries as be discussed neglected.of in [ the A gravity-matter subset system,ity of as those, [ has selected been by studiedin their for the standard presence asymptotically of safe gravitational grav- interactions. For example, this includes a at the interactingoperators fixed of points, mass but dimensionorder this beyond understanding ordering 6 of principle becomecontribution makes gravity-matter relevant. to it systems the Within scale unlikely is dependence thisscalar of that, based scheme, potential). gauge e.g., couplings on a and leading- the Yukawa sectors (as direct well quantum-gravity as the for further details). expect most canonically irrelevantthe (i.e., interacting higher-order) fixed interactions point to sinceO also we be expect irrelevant that at anomalous scaling dimensions are roughly An explicit mass termψ for fermions isis incompatible coupled with to the gravitywith by discrete vanishing means “chiral” torsion, of symmetry can the be vielbein expressed and in the terms spin-connection, of which, the for fluctuation spaces field (see appendix fields) minimally coupled to gravitypatible in the with unimodular the setting symmetries includingsponding with all couplings. operators positive Our com- or truncation vanishing is canonical given dimension by of the corre- see eq. ( det(¯ det ¯ setting the trace mode to zero,the i.e., functional removing measureperformed for over UQG the space can of be traceless fluctuations defined [ in such a way that the integration is version of the path-integral, We apply the background field method by means of the exponential parameterization, The application of functional RG techniques makes it necessary to focus on the Euclidean JHEP09(2019)100 are ζ group (3.12) (3.13) (3.10) (3.11) in order degree of ] remains and µ σ 2 ξ α / TDiff ] carries over 159 1 is required to – , − 80  2 for the Yukawa ]. Furthermore, h ν ) µ ν φ ¯ ]. We expect this 157 ∇ ¯ ∇ , µ 162 A β µ µν d φ∂ ¯ ¯ 80 R ∇ 1+ 158 , σ. µ , µ 2 µν − 61 ¯ ¯ ∇ 61 ∇ ψ∂ g 1 (¯ / µβ ∇ d µν − . ω ¯ h ¯ d g 2 ψ β 1 d √ + ¯ b k ¯ ∇ x 2 − Z ) = . 2 ] of the fluctuation field (note the σ ζ ] = 1 ν ¯ 2 ∇ h ¯ [ ∇ 161 − µ , b µ ( φ/ψ/A ] + 2 F  ¯ ∇ h 0 which further removes the Z [ a k + T ν ln 7→ → t µ F = ¯ ] ξ ∂ α σ ν h [ − ] ¯ – 11 – ∇ T µ = , a F + d 108 ν − µν 2 ξ defines a transverse gauge fixing condition, with ¯ g µ k ω φ/ψ/A ¯ ∇ N η αβ √ G + h x β Z = ¯ being the transverse projector. The parameters ∇ TT µν N h G ν να ¯ ¯ g ∇ = 1 1 πα G − ,µν µν ) T 2 h ] and interactions of the schematic form 32 group. For a discussion of BRST quantization of unimodular gravity P ¯ ∇ ]. = ( µ 159 -part of the Jacobian in the generating functional arising from the York ] for a study of this in standard gravity. This holds, as long as gravity is , ¯ 109 Diff ], as well as non-minimal derivative interactions [ g.f. k ∇ σ – Γ 56 ] = − ]. h 159 [ , 157 , µν 107 T µ , g F 80 80 160 , , = ¯ 103 56 99 In addition, we perform a York decomposition [ The explicit form of the ghost sector will not be relevant for the analysis per- The gauge-fixing part is given by [ We work with dimensionless couplings which are given by ,µν T It is convenient to adoptfreedom the such that Landau it gauge does not contribute towe the employ flow of the matter couplings non-localto [ cancel field the redefinition decomposition. e.g., [ absence of the trace mode due to the unimodularity condition), account for the factinstead that of the symmetry the underlyingsee UQG [ corresponds to the formed in this paper. For further details on the gauge fixing procedure for UQG see, where P gauge parameters for thetransverse gravitational part of and the Abelian usual sectors, gauge fixing respectively. condition The use of the Further, we introduce the anomalous dimensions point. In our context itsub-leading is compared important that to these the higher-order directsee interactions are quantum-gravity [ expected contributions to that be wesufficiently calculate weakly here, coupled such thatat the real induced values. fixed pointsoperators. This in [ motivates our choice of truncation which neglects such higher-order sector [ property to persist in thethe case TT of propagator unimodular at gravity. vanishing Given cosmologicaldard that constant gravity around is in a the the flat same linear in background, parameterization, unimodularto the as the in TT-approximation unimodular of, stan- case e.g., and [ supports the presence of higher-order interactions at the fixed gauge fields [ JHEP09(2019)100 . k (3.16) (3.15) ] for a (3.14a) (3.14b) 166 -modes are, σ to finite order in 0 or a tachyon if , k ,  2 . On the other hand, 4 b < ¯ 0 p # . ) → b d ¯ k b d ¯ b d of the flowing action within 2 1)+ 0. We stress that even for a − − − d 2 d ( 1) + 1) 1 → a − = 0, a Taylor expansion to finite 4 ¯ − k − d . Let us stress that the presence of 2 d 1 0 d p − ( ( , → 2 a truncation a k p 4 ¯ 4  ]. σ = µναβ − TT Z 2 2 P 165 p –
1) − 4 d − – 12 – ( b p ¯ 163 d , 2 -sectors, namely + d σ 1 ). The propagators for the TT- and 2 2)( and ). The first point to be emphasized is the appearance p 130 , − 3.9 . Analyzing the full propagator is beyond the scope of d k C.1 and ( TT 129 1 Z − − b ¯ 1 − ) = ) = # 2 2 = p p ( ( 2 1 p + 1) σσ b G µναβ TT ( defined by eq. ( G σ ) which only features a simple zero at corresponds to either a massive ghost-like particle if 2 Z p 1 ( − b ¯ (2) and − TT = Z 0, therefore leading to unitarity or causality problems. Recent proposals on how to The above propagators results in the presence of the following contributions to the From the FRG perspective, on the other hand, the association of such poles with Let us briefly discuss the structure of the (flat) propagator obtained from the uni- 2 p this work. running of matter couplings and anomalous dimensions function Γ order generically features additionalthe zeros. derivative expansion Therefore are truncationsTherefore not of no suitable Γ automatic to conclusion can addressorder be the terms drawn question in on of a unitarity from instability/unitarity. truncation the for presence Γ of higher- QCD, where higher-order terms areMoreover, certainly already present, but the do analysispedagogical not review, of signal any is classical inconsistency. based instabilitiesIn on the by case the of Ostrogradsky, asymptotic assumption safety see of onethat expects a [ enter infinitely finite many the higher-order number full terms to of propagator be higher-order of present terms. the physical theory at a derivative expansion. Further, weAccordingly, are it only is focusing not onfull clear the non-perturbative whether fixed-point result regime such for at higher-order thehigher-order large effective terms terms action will in Γ the be effective present actionunitarity. or does This not not automatically becomes in result obvious the in instabilities/non- by considering the effective action in cases like QED or avoid such ghosts include, e.g., [ ghosts/tachyons cannot directly beshould be made. analyzed at Thethe the propagators presence level shown above of of were the obtained instabilities within full a effective (non-unitarity) action, i.e., Γ In the framework of perturbativepoles curvature-squared quantum is gravity, problematic. the existence In of particular,at such according to the usual perturbativeb treatment, > the pole with of massive poles in both the TT- and modular truncation given byrespectively, given eq. by ( ¯ JHEP09(2019)100 ] and (3.18) (3.17) (3.19) 6. As ]. / 169 – 170 = 1 , . , 167 φ 2 ) gh. k 141 ν A χ/Z µ + Γ ¯ . = ∇ g.f. k . The first term is a µν χ h g = 0 µ (¯ , see, e.g., [ + Γ b ¯ ¯ g ∇ ]. As a phenomenologically  b d β + √ 4 R x − 1+ 124 ¯ and 2 ψψ Z . φ 1) a − A ζ w 2 χ 2 iy φ Z − µν 0, the graviton propagator becomes + d + h + ( 4 ν 2 → ψ a φ ¯ ∇ 4 / 4 . As in the unimodular case, the ghost ∇ α 4! 0 λ ¯ ¯ g h ψ − µ i √ + 2 ψ ] = x h Z φ Z − [ ν 0 and ˜ and µ – 13 – + d 4 0 α F µ 2˜ β φ ∂ → αβ µ , α ∂ F 2 ] + γ and µν µν h [ , g ν F ν 1 φ ¯ γ F ∇ 2 νβ Z µ g µν ¯ ∇ + µα Y 2 ] 2 g γ h [ C A 4 µ + w Z 1 2 2 ¯ + 1 = 0 and g F + ¯ ∇ , namely  b √ b g µν x ¯ g √ Z 1 x and γ α 1 Z . In the Landau gauge, 2 a 1. The gravitational parameter space in our truncation therefore features two 0 = = µ = ≥ 2 µν , g.f. k WG k 1 Y Γ Γ For our explicit calculation we consider the following gauge-fixing sector As an alternative to the above, one can introduce a Weyl gauge field that allows An interesting point about a conformal gravity description of the fixed-point regime is gauge-fixing term for theparameter Weyl symmetry. Inindependent this of case, the we parameters sector introduce is an not relevant arbitrary forthe mass the gauge-fixing computations procedure performed in in the this framework paper. of For Weyl-squared further gravity, see details [ on where gauge-fixing for the diffeomorphisms.order We to have avoid chosen the a higher-derivative introduction gauge-fixing of in a mass scale in this sector. The second term is the coupling of the non-minimal term canimportant be consequence, arbitrary, this see, allows e.g., to [ absorbof the dilaton spontaneous that scale arises as symmetrymassive the vector, Goldstone breaking boson thereby in evading fifth-force the constraints. longitudinal mode of the corresponding Weyl symmetry requires thewe renormalized will only non-minimal explore couplingpotential the are ˜ Yukawa actually and irrelevant gauge for us. sector, the non-minimalto coupling render and the scalar kinetic term for the scalar field Weyl invariant on its own, such that the teractions, the viabilityparameter of space, a which here UV is completion spanned by for the the coupling that matter if we sector restrict ourselves to inof the the subspace Weyl-invariant of operators. gravitational local terms, Underfor then there this Weyl-squared gravity is assumption, (WG) only a coupled the finite to most number a general fermion, a (local) scalar truncation and a gauge field is given by references therein, do notnot lead provide to any guidance significant as restrictions to on which the side3.3 parameter of space, the poles and Functional is do Renormalization of GroupWe phenomenological explore, for interest. taking Weyl-squared into gravity account Weyl-squared gravity along with Standard-Model-like in- pole lines in The flow cannot cross theseparameter pole space.) lines (or The the current corresponding generalizations experimental in bounds an on enlarged with # JHEP09(2019)100 C (3.21) (3.22) (3.20) Hence, the ) do not con- 7 h 0 entails an in- h. → and ]. α µν σ ¯ g , 1 4 172 µ , ξ , ]. + ). The other relevant ob- 0 and ˜ 2 171 σ 112 p 2 , ( µναβ . → TT ¯ k ∇ P 111 α R ] µν 4 ¯ g µναβ p TT 1 4 P − 4 − 2 p ) σ 2 ν p TT ¯ ( ∇ k Z µ P ¯ [ ∇ = ] of the fluctuation field + ], are all dimension-5 or higher operators, which TT – 14 – µ Z ξ 161 µναβ 159 ν i = – ¯ ∇ TT h + 156 , TT µναβ ν .) ξ )] w µ (2) k,h 2 80 , 2 p ¯ Γ ∇ ( h √ ), for a generic shape function 78 + 2 , TT k p → ( R TT µν [ 74 k κ , h R = 61 , + 2 µν 56 p h ) = 2 p ( k P Results obtained in the last few years indicate that asymptotically safe quantum- We also highlight that higher-order matter interactions, as induced by gravity without Although the model exhibits Weyl invariance, the introduction of the regulator function We perform a York decomposition [ like theories This can be dealt with in principle by introducing a dilaton, see [ 7 these couplings. In thisthis work, problem we by explore inducing the a question (near-) whether perturbative quantum gravity ultraviolet can completion. gravity solve effects might induce a UV completion of the Standard Model and might even allow The Abelian hypercharge sectorhibit and Landau the poles Higgs-Yukawa sector inincomplete. of perturbation the theory, This most Standard is likely Model a rendering ex- consequence the of Standard the Model fact UV that the free fixed point is IR attractive in Weyl symmetry [ introduce explicit mass-terms andthe are flow-equation accordingly setup, incompatible they with areto presumably Weyl modified symmetry. present Ward-identities. as In a cutoff-artifact, and also subject 4 Quantum-gravity induced ultraviolet completion of Standard-Model flow generates couplingsappropriate which modified break Ward the identities.and Weyl we symmetry This restrict and falls theanalogous which outside ansatz to the are for that scope controlled of the other of by flowing local the action symmetries, present to reviewed, paper e.g., be in Weyl [ invariant. The situation is as in the UQG(with case. the These replacement can be obtained from the expressionsin reported the in FRG appendix frameworkdependent breaks mass this term symmetry. and Intuitively, therefore the is regulator is not a invariant under momentum- Weyl rescalings. where jects for computations withinassociated the with Weyl-squared the gravity matter framework, sector such and as the the gravity-matter Hessians vertices, are basically the same taking into account only the TT-mode. The Hessian in the gravitational sector is given by The regulator associated with the TT sector is In terms of York decomposedteresting variables, simplification. the In Landau this gauge case,tribute the to vector the and running scalar of sectors the ( matter couplings. Therefore, all computations can be done by JHEP09(2019)100 (4.2) (4.1) ]. ]. On the other 174 , 175 173 , 87 , ] and as follows from the is the same for all Yukawa y 83 80 , , f 0. In fact, such a linear term A η 82 > , y f 77 which in turn follows from a screen- , 8 ..., 53 + , . i 2 0 and g g 36 i > A ]. Here, we extend this study to unimodular g η f g f − 82 = , – 15 – 2 = g 73 , β grav 55 | – i g 52 β (e.g., gauge and Yukawa), takes the form i g 0 is analogous to an effective dimensional reduction (though > i g f is the same for all gauge couplings and g f 6= 4 spacetime dimensions, where gauge and Yukawa interactions are , is related to the anomalous dimension d g ]. This is a consequence of the fact that a Higgs mass of about 125 GeV is 52 is a function of the gravitational couplings. As gravity is “blind” to internal i g f In the following, we therefore focus on evaluating the leading-order quantum gravity For the Higgs self-coupling, the situation is slightly different: a screening quantum- The quantum-gravity contribution is generically linear in Standard-Model-like matter The exact value features a delicate dependence on the mass of the top quark [ 8 The quantum-gravity contribution to thepling, running denoted as of a minimal (non-)Abelian gauge cou- contribution (i.e., linear(non-)Abelian in gauge the coupling respective and a matter Yukawa coupling. couplings) to the4.1 beta functions (Non-)Abelian of gauge a couplings connected to a near-vanishing Higgs quarticing coupling, quantum-gravity contribution, ashand, found an in antiscreening [ contribution wouldrameter result of in the the theory, such Higgs that self-coupling the being model would a also free be pa- UV complete, albeit less predictive. not marginal. The case not necessarily to an integer spacetime dimension). gravity contribution actually results inmental a value [ prediction of the Higgs mass close to the experi- a dependence of thescalar direct sector. quantum-gravity Thus, contribution oncouplings. the To internal induce symmetries an ultraviolet of completioncontribution a for needs gauge to and Yukawa be couplings, the antiscreening,is gravity i.e., also present in where symmetries, this directcouplings quantum-gravity is contribution independent to ofthe the the direct scale-dependence quantum-gravity gauge contribution of group. to the gauge Similarly, scale-dependence there of Yukawa is couplings, nor no flavor-dependence of couplings, in accordance withcorresponding symmetry diagrammatic considerations, expressions. see Therefore,the [ the running quantum-gravity matter contribution couplings to Standard Model without gravity [ as well as Weyl-squared gravity. For thecoupling latter, the suggests dimensionless that nature of the theimplementation gravitational leading-order of contribution this is calculation universal.one-loop universality in Our by the comparison study FRG with is previous framework, the perturbative providing first results. an explicit test of to predict (or retrodict) the values of several couplings which are free parameters in the JHEP09(2019)100 g ). 0, > 4.3 (4.3) g f ) and the : 4.3 3 ]. If the latter generates an upper 81 , ∗ g 0 in eq. ( 73 , < , indicating a single UV g g 55 f = 0 is IR repulsive. This , 0, corresponding to g 54 < . grav | A η grav . Due to its IR attractive nature it | ∗ A g η + ). The situation may potentially change if the 2 – 16 – 0, this situation does not change once gravity matter π | A (6 < η / 2 is positive for Abelian gauge fields due to the screening = g grav A | = η 0, such that a non-Abelian sector is already UV complete A η matter < : no value above the green line in the right panel can be reached | g A matter η | A matter η | A IR-values of the gauge coupling along UV-complete RG trajectories, that follows from any UV complete trajectory. Larger IR-values of η g finite . We illustrate the beta function (left panel) arising from cannot be crossed. ∗ g an asymptotically safe Abelian gauge coupling with uniquely calculable IR value. an asymptotically free Abelian gauge coupling; For non-Abelian theories with an appropriate matter content, as, e.g., in the Standard This fixed-point structure follows from a competition of the two terms in eq. ( 2. 1. that is reached along the trajectory emanating from the interacting fixed point is the cannot be reached startingfrom from the free fixed point, as the criticalModel, trajectory it holds emanating that without gravity. As long as thereby solving the trivialityantiscreening problem. quantum-gravity contribution and At thein the screening an same matter IR contribution time, results generates attractive the fixed an competition point upper between at the g bound a on finite IR value largest values IR-value of of the gauge coupling: the unique IR value of Due to the quantum-gravityallows contribution, to the reach fixed point at quantum-gravity part is takenadmits into a account region as in discussed thethen space in two of possibilities [ parameters for such a that UV completion are generated, illustrated in figure The first contribution, i.e., impact of charged matter fieldsrepulsive and free starts fixed at quadratic pointsingle order charged in in fermion, the absence of gravity. For instance, in the presence of a on a UV complete trajectory. It receives contributionsquantum-gravity from fluctuations, namely the interaction with other matter fields and also from Figure 3 corresponding RG trajectories (rightbound panel). on The low-energy interaction values fixed of point at JHEP09(2019)100 (4.5) (4.4) 0, the > y ]. , where we f y 81 D , 3 y (1) β for unimodular gravity + g y f y . f 0 − and < = y y 3 f y D ). Thus, the antiscreening nature of y (1) + ( β O + grav | y ψ η  ] for a discussion of such terms. y + D – 17 – 157 , + grav 80 | 0 could result in a UV completion, see [ φ η grav | < 1 2 ψ η = grav + | y f A the structure of the Yukawa beta function is given by ]. Both contribute at η − grav 9 | 80 φ η 1 2  = 0. It also includes the matter contributions to the anomalous dimensions. y β > is the universal (i.e., RG-scheme independent) one-loop contribution from mat- (1) β (1) ]. We will in particular interpret our results in the context of unimodular asymptotic β For Weyl-squared gravity, the absence of mass-scales in theory implies that the one- Our goal in this paper is the first estimation of The resulting fixed-point structure is as in the Abelian gauge sector: for By “leading contribution” we mean that we are not considering contributions coming from induced 176 9 setup. We will thereforeof explore the the beta regulator-,reproduces gauge- functions universal and in results parameterization-dependence the also in Weyl-squared a case gravitational to setting. confirm thatfermion-scalar-fermion interactions, the see refs. FRG, [ as expected, in [ safety, whereas the perturbative literaturefor gravity. does not assume the existence of aloop fixed results point must be universal. It is one of our goals to explicitly test this within the FRG by the unique valuefixed that point. is reached along the trajectory emanatingand from Weyl-squared the gravity interacting fromgravitational contribution functional to RG Yukawa systems techniques. in the For perturbative setting unimodular has gravity, been the studied free fixed point isto IR a competition repulsive, between rendering antiscreening thefluctuations, quantum-gravity fluctuations an Yukawa and coupling IR screening asymptotically matter attractive free.from interacting large fixed values Due of point the exists. Yukawa couplingfixed in It the point IR, shields can i.e., the trajectories only free emanating reach from fixed a the point free finite range of IR values. That range is bounded from above the gravitational contribution can be characterized by the following inequality where ter, with We distinguish between thethe gravitational contribution direct to gravitational the contributionadopt anomalous to the dimensions, the notation and flow from [ of the vertex, encoded in may induce a UVgravitational completion contribution, for the Yukawa coupling. In fact, considering the leading asymptotic freedom is lost, 4.2 Yukawa terms Quantum-gravity fluctuations could also playwe an have important mentioned role before, for an antiscreening the contribution Yukawa sector. coming from As graviton fluctuations is included. For grand unified theories with a large number of matter fields, such that JHEP09(2019)100 0. (5.5) (5.1) (5.2) (5.3) (5.4) 0 are , the G > G > y where the f b ). Within our 0 and and 4.2 a > , g f  ) , and , b  , and explore for which  2 ) 82 ) ) , satisfying this condition. 2 b 4.1 b b ) 2 7 a . − b ) 2 0 b 11 2 − a 0) G, a, b 2 − < a − − a − ) a 246 a 6 21 b a 6 G > 2 6 33 − ) − − 14 b − − 2 − (1 − (see sections (1 − 4 (5 (1 a 2 (31 y 4(5 0. This results in the following constraint a . 0. As all quantities are linear in − D 42 6 − 2 < + b 2, the viable region can be approximated by ) ) ) − − b > / ) b 2 b 13 2 b ) 2 ) and 2 ), we find the following results b grav (1 , as long as gravity is attractive, i.e., – 18 – ) b −
= 1 2 (9 − b y/g ψ G y a 3.9 f b η a D + 6 , (1 + 39 + (1 + ) A + 5 (10 + 7 (1 + b − a 25 (2 + 3 η 2 −  φ 25 (2 + 3 )  , η 5 b (1 π  φ π 1 2 G η π π 90 G G G + 160 20 40 − (1 + ψ 75 (2 + 3 η = = = = = grav grav grav grav y | | | | , with only a sub-leading dependence on y f b φ ψ A η η η does not depend on D − g f and we plot the region where this inequality holds. A gravitational fixed point in that y 4 f 7. . 0 In the Yukawa sector, the viability condition for a quantum-gravity induced UV com- − . In figure region would generate an antiscreeninga contribution to region the at Yukawa beta negative function.Except for There is the vicinity ofb the pole 3 pletion is given by on the space of curvature-squared couplings (assuming it is not meaningful toof combine a regulator determination function, of thedifferent with boundary choice. a within one Of determination specific course,at of choice such least gravitational up unphysical fixed-point to dependences the values must systematic cancel within uncertainties in arising a physical within results, an approximation. conditions required for a UV completionsatisfied of within Yukawa and our gauge truncation. sectors, The shapeadditionally and depend location on of higher-order the couplings boundarya neglected of truncation, these in regions non-universality our can of truncation. gravitational Further,that corrections within the to shape matter and beta location functions of implies the boundary depends on the regulator function. Therefore, In the following, wevalues assume of a these gravitational couplingssign fixed-point it of in results in Accordingly, these results determine regions in the space of parameters truncation for unimodular gravity, eq. ( 5.1 Higgs-Yukawa and (non-)Abelian gaugeIn sector this in section, we unimodular explore gravity Yukawa and where gauge in sectors the in gravitational the parameterquantities framework space of for a unimodular UV this gravity completion is analysis for possible. are The relevant 5 Results for unimodular gravity JHEP09(2019)100 a = 0. y < 0) is (5.6) f y − occurs, < f 0 can be y β > grav | y A f η = 0) does not b = 2. In contrast to = 0 (TT-mode) and / g b = f − a = 1 b . This result can be made + , a 2 µν enters the propagator of the 0 dominates, and yields R > y 2 µν β ) = 0) belongs to the viable region R 2 b ) 7 b b 2 − = a − a a 21 6 − − (1 – 19 – 4 (5 − ) b 2 ) b (1 + 5 (10 + 7 0, is satisfied. The dashed lines indicate the poles 1 + is mostly dictated by the coefficient of < ) is avoided. In the present case, the viable region can be approximated A -mode). The black dot marks the Einstein-Hilbert truncation. η σ ] for a comprehensive discussion. In the unimodular case, the cosmological grav 5.6
), the transverse traceless contribution to y 80 b D 4, except in the neighborhood of the pole line 3 . Similarly to what happens in the Yukawa sector, the sign of the gravitational . = 0 ( 1 + . The red region corresponds to the sub-space of higher curvature parameters, b 5 2 φ − η − 1 2 = 0 = , where the viability condition for a UV completion of the Yukawa sector, i.e., & a b The viable region for a UV complete Abelian gauge coupling ( The point corresponding to the Einstein-Hilbert truncation ( + only appears in the scalar mode. This argument holds in those regions of parameter a 6 b ψ 2 − η R space where an enhancement ofthe scalar inequality contributions ( due toby the nontrivial denominators in the Yukawa sector, the Einstein-Hilbert point ( see figure contribution to plausible by atransverse counting traceless argument: graviton (which the losely speaking coefficient counts of like 5 scalars), whereas that of of higher-order terms opens upcontributions a can larger change parameter from space, screening where to the antiscreening. nature of gravitational characterized by such that the transverse tracelessrealized, contribution see is [ actually subdominantconstant and no longer appearsexpected in to the bevanishing metric similar cosmological propagators. constant to (of course, those Accordingly, the correspondence in the is the results not exact). can linear The be parameterization inclusion for standard gravity at belong to theresult viable in region. the standardcrucial One in gravity might the case: have absencefor of there, expected higher-order the this couplings: presence resultAt at of from sufficiently vanishing the negative cosmological the cosmological cosmological constant analogous constant, (and constant a is reweighing of contributions to and 1 Figure 4 JHEP09(2019)100 = 0 b 0 holds. In -modes, i.e., > 2, the viable σ / g f = 1 0, is satisfied. The b -mode). < σ + a grav | A = 0 ( η b 2 = − g f a 0 can be achieved. The blue region 6 − > − y f 7. This approximated behavior can be . 0 − . 0 holds at vanishing cosmological constant, see, – 20 – b > shows the viable region in the TT-approximation, . g 7 4 f . 1 0 both hold. The dashed lines indicate the poles 1 + = 0 (TT-mode) and 1 − b -mode). 0. Far away from the pole line 3 > σ y > f y = 0 ( f b 2 0 and − a > 0 and 6 g f − > ]. we present the combined constraints on the gravitational parameter space g f 6 177 , . The blue region indicates where the inequality . Combined plot showing the regions where the quantum gravitational contribution to 84 , 54 In figure region can be approximatedunderstood by in terms offor short, a as dominance discussed ofwhich above. is the Figure obtained transverse by traceless neglectingby mode, contributions neglecting from quantum “TT-dominance” diagrams fluctuations containing of the the scalar mode. for a UV completionin of the the standard-gravity gauge case, sector.e.g., where [ Again, this can be plausibilized by thearising results from the sub-space of higher-curvature(with couplings horizontal where lines) only indicatesthe values green of higher-curvature region, couplings(TT-mode) where and only 1 Figure 6 gauge and Yukawa interactions is antiscreening. The red region (with vertical lines) corresponds to dashed lines indicate the poles 1 + Figure 5 JHEP09(2019)100 ]. by y 178 (5.7) (5.8) β 0. The larger , only IR-values > g f = 0. y/g b f and y f 0 (blue, right panel). In both . > 2 Y g f y g . 17 12 2 2 π 3 Y π g 1 16 16 6 41 − 3 + y Y 2 g π 9 g – 21 – f 32 0 (red, left panel) and − + > = y y y Y f f g β − = y ]. Therefore, given a set of values for β 54 , 53 ). It arises from the competition of the screening one-loop term in the 4.4 up to an upper bound can be reached. To illustrate this, we supplement y , the larger the corresponding fixed-point value. This fixed point is necessarily g . We show the regions where f ) and ( and 4.2 So far, we have focused on exploring where in the gravitational parameter space a Y and g y contribution, obtaining This set of beta functions features the following fixed points (we neglect fixed points at Similarly, we supplement thecoupling one-loop in term the in Standard the Model beta (i.e., function including of all the charged Abelian fermions) gauge with the gravity the one-loop contribution for aWe single Yukawa neglect coupling from all the terms Standardfields, Model, related and see only to [ keep the thean other Abelian up-type hypercharge Yukawa flavor contribution, couplings (top, putting and charm, in up), the the hypercharge non-Abelian for gauge beta functions with thef antiscreening gravity term in theIR case attractive. where Accordingly, itfree acts fixed as point, an see upper [ of bound for trajectories emanating from the the thus-defined viable regionmeasurements is of not IR automaticallythere values phenomenologically was of viable, only the given the correspondingby the free the couplings fixed RG point, for flow. thenand the However, Yukawa any for coupling, Standard IR there marginally Model. value is irrelevanteq. of couplings, automatically ( the If a such as couplings second fixed the could point Abelian be of gauge reached the beta functions in differs from the full result. The horizontal dashed line indicates the pole 1 + UV completion of an Abelian gauge sector and a Yukawa sector could be possible. Yet, Figure 7 panels, the gray strip (with vertical dot-dashed lines) show the region were the TT-approximation JHEP09(2019)100 0, > (5.9) (5.11) (5.12) (5.10) y is given f y correspond ], involving Y g 179 below the Planck y and f y , in the IR is tied to a . ) is chosen as the UV g Y f and Y g 2 g g . 5.12 π f g f . 41 16 41 6 · 8 and 6 r y , π , r ), where both = = 0 = 4 = 0 ∗ ∗ ∗ ∗ = 0, where the fixed-point value of Y Y Y Y 5.12 a , g y – 22 – f plane for is given by 0.1 (green, dotted), 0.3 (green, dot-dashed), G Y , + 82 41 g b , g g y f -contours holds for finite values of f y 2 17 ]. The gravitational action was taken to be the (leading) π 9 r 32 depend on gravitational couplings and are proportional to the π ,, g g 3 180 y 4 r , For the particular choice of scheme adopted in [ f start to run as well. Their low-energy values are uniquely fixed = 0 = 0 = = 10 ∗ ∗ ∗ ∗ Y 179 , g y y y y and g must stay at their fixed-point values if eq. ( f 176 and Y g y and ), i.e., the values for the . We show contour lines in the y 5.12 0, Gravitational contributions to the beta functions of Yukawa and quartic scalar cou- The most predictive fixed point is given in ( The unimodularity condition is imposed differently from the procedure we adopt in this work. It is > 10 g plings were computedtive in field the theory context [ Einstein-Hilbert of term. unimodular gravity in the frameworkunclear of if effec- such different prescriptions lead to inequivalent theories at the quantum level. scale. There, by the initial conditionvalues. at the Therefore, Planck a scale,hypersurface specific where in value the they for gravitational have parameter the to space, prediction assume cf. their of figure fixed-point to IR attractive directions.value For an can IR set repulsive direction, indirection). a at deviation This any from is the scale notf fixed-point the (corresponding case to for thefixed an point. free IR Since attractive parameter direction. linkedNewton coupling, to In a the a realistic presence flow relevant will of exhibit a sharp decrease of negative values) by 0.1 (cyan, dotted),contours 0.3 where (cyan, the dot-dashed), fixed-point 0.5 value0.5 of (cyan, (green, dashed) dashed) and andpoint 0.7 0.7 ( (cyan, (green, continuous), continuous). and Both are evaluated at the fully-interacting fixed Figure 8 JHEP09(2019)100 0. > (5.13) (5.14) ]. This grav | 175 φ , η in the action 87 , gV ). For a potential √ 83 , 5.1 82 , . , n 77 , n i ...i 2 n 1 52 i λ . . . φ = 0 [ 1 grav i

2 1 ∗ φ φ V η n ) n ...i i 1 i λ + ] at the level of unphysical quantities, such n ... ,...,i 180 X = 0, which is guaranteed to exist in the absence – 23 – , 1 + i n 1 i ...i 179 ] = , comes through the gravitational contribution to the 1 , i n = ( V λ 176 grav

, . . . , φ n 1 ...i φ 1 [ i λ V β i.e., quantum-gravity fluctuations tend to flatten scalar potentials. In unimodular scalar fields that is Taylor expanded around the origin in terms of the couplings ] for a discussion of this point). The quantum-gravity effects on scalar potentials in , i.e., 11 n 36 . Therefore the fixed point at n φ Note that this can change in the presence of finite fixed-point values for Yukawa and/or gauge couplings, ...i η 1 11 i involved systems despite thenomenological presence consequences, of such additional as fields, a this prediction of could the have Higgswhere potential a mass phe- non-trivial fixed-point in potential thescalar is generated vicinity self-interactions by of like the same the the some terms Higgs in scale. the quartic beta coupling function that in regenerate the Standard Model, even if it is set to zero at bution to the anomalous dimensionof is internal the symmetries. same In for theto each phase of of the unbroken symmetry, scalar thereof fields, is independently explicit no scalar shift-symmetry-breaking contribution Under contributions, the is assumption that infrared this attractive fixed-point structure for is realized in the corresponding more there is a gravitational contribution to the beta functions of the form This expression highlights the “flavor”-independence of gravity: the gravitational contri- and it is ofwith interest to compare theλ two. Our result is given by eq. ( anomalous dimension. This isa very different particular from choice the of standardfrom parameterization gravity a case, and direct where, gauge gravitational unless tadpole is(see [ contribution, made, arising the from main theunimodular contribution term gravity are comes therefore potentially rather different from those in standard gravity implies that (with theall possible terms exception in of the potential thetions, are mass driven term, to which zerogravity, may under the remain the only relevant), impact direct ofmore quantum-gravity gravitational generally fluctua- any contribution scalar to potential the flow of the Higgs potential, or cancel (at least up tocould the become accuracy achievable meaningful. in a given approximations), and comparisons 5.2 The HiggsIn potential standard in gravity, unimodular the gravitational gravity tential contribution is to towards the irrelevance beta at functional the of free the fixed scalar point, po- the beta functions ofas the in Yukawa and the scalar Standardcontributions quartic Model starting couplings. at beyond one two Weresults loop. loops, emphasize to that, Accordingly non-universality agree one just is with shouldas, present those not from in e.g., expect [ gravitational beta our functions. functional RG At the level of physical observables, scheme-dependences must a non-multiplicative field redefinition, a vanishing gravitational correction is obtained for JHEP09(2019)100 0 0, 0, 12 > > > y (5.15) φ g f η f 0, > y 0, necessary for f y, 0. Accordingly, as > , and the effect can TT y | f 9 φ η G > 3 4 = 0 and > G y g 2 f 4, since both are only generated ) ]. ) b / b 182 TT | φ (1 + η 0 holds as long as π 0, as the scalar quartic coupling is then asymp- 15(2 + 3 = > 32 < 0 in cyan as a function of the higher-derivative = TT | TT grav . > | y | ψ φ – 24 – φ η η η  10 grav | TT φ | η ψ η + TT | φ ], the decoupling of the Higgs portal to uncharged scalar dark 0, in conflict with the requirement for a calculable Higgs mass. η 1 2 <  181 , TT = | 4 is a consequence of φ 87 / , η TT | . The left panel shows the full result, whereas the right panel shows the TT- 52 y b β and ], and in general the vanishing of all quartic scalar couplings which could rule 0, showing that scalar fluctuations can play an important role in parts of the a . We show the region where . The origin of this severe restriction can already be seen in the TT-approximation 82 > φ 10 0 impose very significant restrictions on the space of higher-curvature couplings, see η It is intriguing to understand whether the conditions In the absence of higher-order terms, The scalar sector is of course consistent for > 12 φ and gravitational parameter space, see figure totically free. Yet, thevalue, scalar but mass instead is becomes then a no free parameter longer of calculable the in theory. terms of the scalar vacuum expectation where the factor 3 by a tadpole diagram.coupling Hence, becomes the viability conditionBeyond for the TT-approximation, a scalar UV fluctuations completion generate of a the region which Yukawa features find such a regionη in the gravitational parameterfigure space. Theof conditions the beta function for the Yukawa coupling, namely terms, the sign of thebecome gravitational antiscreening. contribution can change, cf. figure UV completions of gauge-Yukawa-systems, canwhich be results combined in with a the flat requirement scalar potential at the Planck scale. Within our approximation, we long as gravity remains attractive, it generatesWe a highlight screening that contribution for although scalar the potentials. “standard” origin (in of linear the parameterization) gravitational and contributionsame, the differs i.e., unimodular between a case, the screening the gravitational result contribution. is In actually the the presence of higher-order curvature approximation. observed value [ matter [ out certain breaking chains in grand unified theories [ Figure 9 couplings JHEP09(2019)100 . µν we h (6.1) (6.2) 11 ) to represent 0 above the dotted ]. In figure = 0. The right-panel λ, a, b > y, ; b  = 0), one can observe β 2 , with the correspond- ) ( 157 grav , λ − 5 | φ a η 6 80 scalar G y . , ). The possibility of a UV λ, a, b − -mode, while in the standard 2 y, ; b ) σ ], f 55 β b D -modes), which depend on the , ( 80 h 53 2 + 3 (1 + scalar π y, = 0 and 1 - and f 15 b 32 σ − = G 2 TT ) , = 0) both cases give the same results, namely λ – 25 – λ 2 b UQG − | y 0 below the continuous line; b β 2 + 3 > = y f (1 + π 15 32  ASQG,TT = | y β grav (for explicit expressions see appendix | y β β ) plane, for several values of the dimensionless cosmological constant (in all cases . We show the boundaries of the regions for a UV completion and predictive Higgs mass: = 0). For vanishing dimensionless cosmological constant ( is the dimensionless cosmological constant. We use a, b β λ 0 above the dashed (thin) line; In the region in gravitational parameter space close to the scalar pole line, on the other Below, we recall the gravitational contribution to the beta function of the Yukawa > g Note that this result isresults rather differ nontrivial, in as the the two settings. various diagrams that contributehand, to the these dominant contribution comesIn from the unimodular the setup, scalar the sector scalar sector of corresponds the to fluctuation the field that there is a coincidenceplausibilized between as the a unimodular consequence and ofourselves the the to standard TT-dominance the setting. in TT-approximation This these (with can results. be In fact, if we restrict gauge parameter completion of the Higgs-Yukawashow sector the was viable studied region for inin a the [ quantum-gravity ( induced fixedwe point set for the Yukawa coupling where the contributions coming from the scalar sector ( We contrast the results on unimodularing gravity, presented ones in obtained section inthe standard framework asymptotically where the safe theory quantum space gravity is (ASQG), defined i.e., bycoupling within full in diffeomorphism standard invariance. ASQG (in the Landau gauge) from [ line. The thick dashedzooms lines in indicates on the pole the overlapping lines region 1 + where the three conditions hold6 simultaneously. Comparison with the Reuter fixed point Figure 10 f JHEP09(2019)100 in = 0 and (6.3) λ a grav 4 3 | 2 g − β b 2 -modes) and is sufficiently = 0, becomes − h b a λ 6 = − . a - and 2 σ G g  ) Since these different setups 13 . ), the definition of which can be λ, a, b h ; β ( (TT-mode) and 1 g and λ λ, a, b f ; 2 σ − β − ( g 2 λ b ) f λ 40 2 in the framework of standard gravity which − − g b – 26 – b , the situation is the opposite. In this regime, scalar (1 + λ 10 + 7 π 1 18  − . Specific gauge choices which deserve further attention include: = = 0, only the trace mode contributes to the results. D β grav | 2 g β , there is a screening behavior of metric fluctuations for values of λ . For negative values of . Viable region (in red) for a quantum-gravity induced fixed point for the Yukawa 11 Regarding the (non-)Abelian gauge field sector, we provide the gravitational contri- For non-vanishing cosmological constant, scalar fluctuations can become more relevant. With the gauge choice close to the scalar pole. This leads to the absence of viable regions in this regime, 13 standard ASQG, and can besecond shown part to corresponds be to exactlyis the characterized the contribution by same from a as gauge-dependent the inextracted function scalar unimodular from sector appendix gravity. ( The previously had only been computed with vanishing higher-order couplings. The first term in the above expression corresponds to the TT contribution to in the enlargement ofnegative, the the point viable corresponding region.part to of the the In Einstein-Hilbert viable truncation, particular, region. we note thatbution if to (non-)Abelian gauge couplings For positive b cf. figure fluctuations contribute in an anti-screening manner to the Yukawa beta function, resulting (trace mode). gravity framework the scalar sectorreceive is contributions composed from of different sectors,neighborhood we of observe the a scalar quantitative pole. disagreement in the Figure 11 coupling for several valuesframework. of the The dimensionless dashed cosmological lines constant indicate within the the poles standard 1 + ASQG JHEP09(2019)100 λ (6.4) . This is a λ = 0 in standard by the inequality λ b , 4 and . Below, we present the gλ enlarge the viable region. a  λ ) 14 λ, a, b ; β ( = 0 we observe the same qualitative 4 λ λ = 0 of ASQG. f λ + 2 = 0, the contribution from the scalar sector ) λ λ , we observe that the overlap with the region 2 b – 27 – − 12 = 0. In this case the gravitational contribution b 2 + 3 β and -mode, resulting in the same expression as obtained in (1 + σ π 11 5 4  = = 0). In the case with β coupling. In the framework of standard ASQG, the running of grav | 4 we show the region in which the ratio of the Higgs mass to the 4 φ λ the results change considerably. In particular, for negative values of 4 β λ 12 λ ) represents terms coming from the scalar modes of the York decom- 0. Similar to the Yukawa case, negative values of : for this gauge choice and for and from the tadpole diagram depicted in figure λ, a, b ; φ η β λ > ( 4 ]). Thus it is rather reassuring to observe that the results at 40 = 0: in this case the contributions coming from the scalar sector vanish and, as λ → ∞ f − regions in the two settings in the subspace consequence, the result is completely determined by the TT approximation. β comes exclusively from the unimodular gravity. Therefore, there is a complete agreement between the viable β b 183 Let us finally highlight that within the truncated theory spaces of UQG and stan- Finally, let us discuss quantum-gravity contributions to the scalar potential, more In order to understand possible effects of a non-vanishing cosmological constant, let • • our truncation is not(see [ sensitive, as weASQG and do the not results in includesimilar. the induced unimodular Such setting matter-ghost a are interactions mildrobustness qualitatively of parameterization as the well dependence as results. could quantitatively be interpreted as a hint for the dard ASQG, one canparameterization-dependence also of interpret the our resultsconsequence results in of in the standard fact the ASQG that unimodularfor at the UQG setting difference vanishing and between as the a tests two slight settings modification of lies of the in the the gauge absence and of Faddeev-Popov ghost sector, to which (in all cases webehavior consider as in UQG.For non-vanishing Once again, thisthe fact cosmological can constant, be theway explained region that in with if terms a we of calculableallowing superpose Higgs TT-dominance. UV fixed figures mass points is in the deformed Yukawa and in (non-)Abelian such gauge a couplings becomes larger. where position. In figure electroweak scale is predicted for several values of the dimensionless cosmological constant dimension result for the gravitational contributionin to standard the ASQG, beta function of the scalar quartic coupling gauge coupling restricts the10+7 space of higher-curvature couplings specifically, to the the scalar quartic coupling receives gravitational contributions coming from the anomalous us consider, forcomes simplicity, from the the case TT-sector and, as consequence, the possibility for a UV fixed point in the JHEP09(2019)100 (7.2) (7.1) . = 0 y D and w y. 2 π . Our truncation for Weyl-squared gravity- , = 0 (trace mode). 15 y 2 λ 128 D π w 4 3 5 – 28 – = − 128 and b 2 = WG ψ | η − y ψ , β a φ 6 η − , η 2 π w 5 32 = φ η (TT-mode) and 1 λ 2 − . Viable region for predictive Higgs mass for several values of the dimensionless cos- b Landau gauge for the one-loop contribution. As the Weyl-squared setting onlyfor features dimensionless the couplings gravity (in sector) particular thetions also leading-order should gravitational be contribution universal. towork, We the show by beta that func- showing this holdsparameterization independence within of of the metric functional the fluctuations RG choice frame- and of of regulator the shape additional function, gauge parameters of in the We explore the one-dimensionalStandard-Model-like gravitational theories parameter space could tosquared be gravity. discover UV whether complete due to the impact of Weyl- The gravitational contribution to the beta function of Yukawa couplings is encoded in • • Thus, the gravitational contribution to the beta-function for the Yukawa coupling is the gravitational contribution to matter systems gives the following results matter couplings. Our aim is twofold: gauge coupling can bypoles achieved along 1 + with a predictive Higgs mass. The dashed lines7 indicate the Results for Weyl-squaredLet gravity us now turn to the discussion of the gravitational contribution to the beta functions of Figure 12 mological constant within theindicate standard the ASQG set framework. of values The where region gravity-induced with fixed diagonal points solid for lines the Yukawa and (non-)Abelian JHEP09(2019)100 , ]. (7.4) (7.3) 196 186 , – 126 188 = 0 holds. , , and is IR repulsive in g TT w η . We explore the perturbative g = 0, ) (7.5) . , 4 ) w g 3 ( w g 5 and ˜ (˜ O TT g O − + 0, as required in the Yukawa sector. At 2 3 + π w η g 2 7 2 ˜ g w < 2 β 576 is expressible in terms of a Taylor series, c + − g – 29 – + 2 = g g 1 and ˜ β ]: two different RG schemes can be understood as two WG ) = ˜ g | g = A (˜ g η 197 g β 0 at the fixed point. In the literature, the only known fixed is the anomalous dimension associated with the TT-fluctuation w < 0 at the fixed point if TT Z < ln t = 0 if TT ]. ∂ η y − 185 = , TT 184 η , For clarity of the discussion, let us repeat the corresponding argument for a theory Let us now discuss a more technical point, namely the universality of the above re- Beyond the leading-order approximation, a non-universal contribution arises that de- For the Abelian sector the situation is more subtle. In this case the leading order grav- . In the above approximation, a necessary condition for a UV complete Abelian gauge ]. Therefore, if we restrict ourselves to this approximation, the screening contribution of 170 [ TT To leading order, the two couplingsorder. are We the now same translate since there the are beta no function quantum expressed effects in to that terms of like matter couplings, which are also dimensionless, should bewith universal a at single leading coupling, order. see,ways e.g., of [ defining theregime, coupling, where the which relation we between will call the canonical dimension of theNewton coupling, couplings gravitational of contributions theory. to beta Dueuniversal functions to already in the at standard gravity dimensionful leading are nature order.non-universality non- of only This the sets is in differentgravitational at for coupling dimensionless 3 is couplings, loops. dimensionless, for the Accordingly, which gravitational in contribution Weyl-squared to gravity, Standard-Model- where the the fixed point that is known in the literature,sults. namely The non-universality (i.e., RG-schemebeta dependence) functions of in gravitational contributions theIndeed to matter beta sector functions has are been never noted universal, in but the the literature, onset see, of e.g., non-universality [ depends on where field. The possibility of anη anti-screening gravitational contribution dependssector on the requires sign of pends on the anomalousresult for dimension the of gravitational gravitational contributionusing to a fluctuations. the regulator anomalous that We dimensionconstructed depends find of using on the the a the gauge Litim-type following wave-function field shape renormalization by function of the metric and is 187 charged matter fields is notitational compensated sector. by an In anti-screening the effect non-Abelianholds, coming case from indicating the the that same grav- asymptotic resultfluctuations. for freedom the in gravitational contribution that sector is not affected by gravitational fixed point at point for thew Weyl-squared theory lies at vanishing valueitational of contribution vanishes, just as it does in the perturbative calculations [ Within this approximation the beta function for the Yukawa coupling has an IR repulsive JHEP09(2019)100 , ν α ] · (7.6) · (7.7c) (7.8a) (7.7a) (7.8b) (7.7b) h e [ ) 4 µα g g can contain (˜ g O = ¯ , 4 3 + ˜ µν Φ  and ˜ g 3 ]. A 2 g
2 π ˜ g w η 198 2 12 5 c + + , ˜ g  ) 5 4 , y p ( ˜ 2 p Φ 0 )) 6 β )) y ( y y r ) − + r ( y , 3 2 r 2 , − ( )
+ ˜ r Φ )   4 2 + 4 3 4 3 y y  ˜ g g ( ( y (˜ ˜ ˜ 2 Φ Φ ( r c 2 O 1 1 TT π : − − + + 2 + 2 + g n n . Typically, the regulator is then chosen to ˜ g 3 2 3 2 24 w η 3 Z 5 ˜ ˜ g Φ Φ 1 ) ˜   – 30 – dy y dy y − 2 β β  ∞ ∞ 2  2 TT TT 0 0 ) 4 3 π + π ˜ Z Z g 2 2 ) ) . 3Φ w η w η c 32 c ) 128 n n 2 2 5 5 1 1 4 − ) 1 Γ( Γ( − 3 2 g − − β ( is expressed in terms of the couplings, it becomes obvious ˜ Φ ( 3 2 3 2 = = + η  Φ Φ O p n p n 1 2 2 2 = (1 β ˜ Φ Φ + π π π w w w g 2 3 c 5 5 5 β 12 64 16 ˜ 2 g ˜ g 2 − ∂ ∂g = = = . Once β η = + + ( ˜ g ∼ 2 2 grav grav grav | | | ˜ ˜ g g k φ ψ A 1 1 η η η β k∂ β = = = ] shows explicitly that to leading order the beta functions are the same for the ˜ g ˜ , and the scale-derivative of the regulator on the r.h.s. of the Wetterich equation β 170 Z independent of the choice of gauge parameter (note that we work off-shell). independent of the choice of shape function independent of the choice of field parameterization In Weyl-squared gravity in the exponential parameterization, i.e., The above argument is for different RG schemes. Losely speaking, the choice of reg- ∼ 3. 1. 2. where the threshold integral is defined as that these terms are higher-order inthese the terms couplings. therefore To have recover to the universal be one-loop neglected, result, see alsothe gravitational appendix contributions C to the of running [ ofcome Yukawa and exclusively (non-)Abelian from gauge couplings the anomalous dimension. Our results are given by systems that the leadingFRG order, results i.e., from a theas truncation universal well that one-loop as includes result, a thebe can wave-function perturbatively be renormalization renormalizable obtained couplings from generates terms In fact, a similardeed, result [ should holdlinear for and the the gravitational exponential betaalso compare parameterization. functions to themselves. results We obtained In- within will perturbation now theory. explore It is the known points from pure 1)-3) matter and gravitational contribution to matter beta functions in the Weyl-squared gravity case is Clearly, the two leading-order termsdo agree, not. while it In the is presence alsothe of obvious scale dimensionful that explicitly, couplings, higher-order such the terms that relation between two-loop universality no longerulator holds. function can beunderstood viewed as in a a different similar choice of way. coupling. Further, Accordingly, field we reparameterizations expect can that the be leading-order into the corresponding expression in terms of ˜ JHEP09(2019)100 2) / +1). (7.9c) (7.11) (7.12) (7.13) (7.9a) (7.9b) n = 1 (7.10c) (7.10a) (7.10d) (7.10b) Γ( ω / =1 , (contributing 4 3 +1 ˜ 2 Φ n n A 2 π . Within this param- w η 12 h 5 . , 4 3  + 4 3 . are universal with respect ˜ Φ  . . In this case, there are ad- ˜ , Φ 5 4 , ) A 2 i ˜ µν 3 Φ 4 3 π grav  i h h 6 | + 2 , ˜ 4 3 ( Φ  i w η ψ . 12 3 2 4 3 + ˜ − 5 Φ  O η ˜ ˜ Φ Φ 4 3 3 2  + 2 µν 4 3 +  + ˜ ˜ Φ Φ g = 0) and the exponential ( 3 2 + 2 ˜ Φ ν   + 2 ˜ and 3 2 Φ α ). It not difficult to verify that the spe- 5 4 ω = ¯ TT 3 2 2 y ˜ + 2 h  Φ ˜ Φ 2 2 ˜ + 2 π TT Φ 3 2  6 µν π k µα 3 2  grav ˜ ( TT Φ g | − ˜ k w y η η Φ 24  w η φ 128 TT 3 2 ω h 5  η R η TT − 15 2 ˜ Φ η 2 TT + − 1 − −  – 31 – π TT h − 2 k −  1 µν π 3 2 2 w η 4 3 1 2 h h TT 128 π Φ h π 16 2 w y ) = 25 w y η 3Φ + 2 2 π y 24 w η gives the same expression as the one obtained in the w 5 ( π w 128 π 15 w y − 5 + r 5 µν 5 3 2 3 2 δ − 32 128 64 − = 15 Φ 3 2 Φ = linear grav 2 = = =  Φ | = w we find π 2 WG A 2 µν | 3 2 η π w g y 25 π 64 w y grav grav grav , WG 5 | | | β 8 | 12 5 − φ ψ A y η η η β = 0 = = = coming from the tadpole diagram represented in figure y y y β linear grav linear grav linear grav | | | φ ψ A ≡ D is independent of the choice of the shape function, namely Φ η η η ], namely +1 n n 142 – tadpole linear | y ). Our results in linear parameterization are given by 139 β y It is worth mentioning that we also have checked the universality with respect to We first observe that To test the universality with respect to the choice of field parameterization, we also As expected, the leading order terms in D tion [ where our results only dependeterization, on the we terms can up interpolate to between quadratic order the in linear ( Using the universality of Φ the choice on the field parameterization by means of an interpolating parameteriza- tial parameterization. Despite thesefunction differences for the the gravitational Yukawa coupling contribution gives to the the same beta result in both parameterizations, namely exponential parameterization, as itthe should gauge in coupling. orderand Furthermore, to we fermions give note are a that different universal the from beta anomalous the function dimensions corresponding for of expressions the obtained scalars in the exponen- employ the linear metric parameterization,ditional i.e., terms in to to the shape function.expected, non-universal The and contributions explicit coming result from depend cutoff on insertions, the however, choice are, of the as shape function. cial case Φ Taking this property into account we find with dimensionless shape function JHEP09(2019)100 µν R µν ]. The R 187 0 (Landau , → 186 , ˜ α α, 126 . Our computations reveal ω . By choosing β the gravitational contribution to the -dependent, however, this dependence 14 ω and 2 γ 1 – 32 – α γ , ˜ α ), by varying the parameter 2 h ( O ]. 199 the following results hold: , in which quantum-gravity contributions to the running of the gauge cou- b 15 are quite different. For the Reuter fixed point, the cosmological constant can also viable range, where quantum-gravity fluctuationsiality could problem solve in the the Landau Abelian pole/ hyperchargeThese triv- and results Yukawa-sector. are very similarfixed to the point results at for vanishing standardis cosmological gravity rather based constant. nontrivial, on as the We the Reuter stress diagrams that underlying the the close results in agreement the two different cases coupling pling and the Yukawa coupling are antiscreening. This is the phenomenologically In the unimodular theory space, there is a restricted range of values for the Within our truncation that includes all local gravitational couplings up to four orders We also compare to the one-loop results from perturbative techniques, and find agree- We briefly highlight the question of gauge dependence: within our truncation, the These are not free parameters in the full dynamical matter-gravity theory but are setWe stress by that demanding our study is subject to systematic errors due to our choice of truncation for the dynamics. i) 14 15 a consistent (asymptotically freeexplore or whether safe) there microscopic dynamics. arepoint regions Here could in we fall. this treat space them into as which free a parameters phenomenologically to viable gravitational fixed in derivatives, of our results for morecontribution general to matter the models. scale-dependencesults of In also particular, all hold the gauge for leading couplingsspanned quantum-gravity non-Abelian is by gauge the the microscopic groups. same, gravitational thereforebeta We couplings our function explore, for re- where the in Yukawa and the the parameter gauge space coupling is antiscreening. spect to their potential observational“standard” viability asymptotically in safe terms of gravity,gravity. unimodular their We work impact asymptotic within on safety a matter, toy andgauge namely model field Weyl-squared for and the a Standard simplegravity Model, Yukawa fluctuations system which are of consists “blind” one of to Dirac an internal fermion Abelian symmetries, and allowing a us real to scalar. deduce Quantum- implications from the results in [ 8 Concluding remarks In this paper, we compare three different quantum field theories for the metric with re- gauge), the results only dependbe on independent the of gauge-independent the TT remaining sector parameters. and thus turnment out with to our results.at The one-loop gravity-contribution even to in the the gaugegravitational more contribution coupling general to case is the of known running curvature-squared to of gravity Yukawa vanish [ couplings at one loop can be extracted that each one ofcancels the out diagrams when discussed we combine aboveand them is (non-)Abelian in gauge order couplings. to compute the beta functions forgauge the fixing Yukawa sector has 5 parameters parameterization, up to order JHEP09(2019)100 (A.2) (A.1) . d ) p d π d (2 . Z x · ≡ ip e ) p p Z Φ( p Z – 33 – and ) = x x d d Φ( -dimensional integrals in position and Fourier space, Z d ≡ , accordingly carry a minus sign in the exponential factor, ¯ ψ x Z . x · ip − e tribution with the functional Renormalization Group,dence explicitly from showing the the regulator indepen- shape function,parameterization as of well as metric the fluctuations. choice ofcoupling, Depending gauge parameters the on and Yukawa the coupling sign canloop of become contribution the to asymptotically Weyl-squared the free. scale dependence The of universal the one- gauge coupling vanishes. creening gravity contributions. The absenceular of asymptotic the safety therefore cosmological leads constant tocouplings, in rather unimod- which severe constraints need on the tocreening higher-order fall gravity contributions. into a rather narrow range in order to achieve antis- be nonzero, opening up a significantly larger viable parameter space region for antis- For the Weyl-squared case, we recover the known universal one-loop gravitational con- We use a shorthand for This motivates several avenues for the future. Firstly, extensions of the truncation are ii) namely respectively In this appendix we listfor the the conventions Fourier and transform notations is used in this paper. Our convention Conjugated fields, such as at the Perimeter InstituteFoundation for under grant Theoretical DNRF:90. Physics and by the DanishA National Research Conventions and thank allduring members the of Heraeus-Klausurtagung. thethe ITP/CP3-Origins G.P.B. grant quantum-gravity is group no. grateful for 88881.188349/2018-01at for discussions and Heidelberg the CNPq University support no. forDFG by under 142049/2016-6 hospitality. grant Capes and no. A.E. under thanks Ei-1037/1. and the A.E. A.D.P. ITP is acknowledge also support partially by supported the by a visiting fellowship parison of unimodular asymptotic safety withallow “standard” us asymptotically to safe disfavor gravity one could of the two quantum-gravity modelsAcknowledgments on phenomenological grounds. We acknowledge helpful discussions with A. Held, M. Schiffer, M. Pauly and S. Lippoldt, of course indicated tomore explore directions whether the are viable addedtational regions to fixed-point we the values find under space herewhether the of open the impact couplings. up fixed of further point matter Secondly, as falls fluctuations the into is calculation the of of viable interest, region gravi- to or show not. Ultimately, the resulting com- JHEP09(2019)100 (B.2) (B.1) (A.7) (A.3) (A.4) (A.5) (A.6c) (A.6a) (A.6b) ) symmetry d ( O , . =0 αβ 2 T . . p P ν # 2 p =0 .  µν T p p µ

µν p , P =0 ) 2 1 p e , = 3 # 1 − δ  =0 ( . µν d 2 L p O . P µν # − δ Diagrams + )  1 d a µ = 0 (Diagrams) να e T − X µ a 2 P . For our purposes it will be sufficient to δ δ ν Diagrams X ψ a µ 2 p 1 2 and µβ e µb T Z p µ e µν T P p 2 X + 1 / – 34 – P Diagrams − 1 / p φ a µ + 1 µ b 2] ) = δe 1 δ − µν i Z νβ L 1 T X p d/ [ d 2 ( P µa + P 2  2]  − e a µ µν 2 2 2 −  µα d/ δ T % [ ∂ , we use the projection rule ∂ 2 d/ P 2 ∂p ∂p µν k ∂ = ( 2 " δ " ∂p 1 2 a µ " φ A = e 1 1 = ψ Z Z = y , we have to express both the vielbein and spin-connection in 1 Z y − − µν T µν D P h = = = µναβ TT φ ψ A P η η η ] 201 , in accordance with the exponential parameterization. µν 200 is the (trivial) flat space vielbein. In order to gauge fix the local h a µ δ We start with the vielbein, denoted as In order to compute the anomalous dimensions of the matter fields we employ the The transverse and longitudinal projectors (on vector fields) are defined, around the where associated with the definition ofgiven the by vielbein, [ we adopt the Lorentz symmetric gauge-fixing terms of expand the vielbein up to second order around a flat background, namely B Fermions and theThe exponential coupling parameterization ofvielbein fermions and to the spin-connection. gravity Sinceof in our the formulation a is fluctuation based setting on field functional with quantization vanishing torsion is through the For the additional contributions for thethe beta diagrams function depicted of in the figure Yukawa coupling coming from following projection rules to the (functional derivatives of the) flow equation In addition, it is useful to define a momentum dependent tensor given by flat background, in the standard way For symmetric rank-2 tensors we use the transverse and traceless projector JHEP09(2019)100 ). (B.4) (B.5) (B.6) (C.1) (B.3a) (B.3b) B , resulting α ν . After some . h = 4. Here, we ) µν 3 d µα h h h ( O = + µν g  . 2 ρ ) σ . , δ α 3 2 . / h  h 1 σ β ( α b νβ ∂ e Z O and c λ δg 0), we introduce the appropri- + . µρ e ). In order to compute Hessians + 7→ h µα µν µν λ µα βλ → α h 3.9 h ν + δg h Γ h µ α ρ = N ac α αβ ∂ δ µα δ h λ α µν h , σ πG + νa h µρ δ νa δg ν b 32 1 2 h δ e TT µν 1 4 β µ √ 1 8 h − ∂ 2 ∂ −  / c ν ] + , + 1 → . For the fermionic sector we use the vielbein TT e – 35 – β µν Z µν g ac µν µν µλ µν , γ 2 δ δ h h h α δg δ 7→  β γ ] νa = [ νa νa ∂ b δ δ δ 2 1 λ TT µν 1 2 α , γ 2 1 2 1 h µν h a + g + γ = = − a µ µα µ a µ a δ dimensions. h = [ e µα β 2 δe = d µ h δ ∂ ] λ ω a µ β ∂ e λ β , γ h α γ − = [ µ ) up to second order in the fluctuation field. For simplicity, (and since in our ]. At the level of our computations both formalisms render the same results. ω 3.9 204 – ate wave function renormalizationnamely: for each one of the fields presented in our setup, eq. ( truncation it yields the samea results as flat technically background more complicated metricformalism choices), adapted ¯ we to use the case of exponential parameterization (see appendix 202 After performing a York decomposition (with Using the exponential parameterization, we expand the flowing action defined in Redefine the fluctuation field as The starting point is the truncation given by eq. ( For the spin-connection, which is not an independent field in our setting, we use the i) ii) iii) report results for UQG in and vertices, which areassociated necessary with to matter fields, compute we beta adopt functions the following and procedure: anomalous dimensions C Unimodular gravity-matter systemsIn in this general section dimensions wematter report systems. on All the some results additional presented in details the and main results text were for restrict unimodular to gravity- With these results we can compute allternative the to fermion-gravity the vertices use used of in vielbein thisism paper. in [ the An description al- of fermion-systems is the spin-base formal- in order to expressmanipulations the we spin-connection arrive at in the terms following of result the fluctuation field in the following expansion for the vielbein expression For the exponential parameterization, we have This condition allows us to obtain the following expressions JHEP09(2019)100 – 205 (C.3) (C.4) (C.5) (C.6) (C.7) (C.8) (C.9) (C.2) , (C.11) (C.10) , ) l , 171 − ,  q . 2 σ 48 − , 2 m p σ 30 − x + ( Z -mode propagator. 2 σ σ , p ) 2 σ l ) ( l m + 2 . The relevant vertices σ σ − 4 ) 0. µν p q q Z , ( h  ) − → φ 1) q ) p ζ p b d ¯ − − − ( ( p d φ 2 ) − 2 l TT αβ d ( 1) + 2)( h σ − − ) , ) l − q − , d ( ) q d ( d q − ( ( φ p TT µν − ) a h − becomes a pole in the p µναβ p − TT ) 4 ¯ ( ( q 2 σ − P φ  ( ( αβ  , ) m TT αβ φ % q ) − h ) TT µν −  l p  2 TT − – 36 – h ( ( 6= 0). In order to accommodate such a possibility, σ p ) = µν m φ L k q µν Z µναβ − 2 β P ( TT % ( + p q φ 1 ζ P β µ 1) 2 ) µν q p p p + % µ TT µν ( − p ν h + φ . να d q µν δ T 2 x ν 4 µ µ να q d Z P p δ p 2)( µ µ b p ¯ modes (for  , p p,q,l γ  p,q − 2 2 TT 2 Z σ p,q,l Z p ψ p p,q d 2 Z m 2 ( Z Z κ φ A TT κ 2 2 Z Z − Z − κ κ TT / 1 2 σ TT σ Z / = = = = = Z Z Z 1 TT ¯ φ φ φ ψ = Z Z Z Z φ k,φφ k,σσ k,ψ 1 2 1 4 1 4 µν k,AA µναβ SB Z Γ Γ Γ i Γ k 2 1 − − − Γ TT = = = = h ]. As an example, the presence of an infrared cutoff terms can generate mass-like TT TT TT φφσ k h φφσσ k k,h (2) Γ sence of higher curvature terms, terms for the TT and we add the following explicitly symmetry-breaking terms to our truncation The numerical factors in the second term were chosen in such a way that, in the ab- FRG formulation breaks the originaland, (volume as preserving) consequence, diffeomorphism the invariance RG-flowunder generates the terms which original are symmetry. notsymmetry At manifestly is the invariant encoded level in modified of207 Slavnov-Taylor the identities (mSTIs) flowing [ action, the aforementioned φφh k TT Γ Γ As a last step, we observe that the introduction of gauge-fixing and cut-off terms in the Gravity-matter vertices can be computed by taking functional derivatives of the fol- After these steps, we can extract the necessary Hessians and vertices. Below we present Γ h φφh k iv) Γ for this work are listed below, In the gauge field sector we employ the Landaulowing gauge terms fixing in the expansion of the flowing action in power of jected to mSTIs, withsubspace no of theory cosmological-constant space. counterpart in the symmetry-preserving the list of Hessians employed in our computations Let us comment that intion the field case arise of from standardnot the gravity cosmological-constant confuse such term. the mass-like generation In termsconstant. the of for unimodular Instead, such the setting, these mass-like fluctua- one are terms must to with be the understood generation as of a a purely cosmological symmetry-breaking effect, sub- JHEP09(2019)100 . ) l − · q  (C.23) (C.24) (C.18) (C.19) (C.20) (C.21) (C.22) (C.12) (C.13) (C.14) (C.15) (C.16) (C.17) ] µ − λ , ) p l ) l ,γ µ − − ) ρ ( − l q γ σ q [ ) − θ − l − q , ( γ p , p ) σ −  − l ) − ( λ p , l ( ) − β ρ ) µναβ l − TT σ δ q − q ( ) , P α − θ q β l ρ − − δ  q βρ βρ (  δ p − ) ν λ p ) δ δ σ α θ 4 − δ p 4 , − δ ) − p p p µλ µλ ( ) ν ( q λ − δ δ ( q ( δ 2 − σ − +2 − ρ ν ν ( p TT αβ ) θ − , 2 q q 2 β αβ ρ q A h k ) +2 p ) ) , δ α α % ( ) − ) 2 θ α 2 λ λ q ) l ) µ p p p θ − ) ) ψ p l νρ k δ p ( ( q δ ( l 2 ] ( ( − ( δ ν α / p , ρ λ − − µλ λ k p µ k δ p δ − λ − δ να ( TT µν  µλ µν A γ TT µν P ν ρ q βρ βρ P p q k δ − δ 1 % h ( , ν ) ( δ µβ h δ δ µ να ( ) ) P ) − b δ k − ¯ ) p δ q να ( q − q µν  (  µβ T q p αλ αλ  ( +2 β ( δ TT µν σ σ ( δ + 2 − δ δ , − P 1 4 ρ λ q ψ − β αβ ) h  Z ψ b d 2 ν ν ¯ ρ p k (    q q % ) A /p ] µλ q q + p 2 1 4 p ] ( α α 2 ) θ θ ρ ) ) q )( µ δ 1) l µ µ p λ ρ µλ α δ δ p 2 ( p p q + − ( 2 γ p p ν − δ ( ν ν ρ ρ ρ γ p A α ν − q − − ,γ ( δ δ 1) + λ ) − − ν ) ) µν λρ A ρ − γ + + ρ k 2 – 37 – νβ q ) d p p ) ) q δ A % p λρ 2 p γ µβ µβ 2 − ρ δ ( , d P 2 p [ νβ βρ βρ   ( δ − δ δ d p p λ − ) ( p µ − θ να d δ ( δ δ k l 2)( ( ( λ 2 2 p p q ( δ νρ νρ γ A − να k µ P k λρ a  − δ δ µν  ) να να β A δ . −  q − −  P δ P 4¯ q )[( % q ) δ δ q λρ   ψ β λ λ λ   N d µλ µλ ν p p δ µ ) ) ) ) q  ( )[ − − Z δ δ l l l q φ A TT µλ µλ νρ p ν p µ − p − p p µ δ β β q − Z Z − Z  ( ( p πG − − − − p − q q q δ q δ − µ − ¯ ¯ − q q q  ( ψ ψ · · ( µλ  ( α α ( p = = 32 ¯ ¯ p p ψ p p − − − p,q,l ψ  ) = ) = ) = p,q µν √ q δ p,q p p p Z TT αβ p,q,l 2 2 p · + + + + µν % Z p,q,l Z p,q ( p,q h p p ) Z 2 p − − − =  Z ) ( ( p,q,l 2 Z κ νρ νρ Z ( ( ( κ βρ βρ µναβ κ 2 l 2 p Z + δ δ δ δ 2 β ψψ k ( α α 2 ρ θ θ κ νρ )] κ κ κ ( φφ k σσ k / κ / 2 δ δ δ δ 2 1 2 1 2 σ TT σ νρ R TT αλ αλ να να ν ν µ ρ ρ κ θ p / R R TT µν / δ δ δ δ δ δ δ AA k δ Z Z Z ( 1 Z µλ 1 TT σ σ TT h λ λ β λ β ψ A A A ) Z R Z Z Z q µβ να µβ q q q q q δ TT k q Z Z Z Z ψ ψ ψ A µ δ δ δ · µ µ µ µ ( p 1 4 2 1 4 1 4 1 p p p p p R Z Z Z Z ψ [ · 1 8 1 4 1 8 1 2 − − − − +2 +2 − − +2 + − − − ======TT TT TT TT ¯ h h ψψσ k AAσ k ¯ k ψψσσ AAσσ k Γ Γ ¯ TT TT ψψh k AAh k Γ Γ We implement the IR cutoff in terms of the following regulator functions Γ Γ ¯ k ψψh AAh k Γ Γ where we have defined JHEP09(2019)100 , , b b ) ) 2 2 (C.34) (C.29) (C.30) (C.31) (C.32) (C.33) (C.25) (C.26) (C.27) (C.28) ]. We d d +2 +2 144 d d 2 2   +(4 +(4 bd bd a a , ) ) , 2 2 2 2 b , 1)+ 1)+ ) − − b d d b − − 2 d d ) 8 8 ) d d 2 d ( ( , d d − − a a , b 4 4 2 d d b +3 ) 2 8 8 +3 b ) − − d 2 d b d − − 2 σ 2 σ ) +2) + , b +2) d + + m m b d 2 + 2 σ (16 (16 2 TT , )+(14+3 ) +(14 , 2 TT +(14 b m − − a 1+ ˜ 1+ ˜ m ) a 2 TT b 0. ) ˜ ) 2 2 +2) m 2 TT   ) ) + b 3 2 2 d d 2 2 d 2 ) m d d  m +4+2( 3 d  → d
− − (1+ ˜ 2 TT +4+2( d d − (1+ ˜ bd 12 d d ζ d bd 12 2
m 2 2 (1+ ˜ 2 − +2 +2 d −

2 − 2  d 2 1)+ 3 2 d − − d 1)+ d − 2 d +2 − 2 +4+2( − 8 8 d 3 − bd − +2 +2  (1+ ˜ d d 2 d
44 d  +6)(2+ ˜ d π G (

Γ 44 d 2 2 d d
( +(4 − d 2 a
2 bd − a ( bd 1)+ 4 − d a π G Γ 32 +2 4 − d/ ) +2 +2 π G π G −
2 Γ Γ +2 d 2 2 d ) 2 d 2 − 1)+ +2 2 2 2 2 d d − – 38 – 32 1)+ 2 d ( (56 ) for the Litim-type shape function [ − d π d/ 3(2 (56 +(10 2 σ − 2 d − 32 32 2 d a 2 σ ) − d/ d/ +2 8 d 3 d π G − Γ 4 − p d ) ) − (4 π G π G ( Γ Γ π m 2 d π G d ( ) ) Γ 2 m − π G a 2 2 π π a Γ 2 2 2 a 2 σ 2 σ − 32 4 ) (4 − 2 d d/ 32 32 4 32 d/ d/ 2 σ π G 3 ) (4 (4 d/ Γ 1+ ˜ 8 2) − 32 m m 2 ) ) 1+ ˜ 2 d/ − ) 2 d π 2 −  ) m k π π  − − 2 σ π 32 2) d 2 σ d/ ( + π (4 d 5 ) 1) θ (4 (4 2 m (4 +2) +4) − m 2 ) 1+ ˜ π (4 d (16 )(2+ ˜ )(2+ ˜ 2 − d d d − +2) 2  2 2 1) 2) d p (4 d d − 1+ ˜ 2) d d 4) 1+ ˜ +5)( +2 2 +4) 2)(  − − +4)  − 1 − − − − d d d d 1)( d d d +1)( +2 − +2)( ( 2 d 5 d d ( 2 +1) +6) − − 2 2 +4) d d d d d 4 4 +4)( − d d d k +2 +2 d − d d d d 10+23 2) − − d d ( 2 )( ( ( +1)( 16 2)( 1)( +6)( +6)( − 1)( d d d − +4)( d + ( − d d 2 − − +1)( +1)( 4 d d − − 12(2 8 ( (12 ( (12 ( 2 d d d d d p ( 8( ( 16( (2 ( ( ( 2( − − + × × × × ======) = 2 p ( sunset sunset sunset sunset tadpole tadpole tadpole tadpole tadpole tadpole k − − − − − − − − − − σ σ σ P | | | σ σ σ TT | | | φ ψ A | TT TT TT φ In the following we report our findings for the gravitational (non-vanishing) contribu- ψ η A | | | η η A η η η φ ψ A η η η η note that the regulatorprojector, associated which is with consistent the with gauge the field gauge choice istions proportional to to the the anomalous transverse dimension of matter fields for arbitrary where JHEP09(2019)100 = , ) 2 TT ¯ (C.35) ψψ m plane. In i yφ 2 σ . ˜ b + m ) 2 × d / Dψ 2 TT ¯ ψ i + 3 m ( d g + (D.1) √ gh. k x Z + (14 + + Γ a ) 2 2 g.f. k αβ d  , the UQG result coincides with F 12 λ b d + Γ µν − F
2 1)+ = 4 d − νβ µν − d
2 σ d g 44 ( R , m a µα − 4 µν + 2 2 2 d g g − b R ¯ and ˜ (56 2 σ √ Γ x π G y + λ − – 39 – 2 Z m 2 2 ) d/ 32 1 4 2 σ − ) π 1 + ˜ a R m + =  (4 + ¯  2 4 2 TT R φ 2) )(2 + ˜ 4 m 2 − 4! λ − d 1) d + 2Λ − − φ d g d ν 4 ( + 6)( √ d . This agreement between exponential and linear parameterization, − φ∂ x d µ Z = 4, identifying the symmetry-breaking masses with the dimensionless ∂ y (12 d 16( y N µν g − × → −∞ 1 1 2 π G ≡ D = β  16 g √ = we plot the viable region for an asymptotically free UV completion of the Yukawa x Z triangle | 13 y + Given the number of free parameters the analysis of results including higher curvature In the main text we have restricted our analysis to the symmetric case with ˜ In addition, since we are interested in the running of Standard Model-like couplings, = 0. The introduction of symmetry-breaking masses in the UQG setting allows us β Stand. k Γ 2 σ ˜ used for these computations D Explicit results forFor the the Reuter sake fixed ofstandard point ASQG completeness framework. in In order this to appendix fix our we notation, report below we some present results the truncation obtained in the simpler, here we switch off thefigure higher curvature terms and focusand on (non-)Abelian the gauge ˜ couplings, amass predicted and ratio the of intersection theof of electroweak curvature scale these to squared three the terms, conditions. Higgs three the conditions As symmetry-breaking can one mass be can terms satisfied, see, just induce as even regions in in where the the all case absence of the linear parameterization. the robustness of theand results. UQG We is highlight only thatgravitational at this couplings the are agreement calculated, between level and standard of where ASQG the the present symmetry-identities truncation, arecoefficients where neglected. and symmetric no breaking beta masses can functions be for rather the cumbersome. In order to make it More precisely, in cosmological constant, namely ˜ the expressions obtained within standardgauge gravity choice with linear metricdespite parameterization differences and at the level of individual diagrams, can be interpreted as a hint for perturbative approximation of the FRG. m to mimic the behavioras of we the do results not obtained in take the into standard account the gravity framework symmetry-identities (as which long differ in the two settings). We highlight that in thesion above contribution set coming of from expressions we cutoff have insertions. neglected This the anomalous approach dimen- is sometimes referred as a the beta functionfrom for the the triangle Yukawa diagram coupling represented receives in figure a gravitational contribution coming JHEP09(2019)100 0, > (D.2) (D.3a) (D.3b) grav | φ + . η 2 ) ) λ λ ) , ) 2 2 ) 4 ) β β 0 (blue region); iii) ν β ) − A β < + , µ 2 29 ¯ λ ∇ 4(3 β grav − | µν 3) − 12 A g 2 η (¯ ) ) − − b (90 λ g 2 2 = ) 2 β − √ 2 ) g ( − f x λ 2 β a Z ) − 1) 6 2 − (369 A ζ β − 2 − 2 Z ) − 4(3 )] + 48(27 β β (1 b − 2 ] + − ) 4(3 h ) [ + 2 β b ν (3 2 − a F − ) + 10( ) ] b − b h (with vanishing curvature squared couplings). From 2 [ a ))(6 – 40 – ((3 13 µ 6 2 σ β − F ( 7 − m 0 (cyan region); ii) − a π µν a 6 − ¯ > G g (1 80 39 and ˜ − g 2 ) (20 √ + − (= 0). For the sake of simplicity, we restrict ourselves to grav (1 β 2 TT | β x 2 2 h φ 7 (5 ) ) Z m − µ η 2 λ β − ¯ ) ∇ N 2 b 0 (dark red region); iv) combined plot (dark green) with 2 ((3 β − β 4 − < 1 1+ b + ((3 [(519 πα G 2 + 3 − 2 grav β )
32 4 y β (1 + µβ 0. = − D h − π G β < + (3 ¯ g.f. k 32 ∇ y (3 φ 25 π Γ f η G 5 − 1 2 − − ] = + = = . Region plots for the sign of the gravitation contribution to SM-like couplings in terms h [ ψ µ η F 0 and grav grav | | Below we report the results for the gravitational contribution to the anomalous dimen- φ = < ψ η η y g f f The running of the Yukawa coupling receives contributions coming from the diagrams sions of scalars and fermions where the four-dimensional case. In addition, we adopt the linear parameterization for the metric. with the gauge-fixing part given by Figure 13 of the symmetry breaking massesleft ˜ to right and− top to down: i) − JHEP09(2019)100 (D.5) (D.6) (D.7) (D.4) gives 2 . 14 )
. λ λ ) 2 # ) 2 ) ) 2 2 β b ) λ around Λ = 0 β ) λ − 2 ) A ) − 295 2 β η b 3 λ β − 4(3 − ) − 2 − a 40(3 − . β a ) 4(3 5 9 − b is the threshold integral ) 885 4(3 ) 2 − − 2 λ − b p n ) − − ) 7 − b 3) λ ) , 2 ) b a 2 − − 2 ) 2 6 N 3)(1 − a 3)(99 (111 λ β 2 G 2 − − − ) a β ) 21 − 2 − ( 6 Λ 2 a β 2 (1 β 6
β β − − β 2 ( 4 3 − ) − 4(3 2 − (5 β (1 2 3Φ (3 3) − (1 ) + 54 2 − 4(3 ) ) 2 β − − . In fact, by expanding b β ) + 2( ) β 3 2 b 2 2 − β β − + 1) irrespective of the choice for the shape ((3 Φ ) − 3) 47 − λ G 4( b n − (3 ( 2 a − – 41 – − β π ≡ ((3 Γ( 6 2 − β / a A ((3 β G y 20 − N a )( = 6 − b π G G 141 − = 1 (1 ) 2 2 − 45 2 ) 2 N − ) -dependent coefficient and Φ 179 ) ) is given by (for a generic shape function) λ G +1 b (1 β β + λ 2 n n − (Λ grav 2 b 2 N | (15 2 λ ) − − a ) G A 2 − β b . η λ 40 β b 2 ((3 A − 2 537 2 + 3 − η 5(2 + 3 ) − b β − (1 + b ((3 (1 + , resulting in the expression " − 4 2 π π = 0), it is possible to verify that the contribution associated with the (3 2(63 G y 4 (1 + 4 10 + 7 b 5 G λ + − π ). Given that Φ = G = = 18 a 2 − 7.8a corresponds to some . Tadpole diagram contributing to the running of the quartic scalar coupling within the Fig. = | β tadpole y | A β 4 λ grav | The complete result for the gravitational contribution to the anomalous dimension of β A η where defined in ( function, the above expression turns outsuch to be a universally result zero. depends It ondiagrams is worth the contributing emphasizing to cancellation that of universal contributions coming from different mensionless combinations such as(and Λ setting dimensionless combination Λ Although this expression isnature of non-universal, the as gravitational onetributions couplings, could (with we observe expect respect that to due it the to is cutoff possible the shape to dimensionful find function) universal appearing con- as the coefficient of di- (non-)Abelian gauge fields, in an approximation containing higher-curvature terms, is Finally, for the scalar quarticthe coupling, following the result tadpole diagram represented in figure Figure 14 standard ASQG framework. represented in figure JHEP09(2019)100 ]. 95 Phys. 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