JHEP07(2017)113 Springer July 6, 2017 July 14, 2017 July 25, 2017 : : : January 19, 2017 : Revised Accepted Published Received , Published for SISSA by https://doi.org/10.1007/JHEP07(2017)113 , [email protected] , . 3 1603.03603 The Authors. c Beyond Standard Model, Higgs Physics, Renormalization Group

In this work we find the minimal extension of the Standard Model’s Higgs , [email protected] Saupfercheckweg 1, 69117 Heidelberg, Germany E-mail: [email protected] [email protected] Max-Planck-Institut f¨urKernphysik, Open Access Article funded by SCOAP Keywords: ArXiv ePrint: vacuum expectation value andminimal therefore model mixes predicts into a theIn sizable physical addition amount Higgs. to of the We mixing physicalheavy find which Higgs, boson. that makes the The the it theory’s heavy testable scalar scalar’s at spectrum properties a contains render collider. one it light a and potential one dark matter candidate. Abstract: sector which can leadsistent to with a the light phenomenological Higgs requirementstheory. boson for a The via low-energy model radiative realization symmetry which ofis turns breaking a an out and conformal to extension is be of con- stable the under Standard renormalization Model group by translations two scalar fields, one of which acquires a finite Alexander J. Helmboldt, Pascal Humbert, Manfred Lindner and Juri Smirnov Minimal conformal extensions of the Higgs sector JHEP07(2017)113 15 18 13 10 6 ]. In recent years those 5 – 1 25 35 32 20 34 30 – 1 – 17 5 3 33 1 27 30 2.2.1 Two2.2.2 real multiplets with zero The vacuum minimal expectation conformal value model 2.1.1 Real2.1.2 multiplet with zero vacuum Real expectation2.1.3 multiplet value with finite vacuum Complex expectation2.1.4 multiplet value with zero vacuum Complex expectation multiplet value with finite vacuum expectation value A radical way of addressing this problem is the assumption that the fundamental theory B.1 SM +B.2 real scalar representation(s) SM + one complex scalar representation A.1 Basics A.2 Calculating the Gildener-Weinberg scale 2.2 SM + two scalar representations 2.1 SM + one scalar representation describing Nature doescan not have be any realized scale.scale-invariant non-linearly models to and In solve explicit such thestudies scales a hierarchy inspired problem conformal can a include number model, appear. [ of the other symmetry Early works addressing works different that open employ questions beyond the theoretical physics. It is aelectroweak naturalness scale problem which, can at be its core, light(SM) asks into in a the spite more question complex why of theory the ato with high-energy other solve heavy the embedding scalar hierarchy of degrees problem of thecontributions is freedom. Standard to the One Model systematic the approach cancellation Higgs ofsupersymmertic mass bosonic within particle and has supersymmetry. fermionic been loop However, observed due yet, to alternative the approaches fact are that appealing. no 1 Introduction The gauge hierarchy problem continues to be one of the most pressing questions of modern B One-loop beta functions 4 Discussion A The Gildener-Weinberg formalism 3 Matching to the semiclassical regime in gravity Contents 1 Introduction 2 Finding the minimal model JHEP07(2017)113 ] for a 37 ] given certain 36 emerge from an embedding ]. Second, the hypercharge 35 ]. A common feature of those may 34 – 6 the Planck scale. In this energy regime below – 2 – If the second possibility is realized, from the point of view of a low-energy theory, this In the Standard Model, the hypercharge gauge coupling is not asymptotically free and We now argue that, if conformal invariance is a fundamental symmetry of Nature, answer to the questionaddress of the what question of is how the anincluding effectively physics conformal gravity. beyond model the The Planck gravityreview. scale. scale itself Of However, can we particularscale will be interest of generated are gravity spontaneously, Yang-Mills in see theories a [ which conformal can set-up. lead We to note, a however, spontaneous that this process can happen and the low-energy theory. Atlies this on point we the emphasize physicsthe that of the theory focus a is of conformal described theof theory present by which paper a is renormalizable expressed quantumare in field necessary terms theory, conditions the of forRG radiative the running any behavior RG up extension running. to of the The the Planck criteria Higgs scale. discussed sector It in in is this order not article to the purpose enable of stable this paper to give a definite means that the electroweak symmetrymass is parameters. broken by radiative Furthermore, corrections thescale, without at theory tree-level which must the allowthat gravitational a in contributions RG the become RG evolution relevant. evolution upscale. no This to Landau means poles Moreover, the in or no Planck particular vacuum explicit instabilities appear threshold below scales the can Planck be located in between the Planck scale values of the couplings ofthis scalar can fields work in and a theis toy curvature model a scalar. set-up. necessary We We will matching argueconformal demonstrate that embedding. condition how this between vanishing of the the low-energy trace of theory the and EMT the UV-complete thus will increase with energy.vanishing trace In anomaly. this First, context,so there the are SM that two gauge the options group corresponding tocontribution is to still coupling embedded the accomplish trace in is anomaly a a asymptotically isas non-Abelian canceled free the by group anomalous the [ gravitational gravitational anomaly. contribution This to is the possible trace can be negative [ physical quantities on the renormalization scale.to Any the quadratically divergent Higgs contributions explicitly mass breaking must the conformal therefore invarianceabsorbed be by by regulators. appropriate purely The counterterms. technical formal divergences and can are be typically introduced by the renormalization-group (RG) running ofof the the parameters energy-momentum tensor leads (EMT), to which a enters non-vanishing the trace divergencethen of the the quantum scale field current. theoryabsence of must explicit have mass a parameters, vanishing thebeta trace trace functions. of anomaly the at The EMT some anomalous is given scale. Ward by identity a In thus weighted the sum allows of only the logarithmic dependence of context of scale-invariant theories;works is see the for need exampletop for mass [ additional does bosonic not degrees permitdifficulty of radiative in breaking the freedom, conformal of as model the building in electroweakmisleadingly is symmetry. called the the classical The nature SM scale of conceptual invariance. alone the This symmetry, the which symmetry is large is sometimes anomalous, since generically SM — like small neutrino masses, the nature of dark matter or baryogenesis — in the JHEP07(2017)113 . In 3 ], the SM extension 2 – 3 – . First, we describe the method used in this paper 2 we discuss important implications of our findings and summarize our results. ]. There, the authors showed that even if a theory possesses a symmetric vacuum 4 38 We present our analyses in section In this paper we revisit several classically scale-invariant models and investigate Our analysis changes the perspective under which the hierarchy problem is viewed. The at tree level, thethen induces one-loop spontaneous effective symmetry breaking potentialdynamically (SSB). generate may In a exhibit other mass words, athis scale radiative non-trivial way corrections in obviously minimum a also classically which cordingly, breaks conformal we expect the model. the (anomalous) theory’s A conformal low-energy scale symmetry phase generated to spontaneously. in contain one Ac- pseudo-Goldstone boson One of the centraltroweak symmetry aspects induced in by theformal a Standard extension negative Model of massspontaneous is the parameter breakdown the SM, must of spontaneous be without the triggered breakingnism Higgs any by was of quantum field. explicit first effects. elec- investigated mass The by InQED corresponding scale Coleman [ a mecha- and present con- E. Weinberg at in tree the context level, of the massless scalar matching of the low-energysection theory to the semi-classical regime in gravity in section 2 Finding the minimal model which can be used to constrain our model’s parameterto space. obtain ourstarting results. with the After extensioninvestigate that of further we scalar the scan extensions SM until through by we one the find additional most a scalar. successful simple model. conformal We We then models, will systematically discuss the vacuum expectation value (vev).group is In not this altered and context,the the minimality scalar additional implies field number without of that the representationsneutrino the is vev can masses minimal. SM be can We gauge a find be viable that our easily dark work matter accommodated is candidate. that in the Furthermore, this physical small Higgs model. will have Another sizable important admixtures of result one of of the singlet scalars whether they canvant be effects low-energy we find realizations thatby of in one contrast a real to scalar previous conformal fieldminimal studies, theory. is conformal for not extension example consistent of Including [ with thedoublet all this SM supplemented requirement. Higgs rele- by Eventually, sector we two to identify real consist the scalar of gauge the singlets, usual complex one Higgs of which develops a non-zero field theory with a giventranslations. set of fields This and RG parametersto stability is analyze stable will whether under be renormalization acertain group our particular representations essential parameter which configuration criterion can is to bescalars, allowed. distinguish fermions added This and to models criterion gauge the and selects bosons SM. can We find lead that to only the the desired interplay RG of stability. with an explicit scale andfact also to induce demonstrate a that gravitational thea conformal trace vanishing anomaly. of of We the the will EMT total use can conformal this vanish anomaly. at aquestion particular is scale, not leading why to in a given model the Higgs mass is light, but rather whether a quantum without further influence on the theory below the Planck scale and gravity might emerge JHEP07(2017)113 . We GW ]. In their for- to be positive. configuration to ) which will play 40 B positive, one has )). By this line of A.5 B A.5 minimal in eq. ( B will inevitably lead to low-scale , positive at low energies. On the Y GW , turns out to be a tightrope walk. B ]. Pl 39 M ]. In the Gildener-Weinberg formalism U(1) 41 )). Consistency requires × degrees of freedom. Note that the scalar is negative such that the effective potential L A.7 SM B – 4 – gauge SU(2) = × B c SU(3) . In particular, we will introduce the loop function A The method used in our analysis is as follows. First, we choose the class of models we In the models under investigation we find that obtaining scalar couplings, which allow The question for the rest of this work will be: what is the It is well known that radiative symmetry breaking `ala Coleman-Weinberg does not From a more technical point of view, determining the effective potential’s minimum is other hand, starting with tooLandau large scalar poles couplings in at the Λ scalar sector. want to investigate. Then,functions. we The unknown derive scalar the couplings corresponding introduced by potential the and potential the constitute our one-loop initial beta expectation values, depending on their quantum numbers. for a successful RG runningOn up the to one the Planckhave hand, mass sufficiently the heavy couplings new scalars need which to then be render large enough at the GW scale in order to is not altered andequal the according additional to number the of aboveIn representations criteria, the is particular, number minimal. of we parameters If willdegrees selects two the of not models minimal freedom add are model. added any to new the model in principle may or may not acquire finite vacuum freedom to the theory. enable radiative SSB withminimality successful implies RG that running the up SM to gauge the group, Planck scale? In this context does not develop ato minimum achieve but a a dominanceargumentation, it maximum. of is immediately bosonic clear In thatby degrees no order fermionic model of to can representations work freedom render only. inneutrinos (see which cannot the In eq. SM facilitate is particular, radiative ( extended SSB. the Hence, SM it supplemented is by necessary right-handed to add bosonic degrees of a central role in our analysis.and It thus quantifies also the the effective PGB potential’s curvature mass at squared its (cf. minimum eq. ( work in the SM duethis to failure the is large reflected in top the quark fact mass that [ malism, minimization conditions manifest themselvesscalar as couplings, implicit equations the for so-calledning, the Gildener-Weinberg these model’s conditions. conditions will only Duetified be to with satisfied the the at scale a couplings’ of particular SSB,review run- energy, henceforth the which basic referred then principles to is and as some to theappendix technical be Gildener-Weinberg details scale iden- of Λ the Gildener-Weinberg formalism in perspective the PGB discussed herefor can a be detailed described discussion by on an the effective phenomenology theory see of [ thetypically dilaton, a challenging taskdue in to models Gildener of and S. several Weinberg which scalars. allows a However, systematic there minimization exists [ a method (PGB) which obtains its finite mass only at loop level. Note that from the low energy JHEP07(2017)113 we (2.1) (2.2) 2.1 , . ) χ UV a T † χ at which at latest are the generators )( φ a UV a T . This set-up contains τ † φ to other scalars is given ( 2.2 2 κ LHC H ) + χ are properly reproduced. As an † χ v )( . φ † ) φ ( 1 κ ,N,Y + – 5 – 2 (1 ) , mainly consists of the field that couples to the χ ∼ a denote the generators of SU(2) in the fundamental LHC χ T † a -dimensional irreducible representation (irrep) under H τ χ ( N 3 λ + 2 and the electroweak scale ) χ † ]. Together with appropriate Gildener-Weinberg conditions, all of -parameter. We then explicitly check whether the observed values multiplet with given hypercharge is added to the SM, χ ρ Higgs ( denotes the usual complex Higgs doublet and L 2 43 m , λ | ) 42 0 + 2 ) , φ 44 [ φ + . † 0 φ φ ( 1 transforms. Accordingly, = ( λ | ≤ β φ χ = In the following we are going to present the results of our study. In section In a first analysis of a given model’s RG running, we apply a “best-case approximation”. In doing so, we ascertain that the well-established physics of electroweak symmetry sin V | where of the SU(2)which Lie algebra inrepresentation. the The scalar potential consistent withour the discussion SM will gauge be symmetries based and on, scale reads invariance, which as follows SM which might allow for radiative symmetrysinglet. breaking is Generalizing obtained this by adding ansatz,colorless a we scalar scalar gauge investigate SU(2) models in which one in general complex, to the Planck scale. 2.1 SM + one scalarIn accordance representation with our discussion in the previous paragraph, the simplest extension of the completely consistent calculation in order to determine the actualdiscuss value the for simplest Λ extension ofsimplest the case SM of by one adding additionalthe two scalar. scalar minimal We extension degrees analyze of of the the freedom next-to- SM in that section leads to correct SSB and successful RG running up Thus, we obtain a conservative estimatean for instability the occurs largest possible or scale thethis Λ theory’s way is couplings significantly turn smaller non-perturbative.with than If the our the Planck previous mass, scale discussion. we found exclude in Otherwise, the we model perform in accordance a numerically more challenging but additional consistency requirement, we makeHiggs sure boson that observed the at scalarSM to the fermions. be LHC, identified with Anby the experimental bound onthe the above constrains mixing will of allowconditions us for to the renormalization limit group the equations model’s (RGEs). parameter space and obtain initial and from this derive the masses of the physicalbreaking scalar (EWSB) modes. is preserved. Forsignificant instance, we shift will of directly the exclude modelsfor which the imply a Higgs mass parameter space. We use the Gildener-Weinberg formalism to obtain the theory’s vacuum JHEP07(2017)113 . (2.4) (2.5) (2.3) How- N at tree 1 2 obtain a ). v as well as 1 ) are those multiplet. λ 2 all 2.2 ) χ 2.2 n = 6 real a ) symmetry. Since 2 Higgs N ...T ( m 1 O a . T × † ) χ χ † (4) represents a χ O χ )( φ ], we will in general ignore those is given by ). † h φ 44 ( 2.20 1 ≡ κ As obvious from the above definition, . χ , + 2 2 LHC v 2 ! = 1 ) H κ . Before we investigate the most general case, χ ∗ † χ B χ C – 6 – ( = 2 multiplet in the sense that it coincides with its ) can only be consistently defined for odd 2 2), 2 χ λ := L √ 2.3 m , and the new scalar’s component fields ˜ + χ φ h/ 2 v ) + φ ≡ † v , always one of its component fields is electrically neutral φ v ( , some operators might be redundant in the sense that they can be χ = 1 . The potential therefore reduces to N λ 0 , see the discussion after eq. ( 1 C φ κ = have zero vacuum expectation value. Then the electroweak vev is ] have found that for the most stable RG running all associated V i and χ 44 2 , λ . Again motivated by the results of [ 1 λ Y vanishes identically for all real fields transforming under an arbitrary irrep is necessarily non-negative at the GW scale. and χ term. 1 . Hence, the only non-zero terms in the general potential eq. ( a κ is a suitable charge conjugation matrix. 2 L N T κ † C for the moment be a real SU(2) χ χ For real scalar multiplets Besides, there exist additional operators, which are only invariant for special combina- Checking the consistency of the models of interest necessarily requires knowledge about Note that further gauge-invariant operators of the form ( Depending on the dimension For the proper definition of and 1 2 3 level. Since all physicalcoupling masses have to be real, the above formulaexpressed as shows a linear that combination of thecoupling operators is containing portal less introduced generator in matrices. those Accordingly, no instances. additional finite mass during EWSB ( Similarly, the mass of the physical Higgs mode only the odd-dimensional irrepssatisfying of the SU(2) reality are condition real in (as eq. opposed ( to pseudo-real),and may multiplets therefore acquire alet finite us vev. assume We that will all discussjust this that case of separately the later. Higgs For doublet, now, proportional to Notice that the above potential enjoys an accidental global where real multiplets necessarily have zeroterm hypercharge. Furthermore, it is easyof to SU(2) show that the let us first restrict the discussion to the2.1.1 situation in which Real multipletLet with zero vacuum expectationcharge value conjugate field, i.e. tions of terms. In cases in which we take them intothe account, corresponding we RGEs. will discussthese Therefore, them models separately. and we list have the results calculated in the appendix one-loop beta functions for the corresponding portal termsever, are, the in principle, authors present ofcouplings in have [ the to potential vanish in eq. theNevertheless, ( infrared, we i.e. include at the theλ Gildener-Weinberg simplest scale representatives in of our context. the above operators, namely the JHEP07(2017)113 ) A.3 (2.9) (2.6) (2.7) (2.8) , which GW ) only guarantees is to be calculated enters the problem 2.9 N . ) that GW . N , we can still insert the 2.5 2 / K, 1 v Landau poles in the scalar GW − − ). Since N N 2 ) and set . and · v B 2
GW / ) now shows that increasing the for given ) 1 add − (Λ Higgs ) and eq. ( Pl B 2.8 GW N K m M . (Λ A.5 ) as well as the renormalization point = 8 ≡ 2 2 1 K v 2 Higgs i π GW K + m ϕ Nκ ) will always be larger than that defined in h 16 (Λ = 125 GeV, since it runs logarithmically and π ) 2 also unavoidably enters in some terms of the 2 λ = GW 2 Higgs – 7 – √ 0, while at the same time allowing small portal add ), i.e. N appropriately. For the following study, we will, Higgs (Λ m B denotes the condensate introduced after eq. ( ). Equation ( 1 add m A.7 2 κ + v B ) := GW B > q Higgs = SM GW v π m i B ϕ (Λ h 1 ln( ) = κ = 8( ∼ GW ) (Λ 0 and , but not the correct overall scale. In a full calculation the latter, 2 Higgs 1 GW κ m > /v ) can now be solved for the unique portal coupling at the GW scale in the above equation is evaluated at Λ 2 v 2.7 Higgs SM , the exact value for m Higgs B N 8 m − of new scalar degrees of freedom implies a smaller value for the portal coupling ) = 0. Accordingly, the tree-level mass of the Higgs vanishes at Λ := N ) and eq. ( ). But the larger the initial portal coupling, the sooner one of the scalar couplings GW is positive, this definition exemplifies the “best-case approximation” in the sense K itself. Note that setting the portal coupling according to eq. ( 2.6 low-scale Landau poles are possible (see also appendix (Λ 2.9 Uniquely solving the given model’s RGEs requires to additionally fix the value of the In order to simplify such a calculation in the class of models under consideration, we Next, the Gildener-Weinberg condition corresponding to the assumed vev configuration 1 K λ GW GW second quartic coupling atΛ the GW scale the proper ratio would have to be set by adjusting Λ As that for given eq. ( will develop a Landau pole. in a non-trivial way, onlyquestion an of explicit whether calculation some Landau of pole the exists RG below running canneglect the shed SM light contribution on to the the Higgs mass to maintain the necessarycouplings. condition One might thereforesector think can that be for entirely largemodel’s evaded. enough beta functions However, leading toΛ a faster RG running such that even for small couplings at Even though measured value of the Higgswe pole always mass assume ln(Λ number for otherwise fixed quantities. In other words, introducing a large scalar multiplet helps Eq. ( which is consistent with the experimental values for where in the appendix. Furthermore, it follows from eq. ( is implies that the physical HiggsHence, is working to in be the identified with GWvia the formalism, the PGB the one-loop of physical formula broken Higgs given scale mass in invariance. at eq. Λ ( JHEP07(2017)113 UV -plet ), no N . The shows 2 L B.2 1 N , figure N which grow as . On the other hand, the 2 1 g ), the plotted values for Λ = 500 GeV. Since we expect 2.9 GW 1 TeV], this approximation will not and the running is not very sensitive , 5 in figure v ≤ N for – 8 – UV at which at least one of the model’s couplings becomes non- ) in the perturbative range and eventually employ the value UV GW to be of the same order as (Λ ). On the one hand, the scalar beta functions receive stabilizing 2 λ GW B.5 6, which is 13 orders of magnitude below the Planck scale. This poor ≈ GeV) . Largest possible UV scale in extensions of the conformal SM by one real SU(2) ) and eq. ( / UV B.3 To understand the effects of including the gauge sector (green triangles) we consult The pure scalar contribution (blue circles) supports running only up to Lastly, we vary . Accordingly, we observe a rise in Λ eq. ( (negative) contributions proportional toLandau the poles gauge in coupling theN scalar RGEs are thus shifted towards larger energies for increasing log(Λ performance can be explained ascancellation follows: can with take the place scalarinitial and couplings the values alone, couplings of see will the eq.top always ( scalar contribution increase couplings into quickly. our the The calculation more larger (red the drastic squares) this makes the effect running becomes. even worse. Including the the largest possible scale Λ perturbative. According to ourare discussion to right be after seen eq. asup an ( to upper the bound Planck for scale. the true values, which is sufficient to exclude running the exact value of Λ on where we preciselysignificantly affect start the in position of the Landau range poles. [100 GeV which allows the farthest extrapolation into the UV. For given order Figure 1 with vanishing vev. Theinto color account. code indicates which set of beta functions and couplingshowever, are ignore taken this additional constraint and choose Λ JHEP07(2017)113 . µ ) prevents , illustrated 1 2.8 λ and a Landau pole N , it is still overpowered explicitly demonstrates 4 2 g is that the relative contribution 2 ) for better comparison with the positive ones. 2 λg -axis. While in the gauge sector the stabilizing − – 9 – ( y 4 y is triggered by the portal term which therefore must − 1 λ ). It shows the running of the contributions from the contributions to the beta function of the Higgs self-coupling. B.1 soon dominates over the positive + 5. which dominates the Yukawa sector for large enough scales 2 relative is about one order of magnitude larger than the non-scalar ones. in the present case. Additionally, figure 2 λg 1 1 κ κ λy − N > beta function becomes positive for large enough 2 g (cf. also appendix . Running of the 2 . Note the logarithmic scale of the µ The results obtained via the full runningWe (orange can diamonds) further shows illuminate the that above we observations cannot by analyzing the interplay between of the portal coupling Correspondingly, the divergence in be kept sufficiently smallsmall in initial order values to for avoidthat any Landau there pole. is However, no eq. ( possibility for complete cancellations between the Yukawa and gauge in figure scalar (blue), Yukawa (red) andscale gauge (green) sectors withnegative respect contribution to the renormalization by the contribution + However, the most important observation from figure reach the Planck scale in this set-up, so that the present classthe of different models contributions must be to discarded. the beta function of the Higgs self-coupling coefficient in the emerges at ever smaller energies.that At of some the point the scalarUV gauge-sector subsystem Landau scale and pole declines thus drops for below becomes the restricting one. Correspondingly, the Figure 2 The different contributions fromand the green, scalar, respectively. Note Yukawa and thatthe negative gauge the contribution dashed sectors proportional red are to (dotted displayed green) in curve shows blue, the red absolute value of JHEP07(2017)113 ˜ ), Φ = ˜ Φ, , it 2 0 U and ˜ m A 2.11 M σ (2.12) (2.14) (2.13) (2.10) (2.11) = χ | | , conve- ˜ Φ ) 2 U 1 2 Φ , ⊇ 1 vev, i.e. V ) and eq. ( 2.10 finite 2. , namely / GW . defined via + 1) ! . | 2 φ ) can only be between zero and N v 1 1 χ 2 1 λ v κ , develops a n 2 h, σ = ( 1 φ 0 . , , v κ 0 σ , σ , − m i 1 1 i + 2 · · 1 ) as 1 χ m ϕ κ λ ϕ 2 κ χ h κ h 2 β β 2 may acquire a non-zero vev. Together ˜ v Φ = ( α 2.4 2). Following the steps of appendix α , the CP-even degrees of freedom 2 √ − ), we will now investigate the situation in = χ χ λ 2 sin v 2 1 r sin cos . The two mass eigenstates (Φ 12 , π/ 2.4 − + cos β = 2 χ h (0 h χ v , n · i ≡ i ≡ · φ χ 1 v ∈ v v – 10 – ϕ ϕ β , we parametrize φ β κ h h and we see that α v 1 2 = 0 ≡ 1 2 1 + A n n κ n 2 1 2 φ α 2 κ = = v = cos = sin − 1 √ φ χ − 1 2 λ 2 tan v v 6 2 Φ Φ = 1 λ

1 2 2 λ n = 4 2 ˜ M if the portal coupling is negative. Combining eq. ( 1 λ GW is symmetric and real it can be diagonalized by an orthogonal matrix 2 = 0 for real multiplets this gives the relation ˜ . Since the symmetry group of electromagnetism is observed to be unbroken at low M Y σ . Furthermore, we have First, however, let us remark that for finite Anticipating the common origin of both vevs in the Coleman-Weinberg mechanism + will in general mix and it is not clear a priori which mass eigenstate is to be identified GW χ Since niently parametrized by a single mixingcan angle then be written as which can be computed from the potential in eq. ( We will use this formulaof in the a two moment vevs. to obtain information about theh relative magnitude with the physical Higgs bosonscalar found mass at matrix the of LHC. the To answer neutral, this CP-even question, modes we consider the one for positive we find the vev alignment angle in terms of scalar couplings at Λ breaking Λ We emphasize that all couplings inΛ the above relations are to be understood as evaluated at thereby defining the vev alignment angle is now straightforward to write downand the deduce GW the conditions following for identities the which model define under the consideration energy scale of spontaneous symmetry v energies, only electrically neutralwith components of and adopting the notation of appendix cancellation must spring from negative contributions within the2.1.2 scalar sector itself. Real multipletStarting with again finite from the vacuum scalar expectation potentialwhich in value one eq. component ( of the real scalar multiplet, say sectors on the one hand, and the scalar sector on the other hand. Consequently, stabilizing JHEP07(2017)113 , ), is β 2 + 2.12 (2.16) (2.15) m explicitly β . The above α . To see which and ). LHC α 1 H . , ) 2 α ˜ 2 M ]. . 0 to obtain expressions for the 0 separately, starting with the det( ) tan 43 1 · , < = 0 , κ 4 1 42 2 − α α β < κ − − α 2 is now to be identified with the PGB 1 2 i λ  tan is the PGB, whereas the diagonal entries ) , m 1 2 2 2 tan 2 tan 2 κ φ 2(3 and the relative magnitude of the two vevs, ˜ . The presence of such a vev will in general 0 and 4) without loss of generality. But then we 0 and v 2 M φ √ ) − − β v 2 1 √ > , since low-energy phenomenology requires the in terms of model parameters by requiring the ) 1 2 tr( . Combining the above identity with eq. ( 4 κ , π/ 1  1 – 11 – 4 1 √ β κ β > λ = − LHC q 1 > π/ − β H λ  corresponds to the PGB (cf. table 2 χ − ) ( ≡ v λ 1 2 ) such that Φ 1 tan 2 ∈ ˜ ) has two distinct mass eigenvalues, which are given by = 4( 2 − M (12 β 2 + , namely = , m and Φ -parameter from its experimentally well-established SM-like 2.13 m 2 + GW ρ β = tr( β m 2  ) in order to introduce the vev alignment angle m tan 2 2 we learn that for positive mixing angle the vev of the additional . ), we can rewrite the above equation as 2.10 1 1 = diag( ]. However, there are exceptions to this. Considering real multiplets, | 2.11 45 to be diagonal. An explicit calculation yields U 2 | 1 [ ˜ U M 2 ≈ U ˜ M ρ ) together with the constraints multiplet is sizable, namely U L 2.11 In order to see whether we can construct a consistent conformal model in the present Next, let us derive an expression for The mass matrix in eq. ( set-up, let usformer. now study From the table twoSU(2) cases significantly shift away the value of which has the two solutions listedwe in can table deduce a relationsee between once the sign more of table depends on the scalarnot couplings. directly translate In into bounds particular, forIn experimental the other contrast, constraints one, using for unless the one all involvedcondition additional angle couplings in are do restrictions known. eq. imposed ( on the scalar couplings by the GW where we used eq.identity shows ( that in a general theory the relation between the angles deduced by simply calculating thewe diagonalized obtain mass matrix for bothare cases. exchanged For for positive negative matrix As expected, the spectrumwith in the vanishing broken tree-level phase mass, stillalways the contains positive. one PGB Which scalar of of degreedepends broken of the on freedom scale mass the invariance. sign eigenstates of Φ In the scalar contrast, mixing angle. The correct assignment procedure can be As before, we caneq. exploit ( the additionalabove relations tree-level masses between at the Λ scalar couplings given in one, note that weimmediately can see the assume necessity ofHiggs Φ state to consist mainly of the SM doublet field [ one of which will have to be identified with the physical Higgs boson JHEP07(2017)113 ∈ add β B )). ). So (2.18) (2.17) , with we can 2 χ 1 2.12 β cv + 2 φ v -parameter can ρ 1 a non-zero vev (cf. table = ). Solving e.g. for χ ) (and eq. ( 1 2.9 . We have used again χ − N >
> v 2.16 GW /v F 1 1 , i.e. the physical Higgs φ φ K. Higgs G . Hence, we can rewrite v v > 2 2 v 2 , m − Higgs √ 2 φ √ √
, 2 v ≤ 2 1 m 2 κ 2 and ) φ r 1 v v = 1). The theory’s spectrum in the − = κ 1 2 2 =: − − ) 2 + r − ) 1 v ) α < α 1 m κ 2 N α 2 λ r ( ( 1 v ( 2 Higgs − β 2 2 1 κ O -parameter. Hence, for positive q 1 m n ρ 2 tan − 2 tan λ 1 Goldstone modes. Consequently, only one 2 2  ( 2 √ √ , implying π 0, we will therefore not only investigate the 2 −
−→ ( 4 1 c 2 1 π ! ) n > π − = N κ 4 – 12 – 2 N − ) and the PGB therefore obtains a negative mass- ( β < = 1 2 O 2 Higgs κ r B add m − B ! 1 LHC LHC = λ ) = H H 2 ( 1 PGB tan 2 1 implies that the physical Higgs is to be identified with the κ 6= n ( 1 β λ 0 0 = (125 GeV) will acquire a non-vanishing mass term. χ β > β < is allowed and the additional vev’s contribution to the ) and the theory’s spectrum contains one additional scalar with unknown ) to eventually arrive at the constraint φ 1 v 2.6  ). But obviously, the LHC Higgs is not heavy enough to compensate the large, χ 4). The assignment in the first column is done according to the discussion right after . Correspondingly, we can calculate ) as a condition on the unknown couplings evaluated at Λ . Summary of differences between positive and negative scalar mixing angle v -sector spontaneously breaks ). The statements in the second (third) column follow from eq. ( + is positive. Replacing the above inequality by an equality corresponds to the “best- be identified with the PGB. Consequently, only the Higgs mass contributes to 2.6 χ , π/ m K 4 2.17 Moving to negative scalar mixing angles, we now have Furthermore, negative Furthermore, a positive mixing angle implies 2.15 π/ , one obtains 1 − that case approximation” in aλ similar sense as discussed right below eq. ( a model-dependent non-negativeeq. constant ( which now depends on the empirically known quantities and use eq. ( The electroweak vev can generically be written as in eq. ( negative top quark contributionsquare. to In other words,minimum the at the one-loop electroweak effective scale potential which exhibits is a clearly maximum unphysical instead and rules of out a this scenario. component field of PGB (cf. table mass it is only therestrict singlet the which discussion does of not the affect additional the real scalar withcannot vev to this case. a priori in principle be sufficientlysinglet case, small. but also For in larger the multiplets. Note atbroken low-energy this phase point will that thus contain for Table 1 ( eq. ( JHEP07(2017)113 ) ), to . real 2.2 1 N κ 2.18 (2.19) ) in the GW being real. . Secondly, r -range (small χ (Λ 1 1 shows a strict . This, in turn, κ κ 1 | ≤ , the dimension 3 1 κ N κ | ) are then fixed by now illustrates the ≤ − 3 GW 1 = 1), all models develop multiplets. Note that a , we can constrain λ (Λ , the RG running starts r 2 L N N λ at which at least one of the in the relevant 1 UV ) and λ GW . = 500 GeV and vary (Λ 60 1 . λ 1 GW and the discussion there can be adopted. ] and are consistent with the conclusions of | ≤ 4 1 1 κ – 13 – ≤ | = 5 in the case without vev, figure 38 , one of the scalar’s components can be electrically N . from the last subsection, figure χ 1 . Except for the singlet case ( 1 representation. The most important result lies in the Pl 2. Using the numerical value for L M )). In the situation without vev the initial hypersurface is r/ 3 ), negative scalar mixing angles imply is assumed to be real, which directly gives B.2 √ 1 λ . This behavior can be easily understood as follows: eq. ( 2.12 ), respectively. ) showing that the initial value for the portal coupling decreases | ≥ N 1 2.9 κ | 2.11 (cf. eq. ( ) we argued that there exist additional operators for special configurations exhibits a peak for with 1 N 2.2 UV , but that it is reasonable to ignore them. For the analysis of the present class Y ) and eq. ( . This can compensate the aforementioned destabilization for sufficiently low and especially the scalar contributions tend to increasingly destabilize the running and N 2.18 After eq. ( Our findings generalize the analysis of [ The relative magnitudes of the calculated UV scales for different sets of beta functions In complete analogy to figure N N complex scalar, as opposedquantum to numbers a are real one, assignedneutral can to and carry may non-zero therefore acquire hypercharge. a If finite appropriate vev. Weof will discuss this case separately later. the next class of models. 2.1.3 Complex multipletIn with this zero section vacuum expectation weCorrespondingly, value drop all the calculations requirement willand of we be the can based drop additional on the scalar restriction the multiplet to potential only introduced odd-dimensional in SU(2) eq. ( determined by eq. ( with Foot et al. Thisscalar multiplet. concludes our Since discussion we of have extensions not of found a the consistent conformal theory SM up by to one now, we move on to which fixes the valid initialnumber parameter of values added in scalar thefrom degrees present the case, of same does freedom. hypersurfaceon not in depend Hence, parameter on for space. the for each The increasing RGEs, however, explicitly depend is very similarNevertheless, to there those are some observed qualitativeinstance, in differences figure between figure the two set-ups.decrease of Whereas, for Λ model’s couplings develops aof Landau the pole additional isfact scalar that plotted SU(2) also as within the aextrapolation present function all class the of of way models, up to therea is Landau no pole representative at which allows even an lower scales compared to the corresponding case without vev. For the following study,allowed we range. will choose The remaining againeq. ( initial Λ conditions results for the present case: the largest possible scale Λ and negative). Furthermore, generalportal arguments coupling. allow to Firstly, constrainas the a valid consequence range of eq. for ( is the only satisfied for where only the solution with the plus sign gives positive JHEP07(2017)113 3 h κ ≡ -plet (2.20) N LHC L H is fulfilled. 1 2 − = in our calculation. φ Y 1 ) symmetry and thus V − N ( = into our analysis because O 1 Y , V 0. This further fortifies our i ≈ 3 , all (complex) component fields κ 2 κ ) + h.c. as long as χ a N and T C 1 ). | κ χ 2.6 )( φ a – 14 – ετ 2, implying that the physical Higgs mode | is the two-dimensional representation of this matrix. φ . We decided to include ∆ √ ( ε during EWSB. However, in contrast to the real case, h N h/ 3 2 k is a matrix in the SU(2) algebra satisfying the defining re- κ are anti-symmetric in all odd-dimensional irreps of SU(2), + m C a φ , and = v a T 1 C V = ∆ 0 φ . The color code indicates which set of beta functions and couplings are for all φ v | -term explicitly violates the formerly exact 2 a 2 κ T √ − < χ = v 1 − C a . Largest possible UV scale in extensions of the conformal SM by one real SU(2) T ) = 0 so that the physical Higgs only becomes massive through quantum effects C -term is only present for even 3 GW For generic values of the portal couplings As in the real case, we will first consider the situation in which only the SM doublet κ will obtain some finite mass (Λ k 1 λ with its one-loop mass squared given by eq. ( χ the presence of the assumption of choosing the specialto couplings zero close in to all zero. formulas For (of better this clarity, we subsection), will even set thoughacquires we a include finite ∆ vev, i.e. is to be identified with the PGB. The associated Gildener-Weinberg condition is again This term then forms aHowever, gauge since singlet the for matrices arbitrary the it is gauge-invariant not only forall one even-dimensional special representations configuration, with but, aout as particular we that have hypercharge. the just learned, best Nevertheless, for it RG turns running is obtained for a value of into the general potential. Here, lation Figure 3 with finite vev taken into account. of models, however, we decided to include the special term JHEP07(2017)113 . , 2 at κ and 1 , we ), we κ 1 2 1 . One (2.22) (2.21) 2.1.2 GW = GW , are best (Λ 2 Y 2 . In conclu- κ κ Y and 7 is obtained for and 3 ≈ λ N and e.g. N transforms is defined in , GeV) 0. 2 2 / χ -term. This term could, in 3 > UV κ Dκ 2 ) 2 π . Note that the dots for even v + .  GW Y 2 2 1 128 κ (Λ 1 v Nκ Higgs κ 4 ↔ − m + 1) π = k Y 4 2 4 k > − m 2 2 ), we can now compute N =1 N ( k X ) by an equality. Given – 15 – Dκ ) then implies i 2.21 − ϕ + 1 . The symmetry of the figure reflects the exchange to the case of a real scalar without vev, figure h 2.22 2.6 κ 2 1 2 1 Y 4 4 π  , it has practically no effect on the RG running since the Nκ 4 1 64 4 4 · = 0, eq. ( q 2 k of the representation under which = 2 axis were obtained including the m , as well as the renormalization point itself. In the following, we turns out to be close to zero and it is multiplicatively renormal- 3 D K > 1 2 λ 3 add . The largest possible UV scale log(Λ − κ B 4 = and . The portal couplings are to be understood as evaluated at Λ 2 Y } λ ). Comparing figure = 500 GeV and vary all unspecified parameters in the perturbative range. 2.2 GW = 6 and small values of on the . Anticipating ,...,N 1 N N B . But as we see from figure ∈ { 1 2 k − The results of the RG running for one additional complex representation with vanishing Uniquely solving the given model’s RGEs requires to fix the remaining couplings at the = 5 or = 2.1.4 Complex multipletAfter with discussing finite the vacuum expectation casethe value of transfer an to additional aa real complex complex multiplet multiplet representation with we is finite naturally straightforward. vev have a in First, different section we normalization mention for that the field for modes, and ized. Also the otherchosen additional near couplings zero with at respect the initial tosion scale. the after eq. Note real ( that case, these findingssee that are their consistent features with are our very discus- scale, similar: which they reaches both its support maximum running atsion up approximately we to the about find same the that values same of this UV model, by far, does not allow for RG running up to the Planck scale. symmetry of the betanumbers functions with respectprinciple, lead to to differences in the UVY scale for even-dimensional multiplets with optimal initial value for GW scale, namely will assume Λ vev are shown in figure N Similar to the previous discussions, we areof only potential interested in Landau an upper poles boundingly, for we and the replace therefore location the employ inequality incan the eq. then “best-case ( simply approximation”. compute the Accord- corresponding value of where the overall factorfields. of The two takes Dynkin into index appendix account the complex nature of the component the Gildener-Weinberg scale. Using eq. ( An explicit calculation yields with can show that requiring real masses for all new scalar particles implies non-negative leads to a mass splitting between the individual components, which is proportional to JHEP07(2017)113 L ). ), 2.2 2.23 (2.23) (see also = 2. The . 1 scalar (start- Y 2.2 real 1. As a consequence -parameter invariant, < ρ χ -parameter. However, for /v ρ φ v α , = , , 0 χ  tan  2 and a septet with 2 Y /v 2 / 2 φ 1 Y 2 it is a second Higgs boson and — κ v √ 3 ) / 2 λ 1 = 1 ) √ := 1 N, 0 = 1 δ Y N, α δ , which we summarized in table − Y − (1 tan – 16 – 1 2 2.1.2 , singlet with zero hypercharge is equivalent to the + (1 + χ 2 1 tells us that v λ κ singlets and will be covered in section 2   1 = 0. The only new aspect is an additional GW condi- 1 4 1 4 √ 2 0 1 doublet with real := := := κ complex 0 2 0 1 L 0 χ − κ v λ 0 2 2 for the electrically neutral mode. As for the complex multiplet = 0, a doublet with and vanishing vev. λ √ 1 Y Y λ σ/ ) = 0. Using the aforementioned replacements, also the scalar mixing + )). We only need to use the primed quantities, defined in eq. ( χ GW v is an SU(2) (Λ = χ 3 2.11 κ 0 m χ . Largest possible UV scale in extensions of the conformal SM by one complex SU(2) ) now reads 4 ]). If 46 20, there exist three complex representations which leave the 2.11 For positive mixing angle, table ≤ -plet with hypercharge N namely a singlet with description of one additional description of twoe.g. additional [ instead of the unprimed ones.eq. By ( this, for instance,tion, the namely Gildener-Weinberg condition from phenomenology is the same as in section the additional vev is sizable and thus will in general tarnish the we obtain exactly the sameing equations from as eq. in the ( case with the additional Introducing the following quantities Figure 4 N especially without vev, we apply the Gildener-Weinberg formalism to the general potential eq. ( JHEP07(2017)113 - ρ N — 4 the are best the case χ v fermionic 2 β κ gauge coupling and Y 3 λ . In order to render )). In summary, the to the beta function ). We conclude that, 2 add 2 B.6 B λy and ) for which the RG running 1 were massless. Here, the addi- 0 and for not too large N,Y m χ χ , we now consider the case in which > v 2.1 φ v ] suggest, the couplings 44 is the PGB and its mass is generated by the are both qualitatively and quantitatively very – 17 – LHC = 2, the beta function of the U(1) 2.1.3 H Y GeV). are added to the SM. In doing so, we will neglect all but and ) symmetry that was spontaneously broken and, by the 7 ξ N ( (10 O O 2.1.1 and explicitly break this symmetry. Consequently, the masses of the χ 2 κ and does not provide us with a consistent, minimal conformal model with a 3 λ β also in the case of a finite vev. We thus conclude that for negative . In particular, there will be no combination ( χ 3 Before we proceed to the next class of models with two additional scalar multiplets, This exhausts all reasonable possibilities in the case of the conformal SM plus a complex For negative scalar mixing angle we have positive, the scalar couplings therefore have to take larger initial values in comparison to supplemented by fermions interacting via Yukawa couplings with the scalar2.2 sector will, too. SM +In two complete scalar representations analogy totwo our real scalar discussion multiplets in section any massive fermionic particle willB give negative contributions to the model without fermion. Secondly, addingRGEs a even fermion more will ultimately due destabilize to theof scalar its a positive generic contribution scalar proportionalif coupling to a (cf. given the theory discussion develops Landau about poles figures well below the Planck scale, then the same theory model up to now. let us comment onrepresentation(s). the conformal One SM can with easilyconformal one see model, additional that since scalar this additional and fermions set-up additional destabilize will the also RG fail running in to two provide ways: a firstly, consistent of the complex multipletin with figure finite vev leadscan to be similar extended results far as beyond in the real casemultiplet shown that develops a finite vev. We have not found a consistent minimal conformal complex multiplet acquiring ais finite further vev substantiated are by comparablethat the to the observation the results — from real stated section similar. case. within Therefore, This the there argument discussion is ofcomplex no figure reason to expect a large difference when going from real to tional couplings charged modes are proportionalprevious to analysis the as symmetry wellchosen breaking as close the parameters. to analysis zero However, of forimately as optimal [ restored. RG our running, With and the the additional symmetry of masses the close potential to is zero, approx- the results of the model with complex scalar multiplet that develops a finite vev. parameter is safe.additional The massive Higgs scalar boson modespotential at possessed the a one-loop global level.Goldstone theorem, In guaranteed the that case all of modes a besides real multiplet the This contradicts our principle ofFinally, minimality and we we investigated will the not septet furtherand model. consider the this relatively In case large this here. hypercharge runs case, into due a to Landau the polecase large before of dimensionality positive reaching the Planck scale (cf. eq. ( without additional assumptions — would have Yukawa couplings to all of the SM fermions. JHEP07(2017)113 ) L , ξ ) N ξ ( † ) = 0. (2.24) (2.25) ξ O )( are nec- are now GW × χ † ) ξ φξ (Λ χ : scalar con- χ φ κ ( N λ ( χξ Left and κ O acquires a finite and χ ξ ) + φχ ξ κ † nor ξ χ )( φ † φ . ). For the model’s RGEs we ( 2 φχ v 2.3 κ φξ κ ) + χ 2 and induces the following masses for = 2 † χ 2 ξ √ )( m . h/ φ † B + φ ( v or – 18 – φχ = 2 κ 0 v φ + φχ 2 κ ) ξ † ξ = 2 ( ξ 2 χ λ m + 2 ) χ ). This is also in line with our notion of minimality discussed earlier. † χ : scalar and top-quark contributions. ( χ λ 2.1.3 + Right 2 ) φ † denotes the SM complex Higgs doublet as before and both φ . Largest possible UV scale in extensions of the conformal SM by two real scalar SU(2) ( φ φ λ new scalar degrees of freedom: = V all Since all physical massesessarily have non-negative to at be the real, GW the scale. two The portal one-loop couplings mass squared of the physical Higgs 2.2.1 Two realSimilar multiplets to with our zero one-scalar vacuumvev. discussion, expectation we In value the first GW assume formalism, thatHence, this corresponds neither the to physical the Higgs renormalization condition symmetry is breaking necessarily then the proceeds PGB via of broken scale invariance. Electroweak where supposed to satisfy realityagain conditions refer like to the the one formulas in given eq. in appendix ( the standard quartic and portalparticular couplings, section in accordance with ourIn previous effect, analyses the (cf. aforementioned in symmetry assumption in introduces the scalar an sector. additional The global associated scale-invariant tree-level potential then reads Figure 5 multiplets with vanishing vevs.tributions The only. results were obtained using RGEs including JHEP07(2017)113 ). ), 0, φχ κ may GW itself. 2.27 (2.26) (2.27) (Λ χξ K > κ φχ GW we obtain κ UV and e.g. ξ ), negative N ), for which consistent , ξ B.2 , as well as negative and χ ). ξ N λ ,N . Accordingly, for negative GW χ ) is not constrained a priori. . N χξ (Λ and κ GW φξ χ φξ . κ v (Λ Higgs λ ) and taking into account κ ξ χξ m 2 φξ 2.6 N κ κ π 4 ξ 2 ), but in the present situation we have N 2 √ ⊇ 2.6 π + > φχ 16 (1) κ 2 φχ = 500 GeV and vary all unspecified couplings 2 φξ β κ κ χ ξ – 19 – GW , as well as the renormalization point Λ N N ), for which also the other free portal coupling, ξ , again exemplifies that including a generic Yukawa χξ + = κ 6 ,N 2 φχ χ add κ and N χ B and ξ N 5 λ ) is replaced by an equation. Given q , χ λ 2.27 , there can be cancellations already within the scalar sector. In ad- χξ κ summarize the findings for the largest possible UV scale Λ 6 . As revealed by scrutinizing the RGEs in eq. ( is again given by equation eq. ( ). Thereby, each of the above panels corresponds to one set of points in χξ h κ and ≡ 5 2.1.1 LHC . On a qualitative level, each individual case gives results resembling those of its (1). However, this is simply a consequence of the constraint derived from eq. ( H 1 O Finally, the most important result from the present paragraph is the following: the We furthermore find that for given dimensions of the scalar multiplets, the farthest Figures Uniquely solving the given model’s RGEs requires to fix the three remaining couplings is of calculation based on theinvestigated full models set can of be RGEs extrapolatednot with all find all the a terms way consistent up included conformal to model shows the in that Planck this none scale. class of of Hence, the theories. we do often sizable keep the scalar part ofcontributions portal through coupling beta mixed functions terms underand like control sufficiently large by in generating negative dition, there exist combinations ( on the left-hand side incoupling figures destabilizes the flowquantitative level, and the thus Landau decreases polesat the in somewhat higher the maximal scales present compared possible study to the of UV corresponding two divergences scale. in additional the scalars one-scalar On develop case. RG a running is obtained for vanishing quartic couplings, the present results to(cf. the section outcome offigure the calculations with oneone-scalar extra counterpart. scalar The without respectivefor vev differences both between set-ups. the In four particular, cases comparing are the panels also on similar the right-hand side with the ones in the perturbative range.due to Whereas the the requirement quartic of couplings vacuum stability, are the confined sign toworking positive in values the “best-caseparticle sectors approximation”. included in In the computation total, of we the RG show running. four It plots, is instructive differing to compare in the the above inequality eq. ( we can then simply compute the corresponding value of at the GW scale,For namely the following study, we will choose Λ For the purpose ofradiative finding symmetry out breaking whether there isevitable. exists possible, To a facilitate explicit pair those, calculations ( we of will the again RG apply the running “best-case are approximation”, in- in which Combining the previous identitywe with arrive equation at eq. ( boson JHEP07(2017)113 . vev. A.2 as we UV finite if such a scale Λ GW . The results were obtained : full RGEs. 5 Right set of RGEs towards the UV starting from the – 20 – , the model allows for an extrapolation all the way full R and S of the Gildener-Weinberg scale in the way outlined in appendix . During the evolution of the early universe, the tree-level potential would GW Λ : RGEs including scalar and SM gauge contributions. . Largest possible UV scale in the set-up described in figure > calculation Left Furthermore, we test whether no GW condition is satisfied at any intermediate energy In the following, we will demonstrate that already for the simplest case with two ad- solution manifold from stepstability one. requirements are At met each by all RG running step, couplings. we checkscale if Λ basic perturbativityhave and developed a non-trivial minimum before reaching the original Λ did before. In particular,mation”. we Instead, will we not will use applythe a the model’s two-step previously procedure: parameter introduced space first, “best-caseare we on determine approxi- satisfied. which the the In hypersurface given in particularsistent low-energy and phenomenology in requirements contrast toSecond, our we analyses will before, numerically we solve perform the a fully con- ditional scalar gauge singlets up to the Planck scale,scale. while Since giving this rise timeleast to our the the goal Planck correct is scale, phenomenology to it at actually is the prove no that electroweak longer the sufficient largest to possible calculate UV an scale upper is bound at for Λ As in the previous section,added we to will the discuss the conformalposed situation SM. to of However, have two real a whereasinvestigate scalar trivial earlier cases multiplets vacuum in both being expectation which additional one value, of scalars we the will were multiplets now has sup- relax a component this that assumption acquires and a Figure 6 using 2.2.2 The minimal conformal model JHEP07(2017)113 . ) ) 2 ), σ Z 065. + . 2.11 iden- 2.28 2.13 S 0 (2.28) (2.29) v < not = | , eq. ( S is φS κ | LHC 2 2.1.2 H R ) φ ). † 1 φ , namely ( . β which transforms non-trivially 3 φR κ S . ), we can therefore directly is a gauge singlet whose vev SR + 6 S GW 2 = 0. (cf. table GW κ . . i 6 S , which therefore might be a viable . (Λ β ) + S
h S φ S 2 S R GW φS at Λ † R λ v v = 3 κ φ ( rather than S S → → SR v 5 at Λ φS R κ 2 κ χ 2 κ R, v + 2 Higgs + be 4 + 2 φ φ 2 m 4 ) v 2 φS λ 2 φ 4 κ , v R + Z † ) is the same as before in section – 21 – φR must , λ −→ − R φ κ ( φ φS λ φS R ) = λ κ h, σ κ ) acquires a finite vev during EWSB, i.e. + SR φS S 4 → → 4 = 2 κ S ) = κ ( 1 1 2 R S S κ λ φS + λ λ m κ 2 ( ) apply to the present situation. In particular, we again have + R 2 as usual, we then obtain the following tree-level mass φ 2 2 λ ) √ S φ 2.16 at the GW scale, namely † ) implies absolute stability of h/ φ SR φ 3 ( ). For given portal coupling κ + λ φ 2.29 would be spontaneously broken by 2 mass matrix of ( φ λ + symmetry to be exact, it 2 v × 2.15 2 Z = ) to eq. ( = Z V 0 = 174 GeV in accordance with the fact that φ v 2.10 does not acquire a finite vev, it does not mix with the other CP-even scalar ≡ φ other fields in the theory left invariant. The three terms in the last line of eq. ( from eq. ( R v + all Next, we use the assumed vev configuration in form of the GW condition in eq. ( Here, we concentrate on the case in which the physical Higgs boson As Let us now first concentrate on the conformal SM extended by two real scalar gauge If we want the m 3 to further reduce the number of free parameters at the initial scale: under it. Otherwise, We set does not contribute to theto electroweak determine scale. the range The of above portal equation couplings can consistent with furthermore positive be used tified with the PGBby of broken scale invariance.calculate In the this value situation, of the Higgs mass is given Using the above replacement rules, itgiven in is moreover eq. straightforward ( to showto that distinguish all positive formulas and negative scalar mixing angle modes. With Furthermore, the 2 upon replacing with are odd under thethe above definition transformation in and eq. ( aredark thus matter forbidden. candidate. Note, furthermore, that In order tosymmetry reduce in the the following number way of free parameters, we impose an additional global assumption inconsistent. singlets (CSM2S), one of whichThe (say most general scalarclassical potential scale which invariance is can consistent be with written the as SM gauge symmetries and existed. Hence, SSB would already have taken place at Λ which would render our initial JHEP07(2017)113 ) is the GW . As Λ PGB / 7 not i m calculate ϕ h and φS κ ). = 1. (grey area on the left). ξ v 2.30 N = χ N , and the PGB mass. In accordance φS κ 15 GeV, we then find a viable region of . by setting B PGB – 22 – m and the electroweak scale . The beta functions for the CSM2S can be obtained Pl GW M ), we immediately discard those parameter points, which RGEs. In each RG step, we check basic perturbativity and 2.8 . In particular, we will show there how to consistently becomes sizable (grey area on the top). For small portal couplings i A.2 ϕ h 0.) The white arrow marks the example point from eq. ( complete one-loop β > . Largest possible UV scale in the CSM2S. (Case in which the physical Higgs is and sufficiently low PGB masses, | The available parameter space can be further narrowed down by noting that the Our calculation for positive scalar mixing angle gives the plot shown in figure Next, we need to clarify whether the model can be consistently extended all the way φS κ parameter space (red area). Into this the regime, Planck a scale fully consistent is extrapolation possible, of while the model reproducing up themixing correct low-energy in phenomenology. thethe Higgs LHC. sector The will currently effect measured signal the strength signal constrains strength the of scalar Higgs mixing angle events to observed at imply a large separation betweenSince Λ the effective potential’sis perturbative too expansion large, is we additionally noand exclude longer the points, condensate reliable for if which| the ln( hierarchy between the GW scale stability criteria of the model’sas any couplings inconsistency and occurs abandon below from the the given parameter general point formulas given as in soon appendix discussed above, we vary one of thewith portal the couplings, discussion after eq. ( presented in appendix the GW scale. up to the Planckthe theory’s scale without any intermediate scale appearing. We do so by solving Figure 7 PGB, i.e. The determination of the remaining parameters’ initial values in terms of JHEP07(2017)113 ) , is , we 2.19 stays φS scalar (2.30) φ κ λ 4 leptons, 2.1.1 ) and ( → 2.8 H parameter are ex- independent can provide a very γ T . While the hadronic 4 4 jets, γ GeV so that the Higgs → → 14 H . H 0018 , we can rule out all points below 2 leptons 2 . 0 01. 7 . is that the pure scalar contribution → − 2 = = 0 H therefore no longer implies that one cou- φS SR ]. Corrections to the B κ 47 [ , κ were obtained for the example point marked in – 23 – β 8 = 313 GeV. Stable RG running up to the Planck , there is now a larger number of parameter are necessarily both loop-suppressed and 015 and R . S m 2.1 = 3 GeV 2 jets 2 leptons, = 0 R PGB → λ m H , γ ) or smaller. Second, since all scalar couplings can now be of the 2 ]. By including this limit in figure − 2 jets 2 43 (10 , O 42 → to be of order one, the corresponding portal coupling in the CSM2S, H 44 [ 1 . , κ 0 , namely γ 4 7 ≤ Now that we have understood how a stable RG running is realized in the minimal We attribute this improved behavior of the scalar contribution to two features. First, Now, the key difference with respect to figure Let us now try to gain further insight on how the scalar couplings can remain free β → scalar contributions. Those cancellations maycouplings help small. to keep the beta functions of the portal conformal model, let usbriefly look discuss for a non-minimal means extension to of achieve the larger conformal PGB SM masses. very similar To to that the end, CSM2S. we compared to the models indegrees section of freedom (multiplets).pling Positivity must of be particularly largerequire at the initial scale:preferably whereas e.g. of equations ( same order of magnitude, there exists the possibility of cancellations between different contribution from thecoupling Yukawa first coupling decreases. for At larger energieswill scales, hence up the ultimately portal develop to a termssmall Landau start 10 pole. up to dominate. to However, our the The calculation Planck coupling shows scale. that Requiring the correct vevscale implies is then e.g. possible for no longer dominates over the whole energy range. Rather, it is exceeded by the stabilizing of Landau polestherefore in compare the the different present contributionscorresponding model. to the results Higgs presented In self-coupling in beta figure figure analogy function. to The our analysis in section mediated by the electroweak gaugeclean bosons. signature Furthermore, and the onlyinteractions. has This the opens background ato window the coming Planck of from scale, opportunity highly directly to at suppressed test the Higgs a TeV symmetry scale. self implemented close symmetry. A further interestingdecays. phenomenological aspect The is the Higgsparticles. existence boson of In exotic can this Higgs H decay decay into chain possible two finaldecays PGBs, states have which contain, a then largewell further background distinguished. decay at In to the particular SM LHC, the the leptons final are states pairwise boosted containing leptons in contrast can to be a decay the dashed black curve.measurements Another performed type of at constraint LEP.their comes However, contributions from as to the the all electroweakproportional precision oblique new to particles the are smallpected scalar to mixing be SM angle negligible singlets, as well since the model’s scalar potential does not violate custodial sin JHEP07(2017)113 The 4 multiplet, L ). demonstrates 2 9 (3) symmetry in the triplet sector. O triplet with vanishing vev. L , two-loop effects will be negligible, since the Pl – 24 – M ) is replaced by a global ). Consequently, potential Landau poles will develop A 2.29 contributions to the beta function of the Higgs self-coupling for , their contributions can only have a significant impact, if ). The different contributions from the scalar, Yukawa and gauge 2 π is exchanged for a real SU(2) relative 2.30 16 R / ), a given PGB mass can now be produced for smaller initial values A.5 symmetry from eq. ( 2 Z 35 GeV becomes accessible in the CSMTS. With respect to the minimal model, . Running of the ) and eq. ( ≈ A.7 Let us finally comment on the robustness of our results under inclusion of higher The imposed 4 PGB provide a perturbative evolutionone-loop up beta to functions arethe already RGEs sizable. of In the contrast, minimalquantitatively. as conformal If there model, two-loop are two-loop contributions mild contributions turned may cancellations out change to in our be results sizeable in some areas of param- by any additional global symmetry. loop orders in thesuppression RG factor running. ofthe 1 Since one-loop higher-order beta terms functions come are with anomalously an small. additional Hence, loop in all the cases that failed to eq. ( of the scalar couplings (cf.at appendix higher scales.PGB A masses would straightforward be and toe.g. replace minimally a the invasive triplet septet. way by Note a to that higher-dimensional generate in real SU(2) this even case larger the dark matter stability does not need to be enforced Here, the singlet scalar resulting model willthat be compared referred to to them as minimal conformal CSMTS model in an thetwo extended heavy region following. of scalar PGB Figure degrees masses of up freedom to are added to the theory’s spectrum. According to Figure 8 the example point insectors are eq. displayed ( in blue, red and green, respectively (cf. also caption of figure JHEP07(2017)113 the (3.1) not ) is the conformal t ( t/a = d η . ) where d i 2 µ µ ~x d T − −h 2 η = – 25 – (d 2 Pl π 2 ) 8 M t ( a R = 2 s 0). denotes the Ricci curvature scalar, which represents the gravitational field. The β > . Largest possible UV scale in the CSMTS. (Case in which the physical Higgs is R the Einstein equation, given by where vacuum expectation value of the scalar-field EMT sources the gravitational field and is stable RG trajectories.the Motivated gravitational by background. thisassume As observation that a we it is model consider a system good aconveniently parametrized description we free for by consider the scalar d early de-Sitter field statetime geometry of in coordinate. our as universe. The we The governing metric equation ansatz is in this highly symmetric system is the trace of In this section wecluding sketch effects how from gravity. our Wein scenario choose curved here might the space-time be semi-classical as embeddedrequires approach it to in concepts quantum is a fields of the broaderprevious general most context section relativity conservative in- that method and small currently quantum or available even field and vanishing theory. only self interactions We in have the seen scalar sector in lead the to destabilize the RGE running. Inexcluded. such However, cases we the expect affected our parameter findings space to would remain need to valid from be a qualitative3 perspective. Matching to the semiclassical regime in gravity Figure 9 PGB, i.e. eter space with mild cancellations between the one-loop contributions, the former might JHEP07(2017)113 φ (3.4) (3.5) (3.6) (3.2) (3.3) is constant the ) can be chosen a R . ( define the vacuum 2 g ) † k ~ . This parametriza- η A eff ( , a ) is as follows m k ~ η ( 2 eff A h m . . can be found in an adiabatic
+ k ~  2 ) f 2 φ x k ( . . We will only sketch the slightly ∗ ~x k ~ ≡ ·  φ f ) in order to cancel out the friction k ~ 2 † k i ~  η ) = 0 and the coupling of the scalar , e ( A 1 6 η k ( /a m h a − = 0 ) ) + ξ  a k ) are the solutions to the equation of motion x (  ( h x R g k ~ 2 k ( ) = 1 3 2 f k ~  a k V ~ – 26 – f ( 1 6 + 1 g 2 A + Ω 2  − 00 k √ φ h ξ 2 k ~ =  X m k 1 2 + f 2 ) = = x m µ ( µ  φ T ) will drop out of the vacuum expectation values of operators is the Ricci scalar of the Friedmann-Robertson-Walker (FRW) ]. The basic idea is that we construct the field operators as + a ( 6, this quantity is zero at tree level and we need to compute its 2 R g / 49 k , . This vacuum state is called the adiabatic vacuum as it is assumed = k ~ 48 = 1 k 2 ξ Ω , the rescaling needs to be η = 0 for all i 0 | k ~ The solution to the equation of motion for each mode A spacetime and we havetion introduced shows the immediately effective the masscurvature-induced special term parameter amounts case to of a the mass de-Sitter correction. spacetime:series. if When substituted in the equation for the EMT between two vacuum states and where the oscillation frequency is defined as In the above equation, it into a harmonic oscillatorcoordinate equation with time-varying mass.term. So choosing We conformal find time that with this parametrization the equation for The rescaling function quadratic in fields and thus hasvalue no physical of meaning. the This EMT, includessuch the which that vacuum expectation we the friction will term compute in below. the general equation At of the motion same is not time present and transforms expansion in the components’ derivatives.into In an our assumption about case the of scaleform de-Sitter factor of geometry and this the its translates solution time for derivatives. the We begin sclar with field a equation general of motion In the above equation thefor mode functions the scalar field inby curved background. The ladderthat operators the components of the metric tensor change in such a way that we can define a sensible vacuum expectation value quantizingtechnical the calculation scalar at field thisgogical point description and [ make reference to the literature for a more peda- As we assume conformalto initial the conditions curvature with given in four space-time dimensions by JHEP07(2017)113 . ) = 0 3.1 (3.7) m ]. Their reheating . ), where 6 we can 6. 50 6 6. Thus / / / 0 eq. (  H 2 H t R = 1 2 i → from 1 exp ( µ µ 6) ξ  / T ∝ h 1 ) t − ( ξ a ( 1 2 from the value 1 0 with corrections of order ξ = 0 and + ≈ R . This is a valid approximation m eff 2 1 m − to the trace of the EMT is positive m ) ξ 2  6) deviates by a value / ξ − 1 1 — is present and would have to be canceled by − − i ξ ( µν 1 3 F (1080 µν + – 27 – F h π R 2 q 18 0. This point in the evolution is special, as the trace pl m = 0. Then, once M + → 2 eff 2 eff ≈ m R m eff , which is called the gravitational conformal anomaly. 1 2 m 2160 R − 1  2160 2 at higher orders, leading to a deviation of )], it unavoidably cancels the contribution of the gravitational conformal π − 1 ξ ) it is clear that this vacuum set-up leads to an inflationary solution with 0 and thus 16 3.7 = is much smaller than the electroweak scale the boundary condition of vanish- 3.1 i → = 0 | i R 5 ) to infer that µ 0 6 which means that | . Under the reasonable assumption that at reheating the space-time curvature µ 12 is the Hubble rate. At first glance this might seem as inflation would continue / µ T 3.1 µ | reheating R/ 0 T = 1 h H √ | ξ We argue that this cosmological scenario is a good motivation for our field theory set- Since the contribution of the non-minimal coupling From eq. ( 2 0 The general form of the trace of theNote energy-momentum tensor that can we for provide example here be a found toy in [ example of a scalar coupled to gravity in order to demonstrate our h reheating = π 5 6 eq. (3.2) reduces to our expression assuming theproposed de-Sitter mechanism. symmetries. Infunction a of full the SM hyperchargethe context — gravitational the proportional conformal contribution to anomaly to as well. the EMT due to the non-vanishing beta but rather used the adiabatic expansion in metric derivatives. 4 Discussion The present study contains thein analysis which of radiative simple symmetry conformal breaking extensions within of the the Higgs Coleman-Weinberg mechanism sector can take ing explicit masses is a goodinvariance approximation turns for out our to study be of an the approximative electroweak symmetry sector with and corrections scale ofup order with a classicallythe vanishing scalar mass fields. and Note asymptotically that we small did quartic not self rely on interactions a of loop expansion to arrive at this conclusion, shows that of the EMT vanishes evento at the quantum FRW level. radiation-dominated Therefore, epochH at takes the place end and of inflationscale a transition later FRW evolution. Itparameter is of important the to scalar evaluateand the field scale during evolution this of process. theuse effective As eq. mass mentioned, ( at thefor beginning a non-vanishing trace of the EMT. At the same time in the limit at vanishing quartic interaction ofinto the a scalar running the of running of the gauge couplings translates definite [cf. eq. ( anomaly. This ends the inflationary epoch and allows therefore the universe to enter its a constant space-time curvature andH a scale factor timeforever, evolution but as thescale space-time and expands induces the a temperature running drops, of which the changes parameters. the As energy we discussed in the introduction, even background As discussed, our16 initial conditions were chosen to be summing over all modes, it leads after renormalization to an expression in the de-Sitter JHEP07(2017)113 2 L + to λ 2 . In ) 4 χ y † . φ − ( 3 2.2.2  4 7 κ = 2 V 8 ]. 24 is a generic gauge coupling. g scalar representation turn out to contains terms proportional to L . Therefore, it is obvious that, with λ 2 , where 2 λy λg ], in which the SM is extended by one real 2 − – 28 – ). gives a negative contribution proportional to 1 2 , y 2 in the potential. The RG running is then highly unstable, , κ (1 ∼ χ , the fermionic contributions always destabilize the system even more. ]. 2 λ with positive coefficients. Contributions from gauge bosons can decelerate the run- -singlet scalar and right-handed sterile neutrinos, turns out to be unstable. Indeed, L 2 κ Other extensions of the Higgs sector by one SU(2) Having excluded those simplest theories, we find the minimal model, which leads to In particular, the simple model discussed in [ We find that none of the conformal extensions of the Higgs sector by one scalar SU(2) As simple extensions of the Standard Model (SM), we consider theories with the same , which is only present for The different treatment of the wave functionIn renormalization order leads to to check deviations this, with we respect have performed to a the fully consistent analysis as presented in section  7 8 mass being strongly suppressed with respect to the vacuumresults expectation of value [ of the new particular, we have taken intoh.c. account the complete scalar potential including the term ∆ correct radiative breaking of electroweak symmetryof and the is Higgs RG sector stable, by among twois scalars. the extensions the To be SM precise, augmented our analysisvacuum by shows two expectation that scalar the value. minimal gauge model fine-tuning. singlets, In one this In of which system addition, has a the to theory light obtain contains Higgs a a non-zero boson pseudo-Goldstone can boson be (PGB) realized with without its unavoidably introduces positive terms scalinggrowing as + be unstable as well, as for example the conformal inert doublet model [ unstable. SU(2) even though a Yukawathe coupling beta function of the Higgs self-coupling, the scalar field wave function renormalization leads to large portal couplings since the beta function forand the Higgs quartic coupling ning, as they contain negative termsHowever, with of the growing form scalar couplings the scalar sector dominates and the system is still multiplet meets the stability criteria.and acquire The a additional vacuum scalar expectationpole can value far be or below the not. either Planck real Inlogical scale. or requirement all The that complex cases the reason the Higgs is boson models that mass in develop is all a roughly parameter Landau half points its the vacuum phenomeno- expectation value, which can only vanishour in necessary the condition UV isand once does that gravity not the contributions develop renormalization-groupWeinberg Landau become formalism, (RG) poles significant. ensuring running below the remains theinto Thus, perturbative account Planck stable nature the scale. complete of one-loop Wefrom our have RG field expansion, used equations. renormalization. and the In have Gildener- particular, taken we include contributions much higher scale, theconsistent, usual the gauge vanishing hierarchy of problemhow the is this trace avoided. scenario anomaly can For at this be the scenario realized high by to a scale be semiclassical isgauge matching necessary. to group. gravity We in discuss Hence, section there is always the beta function of the Abelian gauge coupling place. As a consequence of non-linearly realized conformal symmetry implemented at a JHEP07(2017)113 ] 43 , 42 44 [ . 0 ]. 51 ≤ ] and references β 54 – 52 6 in the far ultraviolet, depending indirectly on – 29 – . symmetry. In contrast, if the second scalar is a septet 4 and 0 2 . Z 48. The mixing can be compared to the SM prediction which leads . Accordingly, in the RG-stable region of parameter space the scalar 9 12 and 0 . scalar content. This is due to the fact that the top Yukawa beta function at L Furthermore, the points of the parameter space in which we find stable RG running Another important phenomenological consequence is that the mass of the pseudo- Within the minimal model, we find that one of the scalar singlets is an excellent dark Our study raises the general question about the stability of a Standard Model extension The exact value of the Higgs quartic coupling at which its beta function vanishes is sensitive to the top 9 quark mass and can have a small positive value or even vanish for some top mass values [ angle between 0 to a constraint on thetherefore mixing already angle. rules out The amight current certain be LHC fraction tested upper of by limit the the of parameter LHC sin space. in the The ongoing complete model run. Goldstone boson is found toas be a always few below half GeV. theHiggs This Higgs width necessarily mass than leads and in is to the preferably additional SM. as Higgs light decays andpredict therefore sizable to singlet scalar a admixtures larger to the physical Higgs state with sines of the mixing overclose the universe. However, abeyond detailed the study scope of of thecrucially this dark relies matter article. on phenomenology the We goes its assumed stress stability again does that not the need stability to of be enforced the by DM any candidate additional symmetry. matter candidate, sincephenomenology it is similar does to the nottherein. Higgs develop portal We model, observe a see the vacuum for dark300 example GeV matter expectation [ mass and value. to 370 GeV. beus confined Its Furthermore, is to we effective a consistent checked rather that with small the region cosmological parameter between observations, space i.e. considered the by scalar field abundance does not UV is maintained. beta functions have veryto small achieve values the at necessaryrealized the anomaly at Planck matching, the scale. thus quantum level. indicating This that is conformal the symmetry desired is behavior observation that in the SM thereself-coupling are approaches no Landau a poles constant belowthe (yet the Higgs Planck negative) scale sector value and portal the inthe couplings Higgs the Higgs are UV. beta necessarily function In present.portal lead our couplings to Their extensions are vacuum positive sufficiently of stability small, contributions at the to all near vanishing energy of scales. the Higgs Additionally, beta if function the in the towards a stable value betweenthe 0 SU(2) one-loop depends on itself andin the the three ultraviolet gauge regime. couplings,quartic which coupling As only can can show a have beYukawa mild a and seen running regime small from portal of the couplings RG-flow appropriate stability to RG the at equations, new finite the scalars. values Higgs given It a is large an interesting top and non-trivial potential. Furthermore, in theRG three-scalar running potential can the be portalthe negative term two and contributions reasons thus to why mutually the the system stabilize remains their stable beta up functions.under to RG the Those translations. Planck are scale. A general observation is that the top-quark Yukawa coupling runs singlet scalar. This turns out to be a natural set-up with no need for large couplings in the JHEP07(2017)113 . to be (A.1) A.2 M y ]. Within 40 in section GW symmetry. Even if such 2 Z , we present our method does not develop a finite vev. R A.1 due to the , R ` φ k φ j φ i φ Additionally, we obtain Dirac-type Yukawa 10 ijk` . f S – 30 – (Λ) are subject to RG running, where Λ is the 1 f 24 = f ~ Φ) = ( V ]. Of course, it remains to be checked whether including the Majorana Yukawa 15 ~ Φ denotes the collection of all real scalar degrees of freedom in a given theory. After discussing the formalism’s basic principles in section To summarize our results, we found that it is necessary to add at least two scalar fields Finally, we would like to remark that in the minimal conformal extension of the SM neu- Note that the right-handed neutrinos, however, do not couple to 10 where Note that the coupling constants a coupling existed, it still would not lead to a Majorana mass term because A.1 Basics Gildener and Weinberg start by introducing a general tree-level scalar potential of the form a potential in abrute-force theory algorithms. with It can multiple bethe scalar one-scalar considered fields case. the without analogue Furthermore, the of havingcouplings formalism the allows to and RG-improved us ensures potential resort to the in derive to conditions applicability numerical on of the the scalar loop expansionto of consistently the calculate effective the potential. scale of spontaneous symmetry breaking Λ A The Gildener-Weinberg formalism In this appendix, wethe review framework the of Gildener-Weinberg the formalism Coleman-Weinberg introduced mechanism, in it [ allows to systematically minimize Acknowledgments We acknowledge very helpful discussionsmanuscript. with We Hermann thank Nicolai Pavel Fileviez and Perez his for comments discussions. on our value to have a modeldiscuss which is is an stable under extensionfound RG of that translations. the the Thus minimal SM the model Higgsmixing minimal contains a in sector model viable the we by dark Higgs two matter sector,the real candidate which realization and singlet might predicts of scalar be sizable conformal a fields. models. powerful We tool have to rule out or get a hint about anism [ coupling negatively influences theeffects RG of running. the top Based quarkcontrollable. on Yukawa coupling our on observations the regarding RGEs, we the expect changes dueto to the Standard Model, one of which has to develop a non-vanishing vacuum expectation trino masses can easily beas accommodated. SM Once gauge we singlets introducetype they right-handed Yukawa coupling neutrino naturally to fields the possess scalar acouplings singlet conformal with and the gauge-invariant, SMthe Majorana- lepton Yukawa couplings and lead Higgs to doublet. a neutrino After mass electroweak matrix symmetry that breaking realizes a type-I seesaw mech- JHEP07(2017)113 (A.4) (A.5) (A.2) (A.3) as the . ] GW 40 ,  flat direction i is the value of the c i − ϕ are given by h ) i B implicitly depends on the ϕ 2 h i i in the above sums runs over in the case of gauge bosons , ~n ϕ i is positive. In particular, it ( m h 5 6 and 2 i  B m ) is given by its field-dependent A 2 = 2 GW ϕ i ln Λ c ~nϕ (  . i ,  ) ) i i m ln is singled out as the actual minimum. ϕ ϕ 4 = 0 h h i ~n ~n ϕ ( ( ~nϕ, counts the particle’s real degrees of freedom Bϕ h ) is assumed to develop a continuous set of GW 4 i 4 i i = + d m m The functions depend on the actual renormalization scheme. A.1 · · 4 Λ=Λ – 31 – , the Gildener-Weinberg scale. These minima lie i

i i flat c 11 ) d d ~ Φ i i Aϕ GW f s s ( 2 2 R 1) 1) ) = − − ( ( gives the position on the ray. Whereas the tree-level po- ~nϕ ( i i ϕ , loop corrections will in general bend the potential along X X (1) eff ϕ ). 4 4 V i i . The coefficient ϕ ϕ A.4 h h GW 1 1 2 2 π π MS scheme, for which one finds to develop a flat direction, the scalar couplings must satisfy certain 64 64 V = = A B ) then implies that the scalar fields corresponding to the non-vanishing com- is different. acquire finite vevs, the relative magnitudes of which are given by the entries along the flat direction at which the one-loop effective potential develops an for scalars or fermions. Finally, as mentioned before, is the renormalization point. ~n ϕ R A.3 3 2 mass evaluated along the flat direction. Note that is a unit vector and GW denotes its spin. The constants = ~n i i s c The one-loop effective potential along the flat direction can be written as [ In order for . Depending on which of the scalar modes acquire a finite vev, the relevant set of Notice that due to dimensional transmutation all dimensional quantities, and in particular masses, ~n 11 parameter extremum. This extremum is a minimum ifwill and be only proportional if renormalization to point the in symmetry eq. ( breaking scale. Hence, it is only reasonable to take Λ tree-level renormalization point Λ and Here, we will useand the A few comments on the notationall are particles in order. in First, the the given index theory. For each particle where Λ ponents of of conditions where tential is minimal forthe each flat direction.Equation ( Thus, a particular value thereby determining the GWWeinberg scale. conditions. The We flat will direction refer can to be conditions parametrized of as this type as Gildener- degenerate minima at a specificon scale a Λ ray through the origin of field space,conditions henceforth which referred can to generically as be the written as renormalization scale. The potential eq. ( JHEP07(2017)113 ) = B ). v ) in A.4 GW (A.8) (A.6) (A.7) Λ GW / i (Λ ϕ S are special h λ 2.2.2 ) and . GW is of the same order as (Λ in a way consistent with PGB A φ ) is the pseudo-Goldstone contains all massive SM λ m R if . ∗ ). As a first step, we isolate 2 m A.3 i , GW . ϕ and 4 i ∗ A.7 h ˜  and m B φS · SM B κ i A 2 GW d , = 8 ∈ i s i − i 2 ϕ h add 1 4 1) = B . The set SM − − ϕ defined in eq. ( i for

(  + ∗ for given ) m flat i SM SM ~nϕ X ϕ ~ ∈ Φ exp – 32 – ( 2 i h B · Higgs 1 ϕ 2 (1) n d m eff = π 1 i GW V 2 64 B m is of the same order as Λ d and i = Λ = = ˜ has to match the Higgs mass according to this equation. ϕ v = i i h B SM ϕ m h ˜ B 2 PGB m ). It is particularly important since it relates the PGB mass to the A.5 ). Now, we show how to calculate Λ GW (Λ contains all additional contributions from the scalar sector, including the one , the loop function , and we can therefore parametrize all SM fermion and gauge boson masses as φS i . There, we have already described how to express κ ϕ add LHC h 1 B H n Before we proceed let us remark that the models we consider in section The crucial quantity in determining a viable parameter point is the loop function The excitation along the flat direction 2.2.2 = . This is a necessary condition to control the loop expansion in powers of ln( φ with appropriate dimensionlessgauge coefficients bosons and ˜ fermions. Defining the function in the following sense: in additionvanishing to vevs. the usual Higgs Hence doublet the onlyv gauge electroweak singlets scale obtain non- originates from the doublet sector only, where from the SM Higgs doublet. terms of the empirically known values of introduced in eq. ( other particles’ masses and thethe condensation contributions scale due via to eq. SM ( fermions and gauge bosons, In this part, weduced enlarge in upon the previous certainthe section. aspects GW of Thereby, scale, we the concentrate whichtion Gildener-Weinberg on we formalism the need intro- consistent in computation our of treatment of the minimal conformal model in sec- Note that in models, inLHC, which the PGB is identified with the HiggsA.2 boson measured at Calculating the the Gildener-Weinberg scale B boson of broken scaleonly invariance. after Massless SSB. at The mass tree of level, the its PGB mass at is one-loop generated level radiatively is given by along the flat direction lies at The above equation shows that is straightforward to show that the minimum of the one-loop effective potential eq. ( JHEP07(2017)113 0 add B ), we (A.9) (A.12) (A.10) (A.11) A.7 ) and one B.1 . Pl . only depend on the < M 2 . However, we can use 2 PGB add v B m depends on SM gauge and 2 1 8 n . We suppress this implicit (Λ) is a strictly monotonously in- . and SM − ˜ SM 1 } B  ˜ B add .... n . 4 B B 4 Higgs A 2 2 v + 2 n . m 2 ) ξ ) − ) + 2 g . 2 SR ( 4 1 π 1 4 add π 2 κ 12 n GW {z B (2) π are reasonably close to each other (cf. our − 64 . Of course, the particular form of ! + = = 0 β (16 (Λ  v 16 2 1 ) + 0 add n SM + B ˜ GW exp ) B 0 – 33 – φR add and 4 1 · 2 g κ ( B | (Λ n π )). G GW = ) is guaranteed, since GW (1) ) with the formula for the PGB mass, eq. ( 16 ) self-consistently, eventually fixes two more variables β Λ ) = defined at the GW scale known at the electroweak scale 1 A.6 0 add A.9 A.11 (Λ) + n GW B itself and ) = A.12 SM = g (Λ ˜ ( B v 4 1 B β GW n , for instance, gives )). ) and eq. ( in the above equation. In contrast, ). The beta functions for one real scalar are obtained from the 2.8 (Λ) := 2.2.2 G ), we can write B.2 GW A.11 = 174 GeV. For a fully consistent calculation, we must therefore ascertain, A.8 v Solving eq. ( In our previous discussion, we have already assumed the electroweak scale to attain its Motivated by combining eq. ( The uniqueness of the root of eq. ( 12 In the followingcomplex we list scalar the ( two-scalar one-loop functions beta by dropping functions every for term with two a real scalarscreasing function ( of the energy scale Λ. In this appendix, we collect theFor renormalization the group equations perturbative used expansion throughout this of work. a given beta function, we adopt the following notation depends on the model under investigation.model An from explicit section calculation in the minimal conformal B One-loop beta functions yields the correct number (cf. eq. ( at the GW scale, namely Λ In addition, we must checkdiscussion whether after Λ eq. ( proper value, if our one-loop effective potential, indeed, possesses an appropriate minimum, i.e. whether Then the Gildener-Weinberg scaleparticular consistent PGB mass with is a defined given via set the condition of scalar couplings and a the RGEs to evolve the gauge and Yukawa couplings todefine any the scale function Λ The above partition isscalar particularly couplings convenient, because whosedependence values on are Λ Yukawa couplings, which are only and using eq. ( JHEP07(2017)113 L and (B.5) (B.3) (B.4) (B.2) χ , respectively scalar SU(2) by exchanging ξ . φχ . , which enters the real κ and i φξ y , χ κ , 4 2 and g φχ , ) + 3 i χξ . κ χ ) . 1) (B.1) ξ 3 κ , 2 2 , 2 λ 4 2 χ g N − y φξ g 4 2 2 + 4 1 N ) 2 g κ g δ + ξ φχ i ξ χ N κ χ χξ ( N C 3 2 ,N + 2 N κ 3 2 g χ ( N ) + 8 4 2 = 6  + 4 g χ N + 1 − ( 12 + 2 ,  6 φχ i ). 19 ξ ξ (1) T ,N κ 2 2 + 3 = φχ λ N χ 2 χξ g β ) symmetry. The one-loop scalar-sector − 3 4 1 κ χ 16 κ B.1 ξ N , g ∆ ) ξ ( ( ξ N + N + h N NC + 2) 3 8 ( + 3) T 2 φξ D , χ , which are, respectively, given by 2 2 1 3 h ξ κ χ 1) O λ g 2 ξ + D N + 2 3 C y 32 = ) + × − χξ ) N χ 2 2 – 34 – 2 ) ξ κ ), i.e. it is not only classically scale-invariant but 2 φχ g + ( + + 2) ) D y χ κ ( N ξ + (4 2 + 2 2 χ χ N . 1 6 − C g 2 1 λ ( ,D + 3 2.24 N  given in eq. ( χ 1)( g φ 2 φχ ξχ + 2 2 1 O + h λ λ ξ κ κ 1) g 2 χ − = χ χ ( + ( χ × + 2) φχ λ D φ χ − ≡ C can be obtained from the ones of C φ 2 N κ χ λ (1) g 2 λ N , respectively. For the following, recall the definition of the 3 2 12 6( 3 (4) β χξ φξ N ξ = 6(2 3 2 N ( + 8) )). Furthermore, we use the abbreviation and κ + 2 κ − − − − O (  h φ χ 2 φ 4 1 χ (1) λ = = = = ) = ( φχ χξ λ B.1 N and β ξ D κ κ and = χ φ χξ φχ ∆ (1) (1) χ λ λ ξ (1) κ (1) κ ,N β β C λ and the Dynkin index β β χ = 8 = 8 = 24 = 8( ∆ ∆ ∆ N ∆ C φ χ χξ ( φχ (1) (1) λ λ (1) κ denote the Casimir invariants of the scalar multiplets (1) κ T β β -plet. For the calculation of the RGEs, we will assume that the scalar β β ξ N C , namely 2 g and χ as well as identifying C ξ Taking into account the scalars’ interactions with the electroweak gauge bosons, the ↔ is that of Besides the Higgs beta function,ξ the only SM RGE which changes in the presence of with the Dynkin indices The only other sizable SMscalar coupling RGEs in is the the following top-quark way Yukawa coupling where (cf. the definition in eq. ( The beta functions of χ above RGEs obtain the following additional contributions for an SU(2) potential is of the formadditionally given enjoys in a eq. global ( beta functions then turn out to be Here, we consider the scalar sectormultiplets, of denoted the as SM supplemented byquadratic up Casimir to two B.1 SM + real scalar representation(s) JHEP07(2017)113 χ , 2 3 (B.6) , κ 4 2  g 4 i N, 13 δ ) and let the 9 2 − , + ) 2 4 2 N g N, i . The scalar sector ,N,Y δ − 1 .  (1 3 − 2 , + g (10 ∼ 2 , 2 1 2.1.3  2 3 , κ N 4 χ 2 3) κ 2 6 h 4 19 2 y − i +2 Cg N, ). Note that we present the − 1) 2 3 , κ δ N , λ 2 2 2 ( 2.2 − + 3 i D 2 3 g h = 6 − 1 3 4 2 1 1 13 N g g 2 2  i ( Cκ , 4 2 + 3 κ − (1) κ 3 8 2 3 ) = Y Y 1 2 2 2 β , κ g + 8 + N 2 2 ∆ 2 2 1 + (1) 2 2 g g 4 16 g 1 − , 3 2 1 2 . g + 12 + 24 , g λ 4 − i Cκ i i i Y 2 2 3 4 2 (10 2 2 2 , β 2 2 2 i Y λ y κ 1 + g g g ) 3 1 N 4 + κ 2 i g h – 35 – 4 + 6 + 12 2 1 y 2 −  g κ + 24 i i + 2) + 2) + 2) 2 + 2 1 − 1) 9 8 2 2 2 2 , 2 3 2 2 2 κ 3 + g g 1 λ 2 3 − λ + 3 λ i N N N 1) 1) NY 4 N 1 + 8 1) 1 3 ( Dκ 33 g − − Cλ 3 + ( + ( + ( 1 2 3 8 − ), which permits any use, distribution and reproduction in 2 2 + λ − 2 2 2 1 1 1 + = 6(2 + + 2 g g g N N C + 1) 2 3 6 2 2 41 1
λ λ (1) + λ − 2 2 2  + ( + ( β g + 1) + 1) + 1) Dκ + 20) 2 2 C D 1 1 2 2 2 = ∆ 1 4 ( g g Cλ + 1) be a complex scalar representation with 2 2 N + 3 Y Y Y 1 ( + ( − + (1) χ g Y Y 2 N 1 CC-BY 4.0 (4 (4 (4 + 8 2 2 2 1 4 4 g N β refer to dimension and invariants of the representation under which h h h λ λ 2 2 h h , since we used them in our analysis in section h 1 2 3 This article is distributed under the terms of the Creative Commons 1 2 3 + ( 3 λ D κ κ κ 2) Nκ + + λ λ λ , let κ 1 3 2 3 2 3 2 3 3 3 1 1 λ − + λ 3 λ − − − − − − +4) 2 1 2.1 h h h and N 3 1 2 λ N ======( κ κ κ C 1 3 2 3 1 2 3 1 , (1) (1) (1) (1) (1) (1) κ λ λ κ κ λ = 4 = 4 = 4 = 24 = 4( = β β β β β β N ∆ ∆ ∆ ∆ ∆ ∆ 3 1 2 1 2 3 (1) (1) (1) (1) (1) (1) κ λ λ λ κ κ β β β β β β Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. The modified one-loop gauge RGEs are and where transforms. The scalar sector isbosons. coupled both The to corresponding SM contributions fermions to and the to scalar the electroweak RGEs gauge are given by As in section scalar interactions be describedRGEs including by the potentialbeta in functions eq. are then ( calculated to be B.2 SM + one complex scalar representation JHEP07(2017)113 B ]. ]. ]. B 751 ]. ]. (1996) 153 ]. SPIRE IN SPIRE SPIRE ]. Phys. Lett. ][ SPIRE IN SPIRE , IN IN SPIRE IN ][ ][ B 379 ]. IN ][ ][ Phys. Lett. (2016) 217 SPIRE ]. ][ (2016) 439 , ]. IN SPIRE ][ ]. (2015) 035 IN SPIRE B 752 SPIRE ][ B 755 IN (2016) 349 Conformal Standard Model 06 Phys. Lett. IN ][ , ][ SPIRE arXiv:1507.06926 IN [ arXiv:0706.1829 ][ JHEP B 754 arXiv:1511.02107 arXiv:0704.1165 [ Neutrino mass in radiatively-broken , Gauge coupling unification in a [ [ arXiv:1507.01755 A new dynamics of electroweak Phys. Lett. arXiv:1508.03031 Phys. Lett. [ , Lepton number violation within the [ , Bosonic seesaw mechanism in a classically arXiv:1505.07453 [ arXiv:1508.02607 Phys. Lett. ]. (2016) 058 (2007) 156 arXiv:0710.2840 [ (2015) 095012 Discriminative phenomenological features of scale , A radiative model for the weak scale and neutrino Electroweak Higgs as a pseudo-Goldstone boson of arXiv:1405.6204 [ The inverse seesaw in conformal electro-weak (2015) 170 ]. (2007) 075014 Neutrino masses and conformal electro-weak [ – 36 – 02 10 SPIRE arXiv:1505.04448 (2015) 075010 (2015) 064 D 92 B 655 [ IN SPIRE D 76 09 ][ JHEP IN (2016) 038 , (2008) 260 JHEP ][ Conformal symmetry and the Standard Model Effective action, conformal anomaly and the issue of quadratic D 92 (2014) 177 ]. ]. ]. ]. , 02 Dark matter and neutrino masses from a scale-invariant Dark chiral symmetry breaking and the origin of the electroweak ]. 10 JHEP Phys. Rev. (2015) 424 , Phys. Lett. B 660 , SPIRE SPIRE SPIRE SPIRE Phys. Rev. , JHEP , IN IN IN IN SPIRE , JHEP ][ ][ ][ ][ P miracle (by scale invariance) `ala self-interaction IN Phys. Rev. , , B 746 m ][ hep-th/0612165 [ The next-to-minimal Coleman-Weinberg model Prospects for discovering the Higgs-like pseudo-Nambu-Goldstone boson of the Phys. Lett. arXiv:1505.06554 , [ View FI Phys. Lett. (2007) 312 , arXiv:1508.03245 arXiv:1503.03066 arXiv:1508.06828 arXiv:1512.05061 hep-ph/9604278 [ classical scale symmetry with an extended scalar sector [ symmetry breaking invariant models for electroweak symmetry breaking (2015) 201 scale symmetry breaking and phenomenological consequences mass via dark matter conformal inverse seesaw conformal extension of the[ Standard Model multi-Higgs portal symmetry breaking with classically[ scale invariance classically scale invariant model divergences broken scale invariance scale-invariant models 648 [ A. Farzinnia, A. Latosinski, A. Lewandowski, K.A. Meissner and H. Nicolai, M. Lindner, S. Schmidt and J. Smirnov, K. Hashino, S. Kanemura and Y. Orikasa, C.D. Carone and R. Ramos, P. Humbert, M. Lindner and J. Smirnov, A. Ahriche, K.L. McDonald and S. Nasri, P. Humbert, M. Lindner, S. Patra and J. Smirnov, Z. Kang, N. Haba, H. Ishida, N. Okada and Y. Yamaguchi, A. Karam and K. Tamvakis, N. Haba, H. Ishida, N. Kitazawa and Y. Yamaguchi, N. Haba, H. Ishida, R. Takahashi and Y. Yamaguchi, R. Foot, A. Kobakhidze and R.R. Volkas, R. Foot, A. Kobakhidze, K. McDonald and R. Volkas, R. Hempfling, K.A. Meissner and H. Nicolai, K.A. Meissner and H. Nicolai, [8] [9] [6] [7] [4] [5] [1] [2] [3] [17] [18] [15] [16] [13] [14] [10] [11] [12] References JHEP07(2017)113 , 08 ]. , JHEP ]. JHEP , , Rev. Mod. Phys. SPIRE A 29 JHEP Eur. Phys. , , , IN (2015) 026 SPIRE , ]. Zh. Eksp. IN ][ [ 09 ][ Physical ]. SPIRE ]. IN Neutrino-axion-dilaton JHEP ]. SPIRE ]. , ][ IN ]. (2015) 344 ][ SPIRE Mod. Phys. Lett. SPIRE IN , SPIRE 120 IN ][ ]. SPIRE IN arXiv:1307.3764 ][ ]. [ IN ]. ][ arXiv:1403.4226 ]. [ ][ Emergence of the electroweak scale Is the Standard Model saved P using scale invariance ]. SPIRE m SPIRE IN SPIRE IN SPIRE IN ][ SPIRE arXiv:1510.03668 IN ][ ]. (2013) 075 [ [ IN arXiv:1301.4224 (2014) 080 [ ][ 10 ]. ]. arXiv:1401.4185 ]. 06 SPIRE [ Cosmological constant in scale-invariant theories Totally asymptotically free trinification arXiv:1412.3616 ]. arXiv:1307.8428 The minimal scale invariant extension of the J. Exp. Theor. Phys. IN [ – 37 – [ , arXiv:1006.5916 (1983) 837] [ ][ JHEP SPIRE SPIRE [ SPIRE , IN IN JHEP (1973) 1888 55 (2013) 060 IN , SPIRE ][ Radiative corrections as the origin of spontaneous ][ (2016) 015009 ][ IN arXiv:1012.4848 Electroweak symmetry breaking induced by dark matter 04 Scalar dark matter in scale invariant Standard Model [ D 7 ][ arXiv:0707.0633 ]. (2016) 558 ]. [ Classical scale-invariance, the electroweak scale and vector dark (2014) 073003 D 93 arXiv:1409.0492 (2010) 021 Agravity Classically conformal radiative neutrino model with gauged B-L Majorana dark matter in a classically scale invariant model (2013) 055020 JHEP Leptogenesis and neutrino oscillations in the classically conformal SPIRE 09 , SPIRE B 760 IN D 89 IN Erratum ibid. [ arXiv:1411.2773 [ Phys. Rev. ][ [ D 88 (2008) 651 , A new type of isotropic cosmological models without singularity (2011) 075010 Classical scale invariance in the inert doublet model JHEP Phys. Rev. arXiv:1304.7006 , (2015) 399] [ arXiv:1409.5776 arXiv:1511.08432 , [ [ [ arXiv:1507.06848 Einstein gravity as a symmetry breaking effect in quantum field theory B 659 (1980) 99 D 84 Phys. Lett. Phys. Rev. 147 [ Is the Higgs boson associated with Coleman-Weinberg dynamical symmetry , (1982) 729 Upgrading sterile neutrino dark matter to FI , (2015) 471 Phys. Rev. , 54 B 91 (2016) 024 (2015) 143 C 75 arXiv:1507.04996 Lett. Phys. symmetry breaking interconnection (2015) 130 (2014) 1450077 [ 04 through the Higgs portal Standard Model naturalness and dynamical breaking of classical scale invariance Phys. Rev. breaking? Phys. Lett. matter asymptotically by conformal symmetry? Teor. Fiz. Standard Model with the Higgs portal symmetry J. 01 S.L. Adler, S.R. Coleman and E.J. Weinberg, S. Bertolini, L. Di Luzio, Malinsk´yand H. J.C. Koleˇsov´a,M. Vasquez, G.M. Pelaggi, A. Strumia and S. Vignali, A.A. Starobinsky, A.D. Plascencia, K. Ghorbani and H. Ghorbani, L. Alexander-Nunneley and A. Pilaftsis, M. Heikinheimo, A. Racioppi, M. Raidal, C. Spethmann and K. Tuominen, C.T. Hill, A. Salvio and A. Strumia, C. Englert, J. Jaeckel, V.V. Khoze and M. Spannowsky, T. Hambye and M.H.G. Tytgat, C.D. Carone and R. Ramos, R. Foot, A. Kobakhidze and R.R. Volkas, A. Gorsky, A. Mironov, A. Morozov and T.N. Tomaras, V.V. Khoze and G. Ro, Z. Kang, S. Benic and B. Radovcic, H. Okada and Y. Orikasa, [37] [38] [34] [35] [36] [32] [33] [30] [31] [27] [28] [29] [24] [25] [26] [22] [23] [20] [21] [19] JHEP07(2017)113 ] , , , , , ]. Phys. (1976) , Chin. ]. Phys. , , SPIRE D 13 IN SPIRE Ph.D. thesis ][ , IN ]. ][ arXiv:0708.1463 [ SPIRE Phys. Rev. , IN [ (2014) 015019 ]. arXiv:1308.0295 (2008) 111802 ]. [ Review of particle physics D 90 ]. arXiv:1505.01721 (1974) 341 ]. 100 [ SPIRE SPIRE IN ]. ]. IN SPIRE D 9 ][ IN ][ SPIRE (2013) 141 Distinguishing the Higgs boson from the IN ][ Phys. Rev. SPIRE SPIRE Gamma-ray excess and the minimal dark matter Scalar dark matter: direct vs. indirect detection ][ Scalar singlet dark matter and gamma lines , IN (2015) 238 IN Planck scale boundary conditions and the Higgs Landau pole in the Standard Model with weakly ][ ][ – 38 – B 727 ]. Phys. Rev. A precision measurement of the mass of the top quark Phys. Rev. Lett. , , B 747 ]. Natural electroweak symmetry breaking from scale ]. SPIRE arXiv:1508.04418 IN A new constraint on a strongly interacting Higgs sector Adiabatic regularization in closed Robertson-Walker universes Symmetry breaking and scalar bosons [ SPIRE arXiv:1510.07562 [ ]. arXiv:1112.2415 Phys. Lett. [ Adiabatic regularization of the energy momentum tensor of a IN SPIRE [ , arXiv:1309.6632 [ IN Higgs partner searches and dark matter phenomenology in a collaboration, K.A. Olive et al., [ Phys. Lett. [ hep-ex/0406031 , [ SPIRE arXiv:1509.04282 [ IN (2015) 119 ][ (1987) 2963 (2016) 008 (2012) 037 ]. 06 (1990) 964 02 (2014) 090001 (2004) 638 Effects of the quantum vacuum in particle physics and cosmology B 751 (2014) 015017 D 36 65 (2016) 152 ]. SPIRE 429 JHEP 06 JHEP C 38 IN , D 89 , [ collaboration, V.M. Abazov et al., SPIRE arXiv:1405.0498 IN JHEP Phys. Lett. mass model quantized field in homogeneous spaces University of Heidelberg, Heidelberg Germany, (2014). Phys. Rev. Towards completing the Standard Model:Rev. vacuum stability, EWSB and dark matter Rev. Lett. interacting scalar fields Particle Data Group Phys. invariant Higgs mechanism classically scale invariant Higgs[ boson sector dilaton at the Large Hadron[ Collider 3333 D0 Nature M. Duerr, P. Fileviez P´erezand J. Smirnov, M. Duerr, P. Fileviez Perez and J. Smirnov, M. Holthausen, K.S. Lim and M. Lindner, M. Duerr, P. Fileviez P´erezand J. Smirnov, J. Smirnov, P.R. Anderson and L. Parker, E. Gabrielli, M. Heikinheimo, K. Kannike, A. Racioppi, M. Raidal and C. Spethmann, M.E. Peskin and T. Takeuchi, L. Parker and S.A. Fulling, Y. Hamada, K. Kawana and K. Tsumura, A. Farzinnia, H.-J. He and J. Ren, A. Farzinnia and J. Ren, E. Gildener and S. Weinberg, W.D. Goldberger, B. Grinstein and W. Skiba, [53] [54] [51] [52] [49] [50] [46] [47] [48] [44] [45] [42] [43] [40] [41] [39]