PHYSICAL REVIEW D 96, 095036 (2017) Precision tests and fine tuning in twin Higgs models

† ‡ ∥ Roberto Contino,1,* Davide Greco,2, Rakhi Mahbubani,3, Riccardo Rattazzi,2,§ and Riccardo Torre2, 1Scuola Normale Superiore and INFN, Pisa IT-56125, Italy 2Theoretical Particle Physics Laboratory, Institute of Physics, EPFL, Lausanne CH-1015, Switzerland 3Theoretical Physics Department, CERN, Geneva CH-1211, Switzerland (Received 27 February 2017; revised manuscript received 7 November 2017; published 29 November 2017) We analyze the parametric structure of twin Higgs (TH) theories and assess the gain in fine tuning which they enable compared to extensions of the standard model with colored top partners. Estimates show that, at least in the simplest realizations of the TH idea, the separation between the mass of new colored particles and the electroweak scale is controlled by the coupling strength of the underlying UV theory, and that a parametric gain is achieved only for strongly-coupled dynamics. Motivated by this consideration we focus on one of these simple realizations, namely composite TH theories, and study how well such constructions can reproduce electroweak precision data. The most important effect of the twin states is found to be the infrared contribution to the Higgs quartic coupling, while direct corrections to electroweak observables are subleading and negligible. We perform a careful fit to the electroweak data including the leading- logarithmic corrections to the Higgs quartic up to three loops. Our analysis shows that agreement with electroweak precision tests can be achieved with only a moderate amount of tuning, in the range 5%–10%, in theories where colored states have mass of order 3–5 TeVand are thus out of reach of the LHC. For these levels of tuning, larger masses are excluded by a perturbativity bound, which makes these theories possibly discoverable, hence falsifiable, at a future 100 TeV collider.

DOI: 10.1103/PhysRevD.96.095036

I. INTRODUCTION where each Λ represents the physical cutoff scale in a different sector of the theory. The above equation is simply The principle of naturalness offers arguably the main dictated by symmetry: dilatations (dimensional analysis) motivation for exploring physics at around the weak scale. determine the scale dependence and the broken shift According to naturalness, the plausibility of specific symmetry of the Higgs field sets the coupling dependence. parameter choices in quantum field theory must be assessed Unsurprisingly, these contributions arise in any explicit UV using symmetries and selection rules. When viewing the completion of the SM, although in some cases there may be standard model (SM) as an effective field theory valid other larger ones. According to Eq. (1.1), any given (large) below a physical cutoff scale and considering only the value of the scale of new physics can be associated with a known interactions of the Higgs boson, we expect the (small) number ϵ, which characterizes the accuracy at following corrections to its mass.1 which the different contributions to the mass must cancel 2 2 2 among themselves, in order to reproduce the observed 3y 9g2 3g1 3λ δm2 ¼ t Λ2 − Λ2 − Λ2 − h Λ2 þ; ≃ 125 h 4π2 t 32π2 g2 32π2 g1 8π2 h value mh GeV. As the largest loop factor is due to the top Yukawa coupling, according to Eq. (1.1) the scale ð1:1Þ Λ 2 NP where new states must first appear is related to mh and ϵ via *[email protected] rffiffiffi † 4π2 2 1 [email protected] Λ2 ∼ mh ⇒ Λ ∼ 0 45 ‡ NP 2 × ϵ NP . ϵTeV: ð1:2Þ rakhi.mahbubani@cern.ch 3yt §[email protected] ∥ [email protected] pffiffiffi The dimensionless quantity ϵ measures how finely-tuned 1 2 2 2 λ 2 2 2 174 2 We take mh ¼ mH ¼ hv = with hHi¼v= ¼ GeV, Λ mh is, given NP, and can therefore be regarded as a which corresponds to a potential 2 measure of the tuning. Notice that the contributions fromffiffiffi g2 λ λ Λ 1 1 pϵ − 2 2 h 4 and h in Eq. (1.1) ffiffifficorrespond to NP ¼ . TeV= and V ¼ mHjHj þ 4 jHj : p ΛNP ¼ 1.3 TeV= ϵ, respectively. Although not signifi- cantly different from the relation in the top sector, these Published by the American Physical Society under the terms of scales would still be large enough to push new states out of the Creative Commons Attribution 4.0 International license. ϵ ∼ 0 1 Further distribution of this work must maintain attribution to direct reach of the LHC for . . the author(s) and the published article’s title, journal citation, Indeed, for a given ϵ, Eq. (1.2) only provides an upper and DOI. bound for ΛNP; in the more fundamental UV theory there

2470-0010=2017=96(9)=095036(31) 095036-1 Published by the American Physical Society ROBERTO CONTINO et al. PHYSICAL REVIEW D 96, 095036 (2017) 2 can in principle exist larger corrections to mh which are not coupling. As discussed in the next section, depending on captured by Eq. (1.1). In particular, in the minimal super- the structure of the model, gSM can be either the top Yukawa symmetric SM (MSSM) with high-scale mediation of the or the square root of the Higgs quartic. As a result, given the δ 2 tuning ϵ, the squared mass of the new colored and charged soft terms, mh in Eq. (1.1) is logarithmically enhanced by RG evolution above the weak scale. In that case, Eq. (1.2) is states is roughly given by modified as follows:   4π2 m2 g 2 m2 ∼ h : : 2π2 1 2 3 2 × ϵ × ð1 4Þ Λ2 ∼ mh yt gSM NP 2 × Λ Λ × ϵ ; ð1:3Þ 3yt ln UV= NP For g >gSM, we could define these model as effectively where ΛNP corresponds to the overall mass of the stops and hypersoft, in that, for fixed fine tuning, the gap between the Λ ≫ Λ UV NP is the scale of mediation of the soft terms. SM-charged states and the weak scale is even larger than However, for generic composite Higgs (CH) models, as that in supersoft models. In practice the above equation well as for supersymmetric models with low-scale media- implies that, for strong g, the new states are out of reach of tion, Eq. (1.2) provides a fair estimate of the relation the LHC even for mild tuning (see Sec. II A for a more between the scale of new physics and the amount of tuning, precise statement). Equation (1.4) synthesizes the potential the Higgs mass being fully generated by quantum correc- relevance of the TH mechanism, and makes it clear that the tions at around the weak scale. If the origin of mh is new dynamics must be rather strong for the mechanism to Λ normally termed soft in the MSMM with large UV work. Given the hierarchy problem, it then seems almost [Eq. (1.3)], it should then be termed supersoft in models inevitable to make the TH a composite TH (although it respecting Eq. (1.2). As is well known, and shown by could also be a supersymmetric composite TH). Realizations Λ ≳ Eq. (1.3), the natural expectation in the MSSM for UV of the TH mechanism within the paradigm of CH models 100 TeV is ΛNP ∼ mZ ∼ mh. In view of this, soft scenarios with fermion partial compositeness [23] have already been were already somewhat constrained by direct searches at proposed, both in the holographic and effective theory LEP and Tevatron, whereas the natural range of the scale of setups [6,9,13,14]. supersoft models is only now being probed at the LHC. It is important to recognize that the factor that boosts the Equation (1.2) sets an absolute upper bound on ΛNP for a mass of the states with SM gauge quantum numbers in given fine tuning ϵ, but does not give any information on its Eq. (1.4) is the coupling g itself. Because of this, strong- nature. In particular it does not specify the quantum dynamics effects in the Higgs sector, which are described numbers of the new states that enter the theory at or below in the low-energy theory by nonrenormalizable operators this scale. Indeed, the most relevant states associated with with coefficients proportional to powers of g=m, do not the top sector, the so-called top partners, are bosonic in “decouple” when these states are made heavier, at fixed fine ϵ standard supersymmetric models and fermionic in CH tuning . In the standard parametrics of the CH, m=g is of models. Nonetheless, one common feature of these stan- the order of f, the decay constant of the σ-model within dard scenarios is that the top partners carry SM quantum which the Higgs doublet emerges as a pseudo-Nambu- 2 2 numbers, color in particular. They are thus copiously Goldstone boson (pNGB). Then ξ ≡ v =f , as well as produced in hadronic collisions, making the LHC a good being a measure of the fine tuning through ϵ ¼ 2ξ, also probe of these scenarios. Yet there remains the logical measures the relative deviation of the Higgs couplings from 2 possibility that the states that are primarily responsible for the SM ones, in the TH like in any CH model. Recent ξ ≲ the origin of the Higgs mass at or below ΛNP are not Higgs coupling measurements roughly constrain charged under the SM, and thus much harder to produce 10–20% [24], and a sensitivity of order 5% is expected and detect at the LHC. The twin Higgs (TH) is probably the in the high-luminosity phase of the LHC [25]. However most interesting of the (few) ideas that take this approach Higgs loop effects in precision Z-pole observables mea- [1–22]. This is primarily because the TH mechanism can, at sured at LEP already limit ξ ≲ 5% [26,27]. Having to live least in principle, be implemented in a SM extension valid with this few-percent tuning would somewhat undermine up to ultrahigh scales. The structure of TH models is such the motivation for the clever TH construction. In ordinary ξ that the states at the threshold ΛNP in Eq. (1.2) carry CH models this strong constraint on can in principle be quantum numbers under the gauge group of a copy, a twin, relaxed thanks to compensating corrections to the Tˆ of the SM, but are neutral under the SM gauge group. These parameter coming from the top partners. In the most natural 4 2 2 twin states, of which the twin tops are particularly relevant, models, these are proportional to yt v =m and thus, unlike are thus poorly produced at the LHC. The theory must also the Higgs-sector contribution, decouple when m is contain states with SM quantum numbers, but their mass Λ m is boosted with respect to NP roughly by a factor 2The factor of two difference between the fine tuning ϵ and ξ is g=gSM, where g describes the coupling strength of the due to the Z2 symmetry of the Higgs potential in the TH models, new dynamics, while gSM represents a generic SM as shown in Sec. II A [7].

095036-2 PRECISION TESTS AND FINE TUNING IN TWIN HIGGS … PHYSICAL REVIEW D 96, 095036 (2017) increased. This makes it hard to realize such a compensa- m, at which new states with SM quantum numbers, color tory effect in the most distinctive range of parameters for in particular, first appear. In order to get a feel for the ∼ 5–10 TH models, where m TeV. Alternatively one mechanism and its parametrics, it is sufficient to focus on could consider including custodial-breaking couplings the most general potential for two Higgs doublets H and H~ , larger than y in the top-partner sector. Unfortunately these ~ t invariant under the gauge group GSM × GSM, with give rise to equally-enhanced contributions to the Higgs 3 2 1 GSM ¼ SUð Þ × SUð Þ × Uð Þ , as well as a Z2: potential, which would in turn require further ad-hoc c L Y cancellations. λ ~ − 2 2 ~ 2 H 2 ~ 2 2 As already observed in the literature [13,14] another VðH; HÞ¼ mHðjHj þjHj Þþ 4 ðjHj þjHj Þ important aspect of TH models is that calculable IR- λˆ dominated contributions to the Higgs quartic coupling þ h ðjHj4 þjH~ j4Þ: ð2:1Þ almost saturate its observed value. Though a welcome 8 property in principle, this sets even stronger constraints on Strictly speaking, the above potential does not have minima additional UV contributions, such as those induced by extra “ ” sources of custodial breaking. In this paper we study the with realistic tunable hHi. This goal can be achieved by correlation between these effects, in order to better assess the simple addition of a naturally small Z2-breaking mass the relevance of the TH construction as a valid alternative to term which, while changing the vacuum expectation value, more standard ideas about EW-scale physics. Several such does not affect the estimates of fine tuning, and hence will studies already exist for standard composite Higgs scenar- be neglected for the purposes of this discussion. Like for – the SM Higgs, the most general potential is accidentally ios [28 30]. In extending these to the TH we shall f encounter an additional obstacle to gaining full benefit invariant under a custodial SOð4Þ × SOð4Þ. Notice how- ˆ from the TH boost in Eq. (1.4): the model structure requires ever that in the limit λh → 0, the additional Z2 enhances the rather “big” multiplets, implying a large number of degrees custodial symmetry to SOð8Þ, where H ≡ H ⊕ H~ ≡ 8.In of freedom. This results in an upper bound for the coupling this exact limit, if H~ acquired an expectation value 4π pffiffiffi that is parametrically smaller than by naive dimensional hH~ i ≡ f= 2, all 4 components of the ordinary Higgs H analysis (NDA); hence the boost factor is similarly would remain exactly massless NGBs. Of course the SM depressed. We shall discuss in detail how serious and g ˆ and SM gauge and Yukawa couplings, along with λh, unavoidable a limitation this is. 8 The paper is organized as follows: in Sec. II we discuss explicitly break the SOð Þ symmetry that protects the the general structure and parametrics of TH models, Higgs. Consider however the scenario where these other couplings, which are known to be weak, can be treated as followed by Sec. III where we discuss the more specific 8 8 class of composite TH models we focus on for the purpose small SOð Þ-breaking perturbations of a stronger SOð Þ- preserving underlying dynamics, of which the quartic of our study. In Secs. IVand V we present our computations λ of the basic physical quantities: the Higgs potential and coupling H is a manifestation. In this situation we can ˆ ˆ reassess the relation between the SM Higgs mass, the precision electroweak parameters (S; T;δgLb). Section VI is devoted to a discussion of the resulting constraints on the amount of tuning and the scale m where new states λˆ model and an appraisal of the whole TH scenario. Our charged under the SM are first encountered, treating h conclusions are presented in Sec. VII. as a small perturbation of λH. At zeroth order, i.e., ˆ neglecting λh, we can expand around the vacuum ~ 2 2 2 hHi ¼ 2m =λH ≡ f =2, hHi¼0. The spectrum consists II. THE TWIN HIGGS SCENARIO H pffiffiffi pffiffiffiffiffiffi pffiffiffi of a heavy scalar σ, with mass mσ ¼ 2mH ¼ λHf= 2, A. Structure and parametrics corresponding to the radial mode, 3 NGBs eaten by the In this section we outline the essential aspects of the TH twin gauge bosons, which get masses ∼gf=2 and the ˆ mechanism. Up to details and variants which are not crucial massless H. When turning on λh, SOð8Þ is broken for the present discussion, the TH scenario involves an explicitly and H acquires a potential. At leading order in exact duplicate, SM,g of the SM fields and interactions, ˆ a λh=λH expansion the result is simply given by substituting underpinned by a Z2 symmetry. In practice this Z2 must be 2 2 2 3 jH~ j ¼ f =2 − jHj in Eq. (2.1). The quartic coupling and explicitly broken in order to obtain a realistic phenom- the correction to the squared mass are then given by enology, and perhaps more importantly, a realistic cosmol- – ogy [19 21]. However the sources of Z2 breaking can have λ ≃ λˆ δ 2 ∼ λˆ 2 8 ≃ λ 2λ 2 a structure and size that makes them irrelevant in the h h mH hf = ð h= HÞmH: ð2:2Þ discussion of naturalness in electroweak symmetry break- ing, which is the main goal of this section. 3Notice that the effective Higgs quartic receives approximately Our basic assumption is that the SM and its twin emerge equal contributions from jHj4 and jH~ j4. This is a well-known and from a more fundamental Z2-symmetric theory at the scale interesting property of the TH, see for instance Ref. [7].

095036-3 ROBERTO CONTINO et al. PHYSICAL REVIEW D 96, 095036 (2017) 2 λ ∼ 0 5 As mentioned above, we assume that mH also receives an than) saturates its experimental value h . for ∼ 3–10 4 independent contribution from a Z2-breaking mass term, m TeV. We can thus write which can be ignored in the estimates of tuning. Note that 3 4 1 in terms of the physical masses of the Higgs, mh, and of its δ 2 ∼ λ 2λ 2 ∼ yt 2 mh ð h= HÞm 2 lnðm=mtÞm heavy twin, mσ, we have precisely the same numerical 2π λH 2 2 relation δm ¼ðλ =2λHÞmσ. The amount of tuning ϵ, 2 2 h h 3y y m2=δm2 ϵ 2ξ 2v2=f2 ≡ t × t ×ln m =m × m2 2:3 defined as h h,isgivenby ¼ ¼ . 2π2 2 ð tÞ ð Þ δ 2 g Our estimate of mH in Eq. (2.2) is based on a simplifying approximation where the SOð8Þ-breaking which should be compared to the first term on the right- quartic is taken Z2-symmetric. In general we could allow λˆ λˆ 4 ~ 4 hand side of Eq. (1.1). Accounting for the possibility of different couplings h and h~ for jHj and jHj respectively, ˆ ˆ tuning we then have constrained by the requirement λ þ λ~ ≃ 2λ . As the h h h sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ 2 ~ 4 rffiffiffi estimate of mH in Eq. (2.2) is determined by the jHj 1 1 ˆ ∼ 0 45 gffiffiffi term, it is clear that a reduction of λ~ , with λ fixed, would m . × p × × TeV: ð2:4Þ h h 2y lnðm =m Þ ϵ improve the tuning, as emphasized in Ref. [22].As t t discussed in Sec. IV, however, a significant fraction of ˆ ˆ Compared to Eq. (1.2), the mass of colored states is on one the contribution to λ and λ~ is coming from RG evolution pffiffiffi h h hand parametrically boosted by the ratio g = 2y , and on due to the top and twin top. According to our analysis, ð tÞ ˆ ˆ the other it is mildly decreased by the logarithm. The λ =λ~ varies between 1.5 in the simplest models to 3 in h h motivation for, and gain in, the ongoing work on the λˆ models where h~ is purely IR-dominated as in Ref. [22]. simplest realization of the TH idea pivot upon the above Though interesting, this gain does not change our para- g=yt. The basic question is how high g can be pushed metric estimates. without leading to a breakdown of the effective description. λ λ The ratio h= H is the crucial parameter in the game. One goal of this paper is to investigate to what extent one Indeed it is through Eq. (2.2) that mH is sensitive to can realistically gain from this parameter in more explicit quantum corrections to the Lagrangian mass parameter mH, CH realizations. Applying naive dimensional analysis or, equivalently, that the physical Higgs mass m is h (NDA) one would be tempted to say that g as big as sensitive to the physical mass of the radial mode mσ.In ∼4π ∼ 10 makes sense, in which case m TeV would only particular, what matters is the correlation of mσ with, and its cost a mild ϵ ∼ 0.1 tuning. However such an estimate seems sensitivity to, m, where new states with SM quantum quantitatively too naive. For instance, by focusing on the numbers appear. One can think of three basic scenarios for simple toy model whose potential is given by Eq. (2.1),we that relation, which we now illustrate, ordering them by λ ≡ 2 can associate the upper bound on H g, to the point increasing level of model-building ingenuity. Beyond these where perturbation theory breaks down. One possible way scenarios there is the option of tadpole-dominated electro- to proceed is to consider the one loop beta function weak symmetry breaking, which we shall briefly discuss at the end. dλH N þ 8 μ ¼ λ2 ; ð2:5Þ dμ 32π2 H 1. Subhypersoft scenario ∼ and to estimate the maximum value of the coupling λH as The simplest option is given by models with mσ m. Supersymmetric TH models with medium- to high-scale that for which ΔλH=λH ∼ Oð1Þ through one e-folding of RG evolution. We find soft-term mediation belong to this class [7], with m representing the soft mass of the squarks. Like in the 2 2 MSSM, mH, and therefore mσ, is generated via RG 2mσ 32π mσ λH ¼ ≲ ⇒ ≲ π; for N ¼ 8; ð2:6Þ evolution: two decades of running are sufficient to obtain f2 N þ 8 f mσ ∼ m . Another example is composite TH models ffiffiffiffiffiffi pffiffiffi [13,14]. In their simplest incarnation they are characterized ∼ pλ ≲ 2π which also gives g H , corresponding to a by one overall mass scale m and coupling g [31], so that significantly smaller maximal gain in Eq. (2.4) with respect ∼ λ ∼ 2 by construction one has mσ m and H g. As dis- to the NDA estimate. In Sec. III B we shall perform cussed below Eq. (2.2), in both these scenarios one then alternative estimates in more specific CH constructions, δ 2 ∼ λ 2λ 2 expects mh ð h= HÞm. It is interesting to compare this obtaining similar results. result to the leading top-sector contribution in Eq. (1.1).For that purpose it is worth noticing that, as discussed in 4For this naive estimate we have taken the twin-top contribu- Sec. IV, in TH models the RG-induced contribution to the tion equal to the top one, so that the result is just twice the SM Δλ ∼ 3 4 π2 Higgs quartic coupling hjRG ð yt = Þ ln m=mt (more one. For a more precise statement see Sec. IV.

095036-4 PRECISION TESTS AND FINE TUNING IN TWIN HIGGS … PHYSICAL REVIEW D 96, 095036 (2017)

It is perhaps too narrow-minded to stickpffiffiffi rigidly to such It should be said that in this scenario there is no extra boost 2 λ 2 estimates to determine the boost that g=ð ytÞ can give to of m at fixed tuning by taking H >yt . Indeed the choice 2 m. Although it is parametrically true that the stronger the λH ∼ yt is preferable as concerns electroweak precision coupling g, the heavier the colored partners can be at fixed tests (EWPT). It is well known that RG evolution in the ˆ tuning, the above debate over factors of a few make it effective theory below mσ gives rise to corrections to the S difficult to be more specific in our estimates. In any case the and Tˆ parameters as discussed in Sec. V [32]. In view of the gain permitted by Eq. (2.4) is probably less than one might 2 2 relation ϵ ¼ 2v =f this gives a direct connection between naively have hoped, making it fair to question the moti- fine-tuning electroweak precision data, and the mass of the vation for the TH, at least in its “subhypersoft” realization. 2 2 twin Higgs mσ. At fixed v =f , EWPT then favor the With this reservation in mind, we continue our exploration pffiffiffi pffiffiffi λ 2 of the TH in the belief that the connection between smallest possible mσ ¼ Hf= , that is the smallest λ ∼ 2 naturalness and LHC signatures is so crucial that it must H yt . The most plausible spectrum in this class of models is roughly the following: the twin scalar σ and be analyzed in all its possible guises. pffiffiffi ∼ 2 Concerning in particular composite TH scenarios one the twin tops appear around the same scale ytf= , last important model building issue concerns the origin of below the colored partners who live at m. The presence of σ the Higgs quartic λh. In generic CH it is known that the the somewhat light scalar is one of interesting features of λ 2 this class of models. contribution to h that arises at Oðyt Þ is too large when g is strong. Given that the TH mechanism demands g as strong as possible then composite TH models must ensure that the 3. Superhypersoft scenario 2 λ leading Oðyt Þ contribution is absent so that h arises at This option is a clever variant of the previous one, where Oðy4Þ. As discussed in Ref. [13], this property is not t below the scale m approximate supersymmetry survives in guaranteed but it can be easily ensured provided the the Higgs sector in such a way that the leading contribution couplings that give rise to y via partial compositeness 2 t to δmH proportional to λH is purely due to the top sector respect specific selection rules. [7]. In that way Eq. (2.7) reduces to   2. Hypersoft scenario λ 3 2 2 3λ δ 2 ∼ h yt 2 yt h 2 mh 2 m ¼ × 2 m: ð2:9Þ The second option corresponds to the structurally robust 2λH 4π λH 8π 2 2 situation where mσ is one loop factor smaller than m. This is for instance achieved if H is a PNG-boson octet multiplet so that by choosing g >yt one can push the scale m associated to the spontaneous breaking SOð9Þ → SOð8Þ in further up with fixed fine tuning ϵ a model with fundamental scale m. Another option would rffiffiffi be to have a supersymmetric model where supersymmetric 1 ∼ 1 4 gffiffiffi m . × p × TeV: ð2:10Þ masses of order m are mediated to the stops at the very 2y ϵ H t scale m at which is massless. Of course in both cases a 2 precise computation of mσ would require the full theory. In principle even under the conservative assumption that pffiffiffi However a parametrically correct estimate can be given by ∼ 2π g is the maximal allowed value, this scenario considering the quadratically divergent 1-loop corrections ∼ 14 ϵ ∼ 0 1 seemingly allows m TeV with a mild . tuning. in the low energy theory, in the same spirit of Eq. (1.1).As It should be said that in order to realize this scenario one λ yt and H are expected to be the dominant couplings the would need to complete H into a pair of chiral superfield analogue of Eqs. (1.1) and (2.2) imply octets Hu and Hd, along the lines of Ref. [7], as well as add     a singlet superfield S in order to generate the Higgs quartic λ 3 2 5λ 2 5 3λ δ 2 ∼ h yt H 2 yt h 2 via the superpotential trilinear g SH H . Obviously this is mh 2 þ 2 m ¼ þ 2 m: u d 2λH 4π 16π λH 12 8π a very far-fetched scenario combining all possible ideas to ð2:7Þ explain the smallness of the weak scale: supersymmetry, compositeness, and the twin Higgs mechanism. 2 Very roughly, for λH ≳ yt , top effects become subdominant and the natural value for mh becomes controlled by λh, like 4. Alternative vacuum dynamics: Tadpole induced EWSB the term induced by the Higgs quartic in Eq. (1.1). In the In all the scenarios discussed so far the tuning of the absence of tuning this roughly corresponds to the techni- Higgs vacuum expectation value (VEV) and that of the ∼ 4π 2 color limit m v, while allowing for fine tuning we ϵ Higgs mass coincided: , which controls the tuning of mh have according to Eqs. (2.4), (2.8), and (2.10), is equal to 2 2 rffiffiffi 2v =f , which measures the tuning of the VEV. This was 1 m ∼ 1.4 × TeV: ð2:8Þ because the only tuning in the Higgs potential was ϵ associated with the small quadratic term, while the quartic

095036-5 ROBERTO CONTINO et al. PHYSICAL REVIEW D 96, 095036 (2017) was assumed to be of the right size without the need for particular it is enhanced at large NTC.EvenatNTC ¼ 2, further cancellations (see, e.g., the discussion in Ref. [33]). staying within the allowed ðS;ˆ Tˆ Þ ellipse still requires a Experimentally however, one can distinguish between the correlated contribution from ΔTˆ , which in principle should need for tuning that originates from measurements of Higgs also be counted as tuning. In spite of this, models with and electroweak observables, which are controlled by 2 2 tadpole-induced EWSB represent a clever variant where, v =f , and that coming from direct searches for top technically, the dynamics of EWSB does not currently partners. Currently, with bounds on colored top partners appear tuned. A thorough analysis of the constraints is at just around 1 TeV [34,35], but with Higgs couplings certainly warranted. already bounded to lie within 10%–20% of their SM value [24], the only reason for tuning in all TH scenarios is to III. THE COMPOSITE TWIN HIGGS achieve a small v2=f2. It is then fair to consider options that reduce or eliminate only the tuning of v2=f2. As argued in In this section and in the remainder of the paper, we will Ref. [18], this can be achieved by modifying the H scalar focus on the CH realization of the TH, which belongs to the vacuum dynamics, and having its VEV induced instead by a subhypersoft class of models. In this simple and well- tadpole mixing with an additional electroweak-breaking motivated context we shall discuss EWPT, fine tuning, and technicolor (TC) sector [36–38]. In order to preserve the Z2 structural consistency of the model. symmetry one adds two twin TC sectors, both characterized Our basic structural assumption is that at a generic UV scale Λ ≫ m , our theory can be decomposed into two by a mass scale mTC and a decay constant fTC ∼ mTC=4π UV sectors: a strongly-interacting composite sector and a (i.e., it is parametrically convenient to assume gTC ∼ 4π). Below the TC scale the dynamics in the visible and twin weakly-interacting elementary sector. The composite sectors is complemented by Goldstone triplets π and π~ sector is assumed to be endowed with the global symmetry a a 8 1 which can be embedded into doublet fields according to G ¼ SOð Þ × Uð ÞX × Z2 and to be approximately scale-     (conformal) invariant down to the scale m, at which it 0 0 develops a mass gap. We assume the overall interaction π σ ~ π~ σ Σ ¼ f ei a a ; Σ ¼ f ei a a ; ð2:11Þ TC 1 TC 1 strength at the resonance mass scale m to be roughly described by one parameter g [31]. The large separation of mass scales Λ ≫ m is assumed to arise naturally, in and are assumed to mix with H and H~ via the effective UV that the occurrence of the mass gap m is controlled by potential terms either a marginally relevant deformation, or by a relevant 2 †Σ ~ †Σ~ deformation whose smallness is controlled by some global Vtadpole ¼ M ðH þ H ÞþH:c: ð2:12Þ 8 1 symmetry. At the scale m, SOð Þ × Uð ÞX × Z2 is sponta- 7 1 ~ neously broken to the subgroup H ¼ SOð Þ × Uð ÞX, m ≪ mH H Assuming TC the vacuum dynamics is not giving rise to seven NGBs in the 7 of SOð7Þ with decay significantly modified, but, for mTC >mh, Vtadpole acts ∼ 1 constant f m=g. The subgroup Uð ÞX does not par- like a rigid tadpole term for H. The expectation value hHi is ticipate to the spontaneous breaking, but its presence is thus determined by balancing such a tadpole against the needed to reproduce the hypercharges of the SM fermions, gauge-invariant jHj2 mass term; the latter will then roughly 2 similarly to CH models. The elementary sector consists in coincide with mh. In order for this to work, by Eq. (2.2) the turn of two separate weakly interacting sectors: one ˆ SOð8Þ-breaking quartic λh should be negative, resulting in containing the visible SM fermions and gauge bosons, ∼ 2 2 3 v ðM =mhÞfTC. It is easy to convince oneself that the corresponding to the SM gauge group GSM ¼ SUð Þc× 2 2 2 1 corrections to Higgs couplings are OðfTCffiffiffiffiffi=v Þ: present SUð ÞL × Uð ÞY; the other containing the twin SM p g bounds can then be satisfied for fTC ∼ v= 10 ≃ 80 GeV. with the same fermion content and a SM gauge group ~ f f In turn, the value of v=f is controlled by f and can thus be G ¼ SUð3Þ × SUð2Þ . The external Z2 symmetry, or ∼ 4π ∼ SM c L naturally small. The TC scale is roughly mTC fTC twin parity, interchanges these two copies. For simplicity, 600–800 GeV, while the noneaten pNGB π in Eq. (2.11) and following [13], we choose not to introduce a mirror 2 ∼ 2 ∼ 2 2 ∼ 400 have a mass mπ M v=fTC mhðv=fTCÞ GeV. hypercharge field. This is our only source of explicit twin- The latter value, although rather low, is probably large parity breaking, and affects neither our discussion of fine enough to satisfy constraints from direct searches. In our tuning, nor that of precision electroweak measurements. opinion, what may be more problematic are EWPT, in view The elementary and composite sectors are coupled of the effects from the TC sector, which shares some of the according to the paradigm of partial compositeness [23]. vices of ordinary TC. The IR contributions to Sˆ and Tˆ , The elementary EW gauge bosons couple to the strong 2 associated with the splitting mπa

095036-6 PRECISION TESTS AND FINE TUNING IN TWIN HIGGS … PHYSICAL REVIEW D 96, 095036 (2017) LV ⊃ α μ μ ~ ~ α ~ μ case of a two-site model developed in Ref. [13]. According mix g2WμJα þ g1BμJB þ g2WμJα; ð3:1Þ to the CCWZ technique, a Lagrangian invariant under the 8 where g1;2 and g~2 denote the SM and twin weak gauge global SOð Þ group can be written following the rules of a μ μ μ μ μ μ 7 couplings, J ≡ J þ J and J , J~ and J are the local SOð Þ symmetry. The basic building blocks are the B 3R X X Σ Π 2 f 2 Goldstone matrix ð Þ, which encodes the seven NGBs, currents associated respectively to the SUð ÞL, SUð ÞL Π Π 1 , present in the theory, and the operators dμð Þ and and Uð ÞX generators. The elementary fermions mix EμðΠÞ resulting from the Maurer-Cartan form constructed analogously with various operators transforming as linear U 1 representations of SOð8Þ that are generated in the far with the Goldstone matrix. An external ð ÞX group is also UV by the strongly interacting dynamics. The mixing added to the global invariance in order to reproduce the Lagrangian takes the schematic form: correct fermion hypercharges [13]. The CCWZ approach is reviewed and applied to the SOð8Þ=SOð7Þ coset in F α A A ¯ α ~ ~ A Appendix A. L ⊃ q¯ Δα O þ t¯ Θ O þ q~ Δα O mix L A R R A L L A R Before proceeding, we would like to recall the simplified ¯ ~ ~ A þ tRΘAOL þ H:c:; ð3:2Þ model philosophy of Ref. [43], which we essentially employ. In a generic composite theory, the mass scale σ where, following, e.g., Ref. [39], we introduced spurions m would control both the cutoff of the low energy -model ~ ~ and the mass of the resonances. In that case no effective ΔαA, ΔαA, ΘA, and ΘA in order to uplift the elementary fields to linear representations of SOð8Þ, and match the Lagrangian method is expected to be applicable to describe quantum numbers of the composite operators. The left- the resonances. So, in order to produce a manageable ~ effective Lagrangian we thus consider a Lagrangian for handed mixings Δα , Δα necessarily break SOð8Þ since q A A L resonances that can, at least in principle, be made lighter only partially fills a multiplet of SOð8Þ. The right-handed that m . One more structured way to proceed could be to mixings, instead, may or may not break SOð8Þ. The consider a deconstructed extra-dimension where the mass breaking of SOð8Þ gives rise to a potential for the NGBs at one loop and the physical Higgs is turned into a pNGB. of the lightest resonances, corresponding to the inverse compactification length, is parametrically separated from We conclude by noticing that g1 2 and g~2 correspond to ; the 5D cut-off, interpreted as m . Here we do not go that far quasimarginal couplings which start off weak in the UV, and simply consider a set of resonances that happen to be a and remain weak down to m . The fermion mixings could bit lighter than m . We do so to give a structural dignity to be either relevant or marginal, and it is possible that some our effective Lagrangian, though at the end, for our may correspond to interactions that grow as strong as g at numerical analysis, we just take the resonances a factor the IR scale m [40]. In particular, as is well known, there is of 2 below m . We believe that is a fair procedure given our some advantage as regards tuning in considering the right purpose of estimating the parametric consistency of the Θ Θ~ mixings A and A to be strong. In that case one may even general TH scenario. imagine the IR scale to be precisely generated by the We start our analysis of the effective Lagrangian with the corresponding deformation of the fixed point. While this bosonic sector. Together with the elementary SM gauge latter option may be interesting from a top-down perspec- bosons, the W’s and B, we introduce the twin partners W~ tive, it would play no appreciable role in our low-energy f 2 phenomenological discussion. to gauge the SUð ÞL group. As representative of the composite dynamics, we restrict our interest to the heavy spin-1 resonances transforming under the adjoint of SOð7Þ A. A simplified model and to a vector singlet. We therefore introduce a set of a In order to proceed we now consider a specific realiza- vectors ρμ which form a 21 of SOð7Þ and the gauge vector 1 ρX tion of the composite TH and introduce a concrete associated with the external Uð ÞX, which we call μ . The simplified effective Lagrangian description of its dynamics. Lagrangian for the bosonic sector can be written as Our model captures the most important features of this class of theories, like the pNGB nature of the Higgs field, and L L LV LV LV bosonic ¼ π þ comp þ elem þ mix: ð3:3Þ provides at the same time a simple framework for the interactions between the elementary fields and the The first term describes the elementary gauge bosons composite states, vectors and fermions. We make use of masses and the NGBs dynamics and is given by this effective model as an example of a specific scenario in which we can compute EW observables, and study the 2 L f μ feasibility of the TH idea as a new paradigm for physics at π ¼ 4 Tr½dμd : ð3:4Þ the EW scale. LV We write down an effective Lagrangian for the com- The second term, comp, is a purely composite term, posite TH model using the Callan-Coleman-Wess-Zumino generated at the scale m after confinement; it reduces (CCWZ) construction [41,42], and generalizing the simpler to the kinetic terms for the ρ vectors, namely:

095036-7 ROBERTO CONTINO et al. PHYSICAL REVIEW D 96, 095036 (2017) 1 1 6 Ψ LV − ρa ρμνa − ρX ρXμν singlet. We denote with 7 the fermionic resonances in comp ¼ 2 μν 2 μν ; ð3:5Þ Ψ 4gρ 4g X the septuplet and with 1 the singlet, both charged ρ 3 under SUð Þc. Together with them, we must introduce a a a b c X X X f 3 where ρμν ¼ ∂μρν − ∂νρμ − f ρμρν, ρμν ¼ ∂μρν − ∂νρμ , analogous composite states charged under SUð Þc;we abc ~ ~ and gρ and gρX are the coupling strengths for the composite use the corresponding notation Ψ7 and Ψ1.Wereferto LV Ψ Ψ~ spin-1 bosons. The third term in Eq. (3.3), elem, is a purely Ref. [13] for the complete expression of 7 and 7 in Λ elementary interaction, produced at the scale UV where terms of the constituent fermions. the elementary fields are formally introduced. Also this The fermionic effective Lagrangian is split into three Lagrangian can contain only the kinetic terms for the parts, which have the same meaning as the analogous elementary fields: distinctions we made for the bosonic sector of the theory:

1 1 1 L LF LF LF : 3:8 V μν a aμν ~ a ~ aμν fermionic ¼ comp þ elem þ mix ð Þ L ¼ − BμνB − WμνW − WμνW ; elem 4g2 4g2 4g~2 1 2 2 The fully composite term is given by: ð3:6Þ LF Ψ¯ − Ψ Ψ¯ − Ψ comp ¼ 7ði=D7 MΨÞ 7 þ 1ði=D1 MSÞ 1 where g1, g2, and g~2 denote the weak gauge couplings. The ~¯ ~ ~ ~¯ ~ ~ LV þ Ψ7ði=∇ − MΨÞΨ7 þ Ψ1ði=∂ − MSÞΨ1 last term in the Lagrangian (3.3), mix, is a mixing term between the elementary and composite sectors originating ¯ i ¯ i ~¯ i ~ þðic Ψ d= Ψ1 þ ic Ψ d= Ψ1 þ ic~ Ψ7 d= Ψ1 from partial compositeness. We have:5 L 7L i L R 7R i R L L i L ~¯ i ~ þ ic~RΨ7Rd= iΨ1R þ H:c:Þ; ð3:9Þ 2 2 M X V Mρ a 21 2 ρ X 2 L ¼ ðTr½ρμT − EμÞ þ ðρμ − BμÞ ; ð3:7Þ mix 2 2 a 2 2 where D7μ ¼ ∇μ þ iXBμ, D1μ ¼ ∂μ þ iXBμ, and ∇μ ¼ gρ gρX ∂μ þ iEμ. We have introduced two sets of Oð1Þ coeffi- 21 cients, cL and cR and their twins, for the interactions where Ta are the SOð8Þ generators in the adjoint of SOð7Þ mediated by the dμ operator. Considering the elementary (see Appendix A). part of the Lagrangian, it comprises just the kinetic terms We now introduce the Lagrangian for the fermionic for the doublets and right-handed tops: sector. This depends on the choice of quantum numbers for the composite operators in Eq. (3.2). The minimal LF ¯ ¯ ~¯ ~ ~¯ ∂~ ~ elem ¼ qLi=DqL þ tRi=DtR þ qLi=DqL þ tRi=tR: ð3:10Þ option is to choose OR and OR to be in the fundamental 8 O representation of SOð Þ,whereastheoperators L and The final term in our classification is the elementary/ O~ L are singlets of the global group. Therefore, the composite mixing that we write again following the elementary SM doublet and its twin must be embedded prescription of partial compositeness. With our choice of into fundamental representations of SOð8Þ,whereas quantum numbers for the composite operators, the spurions ~ the tR and the tR are complete singlets under the in Eq. (3.2) can be matched to dimensionless couplings global SOð8Þ invariance. This choice is particularly according to useful to generalize our discussion to the case of a   fully-composite right-handed top. From the low-energy 00iyL −yL 0 × 4 Δα ¼ ; Θ ¼ y ; perspective, the linear mixing between composite oper- A iy y 000× 4 A R ators and elementary fields translates into a linear L L coupling between the latter and a layer of fermionic ð3:11Þ resonances excited from the vacuum by the operators in the fundamental and singlet representations of and 8 8   the global group. Decomposing the of SOð Þ as 0 40 0 ~ −~ 8 7 1 7 ~ × iyL yL ~ ¼ þ under SOð Þ,weintroduceasetoffermionic Δα ¼ ; Θ ¼ y~ ; A 0 4 ~ ~ 00 A R resonances filling a complete fundamental represen- × iyL yL tation of SOð7Þ and another set consisting of just one ð3:12Þ

5 Notice that in the Lagrangian (3.7), the parameters f, Mρ, 6Notice that in general we should introduce two different M X , gρ, and g X are all independent. It is common to define the ρ ρ singlets in our Lagrangian. One corresponds to a full SOð8Þ parameters aρ Mρ= gρf and a X M X = g X f , which are ¼ ð Þ ρ ¼ ρ ð ρ Þ pffiffiffi singlet, while the other is the SOð7Þ singlet appearing in the expected to be Oð1Þ. In our analysis we set aρ ¼ 1= 2 decomposition 8 ¼ 7 þ 1 of the fundamental of SOð8Þ under the corresponding to the two-site model value (see the last paragraph SOð7Þ subgroup. We will further simplify our study identifying of this section) and aρX ¼ 1. the two singlets with just one composite particle.

095036-8 PRECISION TESTS AND FINE TUNING IN TWIN HIGGS … PHYSICAL REVIEW D 96, 095036 (2017) ’ where we have introduced the elementary/composite mix- Eq. (3.2), the y s can vary from weak to OðgÞ. ing parameters yL, yR and their twin counterparts. These Correspondingly the light fermions vary from being dimensionless y’s control the strength of the interaction completely elementary (for y weak) to effectively fully ∼ between the elementary and composite resonance fields, composite (for y g). For reasons that will become ∼ according to the Lagrangian: clear, given yt yLyR=g, it is convenient to take ≃ ~ ∼ ≃ ~ ∼ yL yL yt, i.e., weak left mixing, and yR yR g. LF ¯ α Δ Σ Ψi ¯ α Δ Σ Ψ ¯ Ψ mix ¼ fðqL αA Ai 7 þ qL αA A8 1 þ yRtR 1 þ H:c:Þ For such strong right-handed mixing the right-handed ¯ α ~ ~ i ¯ α ~ ~ ¯ ~ tops can be practically considered part of the strong f q~ Δα Σ Ψ7 q~ Δα Σ 8Ψ1 y~ t~ Ψ1 H:c: : þ ð L A Ai þ L A A þ R R þ Þ sector. ð3:13Þ The last term that we need to introduce in the effective Lagrangian describes the interactions between the vector Depending on the UV boundary condition and the and fermion resonances and originates completely in the relevance or marginality of the operators appearing in composite sector. We have:

X LVF α Ψ¯ ρ − Ψ α Ψ¯ ρX − Ψ α Ψ¯ ρX − Ψ comp ¼ ½ i 7ið= =EÞ 7i þ 7i 7ið= =BÞ 7i þ 1i 1ið= =BÞ 1i i¼L;R

~¯ ~ ~¯ X ~ ~¯ X ~ þ α~ iΨ7ið=ρ − =EÞΨ7i þ α~ 7iΨ7ið=ρ − =BÞΨ7i þ α~ 1iΨ1ið=ρ − =BÞΨ1i; ð3:14Þ

where all the coefficients αi appearing in the Lagrangian are Alternatively, the limit set by perturbativity on the UV Oð1Þ parameters. interaction strength may also be estimated in the effective We conclude the discussion of our effective Lagrangian theory described by the nonlinear σ-model by determining by clarifying its two-site model limit [44] (see also the energy scale at which tree-level scattering amplitudes Ref. [45]). This is obtained by combining the singlet become nonperturbative. For concreteness, we considered and the septuplet into a complete representation of the following two types of scattering processes: ππ → ππ 8 8 ~ SOð Þ, so that the model enjoys an enhanced SOð ÞL × and ππ → ψψ¯ , where π are the NGBs and ψ ¼fΨ7; Ψ7g 8 SOð ÞR global symmetry. This is achieved by setting cL ¼ denotes a composite fermion transforming in the funda- c c~ c~ 0 and all the α equal to 1. Moreover, we R ¼ L ¼ R ¼ pffiffiffi i mental of SOð7Þ. Other processes can (and should) be have to impose Mρ ¼ gρf= 2, so that the heavy vector considered, with the actual bound being given by the resonances can be reinterpreted as gauge fields of SOð7Þ. strongest of the constraints obtained in this way. As shown in Ref. [44], with this choice of the free Requiring that the process ππ → ππ stay perturbative up parameters the Higgs potential becomes calculable up to the cutoff scale m gives the bound only a logarithmic divergence, that one can regulate by 4π imposing just one renormalization condition. In the sub- Mρ MΨ m ∼ ≲ < pffiffiffiffiffiffiffiffiffiffiffiffi ≃ 5.1; ð3:15Þ sequent sections, we will extensively analyze the EW f f f N − 2 precision constraints in the general case, as well as in the two-site limit. where the second inequality is valid in a generic SOðNÞ=SOðN − 1Þ nonlinear σ-model, and we have set B. Perturbativity of the simplified model N ¼ 8 in the last step. More details on how this result was obtained can be found in Appendix G. Equation (3.15) in In Sec. II A 1 it was noted that a TH construction fact corresponds to a limit on the interaction strength of typically involves large multiplicities of states and, as a the UV theory, given that the couplings among fermion consequence, the dynamics responsible for its UV com- and vector resonances are of order MΨ=f and Mρ=f, pletion cannot be maximally strongly coupled. This in turn respectively. Perturbativity of the scattering amplitude limits the improvement in fine tuning that can be achieved ππ → ψψ¯ compared to standard scenarios of EWSB. In our naive for instead gives (see Appendix G for details) estimates of Eqs. (2.3), (2.7), and (2.9) the interaction pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffi Mρ 12 2π 4π strength of the UV theory was controlled by the σ-model ∼ MΨ ≲ m pffiffiffiffiffiffi ≃ pffiffiffiffiffiffi λ < ; ð3:16Þ quartic coupling H or, equivalently, by mσ=f. By consid- f f f Nf Nf ering the λH one-loop β-function [Eq. (2.5)] we estimated λ the maximal value of H as the one corresponding to an where Nf is the multiplicity of composite fermions Oð1Þ relative change through one e-folding of RG evolu- pffiffiffiffiffiffi pffiffiffi (including the number of colors and families). Our sim- tion. For an SOð8Þ=SOð7Þ σ-model this led to λH ≲ 2π, plified model with one family of composite fermions has or, equivalently, mσ=f ≲ π. Nf ¼ 6, which gives a limit similar to Eq. (3.15):

095036-9 ROBERTO CONTINO et al. PHYSICAL REVIEW D 96, 095036 (2017) 7 MΨ=f ≲ 5.3. A model with three families of composite effects. The second term, generated by loops of quarks and leptons has instead Nf ¼ 24, from which fermions, is Z2 symmetric and explicitly violates the follows the stronger bound MΨ=f ≲ 2.6. SOð8Þ invariance; it thus corresponds to the (UV part As a third alternative, one could analyze when 1-loop of the) last term of Eq. (2.1). Upon electroweak corrections to a given observable become of the same order symmetry breaking, Eq. (4.1) contributes to the physi- as its tree-level value. We applied this criterion to our cal Higgs mass an amount equal to simplified model by considering the Sˆ parameter, the new 3 4 physics contribution to which includes a tree-level correc- 2 yL 2 δm F1f ξ 1 − ξ ; 4:2 tion from heavy vectors given by Eq. (5.6), and a one-loop hjUV ¼ 4π2 ð Þ ð Þ correction due to heavy fermions, which can be found in Appendix E. By requiring that the one-loop term be smaller where ξ controls the degree of vacuum misalignment: than the tree-level correction, we obtain a bound on the 2 strong coupling constant gρ. As an illustration, we considerffiffiffi ξ ≡ 2 hhi v p sin ¼ 2 : ð4:3Þ the two-site model limit cL ¼ cR ¼ 0 and Mρ ¼ gρf= 2 f f and keep the dominant UV contribution to Sˆ in Eq. (E7) which is logarithmically sensitive to the cutoff. By setting Below the scale m an important contribution to the potential arises from loops of light states, in particular from m ¼ 2MΨ, we find: the top quark and from its twin. The bulk of this IR ˆ contribution is captured by the RG evolution of the Higgs ΔS1− Mρ gρ π loop < 1 ⇒ ¼ pffiffiffi < pffiffiffiffiffiffiffiffiffiffiffiffiffi ≃ 2.7: ð3:17Þ potential from the scale m down to the electroweak scale. Δˆ 2 2 2 Stree f log As noted in previous studies (see, e.g., Ref. [13]), for sufficiently large m this IR effect dominates over the UV The perturbative limits obtained from Eqs. (3.15), (3.16), threshold correction and can reproduce the experimental and (3.17) are comparable to that on λH derivedinSec.II A 1. Higgs mass almost entirely. An analogous IR correction to As already discussed there, one could take any of these the Higgs quartic arises in SUSY theories with large stop results as indicative of the maximal interaction strength in the masses, from loops of top quarks. The distinctive feature of underlying UV dynamics, though none of them should be any TH scenario, including our model, is the additional considered as a sharp exclusion condition. In our analysis of twin top contribution. EW observables we will make use of Eq. (3.15) with N ¼ 8 The Higgs effective action, including the leading OðξÞ and of Eq. (3.16) with N 24 to highlight the regions of f ¼ corrections associated with operators of dimension 6, was parameter space where our perturbative calculation is less computed at 1-loop in Ref. [13]; the resulting IR contri- reliable. We use both limits as a measure of the intrinsic 2 bution to m was found to be uncertainty which is inevitably associated with this type of h   estimation. 4 2 2 1− 3y m m δm2j loop ¼ t f2ξð1 − ξÞ log þ log ; ð4:4Þ h IR 8π2 m2 m~ 2 IV. HIGGS EFFECTIVE POTENTIAL t t

As anticipated in the general discussion of Sec. II A, where yt denotes the top Yukawa coupling. The two single- a potential for the Higgs boson is generated at the scale m log terms in parentheses correspond to the IR contributions λ by loops of heavy states through the SOð8Þ-breaking to the effective Higgs quartic h from the top quark and couplings of the elementary fields to the strong sector. twin top respectively. Leading-logarithmic corrections of n Once written in terms of the Higgs boson h (where the form ðα logÞ , arising at higher loops have however an † 2 2 † 2 2 8 α 2 H H ¼ f sin ðh=fÞ=2, H~ H~ ¼ f cos ðh=fÞ=2), at 1-loop important numerical impact. For example, ð logÞ cor- this UV threshold contribution has the form [13]: rections generated by 2-loop diagrams (mostly due to   the running of the top and twin top Yukawa couplings, 3 1 that are induced by respectively QCD and twin QCD) are VðmÞ 2 2 2 2 2 2 h ¼ g gρL1 þðy − y~ Þg L2 sin ∼30% f4 32π2 16 1 L L Ψ f expected to give a reduction in the Higgs mass   for m ≃ 5 TeV. 3 4 yL 4 h 4 h We have computed the IR contribution to the Higgs mass þ 2 F1 sin þ cos ; ð4:1Þ 64π f f in a combined expansion in ξ and ðα logÞ. We have included the LO electroweak contribution, all terms up to NLO in αt where gΨ ≡ MΨ=f, L1, L2, F1 are Oð1Þ dimensionless functions of the masses and couplings of the theory 7 Subleading Z2-breaking terms have been neglected for sim- and the explicit expression of the function F1 is plicity. The complete expressions are given in Ref. [13]. reported in Eq. (C1) of Appendix C.Thefirstterm 8 α 2 4π Here ¼ gSM= , with gSM being any large SM coupling, in the above equation originates from Z2-breaking i.e., gS and yt.

095036-10 PRECISION TESTS AND FINE TUNING IN TWIN HIGGS … PHYSICAL REVIEW D 96, 095036 (2017) energy particle content (SM plus twin states). In particular, considering the aforementioned input value and choosing 5 ξ 0 1 as a benchmark values m ¼ TeV and ¼ . , we find that the contributions of the SM and twin light degrees of freedom account for around 50% and 40% of the Higgs mass squared, respectively, so that almost the entire experimental value of the Higgs mass can be due only to the IR degrees of freedom. Threshold effects arising at the UV matching scale, on the other hand, are model dependent but give a subleading correction. An accurate prediction of the Higgs mass and an assessment of the plausibility of the model thus requires a precise determi- nation of its IR contribution. Indeed the difference between FIG. 1. IR contribution to the Higgs mass as a function of the the IR contribution of Fig. 1 and the measured value mh ¼ ξ 0 1 125 scale m for ¼ . . The dot-dashed (upper) and dashed (lower) GeV must be accounted for by the UV threshold curves denote the LO and NLO result in a combined perturbative contribution in Eq. (4.2); for our previous benchmark expansion in ðα logÞ and ξ. The dotted curve contains the pure choice of m, about 12% of the Higgs mass should be QCD part of the NNLO correction (see Appendix C for details). generated by the UV physics. This translates into a generic The shaded region represents the uncertainty in our estimate. The constraint on the size of y , a parameter upon which central value of this band, given by the solid black line and L electroweak precision observables (EWPO) crucially computed as the average of the LO and NLO results, is the value that we use as our numerical estimate throughout the paper. depend, thus creating a nontrivial correlation between the Higgs mass, EWPO and naturalness. α and S, and some contributions, expected to be leading for V. ELECTROWEAK PRECISION OBSERVABLES α not too large m, at NNLO in S. We report the details in Appendix C. In this section we compute the contribution of the new δ 2 1=2 states described by our simplified model to the EWPO. The value of ð mhjIRÞ is shown in Fig. 1 as a function ξ 0 1 Although it neglects the effects of the heavier resonances, of m for ¼ . . The upper and lower curves represent respectively the LO and NLO calculation in our combined our calculation is expected to give a fair assessment of the ðξ; α logÞ expansion. Numerically, the naive expectation is size of the corrections due to the full strong dynamics, and δ 2 1=2 in particular to reproduce the correlations among different confirmed, as the NLO correction decreases ð mhjIRÞ by ∼35% 5 observables. for m ¼ TeV. The dotted curve, indicated as 2 It is well known that, under the assumption of quark and NNLO , includes NNLO contributions of order αtξ log, α α ξ 2 α α2 3 lepton universality, short-distance corrections to the t S log , and t S log (see Appendix C). Additional electroweak observables due to heavy new physics can α2ξ 2 α2α 3 α3 3 contributions of order t log , t S log and t log are be expressed in terms of four parameters, Sˆ, Tˆ , W, Y, not included. An attempt to include these contributions has defined in Ref. [47] (see also Ref. [48] for an equivalent been made in Ref. [46]. However, the calculation presented analysis) as a generalization of the parametrization intro- there misses some additional contributions from the twin duced by Peskin and Takeuchi in Refs. [49,50].Two GB and does not represent the full NNLO calculation. The additional parameters, δg and δg , can be added to Lb Rb picture that we get from our NNLO calculation and the account for the modified couplings of the Z boson to left- incomplete result of Ref. [46], is that the NNLO gives an and right-handed bottom quarks respectively.9 A naive overestimate of the IR contribution to the Higgs mass as the estimate shows that in CH theories, including our TH aforementioned neglected effects are expected to give a model, W and Y are subdominant in an expansion in the reduction. Given the lack of a complete NNLO calculation, weak couplings [31] and can thus be neglected. The small we estimate our uncertainty on the IR Higgs mass as the coupling of the right-handed bottom quark to the strong green shaded region lying between the LO and NLO results in Fig. 1. In order to choose, within this uncertainty, a value 9 δ δ that is as close as possible to the full NNLO result, in the We define gLb and gRb in terms of the following effective Lagrangian in the unitary gauge: rest of the paper we take as input value for the IR correction to the Higgs mass the average of the LO and NLO results, g2 L ⊃ ¯γμ SM δ 1 − γ eff Zμb ½ðgLb þ gLbÞð 5Þ corresponding to the solid black line in the figure. 2cW The plot of Fig. 1 illustrates one of the characteristic SM þðg þ δg Þð1 þ γ5Þb þ ð5:1Þ features of TH models: the IR contribution to the Higgs Rb Rb mass largely accounts for its experimental value and is where the dots stand for higher-derivative terms and SM −1 2 2 3 SM 2 3 completely predicted by the theory in terms of the low- gLb ¼ = þ sW= , gRb ¼ sW= .

095036-11 ROBERTO CONTINO et al. PHYSICAL REVIEW D 96, 095036 (2017)   dynamics makes also δg small and negligible in our δg g02 y2 v2 y~ 4 Rb Lb ∼ L : ˆ ˆ δ 16π2 2 ð5 4Þ model. We thus focus on S, T, and gLb, and compute them gLb m g by including effects from the exchange of vector and ~ ∼ 0 2 fermion resonances, and from Higgs compositeness. which, even assuming y g,isOðg =ytÞ suppressed with We work at the 1-loop level and at leading order in the respect to the leading visible sector effect we discuss below. electroweak couplings and perform an expansion in inverse Aside the quadratically divergent piece there is also a powers of the new physics scale. In this limit, the twin logarithmic divergent piece whose overall coefficient is states do not affect the EWPO as a consequence of their calculable. The result is further suppressed with respect to being neutral under the SM gauge group. Deviations from 2 2 2 2 the above contribution by a factor ðmt~ =mÞ lnðmt~ =mÞ. the SM predictions arise only from heavy states with SM An additional contribution could in principle come from quantum numbers and are parametrically the same as in loops of the extra three “twin” NGBs contained in the coset ordinary CH models with singlet tR. This can be easily SOð8Þ=SOð7Þ. Simple inspection however shows that there shown by means of naive dimensional analysis and is no corresponding 1-loop diagram contributing to the symmetries as follows. Twin tops interact with the SM EWPO. In the end we conclude that the effect of twin loops fields only through higher-dimensional operators. The is negligible. operators relevant for the EWPO are those involving either Since the effects from the twin sector can be neglected, ˆ ˆ a SM current or a derivative of the hypercharge field the corrections to S, T, and δgLb are parametrically the strength: same as in ordinary CH models. We now give a concise review of the contributions to each of these quantities, 0 g μ ¯ ν 1 ¯ μ distinguishing between the threshold correction generated O ~ ¼ ∂ Bμνt~γ t;~ O ~ ¼ q¯ γμq t~γ t;~ Bt 2 qt 2 L L at the scale m and the contribution arising from the RG mW v ↔ evolution down to the electroweak scale. For recent i † ~¯γμ~ analyses of the EWPO in the context of SO 5 =SO 4 OHt~ ¼ 2 H DμHt t; ð5:2Þ ð Þ ð Þ v CH models see for example Refs. [29,30,45].

~ 10 where t indicates either a right- or left-handed twin top. A. Sˆ parameter The first two operators of Eq. (5.2) are generated at the Sˆ scale m by the tree-level exchange of the ρ . Their The leading contribution to the parameter arises at tree X level from the exchange of spin-1 resonances. Since only coefficients (in a basis with canonical kinetic terms) are 2 2 ~ 2 2 2 2 the (3,1) and (1,3) components of the spin-1 multiplet respectively of order ðmW=mÞðy=gÞ and ðyLv =mÞ × 5 4 ~ 2 ~ ~ ~ contribute, its expression is the same as in SOð Þ=SOð Þ ðy=gÞ , where y equals either yL or yR depending on the composite-Higgs theories11: chirality of t~. The third operator breaks custodial isospin and the only way it can be generated is via the exchange of 2 Δˆ g2 ξ weakly coupled elementary fields at loop level. Given that Sρ ¼ 2 : ð5:5Þ 2gρ the contribution to EWPO is further suppressed by t~ loops, the third operator can affect EWPO only at, at least, two In our numerical analysis presented in Sec. VI we use the loops and is thus clearly negligible. By closing the t~ loops pffiffiffi two-site model relation Mρ ¼ gρf= 2 to rewrite the first two operators can give rise to effects that are 2 schematically of the form BB, Bqq¯ ,orðqq¯ Þ . The formally 2 Δˆ mW quadratically divergent piece of the loop integral renorm- Sρ ¼ 2 : ð5:6Þ alizes the corresponding dimension-6 operators. For in- Mρ stance the second structure gives The 1-loop contribution from loops of spin-1 and fermion   resonances is subdominant (by a factor g2=16π2) and will 0 2 ~ 4 g yL y ∂ μν ¯ γ be neglected for simplicity in the following. Nevertheless, C 16π2 2 νB qL μqL ð5:3Þ m g we explicitly computed the fermionic contribution (see Appendix E) to monitor the validity of the perturbative with C an Oð1Þ coefficient which depends on the details of expansion and estimate the limit of strong coupling in our the physics at the scale m. Using the equations of motion model (a discussion on this aspect was given in Sec. III B). for B, the above operator gives rise to a correction to the An additional threshold correction to Sˆ, naively of the same Zbb¯ vertex of relative size order as Eq. (5.6), arises from the exchange of cutoff modes

10 11 Notice that O ~ can be rewritten in terms of the other two We neglect for simplicity a contribution from the operator Ht μν operators by using the equations of motion, but it is still useful to Eμνρ , which also arises at tree level. See for example the consider it in our discussion. discussion in Refs. [45,51].

095036-12 PRECISION TESTS AND FINE TUNING IN TWIN HIGGS … PHYSICAL REVIEW D 96, 095036 (2017) ˆ at m. As already anticipated, we neglect this correction in The total correction to the S parameter in our model is ˆ ˆ ˆ the following. In this respect our calculation is subject to an ΔS ¼ ΔSρ þ ΔSh, with the two contributions given by Oð1Þ uncertainty and should rather be considered as an Eqs. (5.6) and (5.7). estimate, possibly more refined than a naive one, which takes the correlations among different observables into B. Tˆ parameter account. Tree-level contributions to the Tˆ parameter are forbidden Besides the UV threshold effects described above, Sˆ gets in our model by the SOð3Þ custodial symmetry preserved an IR contribution from RG evolution down to the by the strong dynamics, and can only arise via loops electroweak scale. The leading effect of this type comes involving the elementary states. A non-vanishing effect from the compositeness of the Higgs boson, and is the same arises at the 1-loop level corresponding to a violation of SO 5 =SO 4 as in ð Þ ð Þ CH models [32]: custodial isospin by two units. The leading contribution 4 2 2 comes from loops of fermions and is proportional to yL, ˆ g2 m ΔS ¼ ξ log : ð5:7Þ given that the spurionic transformation rule of yL is that of a h 192π2 m2 h doublet, while yR is a singlet. We find: In the effective theory below m this corresponds to the 2 2 2 2 2 2 2 Δ ˆ yL yLv yt yLv M1 evolution of the dimension-6 operators TΨ ¼ aUVNc 2 2 þ aIRNc 2 2 log 2 ; 16π MΨ 16π MΨ mt ↔ ig † i μ ν i ð5:12Þ O ¼ H σ D HD Wμν; W 2m2 W 1 0 ↔ where aUV;IR are Oð Þ coefficients whose values are ig † μ ν O H D H∂ Bμν : reported in Appendix E and we have defined M1≡ B ¼ 2 2 ð5 8Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mW 2 2 2 MS þ yRf . The result is finite and does not depend on the cutoff scale m . The first term corresponds to the UV induced by a 1-loop insertion of threshold correction generated at the scale μ ¼ M1 ∼ MΨ. The second term instead encodes the IR running from the 1 μ † † O ¼ ∂ ðH HÞ∂μðH HÞ: ð5:9Þ threshold scale down to low energy, due to loops of top H 2v2 quarks. In the effective theory below M1 it corresponds to the RG evolution of the dimension-6 operator Denoting with c¯i the coefficients of the effective operators and working at leading order in the SM couplings, the RG 1 ↔ evolution can be expressed as † μ 2 OT ¼ 2 2 ðH D HÞ ð5:13Þ   v μ c¯iðμÞ¼ δij þ γij log c¯jðMÞ; ð5:10Þ due to insertions of the current-current operators of M ˆ ˆ Eq. (5.11). In particular, ΔT ¼ cTðmZÞξ and one has 2 2 γT;Ht ¼ −γT;Hq ¼ 3yt =4π , γT;Hq0 ¼ 0. Notice that the where γij is the anomalous dimension matrix (computed ˆ size of the second contribution with respect to the first S 2 2 2 at leading order in the SM couplings). The parameter is O½ðy =y Þ logðM =m Þ: for y ∼ y , that is for fully Δˆ ¯ ¯ ξ t L 1 t t L gets a correction S ¼ðcWðmZÞþcBðmZÞÞ , and one has composite t , the IR dominated contribution is formally γ γ − 2 96π2 R W;H þ B;H ¼ g2=ð Þ. An additional contribution to logarithmically enhanced and dominant. the running arises from insertions of the current-current Further contributions to Tˆ come from loops of spin-1 operators resonances, the exchange of cutoff modes and Higgs compositeness. The latter is due to the modified couplings ↔ i μ † of the composite Higgs to vector bosons and reads [32]: O ¼ q¯ γ q H DμH; Hq v2 L L ↔ 3 2 2 0 i μ i † i ˆ g1 m O q¯ γ σ q H σ DμH; Δ − ξ Hq ¼ 2 L L Th ¼ 64π2 log 2 : ð5:14Þ v mh ↔ i ¯ γμ † OHt ¼ 2 tR tRH DμH ð5:11Þ v In the effective theory it corresponds to the running of OT due to the insertion of the operator OH in a loop with in a loop of top quarks. This is however suppressed by a hypercharge. The contribution is of the form of Eq. (5.10) 2 2 γ 3 2 32π2 factor yL=g compared to Eq. (5.7) and will be neglected. with T;H ¼ g1= . The exchange of spin-1 resonances The suppression arises because the current-current oper- gives a UV threshold correction which is also proportional 2 ators are generated at the matching scale with coefficients to g1 (as a spurion, the hypercharge coupling transforms as 2 proportional to yL. an isospin triplet), but without any log enhancement. It is

095036-13 ROBERTO CONTINO et al. PHYSICAL REVIEW D 96, 095036 (2017) thus subleading compared to Eq. (5.14) and we will neglect divergent and encodes the UV threshold correction at the it for simplicity (see Ref. [45] for the corresponding matching scale. The divergence comes, in particular, from computation in the context of SOð5Þ=SOð4Þ models). diagrams where the fermion loop is connected to the b- Finally, we also omit the effect of the cutoff modes because quark current through the exchange of a spin-1 resonance it is incalculable, although naively this is of the same order [29]. A simple operator analysis shows that the threshold as the contribution from states included in our simplified contribution from the vector resonances in the adjoint of model. Our result is thus subject to an Oð1Þ uncertainty. SOð7Þ identically vanishes in our model (see Appendix D The total contribution to the Tˆ parameter in our model is for details). An additional UV threshold contribution to ˆ ˆ ˆ δ therefore ΔT ¼ ΔTh þ ΔTΨ with the two contributions gLb arises from diagrams where the spin-1 resonances given by Eqs. (5.12) and (5.14). circulate in the loop. For simplicity we will not include such effect in our analysis (see Ref. [30] for the corresponding 5 4 C. δg computation in the context of SOð Þ=SOð Þ models). It is Lb however easy to show that there is no possible diagram with In the limit of vanishing transferred momentum, tree- ρ circulating in the loop as a consequence of its quantum δ X level corrections to gLb are forbidden by the PLR parity of numbers, while the corresponding contribution from vector 2 2 the strong dynamics that exchanges SUð ÞL with SUð ÞR resonances in the adjoint of SOð7Þ is nonvanishing in 4 f 2 f 2 in the visible SOð Þ and SUð ÞL with SUð ÞR in the twin this case. SOf ð4Þ (see Appendix A for details). This is a simple The second term in Eq. (5.17) accounts for the IR extension of the PLR symmetry of CH models which running down to the electroweak scale. In the effec- ¯ δ − ¯ protects the Zbb coupling from large corrections [52].In tive theory below M1 one has gLb ¼ ðcHqðmZÞþ c¯0 ðm ÞÞ=2, hence the IR correction arises from the our case PLR is an element of the unbroken SOð7Þ and Hq Z 0 keeps the vacuum unchanged. It is thus an exact invariance evolution of the operators OHq and OHq due to loops of of the strong dynamics, differently from SOð5Þ=SOð4Þ top quarks. In this case the operators that contribute to the models where it is accidental at Oðp2Þ. The gauge running via their 1-loop insertion are those of Eq. (5.11) as couplings g1;2 and yL explicitly break it, while yR preserves well as the following four-quark operators [53]: it. At finite external momentum δgLb gets a non-vanishing ¯ γμ ¯ γ ¯ γμ ¯ γ tree-level contribution: OLR ¼ðqL qLÞðtR μtRÞ;OLL ¼ðqL qLÞðqL μqLÞ; 0 ¯ σaγμ ¯ σaγ 2ξ 2 2 OLL ¼ðqL qLÞðqL μqLÞ: ð5:18Þ f 2 2 yLf δg g α α7 − g α : ð LbÞtree ¼ 8 2 ½ 1ð L þ LÞ 2 L 2 2 2 Mρ MΨ þ yLf 2 2 In fact, the operators contributing at OðyLyt Þ to Eq. (5.10) 5:15 2 ð Þ are only those generated at OðyLÞ at the matching scale; − 0 these are OHt, the linear combination OHq OHq (even In the effective theory below M1, this correction arises from 12 under PLR), and OLR (generated via the exchange of ρX). the dimension-6 operators Notice finally that the relative size of the IR and UV 2 2 2 0 contributions to δgLb is O½ðyt=yLÞ logðM1=mt Þ precisely g μ ν g μ a ν a OBq ¼ ∂ Bμνq¯ γ q ;OWq ¼ D Wμνq¯ γ σ q ;: Δ ˆ m2 L L m2 L L like in the case of TΨ. W W It is interesting that in our model the fermionic correc- ð5:16Þ ˆ tions to δgLb and T are parametrically of the same order and their signs tend to be correlated. It is for example well It is of order y2 =g2 g2=g2 ξ, hence a factor g2=g2 smaller ð L Þð Þ known that a heavy fermion with the quantum numbers of P than the naive expectation in absence of the LR protection. t gives a positive correction to both quantities [28,54–56]. At the 1-loop level, corrections to δg arise from the R Lb We have verified that this is also the case in our model for virtual exchange of heavy fermion and vector states. 13 M ≪ MΨ ∼ Mρ (light singlet). Conversely, a light septu- The leading effect comes at Oðy4 Þ from loops of heavy S L plet (MΨ ≪ M ∼ Mρ) gives a negative contribution to fermions (the corresponding diagrams are those of Figs. 4 S δ ˆ 14 and 5) and reads both gLb and T. Although in general the expressions for Δ ˆ δ   TΨ and ð gLbÞΨ are uncorrelated, their signs tend to be the 2 2 2 2 ρ δ yL yLv m same whenever the contribution from X to Eq. (5.17) is ð gLbÞΨ ¼ 2 Nc 2 bUV þ cUV log 2 16π MΨ MΨ 2 2 2 2 12 0 4 y y v M The operators OLL and OLL are generated at OðyLÞ by the t L 1 ρ ρ þ bIR 2 Nc 2 log 2 : ð5:17Þ tree-level exchange of both X and . 16π M m 13 ˆ Ψ t In this limit one has ΔTΨ ≃ 3ðδgLbÞΨ. 14 1 The existence of a similar sign correlation in the limit of a The expressions of the Oð Þ coefficients bUV;IR and cUV are light (2,2) has been pointed out in the context of SOð5Þ=SOð4Þ reported in Appendix E. The first term is logarithmically CH models, see Ref. [29].

095036-14 PRECISION TESTS AND FINE TUNING IN TWIN HIGGS … PHYSICAL REVIEW D 96, 095036 (2017) subleading. The sign correlation can instead be broken if ρX at the price of some mild tuning by varying the field content contributes significantly to δgLb [in particular, ðδgLbÞΨ can or the representations of the heavy fermions. In order to be negative for αiL ¼ −αiR]. The importance of the above account for these options and thus broaden the scope of our considerations lies in the fact that EW precision data prefer analysis we will treat F1 as a free Oð1Þ parameter. The ˆ a positive T and a negative δgLb. Situations when both value of F1 has a direct impact on the size of the left- δ 2 ∼ 4 quantities have the same sign are thus experimentally handed top mixing yL, since mhjUV yLF1, and hence disfavored. controls the interplay between the Higgs potential and Considering that no additional correction to δgLb arises EWPO. Specifically, as we already stressed, a smaller F1 from Higgs compositeness, and that we neglect as before implies a larger value of yL, which in turn gives a larger Δ ˆ ∝ 4 2 2 the incalculable effect due to cutoff states, the total T yLv =MΨ. This could help improve the compatibility δ δ δ contribution in our model is gLb ¼ð gLbÞtree þð gLbÞΨ, with EWPT even for large MΨ, at the cost of a small with the two contributions given by eqs. (5.15) and (5.17). additional tuning due to the need for a clever maneuver in the S;ˆ Tˆ plane to get back into the ellipse, as well as the fact VI. RESULTS AND DISCUSSION that F1 is generically expected to be Oð1Þ. In the following We are now ready to translate the prediction for the we will thus treat F1 as an input parameter and use Higgs mass and the EWPO into bounds on the parameter Eqs. (4.2) and (B4) to fix yL and yR in terms of the space of our simplified model and for the composite TH in Higgs and top quark experimental masses. Our final results general. We are interested in quantifying the degree of fine will be shown for two different choices of F1, namely 1 0 3 tuning that our construction suffers when requiring the F1 ¼ and F1 ¼ . , in order to illustrate how the bounds mass scale of the heavy fermions to lie above the ultimate are affected by changing the size of the UV threshold experimental reach of the LHC. As discussed in Sec. II, this correction to the Higgs potential. scale receives the largest boost from the TH mechanism The EWPO and the Higgs mass computed in the (without a corresponding increase in the fine tuning of the previous sections depend on several parameters, in par- Higgs mass) in the regime of parameters corresponding to a ticular on the mass spectrum of resonances (see α fully strongly coupled theory, where no quantitatively Appendix B), the parameters ci, i of Eqs. (3.9), (3.14), precise EFT description is allowed. Our computations of and the parameter F1 discussed above. In order to focus physical quantities in this most relevant regime should then on the situation where resonances can escape detection at be interpreted as an educated naive dimensional analysis the LHC, we will assume that their masses are all (eNDA) estimate, where one hopes to capture the generic comparable and that they lie at or just below the cutoff size of effects beyond the naivest 4π counting, and scale. In order to simplify the numerical analysis we thus ~ ~ 2 including factors of a few related to multiplet size, to spin, set MΨ ¼ MS ¼ MΨ ¼ MS ¼ Mρ ¼ MρX ¼ m= . The and to numerical accidents. In the limit where Mρ=f and factor of two difference between MΨ and m is chosen MΨ=f are significantly below their perturbative upper to avoid setting all UV logarithms of the form logðm=MΨÞ bounds our computations are well defined. eNDA then to zero, while not making them artificially large. As a ≡ α α corresponds to assuming that the results do not change by further simplification we set cL ¼ cR c, 7L ¼ 1L, and α α α more than Oð1Þ (i.e. less than Oð5Þ to be more explicit) 7R ¼ 1R. The parameter L appears only in the tree-level δ when extrapolating to a scenario where the resonance mass contribution to gLb, see Eq. (5.15), and we fix it equal to 1 scale sits at strong coupling. In practice we shall consider for simplicity. Even though the above choices represent a the resonant masses up to their perturbativity bound and significant reduction of the whole available parameter 15 space, for the purpose of our analysis they represent a vary the αi and ci parameters within an Oð1Þ range. In view of the generous parameter space that we shall explore sufficiently rich set where EWPT can be successfully our analysis should be viewed as conservative, in the sense passed. that a realistic TH model will never do better. Let us now discuss the numerical bounds on the Let us now describe the various pieces of our analysis. parameter space of our simplified model. They have been Consider first the Higgs potential, where the dependence on obtained by fixing the top and Higgs masses to their physics at the resonance mass scale is encapsulated in the experimental value and performing the numerical fit function F1 [Eq. (4.2)] which controls the UV threshold described in Appendix F. As experimental inputs, we correction to the Higgs quartic. It is calculable in our use the PDG values of the top quark pole mass mt ¼ 173 21 0 51 0 71 simplified model and the result is Oð1Þ [its expression is . . . (see last paragraph of Appendix C), 125 09 0 24 reported in Eq. (C1)], but it can easily be made a bit smaller and of the Higgs mass, mh ¼ . . GeV [57]. Figure 2 shows the results of the fit in the ðMΨ; ξÞ plane for ffiffiffi 0 3 1 15 p F1 ¼ . (left panel) and F1 ¼ (right panel). In both Notice indeed that ðα ¼ 1;c¼ 0Þ and ðα ¼ 0;c¼ 1= 2Þ 0 correspond to specific limits at weak coupling, namely the two- panels we have set c ¼ , which corresponds to the two-site site model and the linear sigma model respectively. This suggests model limit of our simplified Lagrangian. The yellow 2 that their natural range is Oð1Þ. regions correspond to the points that pass the χ test at

095036-15 ROBERTO CONTINO et al. PHYSICAL REVIEW D 96, 095036 (2017)

FIG. 2. Allowed regions in the ðMΨ; ξÞ plane for F1 ¼ 0.3 (left panel) and F1 ¼ 1 (right panel). See the text for an explanation of the different regions and of the choice of parameters.

ˆ 95% confidence level (CL), see Appendix F for details. the sign correlation between ΔTΨ and δgLb. In the case of Solid black contours denote the regions for which the solid line, the signs are anticorrelated (e.g., positive α1 ¼ −α1 ¼ 1, while dashed contours surround the ˆ L R ΔTΨ and negative δgLb), allowing for the compensation regions obtained with α1 ¼ α1 ¼ 1. The areas in blue ˆ L R effect by ΔTΨ. In the case of the dashed line, instead, the are theoretically inaccessible. The lower left region in dark ≡ signs of the two parameters are correlated (both positive), blue, in particular, corresponds to MΨ=f gΨ

095036-16 PRECISION TESTS AND FINE TUNING IN TWIN HIGGS … PHYSICAL REVIEW D 96, 095036 (2017)

FIG. 3. Allowed regions in the ðc; αÞ plane, with c ¼ cL ¼ cR, for F1 ¼ 0.3 (left panel) and F1 ¼ 1 (right panel). The yellow, orange, and red regions correspond to ξ ¼ 0.05, 0.1 and 0.15 respectively. See the text for an explanation of the choice of the other parameters.

increasing F1 (reducing yL) shifts the allowed region given about 1=2 of the plausible choices for both αi and toward positive values of c, since as mentioned above, ci are allowed. smaller yL requires a larger positive c to get a large ˆ enough ΔTΨ. Obviously, larger values of ξ correspond to smaller allowed regions, as is clear from Fig. 2. Finally, VII. SUMMARY notice that the vertically symmetric structure of the In this paper we tried to assess how plausible a scenario allowed regions is due to the quadratic dependence of yielding no new particles at the LHC can be obtained using δ α gLb on . From these plots, one can see that for the TH construction. We distinguished three possible resonances conceivably out of direct reach of the LHC classes of models: the subhypersoft, the hypersoft and ξ ∼ 0 1 and for . , corresponding to about 20% tuning of the the superhypersoft, with increasing degree of technical α Higgs mass, both and c are allowed to span a good complexity and decreasing (technical) fine tuning. We then 1 fraction of their expected Oð Þ range. No dramatic extra focused on the CH incarnation of the simplest option, the tuning in these parameters seems therefore necessary to subhypersoft scenario, where the boost factor for the mass meet the constraints of EWPT. In particular, considering of colored partners [Eq. (2.4)] at fixed tuning is roughly 1 the plot for F1 ¼ (right panel), one notices that the given by bulk of the allowed region is at positive c. For instance by choosing c ∼ 0.2–0.5 the plot in the ðξ;MΨÞ plane for 1 g F1 ¼ 1 becomes quite similar to the one at the left of pffiffiffi × pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð7:1Þ 2y Fig. 3 valid for F1 ¼ 0.3: there exists a “peak” centered t lnðm=mtÞ at MΨ ∼ 2–4 TeV and extending up to ξ ∼ 0.2. The specific choice c ¼ 0 is thus particularly restrictive for Here the gainpffiffiffi derives entirely from the relative coupling 2 F1 ¼ 1 (right panel of Fig. 2), but this restriction is lifted strength g= yt, making the marriage of twinning and ξ ∼ 0 1 compositeness practically obligatory. We attempted a for positive c. Overall we conclude that for . and pffiffiffi 2 for resonances just beyond the LHC reach, the correct more precise estimate of the upper limit on g= yt,as value of the Higgs quartic can be obtained and EWPT compared with previous studies (e.g. Ref. [13]). We found passed with only a mild additional tuning associated with by independent but consistent estimates, that the bound a sign correlation α1L ¼ −α1R, and a correlation between ranges from ∼3 in a toy sigma model [Eq. (2.6)]to∼5 in a c and F1 (e.g. c>0 for F1 ¼ 1). These correlations simplified CH model [Eq. (3.15)], with both limits some- ˆ allow the various contributions to T and δgLb to com- what below the NDA estimate of 4π ∼ 12. Consequently pensate for each other, achieving agreement with EWPT. for a mild tuning ϵ ∼ 0.1 the upper bound on the mass of If forced to quantify the tuning inherent in these effects, the resonances with SM quantum numbers is closer to the we could estimate it to be around 1=4 ¼ð1=2Þ × ð1=2Þ, 3–5 TeV range than it is to 10 TeV. This gain, despite

095036-17 ROBERTO CONTINO et al. PHYSICAL REVIEW D 96, 095036 (2017) being less spectacular than naively expected, is still tumbling SOð9Þ → SOð8Þ → SOð7Þ pattern of symmetry sufficient to push these states out of direct reach of the breaking. The advantage of this approach is that it LHC, provided we resort to full strong coupling. In allows us to remain within the weakly-coupled domain, practice this implies no real computational advantage due to the presence of a relatively light twin Higgs from considering holographic realizations of composite scalar mode σ, whose mass can be parametrically close TH constructions: since the boost factor is controlled by to that of the twin tops, ∼ytf (around 1 TeV for the KK coupling, the 5D description breaks down in ξ ∼ 0.1). As well as giving rise to distinctive exper- precisely the most interesting regime, where the KK imental signatures due to mixing with the SM Higgs coupling is strong. In this situation computations based [64], the mass of the light σ acts as a UV cutoff for the on an explicit 5D construction, such as the ones studied in IR contributions to Sˆ and Tˆ in Eqs. (5.7) and (5.14) Refs. [9,16] for instance, are no better than numerical [12]. For sufficiently light σ then, less or no interplay estimates made in our simplified model. Indeed we have between the various contributions is required in order to checked that EWPT can be satisfied in a sizable portion of pass EWPT. Together with calculability, this property the parameter space, given some interplay among the may well single out the hypersoft scenario as the most various contributions. In particular the IR corrections to Tˆ plausible TH construction. ˆ ξ 0 1 and S are enhanced by lnðm=mhÞ,andfor > . the compensating contribution to Tˆ , which decreases like ACKNOWLEDGMENTS 1 2 =m, is necessary. Given that perturbativity limits m to We would like to thank Andrey Katz, Alberto Mariotti, ξ 0 1 be below 5 TeV for > . (see the upper blue exclusion Kin Mimouni, Giuliano Panico, Diego Redigolo, Matteo region in Fig. 2) this compensation in EWPT can still Salvarezza, and Andrea Wulzer for useful discussions. The ξ take place at the price of a moderate extra tuning. For of Swiss National Science Foundation partially supported the order a few percent on the other hand, EWPT would be work of D. G. and R. R. under Contracts No. 200020- passed without any additional tuning, while the masses of 150060 and No. 200020-169696, the work of R. C. under SM-charged resonances would be pushed up to the Contract No. 200021-160190, and the work of R. T. under 10 TeV range, where nothing less than a 100 TeV collider the Sinergia network CRSII2-160814. The work of R. C. would be required to discover them, and that barely was partly supported by the ERC Advanced Grant so [58,59]. No. 267985 Electroweak Symmetry Breaking, Flavour Although EWPT work similarly in the CH and and Dark Matter: One Solution for Three Mysteries composite TH frameworks, the two are crucially different (DaMeSyFla). R. M. is supported by ERC Grant when it comes to contributions to the Higgs quartic. In the No. 614577 HICCUP High Impact Cross Section CH these are enhanced when g,i.e.m=f,isstrongand, Calculations for Ultimate Precision. as discussed for instance in Ref. [33], in order to avoid additional tuning of the Higgs quartic g cannot be too large. According to the study in Refs. [60–63] the APPENDIX A: SOð8Þ GENERATORS corresponding upper bound on the mass of the colored AND CCWZ VARIABLES ≲ 1 5 top partners in CH reads roughly m=f . ; this should In this Appendix we define the generators of the SOð8Þ ≲ 5 be compared to the upper bound m=f from strong algebra and describe the SOð8Þ=SOð7Þ symmetry-breaking coupling we found in Eq. (3.15). The Higgs quartic pattern, introducing the CCWZ variables for our model. We protection afforded by the TH mechanism allows us to refer the reader to Refs. [41,42] for a detailed analysis of take m=f as large as possible, allowing the colored this procedure and we closely follow the notation partners to be heavier at fixed f, hence at fixed fine tuning of Ref. [13]. ξ. In the end the gain is about a factor of 5=1.5 ∼ 3, not We start by listing the twenty-eight generators of SOð8Þ impressive, but sufficient to place the colored partners and decomposing them into irreducible representations of outside of LHC reach for a mild tuning ξ ∼ 0.1. the unbroken subgroup SOð7Þ: 28 ¼ 7 ⊕ 21. They can be Finally, we comment on the classes of models not compactly written as: covered in this paper: the hypersoft and superhypersoft scenarios. The latter requires combining supersymmetry i ðT Þ ¼ pffiffiffi ðδ δ − δ δ Þ; ðA1Þ and compositeness with the TH mechanism, which, ij kl 2 ik jl il jk while logically possible, does not correspond to any existing construction. Such a construction would need to with i; j; k; l ¼ 1; …; 8. We choose to align the vacuum be rather ingenious, and we currently do not feel expectation value responsible for the spontaneous compelled to provide it, given the already rather breaking of SOð8Þ to SOð7Þ along the 8th component: t epicyclic nature of the TH scenario. The simpler hyper- ϕ0 ¼ fð0; 0; 0; 0; 0; 0; 0; 1Þ . With this choice, the broken soft scenario, though also clever, can by contrast be and unbroken generators, transforming, respectively, in the implemented in a straightforward manner, via, e.g., a 7 and 21 of SOð7Þ, are

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7 i Πj ¼ð0; …; π4 ¼hhiþhðxÞ; …; 0Þ; ðA7Þ ðT Þ ¼ pffiffiffi ðδ8γδβρ − δ8ρδβγÞ; unitary gauge β γρ 2 21 i ξ 2 h =f v2=f2 Σ ðTαβÞγρ ¼ pffiffiffi ðδαγδβρ − δαρδβγÞ; ðA2Þ where ¼ sin ðh i Þ¼ . In this case the matrix 2 simplifies to: 2 3 where α; β ¼ 1; …; 7 and γ; ρ ¼ 1; …; 8. It is useful to I 000 4 ∼ 2 2 6 3 7 identify the subgroups SOð Þ SUð ÞL × SUð ÞR and π π pffiffi 6 4 4 7 2 7 0 cos 0 sin f 4 ∼ f 2 f 2 8 i π4T 6 f f 7 SOð Þ SUð Þ × SUð Þ of SOð Þ; they are generated Σ Π f 4 6 7 L R ð Þjunitary gauge ¼ e ¼ 6 00I 0 7: by: 4 3 5 0 − π4 0 π4     sin f cos f α 0 α 0 α tL α tR ðTLÞ ¼ ; ðTRÞ ¼ ; 00 00 ðA8Þ     00 00 ~ α ~ α Given the above symmetry breaking pattern, it is ðTLÞ ¼ α ; ðTRÞ ¼ α ; ðA3Þ 0 tL 0 tR possible to define a LR parity,

tα tα 4 4 where L and R are × matrices defined as PLR ¼ diagð−1; −1; −1; þ1; −1; −1; −1; þ1Þ; ðA9Þ   i 1 β γ β γ α − ϵαβγ δ δ − δ δ δαδ4 − δαδ4 which exchanges SUð2Þ with SUð2Þ inside SOð4Þ and ðtL;RÞij ¼ 2 2 ð i j j i Þð i j j i Þ L R f 2 f 2 f 4 SUð ÞL with SUð ÞR inside SOð Þ. The corresponding ðA4Þ action on the fields is such that π4 is even, while all the other NGBs are odd. As already noticed in Sec. VC, PLR is with α ¼ 1, 2, 3 and i; j ¼ 1; …; 4. The elementary SM and an element of both SOð8Þ and SOð7Þ, which means that it is twin vector bosons gauge, respectively, a subgroup an exact symmetry of the strong dynamics and acts linearly 2 1 4 1 ≡ 1 SUð ÞL × Uð ÞY of SOð Þ, with Uð ÞY Uð ÞR3, and on the physical spectrum of fields. f 2 f 4 the subgroup SUð ÞL inside SOð Þ. This choice corre- The CCWZ variables dμ and Eμ are defined as usual sponds to having zero vacuum misalignment at tree level. through the Maurer-Cartan form, The spontaneous breaking of SOð8Þ to SOð7Þ Σ† Π Σ Π ≡ i Π 7 a Π 21 delivers seven NGBs, that we collect in the vector ð ÞDμ ð Þ idμð ÞTi þ iEμð ÞTa ; ðA10Þ t Π ¼ðπ2; π1; −π3; π4; π~ 2; π~ 1; −π~ 3Þ . The first four transform 4 as a fundamental of SOð Þ and form the Higgs doublet; the as the components along the broken and unbroken remaining three are singlets of SO 4 and are thus neutral ð Þ generators of SOð8Þ respectively. The derivative Dμ is under the SM gauge group. All together they can be covariant with respect to the external SM and twin gauge arranged in the Goldstone matrix fields:

pffiffi 2Π 7 Σ Π i f ·T A A ð Þ¼e Dμ ¼ ∂μ − iAμ T ; with 2  pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 3 ΠΠt Πt·Π ffiffiffiffiffiffiffiffiΠ Πt·Π A A α α 3 ~ ~ α ~ α I7 − t 1 − cos p sin Aμ T ¼ g2WμðTLÞ þ g1BμðTRÞ þ g2WμðTLÞ : ðA11Þ 6 Π ·Π f Πt·Π f 7 ¼ 4 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 5: Π Πt Π Πt Π − pffiffiffiffiffiffiffiffi sin · cos · Πt·Π f f Under the action of a global element g ∈ SOð8Þ, dμ and Eμ transform with the rules of a local SOð7Þ ðA5Þ transformation: The latter transforms nonlinearly under the action of an ≡ i 7 → Π † Π SOð8Þ group element g, according to the standard relation: dμ dμTi hð ;gÞdμh ð ;gÞ; a 21 † Eμ ≡ EμTa → hðΠ;gÞðEμ − i∂μÞh ðΠ;gÞ: ðA12Þ ΣðΠÞ → g · ΣðΠÞ · h†ðΠ;gÞ; ðA6Þ It is straightforward to derive the explicit expressions of dμ where hðΠ;gÞ ∈ SOð7Þ depends on g and ΠðxÞ.We and Eμ from the Maurer-Cartan relation in Eq. (A10). They identify the Higgs boson with the NGB along the generator are however lengthy and not very illuminating, so we do not 7 T4; in the unitary gauge, all the remaining NGBs are report them here. We can easily obtain the mass spectrum of nonpropagating fields and the Π vector becomes the gauge sector of our theory from the NGB kinetic term;

095036-19 ROBERTO CONTINO et al. PHYSICAL REVIEW D 96, 095036 (2017) in the unitary gauge and after rotating to the mass eigenstate respectively; their mass is exactly given by the Lagrangian ~ basis, we find: parameters MΨ (for the first one), and MΨ (for the last two). The remaining sectors have charge −1=3, 0, and 2=3 and 2 2 2 2 f i 2 g2 2 þ μ− ðg1 þ g2Þ 2 μ because of the elementary/composite mixing the associated L ¼ ðdμÞ ⊃ f ξWμ W þ f ξZ Zμ mass 4 4 8 mass matrices are in general nondiagonal and must be 3 g~2 X diagonalized by a proper field rotation. The simplest case is 2 2 1 − ξ ~ i 2 −1 3 þ 8 f ð Þ ðWμÞ : ðA13Þ the ð = Þ-charged sector, containing the bottom quark i¼1 and the heavy B field; the mass matrix in the fb; Bg basis is   0 fyL APPENDIX B: MASS MATRICES M−1=3 ¼ : ðB1Þ AND SPECTRUM 0 −MΨ

In this Appendix, we briefly discuss the mass matrices of After rotation, we find a massless bottom quark (it has no the different charged sectors in the composite TH model mass since we are not including the bR in the model), and a and the related particle spectrum. We refer to Ref. [13] for 2 2 2 2 ~ massive B particle with mB ¼ MΨ þ yLf . the expressions of the Ψ7 and Ψ7 multiplets in terms of their As regards the sector of charge 2=3, it contains seven component heavy fermions. different particles: the top quark, the toplike heavy states T We start by considering the fields that do not have the and X2=3 and four composite fermions that do not right quantum numbers to mix with the elementary SM and 1 4 twin quarks and whose mass is therefore independent of the participate in the SM weak interactions, S2=3; S2=3.In ~ 1 … 4 mixing parameters yL;R and yL;R. These are the composite the ft; T; X2=3;S2=3; ;S2=3g basis, the mass matrix is ~ ~ fermions X5=3, D1 and D−1, with charges 5=3, 1 and −1 given by:

0 ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi pffiffi 1 1 p 1 p ξ 0 fy ð 1 − ξ þ 1Þ − fy ð 1 − ξ − 1Þ 0 − fypLffiffi B 2 L 2 L 2 C B C B 0 −MΨ 000C B C M2=3 ¼ B 00 −MΨ 00C: ðB2Þ B C @ A 00 0−MΨ × I3 0

fyR 000−MS

1 2 3 The states S2=3;S2=3;S2=3 completely decouple from the elementary sector and do not mix with the top quark. Their mass is therefore exactly given by the Lagrangian parameter MΨ. The remaining 4 × 4 matrix is in general too complicated to be analytically diagonalized, but one can easily find the spectrum in perturbation theory by expanding M2=3 for ξ ≪ 1, which is in general a phenomenologically viable limit. The leading order expression for the masses is then:

4 2 2 2 ≃ f yLyR ξ ξ2 mt 2 2 2 2 þ Oð Þ; MS þ yRf 2 2 m ¼ MΨ; X2=3   ξ 2 ≃ 2 2 2 1 − ξ2 mT MΨ þ yLf 2 þ Oð Þ; 2 2 2 2 ≃ 2 2 2 yLf MS ξ ξ2 m 4 MS þ yRf þ 2 2 2 þ Oð Þ: ðB3Þ S2 3 2 = ðMS þ yRf Þ

The X2=3 fermion can be also decoupled and its mass is exactly equal to MΨ. On the contrary, the other three particles mix with each other and their mass gets corrected after EWSB (as expected, the top mass is generated for nonzero values of ξ). Finally, we analyze the neutral sector of our model. It comprises eight fields, the twin top and bottom quarks, and six of ~ ~ ~ 1 ~ 2 ~ 1 ~ 4 the composite fermions contained in the Ψ7 multiplet. Working in the field basis ft;~ b; D0; D0; U0; …; U0g, the mass matrix reads:

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0 ffiffiffi ffiffiffi pffiffiffiffiffiffi 1 1 p 1 p ~ 1−ξ~ 00− f ξy~ f ξy~ 00− ifpyffiffiL − f pffiffi yL B 2 L 2 L 2 2 C B ~ ~ C B 00 0 0 − ifpyffiffiL fpyffiffiL 00C B 2 2 C B C B ~ C B 00 −MΨ 00000C B ~ C B 00 0 −MΨ 000 0C M0 ¼ B C: ðB4Þ B ~ C B 00 0 0 −MΨ 00 0C B C B ~ C B 00 0 0 0 −MΨ 00C B ~ C @ 00 0 0 0 0 −MΨ 0 A ~ fy~R 00 0 000−MS

After diagonalization, we find one massless eigenvalue We conclude by noticing that the masses of the corresponding to the twin bottom quark (it does not acquire ~ particles in different charged sectors are not unrelated mass since we are not introducing the bR field). Four of the to each other, but must be connected according to the neutral heavy fermions completely decouple and acquire action of the twin symmetry. In particular, it is obvious the following masses 4 ~ 4 that the two singlets S2=3 and U0 form an exact twin pair, as it is the case for each SM quark and the corresponding ~ 2 ~ 2 2 2 m ~ 1 ¼ m ~ 3 ¼ m ~ 2 ¼ MΨ;m~ 2 ¼ MΨ þ y~ f : ðB5Þ U0 U0 D0 U0 L twin partner. The remaining pairs can be easily found from the spectrum and correspond to the implementation The elementary/composite mixing induces instead correc- of the twin symmetry in the composite sector as defined tions to the masses of the remaining neutral particles; at in [13]. leading order in ξ we find:

f4 y~2 y~2 APPENDIX C: RG IMPROVEMENT OF THE m2 ≃ L R ð1 − ξÞþOðξ2Þ; t~ 2 ~ 2 ~2 2 HIGGS EFFECTIVE POTENTIAL MS þ yRf 1 2 ≃ ~ 2 ~ 2 2 1 ξ ξ2 In this Appendix we describe the computation of the m ~ 1 MΨ þ yL f ð þ ÞþOð Þ; D0 2 Higgs effective potential and its RG improvement. First of ~2 2 ~ 2 all, the UV threshold correction can be computed with 2 ≃ ~ 2 ~2 2 yLf MS 1 − ξ ξ2 m ~ 4 MS þ yRf þ ~ 2 2 2 ð ÞþOð Þ: a standard Coleman-Weinberg (CW) procedure, from U0 2ðM þ y~ f Þ S R which we can easily derive the function F1 of Eq. (4.2). ðB6Þ We find:

 1 4 2 2 − 2 2 2 MS MS þ MΨ f yR m F1 ¼ − 1 þ − log 4 ðM2 þ f2y2 Þ2 M2 − M2 þ f2y2 M2 S R S Ψ R Ψ  M2ðM2ðM2 þ f2y2 Þþ2f2y2 M2 þ M4Þ m2 S S Ψ R R Ψ S : þ 2 2 2 2 2 − 2 2 2 log 2 2 2 ðC1Þ ðMS þ f yRÞ ðMS MΨ þ f yRÞ MS þ f yR

ξ α α 2 4π The IR contribution to the Higgs mass can be organized using a joint expansion in and ð i logÞ, where i ¼ gi =ð Þ for couplings gi. Schematically:  α α α α α2 2 2 t t S 2 S S 2 δm m t a1 b1ξ b2 t b3 t c1ξ c2ξ t c3 t hjIR ¼ t 4π þ þ 4π þ 4π þ þ 4π þ 16π2  α α2 α α α t t 2 t S 2 2 EW c4ξ t c5 t c6 t − a2m t ; þ 4π þ 16π2 þ 4π 4π W 4π þ twin ðC2Þ

095036-21 ROBERTO CONTINO et al. PHYSICAL REVIEW D 96, 095036 (2017) 2 2 where t ¼ log m=mt and all the couplings are evaluated at On integrating out the heavy degrees of freedom, the 1 the scale m. Here ai, bi, ci are the Oð Þ coefficients of the composite twin Higgs model at the scale m can be LO, NLO, and NNLO terms respectively. The twin con- represented by an effective Lagrangian that is invariant α 3 2 1 tribution is obtained by substituting all i, ai, bi, ci with the under a global SUð Þc × SUð ÞL × Uð ÞY symmetry, iden- ~ 2 2 corresponding tilded quantities and t with t ¼ log m=mt~ . tified with the gauge group of the SM, as well as an The calculation of the LO and NLO terms is straightfor- g3 g2 g3 additional SUð Þc × SUð ÞLþR. SUð Þc is the gauged twin ward and all these contributions are included in our g color and SUð2Þ is a global twin custodial group, calculation. The NNLO terms indicated in blue are also LþR broken only by fermion interactions, under which the twin simple to evaluate, and are included in our final result, ~ indicated as NNLO. The calculation of the remaining gauge bosons (W) and Goldstone bosons (GBs) transform 0 ~ 0 NNLO terms is more complicated, since it involves the as triplets. We work in the gaugeless limit g ¼ g ¼ g ¼ , running of several higher dimensional operators involving since we are only interesting in capturing the leading both the SM and twin fields. An attempt to compute the full contributions to the Higgs potential that parametrically ~ ~ potential at NNLO has been presented in Ref. [46]. depend on yt, yt, gS, and gS. The relevant light degrees of freedom are the SM fermion However, while that result includes the contribution of ~ doublet, QL, and singlet, tR, and their twin counterparts tL the SM Goldstone bosons, additional contributions induced ~ by the twin Goldstones, which are expected to arise at and tR, the gluons and their twins; the Higgs doublet NNLO, are not included. Since the full calculation at (including the massless GBs) and the twin GBs. For NNLO is beyond the scope of this paper, we limit ourselves economy of notation we define a twin Higgs doublet in to only include the colored contributions in Eq. (C2). analogy with the SM one, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Comparing with Ref. [46] suggests that our NNLO result  pffiffiffi  1 2π~ þ 2H†H þ π~ 2 gives an overestimate of the IR Higgs mass; that this is the H~ ffiffiffi ; h~ f 1 − ; ¼ p ~ 0 with ¼ 2 case at large m is also evident from Fig. 1. For this reason 2 h þ iπ~ f we consider the NNLO curve only as an indication of the ðC5Þ importance of the NNLO correction, and use as final input for our numerical analysis the average of the LO and NLO making explicit the dependence on the twin Higgs which is results as described in Sec. IV. integrated out at tree-level, thus realizing the SOð8Þ We present now a procedure for computing the afore- symmetry nonlinearly. mentioned contributions to the Higgs mass based on the Since the background field calculation at leading-log approach of Ref. [46]. The RG improvement of the Higgs order requires corrections to the fermion masses and Higgs effective potential can be obtained by solving the one-loop wave function at only one loop, we can neglect in the β-function of the vacuum energy in the background of the effective Lagrangian at the scale m, any operator that has Higgs field. The fermion contribution to the vacuum energy vanishing coefficients at tree level. We also omit all is given by operators that cannot be predicted solely in terms of the low-energy parameters of the theory (yt and y~t) and f, impr dVfðhc;tÞ Nc 4 4 including products of Higgs and fermion currents that arise ¼ ½M ðh ;tÞ þ M~ðh ;tÞ ; ðC3Þ 17 dt 16π2 t c t c from integrating out composite fermions. We make the following field redefinition: whereas the leading gauge contribution is given by Π sinðj jÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f i i † 2 3 f Π → Π ; for jΠj¼ 2H H þ π~ ðC6Þ 2 2 2 jΠj V h ;t 2M h M h 3M ~ h t: gð c Þ¼16π2 ½ Wð cÞ þ Zð cÞ þ Wð cÞ which allows the effective Lagrangian, containing a pure ðC4Þ scalar sector LS and a fermionic sector LF, to be expressed in an especially compact form. The relevant operators in In order to compute the improved potential to NNLO each sector are given below: accuracy, we must input the gauge boson masses at tree- level and the renormalized top and twin top masses with † μ 1 μ cHOH þ cHπ~ OHπ~ þ cπ~ Oπ~ L ¼ DμH D H þ ∂μπ~∂ π~ þ leading-log accuracy in the corresponding Yukawa cou- S 2 2f2 plings and with next-to-leading-log accuracy in the strong † 2 H H O gauge couplings. After solving Eq. (C3) for the effective þ dH 2 H; ðC7Þ potential, one must properly account for the Higgs wave f function renormalization before expanding around the minimum of the potential in order to compute the physical 17In any case, these do not contribute to the effective potential Higgs mass. We fill in the salient details below. at this order.

095036-22 PRECISION TESTS AND FINE TUNING IN TWIN HIGGS … PHYSICAL REVIEW D 96, 095036 (2017) and The Wilson coefficients at the scale m have the boundary values: ¯ ¯ ¯ ¯ LF ¼ ibL=∂bL þ ibR=∂bR þ itL=∂tL þ itR=∂tR 1 ¯ c cHðmÞ¼cHπ~ ðmÞ¼cπ~ ðmÞ¼dHðmÞ¼ ; ðC11Þ − yt½QLH tR þ H:c: ¯ ¯ ~ ~ ~ ~ ~¯ ~ ~¯ ~ while the Z2 symmetry enforces þ ibL=∂bL þ ibR=∂bR þ itL=∂tL þ itR=∂tR ¯ − ~ ~ ~ c~ ~ α αe yt½QLH tR þ H:c:: ðC8Þ ytðmÞ¼ytðmÞ and sðmÞ¼ sðmÞðC12Þ where We now expand the effective Lagrangian in the back- ˆ ground of the Higgs field h ¼ hc þ h to obtain: O ∂ † ∂μ †   H ¼ μðH HÞ ðH HÞ; 1 0 μ 0 þ μ − 2 μ 2 L ¼ Zπðh Þ ∂μπ ∂ π þ ∂μπ ∂ π Oπ~ ¼ ∂μπ~ ∂ π~ ; S c 2 O ∂ † ∂μπ~ 2 1 1 Hπ~ ¼ μðH HÞ : ðC9Þ ˆ μ ˆ μ Z ˆ h ∂μh∂ h Zπ~ h ∂μπ~∂ π~; þ 2 hð cÞ þ 2 ð cÞ ðC13Þ We have defined a twin fermion doublet Q~ ¼ðt~ ; b~ ÞT, L L L where we defined transforming under the twin-sector symmetries, and the Yukawa couplings are defined at tree-level as 2 4 hc hc Zπðh Þ¼1;Zˆ ðh Þ¼1 þ c þ d ; c h c H f2 H f4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiyLyRf ~ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiyL~ yR~ f ytðmÞ¼ 2 2 2 and ytðmÞ¼ : ~ 2 2 2 Zπ~ ðh Þ¼1; ðC14Þ MS þ yRf M y f c S þ R~ ðC10Þ and

X ¯ L ψ¯ ∂ψ − ¯ − ¯ − ~¯ ~ − ~ ~ F ¼ iZψ ðhcÞ = mtðhcÞtt mbðhcÞbb mt~ðhcÞt t mb~ ðhcÞb b ψ y ðh Þ iy ðh Þ iy ðh Þ iy ðh Þ − tpffiffiffic hˆ tt¯ − pt ffiffiffic π0t¯γ5t − t c π−b¯ð1 þ γ5Þt þ t c πþt¯ð1 − γ5Þb 2 2 2 2

y~ ðh Þ ¯ iyπ~ ~ ~ðh Þ ¯ iyπ~ ~ ~ ðh Þ ¯ iyπ~ ~ ~ ðh Þ ¯ þ tpffiffiffic hˆ t~ t~ − pt t ffiffiffi c π~ 0t~γ5t~ − t b c π~ −b~ð1 þ γ5Þt~ þ t b c π~ þt~ð1 − γ5Þb:~ ðC15Þ 2 2 2 2 ~ ~ ~ ~ Here ψ runs over all the Weyl fermions ftL;tR;bL;bR; tL; tR; bL; bRg and we defined

ythc Zψ ðh Þ¼1;yðh Þ¼y ;mðh Þ¼ pffiffiffi ;mðh Þ¼0; c t c t t c 2 b c sffiffiffiffiffiffiffiffiffiffiffiffiffi 2 y~thc 1 y~tf hc ~ ~ ~ qffiffiffiffiffiffiffiffiffiffiffiffi ~ ffiffiffi 1 − ~ 0 yπ~ t~ bðhcÞ¼yt; ytðhcÞ¼ 2 ;mtðhcÞ¼p 2 mbðhcÞ¼ : ðC16Þ f 1 − hc 2 f f2

We can now compute the running masses of the top and twin top in the background:     g2 3y ðh Þ2 22g4 M h ;t ≈ m h 1 S − t c t S t2 ; tð c Þ tð cÞ þ 4π2 4 4π 2 þ 4π 4 ðC17aÞ ð Þ Zhˆ ðhcÞ ð Þ     ~2 3~ 2 22~4 gS ytðhcÞ gS 2 M~ h ;t ≈ m~ h 1 − t t ; tð c Þ tð cÞ þ 4π2 4 4π 2 þ 4π 4 ðC17bÞ ð Þ Zhˆ ðhcÞ ð Þ 2 μ2 with t ¼ logðm= Þ. The values of the parameters in the Higgs background are simply read off from the effective ~ “ ” Lagrangian at the scale m. Notice that in the background, only yt and yt are scale-dependent, while hc is frozen ; hence the running of the quark mass is identical to that of the corresponding Yukawa coupling.

095036-23 ROBERTO CONTINO et al. PHYSICAL REVIEW D 96, 095036 (2017)

We can now substitute Eqs. (C17) into the RG equation (C3), integrate the result and expand at the required order in hc=f to get       N h2 2 y4h4 g2 3y2 V ðh ;tÞimpr ¼ c y4h4 þ y~4f4 1 − c t þ t c S − t t2 f c 64π2 t c t f2 2 π2 ð4πÞ2     23 h2 2 þ g4y4h4 þ g~4y~4f4 1 − c t3 96π4 S t c S t f2       y~4f4 h2 2 h2 h2 þ t 1 − c 16g~2 − 3y~2 c 1 − ð1 þ c Þ c t2 : ðC18Þ 32π2 f2 S t f2 H f2

This is the improved effective potential written in terms of that is not logarithmically divergent. After normalizing hc parameters computed at the scale m. We can now treat hc using Eq. (C19), including a Z2 breaking mass term as a quantum field, fluctuating around its minimum necessary to achieve a viable minimum and minimizing hc ¼ h þhhi. We then have to take into account that the the potential, hc acquires a vacuum expectation value potential (C18) has been computed in a noncanonical basis hc ¼ h þhhi. This makes the wave function of h (the for hc, where its kinetic term coefficient, including only the Higgs field in the minimum) noncanonical again, due to the leading logarithmic running, is presence of cH:

3y2 hhi2 Z ðtÞ¼1 þ t t: ðC19Þ Z ðhhiÞ ¼ 1 þ c : ðC20Þ hc ð4πÞ2 h H f2

Notice that only the top contributes to the one-loop running After normalizing h taking into account also this last effect, of the hc wave function, since the twin top only couples and including the LO gauge contribution, the desired quadratically to hc, giving rise to a one-loop contribution contribution to the Higgs mass reads:

 3v2 1 δm2j ¼ ðy4t þ y4t~Þð1 − c ξ þðc2 − d Þξ2Þ − ½ð3g4 þ 2g2g2 þ g4Þt þ 3g~4t~ h IR 8π2 t t~ H H H 16 2 1 2 1 2 1 þ ð16y4g2t2 þ 16y4g~2t~2Þð1 − c ξÞ 32π2 t S t~ S H 1 þ ð−15y6t2 − 12y2y~4t~2 þ 3y6t~2ð1 þ c ÞÞ 32π2 t t t t~ H  23 þ ðy4g4t3 þ y4g~4t~3Þ ; ðC21Þ 96π4 t S t~ S where we have highlighted the various contributions present in Eq. (C2) with the corresponding colors and set 2 2 ~ 2 2 t ¼ log m=mt , t ¼ log m=mt~ . The Higgs vev has been defined by fixing the W mass according to 2 2 1 2 g hhi ⇒ 2 2 ffiffiffi 246 mW ¼ 4 ; hhi ¼ v ¼ p ¼ GeV: ðC22Þ 2GF δ 2 A numerical determination of the IR contribution mhjIR can be obtained by making use of the experimental value of the ~ top quark mass to fix ytðmÞ and ytðmÞ. In fact, the 1-loop RG equation for yt is decoupled from the twin sector at the order we are interested in and can be easily solved. Including only the orders required to match our calculation of the Higgs mass squared in Eq. (C21) we get:     2 9 2 2 22 4 2 gSðmtÞ ytðmtÞ m gSðmtÞ 2 m y ðm Þ¼y~ðm Þ¼y ðm Þ 1 − − log þ log ; t t t t 4π2 64π2 m2 ð4πÞ4 m2   t t 7 2 2 ~ 1 − gSðmtÞ m gSðmÞ¼gSðmÞ¼gSðmtÞ 2 log 2 ; ðC23Þ 32π mt

095036-24 PRECISION TESTS AND FINE TUNING IN TWIN HIGGS … PHYSICAL REVIEW D 96, 095036 (2017) 7 which fixes ytðmÞ in terms of ytðmtÞ. As an input to our SOð Þ (where the 7 is complex), and its formal trans- numerical analysis we use the PDG combination for the top formation rule is quark pole mass mt ¼ 173.21 0.51 0.71 GeV [57]. † This is converted into the top Yukawa coupling in the χL → hðΠ;gÞχLh ðΠ;gÞ;h∈ SOð7Þ: ðD3Þ MS MS scheme yt ðmtÞ¼0.936 0.005 by making use of As a second ingredient to build the effective operators, we Eq. (62) of Ref. [65]. We then use y ðm Þ¼yMSðm Þ.For t t t t uplift the elementary doublet q into a 7 þ 1 representation the strong interaction, we run the parameter measured at the L of SOð7Þ by dressing it with NGBs: scale of the Z boson mass, gSðmZÞ ∼ 1.22, to the scale mt to obtain gSðmtÞ. Σ†Δ† QL ¼ð qLÞ: ðD4Þ

APPENDIX D: OPERATOR ANALYSIS OF THE ð7Þ ð1Þ We will denote with QL and QL respectively the septuplet HEAVY-VECTOR CONTRIBUTION TO δgLb ð1Þ and singlet components of QL. Since QL does not contain 7 In this Appendix we discuss the UV threshold contri- b (it depends only on t ), only Qð Þ is of interest for the δ L L L bution to gLb generated by the tree-level exchange of the present analysis. Under a SOð8Þ transformation composite vectors ρ [adjoint of SOð7Þ] and ρX [singlet of SO 7 ] at zero transferred momentum. This effect arises at ð7Þ ð7Þ ð Þ Q → hðΠ;gÞQ : ðD5Þ leading order from diagrams with a loop of heavy fermions, L L as in Fig. 5. Our simple effective operator analysis will The effective operators contributing to δg can be thus ρ Lb show that the contribution of the identically vanishes, in χ ð7Þ agreement with the explicit calculation in the simpli- constructed in terms of L, dμ, and QL . We find that the ρ fied model. exchange of μ in the diagram of Fig. 5 can generate two An adjoint of SOð7Þ decomposes under the custodial independent operators, 2 2 SUð ÞL × SUð ÞR as: ¯ ð7Þγμ a ð7Þ χ a O21 ¼ QL T QL Trðdμ LT Þ; 21 3; 1 1; 3 3 2; 2 3 1; 1 : 7 7 ¼ð Þþð Þþ × ð Þþ × ð Þ ðD1Þ 0 ¯ ð Þγμ a ð Þ aχ O21 ¼ QL T QL ðdμT LÞ88; ðD6Þ The first two representations contain the vector resonances where Ta is an SOð8Þ generator in the adjoint of SOð7Þ; the that are typically predicted by ordinary CH models, namely X 18 exchange of ρμ gives rise to other two : ρL and ρR. They mix at tree-level with the Z boson and in general contribute to δgLb. The remaining resonances do ¯ ð7Þγμ ð7Þ χ 0 ¯ ð7Þγμ ð7Þ χ not have the right quantum numbers to both mix with the Z O1 ¼ QL QL Trðdμ LÞ;O1 ¼ QL QL ðdμ LÞ88: boson and couple to the left-handed bottom quark due to ðD7Þ isospin conservation. As a result, only the components ρL ρ 21 δ and R inside the can give a contribution to gLb at the Simple inspection reveals that only the septuplet compo- 1-loop level. nent of χL contributes in the above equations. One can In order to analyze such effect, we make use of an easily check that the operators of Eq. (D6) give a vanishing operator approach. We classify the operators that can contribution to δgLb. In particular, the terms generated by be generated at the scale m by integrating out the the exchange of the (2,2) and (1,1) components of the ρ composite states, focusing on those which can modify give (as expected) an identically vanishing contribution. ¯ a the Zbb vertex at zero transferred momentum. In general, Those arising from ρL and ρR [obtained by setting T in since an exact PLR invariance implies vanishing correction Eq. (D6) equal to respectively one of the (3,1) and (1,3) δ to gLb at zero transferred momentum, any gLb must be generators] give instead an equal and opposite correction to generated proportional to some spurionic coupling break- gLb. This is in agreement with the results of a direct ing this symmetry. In our model, the only coupling calculation in the simplified model, from which one finds breaking PLR in the fermion sector is yL, and a non- δ 4 vanishing gLb arises at order yL. The effective operators 18 Additional structures constructed in terms of dμ and χL can be constructed using the CCWZ formalism in terms of can be rewritten in terms of those appearing in Eqs. (D6) and the covariant spurion (D7), hence they do not generate new linearly independent opera- χ a ∝ aaˆ bˆ aˆ χð7Þ bˆ aχ ∝ tors. Notice that Trðdμ LT Þ f dμð L Þ , ðdμT LÞ88 † † ˆ 7 ˆ 7 7 χL ¼ Σ Δ ΔΣ; ðD2Þ aaˆ b aˆ χð Þ b χ − aˆ χð Þ aˆ χ − aˆ χð Þ aˆ f dμð L Þ , ðdμ LÞ88 ¼ dμð L Þ ,Trðdμ LÞ¼ dμð L Þ þ aˆ ð7Þ aˆ ð7Þ dμ χ χ χ Δ χ ð L Þ , where L denotes the component of L transforming where is defined in Eq. (3.11). By construction L is a as a (complex) fundamental of SOð7Þ. A similar classification in Hermitian complex matrix. Under the action of an element the context of SOð5Þ=SOð4Þ models in Ref. [29] found only one 0 − g ∈ SOð8Þ, it transforms as a 21a þ 27s þ 7 þ 1 þ 1 of operator, corresponding to the linear combination O1 O1.

095036-25 ROBERTO CONTINO et al. PHYSICAL REVIEW D 96, 095036 (2017) ρ ρ ˆ that the contributions from L and R cancel each other. Let us start considering the T parameter. For conven- δ Finally, a nonvanishing gLb arises from the operators of ience, we split the UV contribution into two parts, ρX Eq. (D7) generated by the exchange of . Upon expanding redefining aUV as: 0 in powers of the Higgs doublet, O1 and O1 both match the   dimension-6 operator O of Eq. (5.11) and differ only by 2 Hq Fin Log MΨ higher-order terms. aUV ¼ aUV þ aUV log 2 2 2 : ðE1Þ MS þ f yR

APPENDIX E: EXPLICIT FORMULAS The coefficients aFin and aLog are obtained through a FOR THE EWPO UV UV straightforward calculation, but their expressions are In this Appendix we report the results of our calculation complicated functions of the Lagrangian parameters. We of the electroweak precision observables, in particular we thus show them only in the limit cL ¼ cR ≡ c and ≡ collect here the explicit expression of the coefficients aUV, MΨ ¼ MS M, for simplicity. For aIR we give instead aIR of Eq. (5.12) and bUV, cUV, bIR of Eq. (5.17). the complete expression. We find:

  1 12M4 6f2M2y2 9M6 Fin − R − 8 aUV ¼ 12 4 4 þ 2 2 2 2 þ 2 2 2 3 f yR ðf yR þ M Þ ðf yR þ M Þ cð−5f8y8 − 2f6M2y6 þ 7f4M4y4 þ 12f2M6y2 þ 4M8Þ þ R pffiffiffi R R R 2 4 4 2 2 2 2 f yRðf yR þ M Þ c2ðf2y2 þ 2M2Þ2ð3f2y2 − 5M2Þ R R ; þ 2 4 4 2 2 2 f yRðf yR þ M Þ f6y6 − f4M2y4 − f2M4y2 − 2M6 aLog R R R UV ¼ 2 6 6 f yR pffiffiffi 2 2 6 6 − 3 4 2 4 3 2 4 2 2 6 2 2 4 2 − 10 6 cð f yR f M yR þ f M yR þ M Þ c ðf M yR M Þ þ 6 6 þ 6 6 ; f yR f yR 1 2 2 pffiffiffi 2 2 2 MSMΨ 2 cLMSMΨ þ cRf yR aIR ¼ 2 þ 2 2 2 2 2 þ 2 2 2 : ðE2Þ ðMS þ f yRÞ MS þ f yR

δ 2 2 2 The derivation of g at 1-loop level is more pffiffiffi g X f bL α α α 1 2 ρ MΨ involved and requires the computation of a series of cUV ¼ 7Lð 1R þ 7RÞð þ cRÞ 2 2 2 2 : M X 2ðM þ y f Þ diagrams. As explained in the text, we focus on those ρ Ψ L featuring a loop of fermions and NGBs (see Fig. 4), ðE3Þ and that one with a loop of fermion and the tree-level exchange of a heavy vector (see Fig. 5). The coef- We remind the reader that in our numerical analysis we ficients cUV is generated only by the latter diagram; use MρX =ðgρX fÞ¼1, see footnote 5. We redefine the we find: other two coefficients as

¯ i FIG. 4. Diagrams with a loop of fermions and NGBs contributing to the ZbLbL vertex. Here f indicates any fermion (both heavy and light) in our simplified model; π are the charged NGBs.

095036-26 PRECISION TESTS AND FINE TUNING IN TWIN HIGGS … PHYSICAL REVIEW D 96, 095036 (2017) b ¼ δ þ δ¯ IR IR IR   M2 δFin δ¯Fin δLog δ¯Log Ψ bUV ¼ð UV þ UVÞþð UV þ UV Þ log 2 2 2 ; MS þ f yR ðE4Þ δ δFin δLog where IR, UV, and UV are generated by the diagrams in δ¯ δ¯Fin δ¯Log Fig. 4 only, whereas IR, UV, and UV parametrize the correction due to the tree-level exchange of a heavy spin-1 singlet in Fig. 5. As before, we report the expression of the ¯ UV parameters in the limit cL ¼ cR ≡ c, MΨ ¼ MS ≡ M, FIG. 5. The diagram contributing to ZbLbL with a loop of for simplicity; in the case of the coefficients with a bar, fermions and tree-level exchange of a heavy vector. Here fi and fj denote generic fermions in the theory, both heavy and light, generated by the diagram of Fig. 5, we further set α7L ¼ whereas ρν indicates the heavy vector. α1L and α7R ¼ α1R. We find:

−2f6y6 − 4f4M2y4 − 4f2M4y2 M6 c f6y6 4f4M2y4 − 2f2M4y2 M6 δFin R R R þ − ð R þ ffiffiffi R R þ Þ ¼ 2 2 2 3 p UV 12ðf y þ M Þ 6 2f2y2 ðf2y2 þ M2Þ2  R  R R 1 3 2 c3M2 þ c2M2 − − pffiffiffi ; 6 2 2 2 2 2 3 2 2 2 f yR þ M f yR f yR 1 M2 c3M4 c2M4 cð−f4y4 þ 2f2M2y2 þ M4Þ δLog ¼ − − pffiffiffi − − R pffiffiffi R ; UV 6 6 2 2 3 2 4 4 3 4 4 6 2 4 4 f yR f yR f yR f yR 2 2α 2 4 4 α 2α 2 2 2 3α 8α 2 4 α α f M 1Lg X ðf y ð 1L þ 1RÞþf M y ð 1L þ 1RÞþ M ð 1L þ 1RÞÞ δ¯Fin ρ R R UV ¼ 4 2 2 2 2 2 2 2 2 MρX ðf yL þ M Þðf yR þ M Þ 2 2α 2 2 2 2α α 2 2α cf M 1LgρX ðf yRð 1L þ 1RÞþ M 1LÞ þ pffiffiffi ; 2 2 2 2 2 2 2 2 MρX ðf yL þ M Þðf yR þ M Þ 4α 2 α − α 2 2α α 2 cM 1LgρX ð 1L 1RÞ f M 1L 1RgρX δ¯Log ¼ pffiffiffi þ : ðE5Þ UV 2 2 2 2 2 2 2 2 2 2 2 yRMρX ðf yL þ M Þ MρX ðf yL þ M Þ For the IR coefficients we give instead the full expressions. We find: 2 2 ffiffiffi ffiffiffi 2 2 1 M M p c M MΨ p c f y δ S Ψ 2 L S 2 R R ; IR ¼ 6 þ 6 2 2 2 2 þ 3 2 2 2 þ 12 2 2 2 ðMS þ f yRÞ ðMS þ f yRÞ ðMS þ f yRÞ g2 4 2 2 ¯ ρX f MΨyR δ α7 α1 : IR ¼ L R 2 2 2 2 2 2 2 2 ðE6Þ MρX ðf yL þ MΨÞðf yR þ MSÞ

Notice that the IR corrections aIR and bIR are related to each other and parametrize the running of the effective coefficients ¯ ¯0 ¯ cHq, cHq, and cHt, as explained in the main text. Finally, we report the contribution to Sˆ generated in our simplified model by loops of heavy fermions. We do not include this correction in our electroweak fit, because in the perturbative region of the parameter space it is subdominant with respect to the tree-level shift of Eq. (5.6). Rather, we use this computation as an additional way to estimate the perturbativity bound, as ˆ ˆ discussed in Sec. III B. Analogously to what we did for T and δgLb, we parametrize the fermionic contribution to S as:   2 2 2 Δˆ g2 ξ 1 − 2 − 2 m 1 − ~2 − ~2 m SΨ ¼ 2 ð cL cRÞ log 2 þð cL cRÞ log ~ 2 8π MΨ MΨ      2 2 ~ 2 g M MΨ þ 2 ξ sFin þ sLog log Ψ þ s~Fin þ s~Log log 16π2 UV UV 2 2 2 UV UV ~ 2 2 ~2 MS þ f yR MS þ f yR 2 2 2 2 g2 ξ yLf M1 þ 2 sIR 2 log 2 : ðE7Þ 16π MΨ mt

095036-27 ROBERTO CONTINO et al. PHYSICAL REVIEW D 96, 095036 (2017) Terms in the first line are logarithmically sensitive to the UV cutoff, the second line contains the UV threshold corrections, ~ ~ while the IR running appears in the third line. The UV thresholds include a contribution from the twin composites Ψ7 and Ψ1, ~Fin ~Log parametrized by sUV and sUV . At leading order in yL, by virtue of the twin parity invariance of the strong sector, such a Ψ Ψ Fin Log contribution can be obtained from that of 7 and 1 (i.e. from sUV and sUV ) by simply interchanging the tilded quantities with the untilded ones. Higher orders in yL break this symmetry and generate different corrections in the two sectors. We performed 0 2 the computation of the UV coefficients for yL ¼ , whereas sIR isderiveduptoorderyL.Wefind:   2 2 2 2 2 2 2 2 1 6c c M MΨðf y þ M þ M Þ ðc þ c Þ 24M M sFin − L R S R S Ψ R L S Ψ − 7 ; UV ¼ 2 2 2 2 − 2 2 þ 6 2 2 2 − 2 2 ðf yR þ MS MΨÞ ðf yR þ MS MΨÞ 2ðM2 þ f2y2 Þ sLog − S R 6c c M M3 c2 M2 f2y2 M2 − 3M2 UV ¼ 2 − 2 2 2 3 ½ L R S Ψ þ R Sð R þ S ΨÞ ðMS MΨ þ f yRÞ 2 2 2 2 2 2 2 − 3 2 þ cLðf yR þ MSÞðf yR þ MS MΨÞ; M2M4 − f2y2 ððf2y2 þ M2ÞðM2 − f2y2 ÞþM2M2 ÞþM2 ðf2y2 þ M2Þ2 s S Ψ L R S S R S Ψ Ψ R S IR ¼ 6 2 2 2 2 2 MΨðf yR þ MSÞ pffiffiffi pffiffiffi 2 2f2y2 ðM2 − f2y2 Þ 2M ðM2 − f2y2 Þ − c R Ψ L − c S Ψ L : R 3 2 2 2 2 L 3 2 2 2 ðE8Þ MΨðf yR þ MSÞ MΨðf yR þ MSÞ

APPENDIX F: THE EW FIT

For our analysis of the electroweak observables we make use of the fit to the parameters ϵ1;2;3;b [66–68] performed in Δϵ ≡ ϵ − ϵSM Ref. [69] (see also Ref. [26]). The central values there obtained for the shifts i i i and the corresponding correlation matrix are 0 1 Δϵ1 ¼ 0.0007 0.0010 10.80.86 −0.33 B C Δϵ2 ¼ −0.0001 0.0009 B 0.810.51 −0.32 C ρ ¼ B C: ðF1Þ Δϵ3 ¼ 0.0006 0.0009 @ 0.86 0.51 1 −0.22 A

Δϵb ¼ 0.0003 0.0013 −0.33 −0.32 −0.22 1

ˆ ˆ We can directly relate Δϵ1 to ΔT and Δϵ3 to ΔS by using APPENDIX G: ESTIMATES OF THE Δϵ −2δ the results of Refs. [45,47], and furthermore b ¼ gbL . PERTURBATIVITY BOUND Δϵ 0 We set 2 ¼ in our study, since its effect is subdominant This Appendix contains details on the derivation of in our model as well as in CH models [47]. We thus make the perturbative limits discussed in Sec. III B. As explained use of Eq. (F1) to perform a χ2 test of the compatibility of there, we consider the processes πaπb → πcπd and πaπb → our predictions with the experimental constraints. The χ2 ψ¯ cψ d ψ Ψ Ψ~ function is defined as usual: , where ¼f 7; 7g and all indices transform X under the fundamental representation of the unbroken χ2 Δϵ − μ σ2 −1 Δϵ − μ σ 2 σ ρ σ ¼ ð i iÞð Þij ð j jÞ; ð Þij ¼ i ij j; SOð7Þ. In order to better monitor how the results depend ij on the multiplicity of NGBs and fermions, we perform the ðF2Þ calculation for a generic SOðNÞ=SOðN − 1Þ coset with Nf composite fermions ψ in the fundamental of SOðN − 1Þ. where μi and σi denote respectively the mean values and the Taking N ¼ 8 and Nf ¼ 2 × 3 ¼ 6 thus reproduces the standard deviations of Eq. (F1), while Δϵi indicates the theoretical prediction for each EW observable computed simplified model of Sec. III A. in terms of the Lagrangian parameters. After deriving The perturbative limits are obtained by first express- the χ2, we perform a fit by scanning over the points in ing the scattering amplitudes in terms of components our parameter space keeping only those for which with definite SOðN − 1Þ quantum numbers. In the case Δχ2 ≡ χ2 − χ2 7 82 of SOð7Þ the product of two fundamentals decomposes min < . , the latter condition corresponding to the 95% confidence level with 3 degrees of freedom. as 7 ⊗ 7 ¼ 1 ⊕ 21a ⊕ 27s, where the indices a and s Using this procedure, we convert the experimental con- label respectively the antisymmetric and symmetric straints into bounds over the plane ðMΨ; ξÞ. two-index representations. A completely analogous

095036-28 PRECISION TESTS AND FINE TUNING IN TWIN HIGGS … PHYSICAL REVIEW D 96, 095036 (2017) decomposition holds in the general case of p-wave. At energies much larger than all masses the SOðNÞ=SOðN − 1Þ,19 but for simplicity we will use amplitudes read 7 the SOð Þ notation in the following to label the various s t u M πaπb → πcπd δabδcd δacδbd δadδbc components. The leading tree-level contributions to the ð Þ¼ 2 þ 2 þ 2 ; scattering amplitudes arise from the contact interaction f f f s generated by the expansion of the NGB kinetic term of M πaπb → Ψ¯ c Ψd θ δacδbd − δadδbc ð 7 7 Þ¼ 2 sin ð Þ; Eq. (3.4) and from the NGB-fermion interactions of L L 2f s Eq. (3.9). The structure of the corresponding vertices M πaπb → Ψ¯ c Ψd θ δacδbd − δadδbc ð 7 7 Þ¼ 2 sin ð Þ: implies that the four-NGB amplitude has components in R R 2f − 1 all three irreducible representations of SOðN Þ and ðG1Þ contains all partial waves. The amplitude with two NGBs and two fermions, instead, has only the anti- They decompose into irreducible representations of symmetric component of SOðN − 1Þ andstartswiththe SOðN − 1Þ as follows:

Mð1Þ πaπb → πcπd − 2 s ð Þ¼ðN Þ f2 ; ð21Þ a b ¯ c d s M ðπ π → Ψ7 Ψ Þ¼ 2 sin θ; L 7L 2f Mð21Þ πaπb → πcπd s θ ð Þ¼f2 cos ; ðG2Þ ð21Þ a b ¯ c d s M ðπ π → Ψ7 Ψ Þ¼ 2 sin θ: R 7R 2f Mð27Þ πaπb → πcπd − s ð Þ¼ f2 ;

π π Performing a partial wave decomposition we get δðrÞ ⇒ ðrÞ j j j < 2 jaj j < 2 ðG6Þ X X∞ r r MðrÞ ¼ Mð Þ ¼ 16πkðiÞkðfÞ að Þð2j þ 1Þ r 1 λi;λf j Let us consider first the case ¼ , corresponding to the λ ;λ j¼0 − 1 i Xf amplitude singlet of SOðN Þ. The only contribution j comes from the four-NGB channel. Since the helicities of × Dλ λ ðθÞ; ðG3Þ i; f the initial and final states are all zeros, in this particular case λi;λf the Wigner functions Dj ðθÞ reduce to the Legendre λi;λf where λ , λ are the initial and final state total helicities, and i f pffiffiffi polynomials: ðiÞ ðfÞ 2 k ðk Þ is equal to either 1 or depending on whether Z the two particles in the initial (final) state are distinguish- 1 π ð1Þ θ θ Mð1Þ ðrÞ aj ¼ d Pjðcos Þ : ðG7Þ able or identical respectively. In the above equation M 64π 0 should be considered as a matrix acting on the space of ðrÞ The first and strongest perturbativity constraint comes from different channels. The coefficients aj are given by Z the s-wave amplitude, which corresponds to j ¼ 0.Wefind: π X ðrÞ 1 j ðrÞ a ¼ dθ Dλ λ ðθÞMλ λ : ðG4Þ j ðiÞ ðfÞ i; f i; f 1 N − 2 s 32πk k 0 λ λ ð Þ i; f a ¼ ; ðG8Þ 0 32π f2 and act as matrices on the space of (elastic and inelastic) channels with total angular momentum j and SOðN − 1Þ where N ¼ 8 in our case. From Eqs. (G6) and (G8),one irreducible representations r. They can be rewritten as a obtains the constraint of Eq. (3.15). function of the scattering phase as We analyze now the constraint from the scattering in the antisymmetric representation, r ¼ 21. In this case, both the r 2iδð Þ r e j − 1 r NGB and the fermion channels contribute; the process að Þ ¼ ∼ δð Þ: ðG5Þ ππ → ππ j 2i j is however independent of the fermion and Goldstone multiplicities and can be neglected in the Our NDA estimate of the perturbativity bound is derived by limit of N and Nf. The process involving fermions is a requiring this phase to be smaller than maximal: function of Nf and generates a perturbative limit which is comparable and complementary to the previous one. 19 N ⊗ N 1 ⊕ N N − 1 2 ⊕ N N 1 2 − 1 One has ¼ ½ ð Þ= a ½ ð þ Þ= s. We have:

095036-29 ROBERTO CONTINO et al. PHYSICAL REVIEW D 96, 095036 (2017) X Z 21 1 π 21 N ð Þ θDj θ Mð Þ ð21Þ fffiffiffi s aj ¼ d 0;λ ð Þ 0;λ : ðG9Þ a1 ¼ p 2 : ðG10Þ 32π 0 f f 24 2π f λf¼1

As anticipated, this equation vanishes for j ¼ 0, so that the strongest constraint is now derived for p-wave scattering, From Eqs. (G6) and (G10) follows the constraint with j ¼ 1. We have of Eq. (3.16).

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