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Title Minimal mirror twin Higgs

Permalink https://escholarship.org/uc/item/5pn7r8vm

Journal Journal of High Energy Physics, 2016(11)

ISSN 1126-6708

Authors Barbieri, R Hall, LJ Harigaya, K

Publication Date 2016-11-01

DOI 10.1007/JHEP11(2016)172

Peer reviewed

eScholarship.org Powered by the California Digital Library University of California JHEP11(2016)172 Springer November 19, 2016 November 29, 2016 September 29, 2016 : : : breaking, can generate 10.1007/JHEP11(2016)172 2 c,d Z Received Accepted Published doi: , Published for SISSA by breaking from the vacuum expectation 2 Z and Keisuke Harigaya [email protected] c,d , . 3 Lawrence J. Hall a,b 1609.05589 breaking fields are also discussed. The Authors. L c Beyond Standard Model, Cosmology of Theories beyond the SM

In a Mirror Twin World with a maximally symmetric Higgs sector the little , − parity exchanging the Standard Model with its mirror be broken in the Yukawa [email protected] 2 B Z breaking in the Higgs sector necessary for the Twin Higgs mechanism. The theory 2 Z [email protected] Department of Physics, UniversityBerkeley, of California California, 94720, U.S.A. Theoretical Physics Group, Lawrence BerkeleyBerkeley, National California Laboratory, 94720, U.S.A. E-mail: Institute of Theoretical Studies, ETHCH-8092 Zurich, Zurich, Switzerland Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy b c d a Open Access Article funded by SCOAP ArXiv ePrint: neutrons and asignals variety and of the self-interacting effects mirrorvalues of of atoms a possible are additional considered. Neutrino mass Keywords: couplings. A minimal suchthe effective field theory, withhas this constrained sole and correlated signalsDark Radiation, in all Higgs within decays, reach directwhere of Dark the foreseen Matter experiments, fine-tuning Detection over for and a the region of electroweak parameter scale space is 10-50%. For dark matter, both mirror Abstract: hierarchy of the Standardcutoff Model scale above can the bethat LHC the significantly reach. mitigated, We perhaps show displacing that the consistency with observations requires Riccardo Barbieri, Minimal mirror twin Higgs JHEP11(2016)172 9 U(1), × 29 ) interact via SU(2) 0 × , l 0 q 6 17 25 2 H 10 17 12 m vevs 12 3 L ) and gauge interactions SU(3) 25 − 10 – 1 – q, l spectrum 23 B 0 q 20 8 13 18 20 20 symmetry breaking in Froggatt Nielsen sector 2 Z 25 16 19 breaking from symmetry breaking in Yukawa couplings 2 2 Z Z 1 27 and constraints on the 0 8.1 Dark radiation 8.2 Additional dark matter candidates 6.3 Milli-charged particle 7.1 Dark matter7.2 candidates Direct detection of dark matter 5.3 Dark radiation 5.4 Cosmological signals of mirror neutrino masses 6.1 Decoupling temperature6.2 Dark radiation 4.3 Mirror QCD phase transition temperature 5.1 QCD 5.2 Decoupling temperature 4.1 Constraint on4.2 Yukawa couplings from Standard ∆ Model like Higgs decays 1 Introduction An intriguing idea, that has persistedStandard Model, over several with decades, quarks and is leptons thatis ( of supplemented the by Mirror an World: identical the sector where mirror quarks and leptons ( A Heavy axion and 8 Non-minimal 9 Conclusions 7 Mirror baryon dark matter 6 Thermal history including kinetic mixing 5 Thermal history with Higgs exchange 3 Necessity of 4 Minimal mirror twin Higgs Contents 1 Introduction 2 Review of the twin Higgs mechanism JHEP11(2016)172 . ] P y 6 , 6= 5 0 y loops. 0 q thermally 0 ]. Secondly, H ]. How is this 2 0† , 4 [ 1 , is large relative λ P HH † ]. As the mirror Higgs H 11 . There are two prime motivations for 0 ] and figure 10 of [ , it could be beyond the LHC range, and 4 λ breaking in the Yukawa couplings to obtain U(1) P × – 2 – ; see [ 0 violation leads simultaneously to three key results ZZ P or SU(2) , allowing a neat restoration of parity [ × , the Mirror World reduces the amount of fine-tuning by P 0 ], but note that both account for the approximate SO(8) WW SM 10 λ – 7 13 is small, this improvement can be very significant, allowing a Little to reach any particular UV cutoff of the effective theory. Since we now . ) possess a potential with maximal SO(8) symmetry at leading order, 0 term necessary for the Twin Higgs mechanism is generated via SM = 0 2 H H,H m SM λ/λ λ ]. Furthermore, if the symmetric quartic coupling of this potential, The mirror baryon mass is raised, allowing viable dark matter. The ∆ The mirror QCD phaseperature transition of temperature the is two sectors, raised solving above the the problem decoupling of tem- excessive dark radiation. . We find this term by itself to be insufficient to solve the dark radiation problem, 3 A striking signature would be the discovery at the LHC, or a future collider, of the Hence we introduce an effective theory, Minimal Mirror Twin Higgs, where all The SO(8) invariant quartic interaction contains an interaction In this paper we formulate a minimal, experimentally viable, low energy effective theory A third, more recent motivation for the Mirror World arises from the absence so far of H 2 • • • m mirror Higgs itself, decaying to mass depends on thehere SO(8) we invariant focus quartic, on other signals. The size of the two sectors. This single source of ∆ nomatter what other interactions connect thepercent two sectors, level. at This least then for fine-tunings implies above that the the Yukawa couplings of theviolation two arises sectors from differ, a breaking of flavor symmetry, yielding different Yukawa couplings in symmetry of the Higgs potential. coupling the two sectors aton cosmological dark temperatures radiation above provides a a few verymechanism GeV, requires severe so a constraint that parity on breaking the Mirror contribution bound Twin to Higgs. the Higgs The mass Twin terms Higgs in the potential, for this idea, Minimalmirror Mirror baryon Twin dark Higgs, matter, and thelimits Twin study on Higgs its the mechanism signals. amount and ofto consistency This be dark with is cosmological accomplished? radiation a all We pressingor do require issue: not a with attempt breaking composite a of Higgs UV parity, completion, [ whether supersymmetric [ to the SM quartica coupling, factor of 2 know that Hierarchy between the weak scale and the UV cutoff, which may be beyond the LHC reach. new physics at colliders totwo explain the sectors origin ( of thethe weak observed scale. Higgs If boson theinsensitive Higgs becomes to doublets a the of pseudo-Nambu-Goldstone usual the Standard boson Modelidea with (SM) [ a quadratic divergences; mass this that is is the Twin Higgs this idea. The discrete symmetryinterpreted that as interchanges spacetime the parity, ordinarymirror and baryons mirror worlds are can expected be stable to to yield be dark matter, producedenergy and in this densities may of the lead baryons to early and an universe understanding dark and of matter why are to the comparable. cosmological be sufficiently mirror gauge interactions SU(3) JHEP11(2016)172 we (2.1) , and 5 in the 2 H and are P 20) GeV, m L − we examine 6 . For dark matter, 0 inv) and ∆ q , is embedded into a mass of (2 0 . , we demonstrate that → can be. In section 0
. We define the Minimal H 2 q h 0 2 c | 3 0 T H | inv), of the Higgs boson. + is of order 10 TeV. → 2 | a h and dark matter candidates when f H | eff N 2 H interaction, together with the enhanced and find that the decoupling temperature by the vacuum expectation value (VEV) m 0 0 H Y + H 0† 2 0†
2 . | – 3 – U(1) 0 HH eff HH † † H × Yukawa couplings from Γ( | N H , and discuss the consequences of breaking to lie inside the observational limit. We predict the 0 L H q + 4 2 eff , together with a scalar field | N H H | we briefly study ∆ 8 λ , and the invisible width, Γ( = µ , leading to the possibility that sym 3 V , allowing ∆ 0 c , and find that the T in the Yukawa couplings is necessary in section 7 P breaking arises from the absence of Majorana masses for right-handed mirror P The effective sum of neutrino masses affecting large scale structure and the CMB. The signal strength, The amount of dark radiation, ∆ The direct detection rate for mirror baryon dark matter from Higgs exchange. . Four of the pseudo-Nambu-Goldstone bosons form a doublet of SU(2) 0 Let us take a closer look at the Higgs potential. A global symmetry preserving potential After a brief review of the Twin Higgs mechanism in section • • • • H Yukawa couplings, will allow direct detection at planned experiments over a large part 0 identified with the Standardcomparison with Model the Higgs scale doublet. of the Higgs The potential lightness canis of be given understood by the in Higgs this mass way. in In this section, weStandard review Model the Higgs Twinrepresentation doublet Higgs of mechanism some usingdown approximate a to global linear a sigma symmetry. subgroupof model. containing SU(2) The The global symmetry is broken neutrinos. In the appendixa we solution show to that the aby strong PQ a CP symmetry factor problem, common of with to order the both 10 axion sectors mass allows enhanced2 by the mirror sector Review of the twin Higgs mechanism mixing of the hypercharge gaugematter bosons. in We section study aq variety of candidates for mirrorof dark the mass range.additional In section study the cosmological historythem of arises from the the Higgs two interaction sectorscan when be the lower than onlyamount communication of between dark radiation and thean effective alternative sum cosmology of when neutrino communication masses. between the In sectors section is dominated by kinetic the breaking of Mirror Twin Higgs theory inYukawa couplings. section We constrain the study how large the mirror QCD phase transition temperature These signals are tightly correlatedsectors, as they the all ratio depend of onboth the the mirror Higgs weak portal neutrons scales, between and the and a two on variety of the self-interacting masses mirror of atoms the are light considered. leading us to compute signals for the following quantities the above three results leads to a preferred range of the lightest JHEP11(2016)172 (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) (2.2) (2.3) is given by i 0 1 H h ), with Λ the cut off A . ) + .  symmetric mass term is 2 t 2 2 0 2 2 H  /m “ to denote mirror objects. Z 2 λv 0 m /v 3 0 ∆ v 2∆ . ), we obtain )  (log(Λ . − 4 2 | 0 2.3 . × π 1 ! 0 0 2 . H 2  H 64 v v | 2 2 / h λ 2 h | , 2 . 4 m t 0 2 2 2 √ √ 2 + y v 0 H − = 0 initially, the VEV of | 4 ) and ( | i , is λ v sin cos ' 0 2 H = 3 = λ v may exist, as, e.g., from a supersymmetric D-term. H h 0 H 4 | 2.2 2 m h λ i the mirror Higgs. We also introduce appropri- v 0 8∆ ( λ i 0 2 v 0 h breaking is given by ' (200 GeV) – 4 – λ h ' breaking in the mass term, breaking terms to obtain a correct electroweak 0 H √ 2 H 2 h = ∆ ' → 2 h , the pseudo-Nambu Goldstone boson, namely the 2 2 Z 2 2 2 h Y m Z = ∆ m Z ≡ h Z ! Z Z sin 0 m 2 V 2 0 0 X 2 H . Setting 2 H 0 X v v 2 v U(1) v i

SO(8) X and call = 1 4 × V 0 H 2 given by eqs. ( h L i ' H , is given by h > a finite UV-dependent term. To obtain the observed Higgs mass requires h H 2 H ↔ 2 A i m 0 ), the required ≡ h H ∆ H 2 h 2.7 v breaking mass term is explained in the next section. 2 Z 7. Alternatively a tree level ∆ ≈ is given by ) and ( 2 h symmetry 2.6 A/ 2 Z 0, we expect ) + t > ] for the case of a composite Twin Higgs model. /m Let us derive the VEVs of the Higgs fields in the small SO(8) breaking limit. Assuming 9 2 H Quantum corrections from top quarks gives ∆ 1 m scale of the Higgslog(Λ sector and See [ whereas the mass of the mirror Higgs boson, From eqs. ( The mass of Standard Model like Higgs Minimizing the potential of ∆ In the unitary gauge of SU(2) As we will seesymmetry breaking later, scale. we We assume need small The origin of the ate mirrors of otherThe SM global particles. symmetry In of theaccidentally the following, SO(8) Higgs we symmetric, potential use and is “ an now SO(8) SO(8). breaking potential A is given by models, as they areexplicitly irrelevant broken for by Yukawa the couplings followingbreaking and discussion. the of electroweak the gauge Since global interaction, theThe symmetry we global quantum in expect symmetry correction the a is Higgs toassume potential, the a at mass least term by is quantum the corrections. most dangerous, and to suppress it we We have neglected possible higher order terms which are expected in composite Twin Higgs JHEP11(2016)172 ] 16 3 at (2.14) (2.15) (2.10) (2.11) (2.12) (2.13) /v > 0 v counting [ can be above ] require that is the coupling 13 N ∗ TH g ]. The top Yukawa , which depends on the /v 12 0 , v 9 , where . The standard fine-tuning 0 2 v 0 ∗ g ] and the large λ v , 2 ∼ . 15 . Λ , 2 2 . 0 symmetry [ , ,
0 v v 2 t , imply an overall reduction of the 14 h 2 2 SM 2 TH 0 y 2 TH 1 2 2 SM m Z Λ Λ h 2 Λ 2 Λ − v 0 ∆ 2 t = 2 t 0 λ y 2 is the cut off scale of the SM. SM t y

2 × 2 λv SM y 2 λ and 2 2 H π λ π 2 2 0 2 8 SM 2 8 2 = h v m v v / / 3 π 3 – 5 – 1 2 3 ln 8 Hence we need a tuning of at least 50%: this is TH ln∆ ∂ ≈ = = 2 ' ∆ ∂ 0 ). / t h

y TH | SM m SM ≡ 2 H ∆ ∆ must be tuned against ∆ 4.2 ∆ ∆ m ∆ is the size of the confining gauge group of the composite 2 H ∆ = 1 TeV, considering the IR contribution only, one obtains m N 0 h , ∆ m v ). The precision measurements of these couplings [ 2 ) , where and 0 0 N v v/v √ it is 2( 3 π/ , / 4 1 13, this improvement is significant: for a moderate tuning Λ . gives a one-loop quantum correction, TH is the SM quartic coupling and Λ ∼ − 0 1 this requires a suppression of the Higgs loop contribution in the UV completion of the Twin 1, the overall fine tuning in Twin Higgs is given by t ∗ y ' g > SM 2 at 95% C.L. (see section λ > 0 λ h SM Let us comment on the required quality of the Thus the fine tuning in TH relative to the SM is In general we expect a fine tuning also in the mass of the mirror Higgs boson. Assuming Let us comment on the fine-tuning in the Twin Higgs model. In order to obtain the m The electroweak precision measurement as well sets an indirect bound on For λ 2 3 /v > 0 suggest mass of the mirror90% Higgs. C.L. For Higgs model considered here. where Λ is the cutof off higher of resonances, the which Higgsstrength is sector. of expected hadrons. In be composite The as Twin naive-dimensional analysis Higgs large [ models, as Λ Λ is the scale As the scales directly explorable at the LHC. coupling where that can be compared with the fine tuning in the SM at a scale Λ If ∆ couplings of the Standardamount (1 Model higgs tov any Standard Modelthe particles unavoidable, minimal by fine-tuning the in relative Twin Higgs models. that the dominant contribution to this fine tuning comes from the top loop suitably cutoff hierarchy between measure is given by The mixing in the physical higgs bosons, JHEP11(2016)172 . ) f y 2.18 6= (2.20) (2.16) (2.17) (2.18) (2.19) 0 , below f d ), ( y T symmetry symmetry breaking is 2 2 2.16 2 Z Z Z is determined by Higgs d symmetry is via T . 2 . . 2 Z 2 2  ,   2 symmetry in Yukawa couplings , Λ 2 2
Λ Λ Λ Λ Z 2 0 3
5 TeV 5 TeV 5 TeV 2 g 0 2 symmetry shown in eqs. (  g   2 4 − 2 2 symmetric, with a complete copy of all 2 2 π π 2 3 −  Z 8   g 2 2 2 64 g Z /v 2 t 3 /v /v 0 3 3 0 0 y v 9 v v 3  – 6 –   ' ' 2 5 . . 03 . 0 0 0 weak . . | breaking necessary to construct a fully realistic theory. . strong

| 2 H

2 2 2 2 3 t 2 H m g g Z y 2 2 m 2 3 ∆ t − − g g − y ∆ 2 2 0 0 0 2 3 t g g y

symmetry breaking in Yukawa couplings ), we obtain 9 2 . 2 interaction gives a one-loop quantum correction, Z L ) is that the Lagrangian is precisely ) so that in the early universe at temperatures of order the weak scale the two sectors 2.20 2.1 The Standard Model and mirror particles interact with each other by the interaction in A natural explanation for the quality of the eventually annihilate/decay into mirrorcomponent photons of relativistic (and particles, which mirror isof neutrinos), often dark referred radiation giving to depends as an on dark the extra radiation.which decoupling the temperature The interaction between amount the rate two between sectors, StandardHubble Model expansion and rate. mirror particles Without is introducing smaller extra than interactions, the to suppress the abundanceand of no dark matter radiation, what noThus interactions matter the might what minimal be phenomenologically the viable added origin way to of to break the the theory toeq. couple ( the two sectors. are kept in equilibrium via Higgs exchange. At lower temperatures the mirror particles As reviewed in theunder previous which section, the the Standard Twinmust Model Higgs be and mechanism broken mirror requires eventually to particlesthis a obtain are section, the we exchanged. show correct that The electroweak it symmetry is breaking mandatory scale. to In break the and ( Standard Model particles — thiscomes is the nothing form but and the origin Mirror of World. the A key question then3 be- Necessity of requiring Finally, the SU(2) The strong interaction gives a two loop quantum correction, leading to the requirement multiplied by a log enhancement factor.required one Requiring in that this eq. correction ( does not exceed the Twin Higgs model. In the supersymmetric Twin Higgs model, Λ is the stop mass scale JHEP11(2016)172 = as 0 γ eff (3.1) (3.2) ]. We is the N 17 0 r and g 0 τ ), indicated , σ 0 µ , 0 . , make the claim e , 40 (1 7 . , which is given by eff to ) + 1 2 2 N ) + 1 v 4 / x d v 3 , 2 d
3 / ) and 0 1 /T / 1 0 σ 4 as a free parameter. /T
0 v ) is extracted from [ − 1  d f v , respectively, and T 2 ) f T ( T − d x g 65 (2 0 r ' γ e',τ', μ', T 2 xm . g ( xm 0 x 0 100 g ]. In this section, to be general, we < exp(  5 , exp( x eff 4 × x d N d 3 / ∞ 10 4 , 1 ∞ 40 / GeV ) 1 Z  d d Z ) 5 4 symmetric, the mirror charged lepton masses T T d gauge coupling.) As their masses are relatively 4 75 (  2 . 0 T – 7 – c 0  ( p Z v between the Standard Model and mirror sectors. 0 v 10 g 4 1 v d v v'/v=3 d  T  = 2 and T  ) + 4 f . (The masses of mirror hadrons could be also affected 0 r d 4 f × g m T /v 0 r ( 0 m 0 g v ρ 4 7 2σ 1σ e,µ,τ 2 0.1 X e,µ,τ = π = 45 X f = 2 f 2

eff

0.7 0.6 0.5 0.4 0.3 0.9 0.8

2 2 eff π Δ N 2 N π 3 ∆ ) are the effective entropy degrees of freedom (d.o.f) of the Standard , we show the prediction for the abundance of dark radiation, which by ) = ) = ) = 2 + d d d T 1 ( T T T 0 ( ( ( 0 0 0 g p g ρ ], which leads to the upper bound of ∆ ) and symmetry breaking in the SU(3) 21 . Prediction for the amount of extra dark radiation arising from only T 2 ( 5 [ Z g . 0 The Planck collaboration puts a bound on the effective number of neutrinos, Assuming the Yukawa couplings are  2 . 3 heating of the Standard Modeldark neutrinos, radiation. and The the d.o.f thirdassume of factor expresses that the the Standard only heating Model the ofthe particles mirror the dark radiation. electron, muon, Light mirror tauin neutrinos, this and as section considered photon even in are sections stronger. light Then and contribute to where Model particles andd.o.f. the of mirror the particles radiation at component temperature of the mirror sector. The second factor expresses the are determined solely by the ratio by the small, they remain in the thermalradiation. bath until In low temperature, figure and contribute tooconvention is much dark expressed as an effective extra number of neutrinos, ∆ a function of the decoupling temperature exchange which depends on theallow Yukawa additional couplings interactions [ coupling the sectors and treat Figure 1 JHEP11(2016)172 0 . ). λ H 0 µν 0 (4.1) (4.2) (4.3) B N 2.20 0 bound µν L σ , so that ), ( B t + 2 W 6= θ .  2.18 t 1% to obtain . y f : ) are not exact cos LNH ), ( = 1 2 TH 0 4.2 t , for + y 2.16 0 0 f y H 0† ]. Once these fields take 22 HH † i H .... g 00 λ + t. = 2 0 i 6= symmetry breaking if they arise from ) + 2 f /M symmetries) play an important role in 0 2 ) 0 Z H L H 0 − , m 0 L B )( , λ . We restrict the size of 0 symmetric set of dimension 5 operators and are – 8 – f , g 2 TH LH f , y H Z 0 y i m for Minimal Mirror Twin Higgs is g + ( ( 6= 0 bound can be only marginally satisfied. The 1 1 is not too large, the theory may have neutrino oscillation signals. = 0 1 0 f TH 0 2 σ 2 ) from a naive stepwise function. We conclude that it is 0 y . Including all operators consistent with gauge invariance /M H 0 3 d M 2 2 , T ) L 1 0 ( 40, which requires a fine-tuning of more than 0 0 and M g H 0 symmetry in the Yukawa couplings to further raise the mirror & are excluded from limits on oscillation to sterile neutrinos, sug- ) + symmetry of the Yukawa couplings is spontaneously broken. As . 1 . The 2 L  2 H 2 2 symmetric neutrino Yukawa couplings and right-handed neutrino M , ) describes the Standard Model including all operators up to di- 1 /v Z 1 Z 0 2 H + ( similarly describes the mirror sector. The lepton-Higgs interactions v Z M 1 0 is hard, meaning that the boundary conditions of ( 1 , λ, m 0 2 We assume these symmetries lead to neutrino couplings 2 f 0 /M Z λ, m , λ, m 0 3 2 4 , y f ) i L = g , y ( 0 . i f λ LH g ( y 321 = L + ( 6= 321 00 0 = L λ f y The Twin Higgs mechanism is imposed by boundary conditions at Λ The Yukawa couplings can naturally acquire If the hierarchy between breaking is the minimal consistent with the requirement of the previous section, namely 4 EFT 2 L saw mechanism with masses. Comparable gesting that lepton numberneutrino symmetries masses. (or This breaking of and are broken bythese loop corrections corrections maintain the at Twin Λ HiggsThe mechanism dimension and satisfy 5 eqs. operators ( have flavor structure suppressed and would arise from the see- As discussed in footnoteZ 1, there could also be a boundary condition giving non-zero ∆ mension 4 and of the second line aresuppressed the by most large general masses allows kinetic mixing via where with 4 Minimal mirror twinThe Higgs effective field theory below Λ VEVs of fields, asasymmetric VEVs, in the the Froggatt-Nielsenlong (FN) as mechanism the [ topmechanism Yukawa is coupling maintained. does In this notbut paper depend study we do the on not physical these specify consequences the field of model values, of a the the low FN Twin energy mechanism, Higgs effective field theory for Twin Higgs can be satisfied when the electroweak symmetry breaking scale.the Note deviation the of importance the tonecessary actual reach to this break conclusion of the charged lepton masses. by dashed lines in figure JHEP11(2016)172 2 µ ). Z 2.9 (4.6) (4.7) (4.4) (4.5) ; the 2 M ). The  4.7 1 . M 2 and is negligible.  or 0 4 f 1 y 1. In the following we , Standard Model and Λ 0 M , ' 2 5 TeV   Λ ,  NN 0 l 2 4 δ  M Λ , 2 breaking Higgs mass term 5 TeV /v 3 0 2 v  is UV sensitive and hence involves ; 3 for mirror quarks and one for Z 0 H 2 2  (Λ) Λ f  . 0 2 m 2 ν f Yuk | y , so that /v 3 0 m 0 . 0 2 H 2 2 v ν f N ) 10) GeV, and π m  8 N m − /v by requiring that quantum corrections to the t ), the sum of the square of the mirror Yukawa 0 04 from the invisible decay width of the SM-like = 0 v 6= and . f f 0 X 0 – 9 – 0 (100 GeV) 2.9 y ν f = ( N y ' m 0 ' ' = (100 ν 0 0 q mass mixing is absent, neutrinos are Majorana with m 2 f Yuk of eq. ( as required for the Twin Higgs mechanism in eq. ( | m 0 (Λ) m 0 2 H 2 H 2 f 2 H y ! m symmetry breaking in the Yukawa couplings by requiring m NN m Λ breaking in 5 for  , 0 ∆ . 2 0 0 f ) encodes the effect of the renormalization between a scale 2 2 1 f f Z 3 δ µ Z N − ( N 3 0 3 f . 

t /y 6= ) scale. 0 f X f 0 , arising if 1 m X u,d,c,s,b ( M 0 = f is about 1 f 4. Note that the estimation of ∆ , arising if the only right-handed neutrino mass is y . 2  Λ , 0 is the multiplicity of the mirror fermion ≡ 2 M q = 1 0 δ ,µ f M 0 . Λ  , 0 f N 0 f 1 δ q δ m Next we constrain the For M (1) uncertainty, which we formally treat by varying the value of Λ in eq. ( Here, and take O breaking correction to the Higgs quartic couplings is proportional to where mirror leptons. It shouldFor be this correction remarked to that explain the ∆ couplings, sign and of hence the the mass mirror term fermions is masses, the are required determined one. assume the top Yukawa couplingsbut of allow the other Standard Yukawa couplings Model to and differ, mirror inducing sectors a are identical maximize the QCD 4.1 Constraint on YukawaWe couplings derive from a ∆ constraintthat on quantum correction to the soft Higgs masses yields the Twin Higgs mechanism. We tion in the mirror scheme. Higgs masses yield non-zero ∆ We then place experimentalHiggs. bounds on Finally, we study mirror fermion spectra, consistent with these constraints, that For mirror left-handed neutrinos obtain Dirac masses, so that Such Dirac masses from a seesaw are an interesting possible consequence of parity restora- light neutrinos are then Majorana or Dirac, respectively. light mirror neutrino masses given by and masses for right-handed neutrinos, JHEP11(2016)172 . , 2 ], 2 = / ) inv 0 h 23 h h (4.8) (4.9) , (4.11) (4.10) 5 . 0 0 , Br around 93 [ c /m 0 . < m 0 H T h 0 q 0 . m f v/v m ], as it has h . Since the 0 m c , and µ < ,m 13 2 T 0 0 R ¯ − f f and ( 3 for δ H 0 . L 2 4 ) ) . However, there is 0 hf 0 h f 2 and degenerate, and 5.3 v/v / ,m m 0 0 h f f 10GeV δ , . m ( 2 0 0 , is about 1 f vm v inv h 3 2 sum 2 N ,m 0 m √ h q δ of the Higgs couplings to all pairs + Br − ≡ ) above , c H ], which probes ≡ h T m 0 f 24 2 γ y s = Br( above the decoupling temperature of the two sectors is critical to obtaining 0 inv L ⊃ − c , with 0 T Br H γ 4 can be probed. The ILC is expected to measure the Higgs signal-strength with s below the current bound, as specifically illustrated in section + First, we take the masses of Higgs decays to mirror fermions leads to an invisible branching ratio eff In the following, we take 5 H N /v < 0 γ a limit to how much determine their mass according to eq. ( following discussion. The larger Yukawa couplingsquarks, of and the hence mirror tonumber quarks of a degrees leads of larger freedom to mirror ofincreasing the a QCD mirror larger sector phase changes mass rapidly transition near∆ of temperature, the phase the transition, mirror fermion heavier than which is shown byv a dashed line inan accuracy figure of 1% [ 4.3 Mirror QCD phase transition temperature We adopt the bound the smallest uncertainty. Thegive bound sub-dominant contributions is to so ∆ strong that mirror fermions with SM final state, versus the relevant combination of mirror fermion masses where phase space has beenand neglected. the Figure universal deviation from unity of the Higgs signal-strengths at the LHC into any where the QCD running factor for10 a GeV, mirror and quark the precise value depends on the details of the mirror quark spectrum. The Standard Model likec Higgs In turn this leads toof a SM universal particles reduction as by well a as factor to a coupling of the same Higgs to the mirror fermions 4.2 Standard Model like Higgs decays JHEP11(2016)172 18 ) and is well 16 0 v'/v=3 4.7 q 75. With 1 by dots Λ=3TeV . . m 0 0 14 , 05 . µ > 0 12 , . The figure shows 02 / GeV . 3 10 q' m saturating these bounds 60 8 − 0.2 0.1 inv) = 0 m =0.05 inv 6 50 → Br and h 4 3 2 1 0 =5 + + 4 μ<0.75 N 40 m ], matching the renormalization scale , as shown in figure

0

25

c 3.0 2.5 2.0 1.5 1.0

c / GeV ' /GeV T T 30 HL-LHC sum m – 11 – ), and constrained by the bound 20 0 18 5 q m 4 , the universal 16 10 . The dotted lines show contours of the invisible branching + v'/v=3 Λ=5TeV N /v 0 =0.1 v v'/v=3 14 0 inv can be read off from the right hand end of each line. 0 Br c

T

0.20 0.15 0.10 0.05 0.00 0.30 0.25 12

μ - 1 / GeV 10 q' ), for given m 0.050.02 4.11 8 , is saturated. We also show values of Br( µ : the maximal 6 0 c T 4 3 2 1 0 . The prediction of the mirror QCD phase transition temperature as a function of the =5 . Prediction for the universal Higgs signal-strength as a function of a sum of mirror fermion can be well above 1 GeV, and reach a few GeV. Here the right end of each line + mirror quarks to be the same ( 4 0 N c 75, while the dynamical scale of mirror QCD depends on the product of the mirror . + T 0 Note that the sum of the square of Yukawa couplings is bounded as eq. ( N

1.0 2.0 1.5 2.5

c / GeV ' T − on each line, from left to right. µ > quark masses. Thus, formaximize a given the pure SU(3) Yang-Millsand gauge the theory inverse of in thethat ref. lattice spacing, [ we obtain shows the point wherethe the limit bound on from the invisible decay of the Higgs, as inferred from 5 these masses, we solvegauge the coupling two-loop constant. renormalizationabove We group the find equation scale that, at of for which the a the mirror wide gauge range QCD coupling diverges. of Adopting parameter the space, lattice calculation for Figure 3 light mirror quark mass. ratio, and the dashed line shows the sensitivity of the high-luminosity running of the LHC. Figure 2 masses defined in ( JHEP11(2016)172 ) d In T ( 10 (5.1) (5.2) g − . In the . Hence, 0 c T eff the glueball N flavors. The ) drops from . 0 c 0 T in the effective 8 T q (   , which is at most 0 c contribute to 0 0 QCD q g m < T , giving a decay rate 0 d q 20 GeV is large. In this case, the T 0 m  , the mirror glueball is not g u, d, s, g 9 3 ) that are excluded by bounds  is lower than computed in the d 0 ). To satisfy observational con- T c ) has been accurately computed 0 ( T 0 T Λ g 3.1 ( , 2 GeV 0  F 0 0 QCD , then g 0 F 0 = 5 in figure via GeV G ); roughly speaking, decoupling must occur 0 0 + , much of the entropy of the mirror gluon d 0 γ c the colored states 0 G 17 N T T ( d γ − spectrum 0 scale Λ 0 α 4 q 1 T – 12 – 0 g 0 s . 10 1 m q α contributions to ∼ 0 > where its contribution to 0 and any 4-loop contribution would be of order 10 9 8 q 0 0 c d L ∼ 6 T Λ decay to m T 2, labelled . 0 2 3 / 3 S h α π becomes strongly coupled, and is comparable to ) so that at ; immediately after the phase transition at , so that the sectors are coupled only via Higgs exchange, and 2 s 0 0 c d phase transition is purely gluonic with zero 5 m 64 α T T − 0 ( g 10 ) = 0  γ 0 )  < ) is 0.6. If 0 γ d c T T ( ( vanish at 3 loops, → 0 0 ]. As the temperature decreases from high values, g 0 g heavier than and QCD phase transitions, and we explore this further below. 26 S TH 0 0 and constraints on the q , mirror glueballs 0 Γ( c 0 T states contribute very little to are light compared to the QCD 0 . Hence the QCD 0 q eff ) in the limit of N Below Candidates for such three-loop corrections are diagrams where a SM photon is connected with two 4.1 is the scale at which QCD 6 , the interaction between the sectors becomes inefficient and they evolve independently. 0 d in thermal equilibrium just above Higgs with a loopmanner, of and SM the two particles,involving Higgs and a connect photon a to and mirror two twosymmetrized photon mirror Higgs two is vanishes Higgs Higgs due via connected have to vanishing Higgs with the angular mixing. two Bose momentum, mirror statistic and We of find Higgs vanishing the correlators that in Higgs. with the a a Intuitively sub-diagram similar photon. speaking, the Λ case with all generated by integrating out the lightest mirror quark, of mass to the critical temperature contribution to plasma is distributed solely toin the the mirror next particles, sub-section which we(1–2.8) leads seek GeV, regions to as of too shown parameter large in space ∆ figure where previous section resulting in large QCD on ∆ temperature dependence of this zeroon flavor the QCD, lattice [ its large perturbative value of 16 only very gradually, and then sharply drops very close but QCD between QCD 5.1 QCD If some particles interact with each other and haveT the same temperature. Below someHeavy temperature, mirror particles eventuallywhich decay are or observed annihilate as into darkstraints mirror requires radiation photons estimated and in neutrinos, eq. ( In this section, weof discuss ( the thermal historyfocus of on the the amount Minimal oftheory Mirror dark below Twin Λ radiation. Higgs Note theory thatthe radiative early universe, corrections at to temperatures larger than several GeV, the Standard Model and mirror 5 Thermal history with Higgs exchange JHEP11(2016)172 (5.5) (5.6) (5.7) (5.3) (5.4) in the 0 , f 4  and standard spectrum, the /v ) less than the 3 0 0 0 0 ; for one lepton v q f γ f . 0 0 = 1-4, there must  γ 2 f 4 p 4 h + 2 cm  are indeed in thermal → / N m p 3 0 0 c 0 ) 0 S T f 2 f 4 h 0 q f m m m m 0 m . is given by 3 f + 20 GeV f −
) 0 0 f m f f  m 2 f f 9 m , m 0 m /m  2 ( 0 f and Standard Model particles can G  f . Mirror fermions 0 2 0 m 0 d 0  f G m 2 γ 0 T 0 is the momentum of the fermion in the √ Λ ). f D 0 v 2 + 0 vm 2 GeV γ + 0 v 0 ). Cases with Γ( cm 0 0 σν f  c vm  γ p † f 2 T  ν into a pair of (  m 0 /m 2 0 → 0 g f f  0 2 Z f 2 + v computed below, and hence is excluded by the GeV – 13 – f m S m α m f v d 0 s  m 20 2 q via T m α −  0 ( π m 2 + 1 ¯ ν 8 f T 0 10 π 4 ν 3 4 N in the center of mass frame. In the thermal bath, it is ∼ L ∼ is the multiplicity of the Dirac fermion ) = 0 = 0 0 = f 4 Z f v 9 must be discarded as they give too much dark radiation. ff 2 cm ) 0 N m ¯ p 0 0 f c Λ 4 q is heavy and its number density is much smaller than that of f → T 2. 0 shows that, in all the remaining cases of m 0 f 2 2 3 → 3 T/ ff 0 0 α π ( ¯ 2 f s f 0 64 α , and contribute 0.6 to , thermal equilibrium between m 0 f σv 0 by Higgs exchange is given by ( 3 ) is fast enough to ensure that the glueballs at γ γ 0 σ f ) = ' 0 5.2 . Figure 0 ¯ ν 0 2 f eff = 1(3). ν p interact with each other by the exchange of the standard model Higgs. Since and N f lighter than 22 GeV. Over much of the parameter space of the is the momentum of f N f → 0 0 0 q f S p Let us first estimate the interaction rate assuming that the dynamics of Mirror glueballs also decay to Γ( interact with 0 Here as large as (quark), The annihilation cross section of a pair of where we assume a non-relativisticcenter limit. of Here mass frame. In the thermal bath, it has a typical size relativistic particles. We discuss the thermal equilibrium of mirrorthermal neutrinos bath later. is described byleptons. that We of comment free on fermions.between the This case is with certainly mirror correct for quarks the later. mirror The scattering cross section 5.2 Decoupling temperature Now, let us estimatefermions the decoupling temperature f be maintained, even if which is negligible in comparison with Γ( giving a decay rate limit on ∆ be one decay rate of ( equilibrium with Hubble expansion rate at mirror glueballs decay late, well after JHEP11(2016)172 ]. in 0 24 (5.8) . We 0 ff ff → is smaller 0 0 → 0 as estimated E ff /ρ 0 c ) t ∆ ff T 3 mirror quarks d / 0 5 and Λ = 3 TeV, × / 0 f , ) ρ 0 N = 3 ff in the thermal bath at . /v → 0 m 0 when f 0 v , so that the energy of the 0 c 0 . c 0 T ff T f ( with various colors for different m 0 σv 2 T < m f are depicted by dotted lines. The

m × 0 , below which (d of the mirror QCD interaction. As c ) d ¯ ,T , for f 0 T D f f T > T B m ∼ → d 0 T ¯ F E f , is transferred into Standard Model particles is smaller than 0 0 n 0 f ρ assuming that the dynamics of mirror fermions d f ( 75 is saturated. (For – 14 – T . d N 0 T σv 2 ) )) is more important than the scattering 93, whose corresponding invisible branching ratio is µ > ¯ f . ,T 2 with the same mass and determine their mass so that ,T 0 = 3-5, 0 which yield f / f f h 0 m m f /v → ( 0 ( m 3, the deviation of the Higgs signal-strength from unity as 0 is a typical energy transfer by the scattering v are determined in the following way. We first put as many m F F ¯ µ < 3, and hence the second step is not necessary.) On the right f 0 ∼ n 0 c n / T 0 0 f , with their color chosen to match those of the corresponding f T f 0 f ' /v f can be probed by the Higgs-signal strength as well as the invisible N N 0 N 0 c v m ) above E 4 ( f ) is saturated. We then determine the masses of remaining quarks − + 15. The black and red dashed lines show the maximal X f , N < T 4.7 X + d 12 = 5 T , ]. For = by Higgs exchange. This is because the energy transfer per annihilation is 9 0 + 3) so that the bound , we show the predictions for the invisible branching ratio of the Standard , solid curves show the temperature , and ∆ , d 23 ρ ) is the number density of a Weyl fermion of mass / N 0 6 T . Other dashed lines shows the maximal possible 5 4 t T saturate the current bound. For a mirror fermion mass in the range between f , d d 3 0 N , 4.3 f m, T 2 ( − , F + n and is larger than that per scattering, ∆ N In the above analysis we calculated In figure In figure The energy density of mirror particles, = 1 0 f 0 − f m is described bytemperature that is of smaller free thanthe fermions. the temperature binding drops This below energy namely the is mirror quarkonia. binding not energy, This correct effect some is mirror for expected quarks to mirror enhance form the quarks bound energy transfer when states, rate the by the ratio of 10% [ well as non-zero invisible branchingof ration the may LHC. be The detected ILC inThe is the sensitive region high-luminosity to with running an invisible branchingdecay fraction of of the sub-percent Higgs level [ at the high-luminosity LHC and the ILC. Model Higgs. Theshaded ranges region of is excludedluminosity by LHC the can measurementshown probe of by a the dashed Higgs-signal line strength. in the The right high- panel. It is also sensitive to the invisible branching (5 we find that end of each dashed line,fermions the invisible decay width ofabound the 5 Standard Model and Higgs 20-30 GeVmirror into mirror gluon for plasma is efficiently transferred into the Standard Model particles. in section have a common mass solid lines. Thequarks maximal as possible ( the bound in eq. ( determining 2 than the expansion rate of theN universe, as a function of Here temperature find that the annihilation at a rate JHEP11(2016)172 30 30 (5.9) (5.10) (5.11) 25 25 v'/v=5 Λ=3 TeV HL-LHC . The ratio T 20 20 ) is the number , / GeV 0 , q f' / GeV 3 15 f'
m m, T m 15 0 3 ( 2 m α B . 0 × n 2 f 10 / ) 5 ¯ m f 10  . In the parameter space we f B π v'/v=5 1 5 , and T 4 0 → Λ=3 TeV F,B E q 0 n 5  q × m η

0 3

∼ α 0.08 0.06 0.04 0.02 0.00

) → ( inv h Br 0 q

2

)Γ( 4.0 3.5 3.0 2.5 2.0

/ d m / GeV T 5 ' ,T 0  0 f q 2 p 0 3 | 2 3 6 9 m α 2 3 6 9 v 12 15 12 15 0 =1 =1 ) T f' f' q ¯ (2 f N N B m f – 15 – n  , between the two sectors as a function of the mirror is comparable to the temperature, and hence the d → f T 0 ∼ B in the thermal bath with a temperature X ¯ q 0 0 E q q m ( N σ degenerate states labelled by color. The dashed lines show the v'/v=3 free v'/v=3 ∼ | 0 Λ=3 TeV ∼ t Λ=3 TeV f 15 15 quarkonia d ) | N ¯ / t f 0 are depicted by solid (dotted) lines. d f ρ 0 / c 0 d T ρ ) → d 0 > / GeV / GeV ¯ annihilation q ( | 0 f' 0 f' 10 q 10 m ρ < m t d Γ( d , for various d 0 T f m is the number of mirror quarks with a mass 5 5 0 . The decoupling temperature, . Predictions for the invisible branching ratio of the Standard Model Higgs. The ranges q μ<0.75 N that yield 0 f

0.05 0.00 0.20 0.15 0.10 1.5 3.0 2.5 2.0

m d ) → ( / inv h Br GeV T Here we have used thehave non-relativistic discussed, approximation the for binding energy where density of a real scalar withof a the mass energy transfer rateinside by quarkonia the is annihilation of free fermions to that by the annihilation where the second factorby is annihilation the is inverse given volume by of a quarkonium. The energy transfer rate Figure 5 of annihilation of mirror fermions. The annihilation rate of mirror fermions inside quarkonia is maximal possible mirror QCD phase transition temperature. Figure 4 fermion mass, JHEP11(2016)172 , . d 0 d in T ¯ ν T 0 d ν T (5.15) (5.16) (5.12) (5.14) (5.13) ↔ . Chem- 0 d ¯ f 0 T f are depicted 10 and 0 c / , 0 f 3 / > T 4 , m , is smaller. Lines in 0 d  3 2 0 4 Z T / ) 4 T d significantly. But we . m s,f 0 0 T ! g 10. Kinetic equilibrium 80 4 2 f ( 4 Z )  / g 0 d 1 4 m m f T .  , we show the prediction for 0 ( 2 m + f 0
f 6 28) GeV. × 0 g 2 25 N f
. − 0 29 7 Q f × . . Taking account mirror photons 2 w 0 2 c Q 20. Comparing s 25 + ) . / 2 2 w 65. = 0 0 7 . s f ,T 3 0 < T 2 − 0

/ . In figure 0 f m 4 0 d . Regions that give 3 f − < 0 , with a cross section c / f T 3 0  m 0 4 I ( ) f ν eff 0 3 d  F 75 I f . < T ) T N n 4 w ( d π d 10 g T is well approximated by that of the ideal gas of  × → /c 80 T – 16 – ( 4 0 2 64 0  v g 4 w g is ν ) π 0 0 s,f  × /c ¯ ν f g = 4 0 2 32 eff is achieved for a wider parameter region.  × g ν v ), we find that chemical equilibrium as well as kinetic N 3 0 0 c ) 0 ν 29 = → ¯ × . ν ( that give 0 0 v n , some mirror fermions must have masses not far above < T 0 7 4 ν ¯ 0 ) c f 0 f 0 d = 0 T × ν f 0 T m 0 → ( f f 2 + 0 H 0 σ . Here we have neglected the contribution from the mirror gluon ¯ ν f 0  0 0 f → f ,γ × 0 ( m ν eff σ 4 7 0 N f ( = ∆ σ 0 ν smaller than 0 d ,γ T eff N is the effective d.o.f. of mirror fermions ∆ 0 f , thermal equilibrium of mirror neutrinos is also maintained down to temperature g as a function of 4 To summarize: for the amount of dark radiation to be below the experimental bound, To keep Here we show that mirror neutrinos can be in thermal equilibrium down to eff . This is correct for mirror leptons; for mirror quarks, the actual N 0 f the left panel are terminatedamount if of the the Higgs coupling-strength dark falls radiation below can 0.75. be The consistent predicted withthere the must experimental be bound. a mirror fermion with mass in the range (4 where ∆ plasma, which is correct for by dotted lines. We also assume that which is consistent with the upper bound, ∆ and so they also contribute to dark radiation, giving Standard Model particles downand to neutrinos, temperature the prediction for ∆ and is effective downfigure to a temperature of about 5.3 Dark radiation As we have shown, mirror photons and mirror neutrinos can be in thermal equilibrium with Comparing this rate with equilibrium are maintained downalone to is a maintained temperature by of the about scattering The number of mirror neutrinos produced/annihilated per unit volume and time is given by note that it is possible that ical equilibrium of mirror neutrinoswith is cross maintained section by the annihilation process formation of the bound state does not change the result in figure JHEP11(2016)172 , ν m 20 2 ) (5.17) (5.19) (5.18) v'/v=5 /v 0 v = ( . 15 0 ν ν can be larger or ν m m , for various multi- 0 m / GeV eff f f' ) X m ν m . 3 X . ν m 10 2 n ' 4 P > / ! 3 4 ! /  2 3  eff  0 5 v N v eff 3, and the experimental constraint ∆ .  N 0 4 7

∆

/ 29 0.7 0.6 0.5 0.4 0.3

are depicted by solid (dotted) lines.

eff 3 Δ N > 0 7 c  29  T eff )  = eff > 2 3 6 9 N 12 15 ν ( =1 N f' n N 1 + ∆ < , ∆ ×

d 7 , both parameters may play a comparably im- – 17 – 6 29 0 T ) ν × ν d eff  0 ) r n n T  ν g 0 ( 14 ν 0 ν g m 1 + m m

× P v'/v=3 ) × X that yield 12 X d 75 as a function of the mirror fermion mass  0 .  T f + ν ( = eff 10 g m ν m N , the dimension 5 operators of the Minimal Mirror Twin Higgs m eff = 10 . Such masses can have significant effects on structure formation  0 than on ( 4 number density is ν X ν ν 0 / GeV  X , so that Higgs exchange is the unique interaction coupling the two m n ν m 5 eff f' − ≡ = 8 = m N 0 symmetry is also broken in the Dirac mass term of neutrinos. X 10 eff ν  2  m ν Z ) result in neutrino masses that are either Majorana, with  < 6 have a small effect on structure formation and the CMB spectrum. m . Thermal equilibrium between the two sectors is maintained by the exchange of the 0 0 4.1 f ν . Prediction for ∆ X N m 3, to obtain the last inequality. Although the current cosmological data are more  4 For the case of Dirac neutrinos, the effective total mass of neutrinos is For the case of Majorana neutrinos, the effective total mass of light neutrinos, con-

0.3 0.7 0.6 0.5 0.4

eff Δ N /v > 0 so that 6 Thermal history includingIn kinetic the mixing last section,theory we with discussed the thermal history of the Minimal Mirror Twin Higgs smaller if the Here we have used thev prediction of figure constraining on ∆ portant role in observations in the near future. We note that ( strained by data on structure formation, is or Dirac with in the universe asModel well particles, as the the CMB spectrum. Taking into account dilution by Standard 5.4 Cosmological signalsAs of discussed mirror in neutrino section masses theory of ( Figure 6 plicities, SM-like Higgs. The ranges of JHEP11(2016)172 . Y kin , . If 0 0 (6.2) (6.3) (6.4) (6.1) γ 0 f , which 0 d,ν , so that , f T 0 is given by, TH ≡ d,γ γ 0 T f 20 / 0 . The scattering 0 f ↔ f 0 m . γ 0 0 ). As we will see, the 2 4 Z f T 4.1 m . 2  , 3 0 2 / 4 w 4 , c α d,ν 0  2 f T 2 f . In this basis, Standard Model q 1 /v m 3 π 0 A 20. 0 3 v 4 f / . 16 0 + q 2  term of eq. ( f 0 2 −  3 ' α A m / 10 2 0 µν boson and the Standard Model photon, 2 v 0  ) 0 − & B f 0 → π 3 0  ¯ ν . Kinetic equilibrium of mirror neutrinos is 0 Z 8 0 is large enough, kinetic as well as chemical m µν 2 A ν 0 d,γ − che  = , ¯ ν T 0 0  B , from the UV completion above Λ 10 γ ↔ v 20 + 2ln ν 0 – 18 –  ) ¯  f γ d,ν 0 ' f ↔ T ( 0 f is the electromagnetic charge of 0 σ ¯ f 0 ≡ d,γ 0 ↔ f , so that T ' f q 3 GeV 0 2 . 10 γ v 0 − 0 / ) 0 0 f f 10 ' ( 0 σ fν m , and . 0 f d,ν , which is efficient down to a temperature of  → 0 T 0 m ν 0 f fν  ( σ → T 0 ν 0 f gauge bosons, described by the 0 Y must be larger than the QCD phase transition temperature, otherwise the mirror Mirror photons are in the thermal equilibrium with mirror charged fermons U(1) kinetic mixing also mixes the mirror Mirror photons are in thermal equilibrium with mirror charged fermions 0 d,ν the scattering rate ofequilibrium the of process mirror neutrinos isdown maintained. to As the we temperature have of seen,maintained this by interaction is effective T neutrino abundance exceedsbound the on upper kinetic mixing, bound on dark radiation. This gives an upper This scattering becomes ineffective below the temperature allowing mirror neutrinos to interact with Standard Model fermions with a cross section, where we assume rate is smaller than the expansion rate of the universe below a temperature photons and mirror photons. also interact with photons andStandard Model through particles mixing, and maintaining mirror thermal photons. equilibrium The between cross section for 6.1 Decoupling temperature Here we list the decouplinglibrium temperatures of of mirror various processes photonsphoton that and/or is maintain shifted neutrinos. thermal to equi- eliminatecharged We kinetic take particles mixing, a interact field only basis with such photons, that the while mirror mirror particles interact with both In this section wethe allow Standard larger Model valuesand of and U(1) mirror sectorsallowed also range of interact mirror fermions by masses kinetic are wider mixing than the between case without U(1) the kinetic mixing. sectors, and found that light mirror fermion must be in the range of about (4-20) GeV. JHEP11(2016)172 .  for . In ) (6.5) 0 0 ) with is well d,γ d,γ 4.3 T T eff ( 5.15 N g ) + 2 = − d che , ) T 0 is determined by che d,ν , 0 T ( eff , they can be lighter d,ν 0 g c N T T ) with ( g . This requires a charged 94, so that ∆ 0 c . 2 is given by eq. ( 0.010 0 T 5.15 (3) d,ν' 6 eff ≥ =20 GeV T ζ π q' N E 7 m 4860 ' c − T 0.001 , we show the decoupling temperatures 1  7 3 ϵ d,γ' / T 4  ) 0 . In region D, ∆ -4 0 – 19 – 75 ) must be smaller than d,γ . 10 d,γ . In figure T . We also show the mirror QCD phase transition 0 c T ( 10 kin  is given by that in eq. ( , T g 0 γ = 0  eff = 20-50 GeV. Charged mirror fermion masses are also d d,ν T N 0 c ABCD T d,ν'γ',kin + 3 d,ν'γ',che ,T , giving -5 0 T T is too large unless charged mirror leptons have masses larger 3 ν for a representative point of parameter space, illustrating the / che 10 T 4 , eff 0 ) with eff  N

) d,ν

N 0

3.0 2.5 2.0 1.5 1.0 0.5 0.0

T / GeV T 75 5.15 d,γ . ≡ T or max( ( 10 ) 0 0 g γ T  d,ν 2 ,T × che , 0 4 7 γ 0 . Decoupling temperatures of various process when the lightest mirror charged fermion is ' d,ν T = 3 and Λ = 3 TeV. In region A, kinetic mixing is insufficient to transfer the energy eff Here we estimate ∆ ( N /v 0 ∆ The last factor indensity the of second mirror term accounts neutrinos.approximated for However, by the eq. we conservation ( find of this the factor comoving is number of the mirror gluonHiggs plasma exchange. into In Standardregion Model region C, the particles, B number ∆ and densitymin of ∆ mirror neutrinos per comoving volume is conserved below that the lightest mirror charged fermiondecay is into a mirror quark photons of just massof below 20 GeV, various so processes that mirror astemperature, glueballs a which function we of v assume to be the maximal one we estimated in section than 2 GeV. Since mirrorthan quarks 2 are GeV bound as tofermion long form masses as mesons is the below wider meson than the masses case are without above theimportance 2 kinetic GeV. of mixing. a The variety allowed of range reactions of between mirror mirror and QCD phase transitions. Suppose fermion masses. In orderModel particles, for the energymirror of mirror fermion gluons lighter to thanbounded be 20 from transferred below. into Even Standard QCD if phase mirror transition, and ∆ Standard Model sectors decouple just before the Figure 7 a quark of mass 20 GeV. 6.2 Dark radiation From the above consideration on decoupling temperatures, we obtain a bound on mirror JHEP11(2016)172 0 , ]. ν eff 30 [ ] for N (6.6) 3 and , while 34 − 0 – d che 10 , , symmetry 32 0 . 0 − γ 2 u 3 0 2 / Z is larger than 4 − d,ν  T eff ) 0 ] (see [ N = 10 as possible compo- 75 d,ν . 31  0 T e ( 10 g  , it decays into can be understood to carry 0 , number changing processes e 0 0 + 3 f d m q to be within the allowed range. 3 , / 0 4 u eff and electron  . N 0 (1-30) GeV. 2 d 25 . are both stable and have separately O − 25 0 . 10 d (1-10) GeV, the most stringent constraint ) + 5 ' , if a charged mirror lepton is light enough.  O 0 ) + 5 c kin 0 and , 0 T is about 0.3. In region E, ∆ = γ 0 for 0 d,ν – 20 – 0 u T f eff ( d,ν eff , down quark g 0 T N m ], which is much weaker than the bound from ∆ ( N u g 29 ,  , giving 3 0 28 / 4 d,ν  T ) 0 75 d,ν . . Below we consider several interesting possibilities. T 0 ( 10 larger than the dynamical scale of mirror QCD, so that the masses . In the range e 0 g 0 f : mirror neutron d  0 q , but decay later. This is excluded if kinetic mixing is absent, because d 0 and c 2 , 0 T and 0 × d 0 u , 7 4 u 0 u symmetric, and the dark matter mass is = are close to each other. With kinetic mixing, the energy of mirror photons can exchanges are absent, so that 2 . In region E, mirror photons decouple from the thermal bath at sufficiently heavy, as the temperature drops below 0 d eff . A proposed search at the LHC can search down to mixings of 0 Z 0 2 T e N d,ν − W T ∆ We consider the mirror up quark If the lightest mirror quark is heavier than 20 GeV, mirror gluons do not decay im- 10 and = 0 . c d 7.1.1 Light Taking and its abundance becomes negligible.from For the remaining of mirror baryons mainly originate from mirror quark7.1 masses. Dark matterWe candidates consider dark matter candidates in Minimalis Mirror Twin broken Higgs, solely where the spectrum by of Yukawa couplings. A dark matter candidate depends on the mass understood if the sectorsclose which to generate the Standard Model and mirrornents asymmetry of are dark matter.the To masses simplify of the discussion, and motivated by section IVC, we take The mirror sector, like thebaryons SM, possesses and accidental baryon leptons andearlier may lepton work symmetries. account on Mirror for astrophysical considerationsnon-zero dark of mirror mirror matter matter-antimatter baryons asymmetry in and with the leptons.).dark the asymmetric matter. We component universe assume comprising The a [ proximity of the energy densities of baryons and dark matter could be comes from collider experiments [  7 Mirror baryon dark matter sufficiently light mirror charged leptons can suppress ∆ 6.3 Milli-charged particle With U(1) kinetic mixing, mirrorSM fermions electric of mirror charge charge 0.3, and can saturate the bound on ∆ mediately below T be transferred into SMAs particles well long below as mirror glueballs decay into mirror photons before the QCD phase transition, mirror neutrinos decouple at We find that in region B, C, and D, ∆ T JHEP11(2016)172 0 2 / B 0 uud (7.1) , the B 0 d scatter ] or the 0 m B is spin-3 36 , 9 GeV. The . Below the by 2 0 , 35 > 0 u ddd 4 n B 0 2 udd D R B m + 1) = 2 16 R 0 ] ( d n 3 38  eventually “recombine” into . For example, the scattering 10 GeV. The upper bound on 0 D , e ¯ d 0 udd m ∼ 0 ) u . 0 B ) ] requires that 10 GeV 0 γ D γ  39 m ,M (+ are combined into mirror baryons × (+ g [ 0 as the mirror baryon 0 and disappears from the thermal bath. ddd / d g is the mass of the dark atom. Here we 0 2 udd 0 / 0 udd 2 ν B 0 udd D ,B 2 B and B m = 1, which we assume in the following. The and positrons ¯ . To preserve charge neutrality, there are the 0 is larger than the mass of 0 udd 10 cm 6 cm → and → u is captured by mirror baryons, e.g. . R 0 0 ddd ¯ 0 – 21 – ddd < d e 0 ddd ,B 0 B u 0 0 ddd ddd B , = 6 D 0 ) and B B B M d 0 uud 3 D 0 ddd 1 + + B m ¯ σ/m d and ,B 0 u 4 ¯ ]. In our case, mirror pions are also heavy due to large 0 d uud /m 0 u 0 M 0 uuu B 2 e 37 M + 1) B R , m R 0 ( combines into e from the thermal bath. Note that the sum of the masses of 0 2 d is now stable, as its decay is kinematically forbidden. After the /m α 0 100 ddd 0 e : mirror atom B 0 ddd 0 ' B e . The mirror baryons 0 D 1 and the kinetic energy of the dark atom is much smaller than the binding m , e 0 and . The stable hadrons are d 0 ∼ σ/m 0 uud R M disappear from the thermal bath by this capture process. Finally, B ¯ d is larger than twice of the mass of 0 = max( u ), and does not affect structure formation. The cross section can be enhanced M R phase transition, 0 ddd 0 2 udd 0 is sufficiently heavy, it decays to − B B The self scattering cross section of mirror atoms is given by [ Without any IR effects, the scattering cross section between mirror neutrons is 0 m u ( cored dark matter halo profilesthe could cross be section in explained dwarf by constraint galaxies, from galaxy clusterslarge is that weak, its since kinetic thecross energy velocity section is dispersion is comparable of suppressed. to dark the matter binding is energy, so and the self interaction where assume that energy. The cross section is minimized for QCD same number of ¯ mirror atoms. mirror quark masses, and those effects are expected7.1.2 to be suppressed. Light If The mirror electron and has a contribution to its mass from aO spin-spin interaction. up to the unitarityexistence of limit resonance by states some [ IR effects, e.g. the Sommerfeld effect [ with each other and almost all becomeprocess the mirror neutron eliminates and As the sum ofmeson the masses of mirror QCD phase transition temperature, and mesons with an obvious notation. The meson conserved comoving numbers. Mirror charge neutrality implies JHEP11(2016)172 , for (7.3) (7.2) 0 e ]. , where 0 44 m are stable, and mirror 8 0 . d 0 ddd T , 0 B u The upper bound , 0 7 e and ]. 42 , ). The constraints from 0 uuu 0 , the ionized components e B 41 6 + ¯ . 2 , ) . 0 ddd 0 T GeV is the energy density of dark R ( remove B 9 + 1) 4 result in the removal of the 0 , e − H 0 R B . (Other mass spectra belong to one ( 0 10 0 ddd ] which, together with the limit from d W 2 ) = ×  43 6 T ,B . ( mediated interaction may freeze out before 3 > m s 0 D 0 0 01 [ udd . ' u /s 0 0 W m m m ,B /s 10 GeV – 22 – DM < are stable and eventually form a mirror atom, +  ρ 0 0 0 uud DM e e × ρ × m atom 0 ,B 2 02 , 0 Ω . 4 W 0 / and 0 u m 0 uuu m ' B 4 2 π ] and the scattering of clusters. The estimation of these and 0 g are all light, with a spectrum such that 0 8 uuu e > m ionized 0 x B 40 0 e d , 0 m : mirror neutron 0 d e . Scattering among the various + , 0 0 0 , : mirror atom ] we obtain the ionized fraction, d e u 0 0 0 all light, reactions mediated by ν d e 40 m 0 , , e , 0 0 0 → e u u 0 u 0 e ) with a binding energy larger than that of ( , and 0 > m 0 ) is the entropy density, and e 0 d d T , ( 0 m s + 2 u is sufficiently heavy, In the expanding universe, however, the With the sizable kinetic mixing that we considered in section The recombination of mirror baryons and electrons is incomplete. From the numerical + 0 Mirror baryons and electrons interact with SM particlesThe via galactic kinetic magnetic mixing field and prevents are the heated, ionized which component may from entering the disk, and ionized compo- is the mass scale of mirror electrons and mirror protons, the freeze-out temperature is 0 0 d uuu 7 8 0 u B change the ionized fraction. nents initially in the diskFurthermore, are the likely to magnetic escape field theHence, disk the of owing ionized to the component Fermi Earth acceleration may by prevents evade Super-Nova direct any remnants. detection ionized experiments component performed from on the reaching Earth the [ Earth. given by solving where matter divided by the entropy density. The first factor in the left-hand side is the cross example by charge neutrality implies that only the mirror neutron remains. mirror protons and mirror electronsm are removed from the thermal bath. At m of the fore-mentioned three cases.) Then the following mirror particles are stable, With precise determination of the constraints is beyond the7.1.4 scope of this paper. Light Let us assume that 7.1.3 Light If ( self interactions and the ionized components of the atomic dark matter are weakened. The and may affect halo shapeother [ constraints on the ionized fraction is beyond theparticipate scope in acoustic of oscillations this and affect paper. on the CMB the spectrum [ ionized fractionself is scattering Ω in dwarfs, disfavors atomic dark matter for such kinetic mixing. estimation in ref. [ The ionized components interact with each other and with atomic dark matter strongly, JHEP11(2016)172   (7.5) (7.6) (7.7) (7.8) (7.9) (7.4) aµν 7.1.2 G 10 GeV. µν a , through & G N 0  , i ) m q N | m π ¯ q ( , 3 q q α ],   , mirror protons and m 0 n | 4 49 11 aµν m N h G . − 2 1 + a µν . 0 e ].  ,  4 u,d,s G 0 X . m = 2 N  0 52 π π 0 3 3 N q , 2 N m 9 α N α + 12 m /v 12 51 3 + 2 0 ], 0 p m m v i + × × 50 N m  c,b,t N 5 1 N | = 2 . X . m = q 0 i m 0 0 aµν + 3 +  m N through Higgs exchange is given by G | ¯ ¯ q q i ' 0 2 N 5 GeV is given by q q ¯ h 4 f a µν N q q 0 N 0 m | m × G 0 f ¯ q m m | 0 4 N – 23 – 0 2 N q f v q N and h m m is mainly given by the masses of mirror fermions. 0 | m u,d,s u,d,s 3 0 | , interact with Standard Model nucleons, X X 0 π 0 = = α N 8 q q 1 GeV N N π 9 028 N N h . h 0 − − 0 ' − N 0 0 0 ¯ ¯ ∈ f f 0 = u,d,s = X 0 0 X f ,W 0 = i f f q 0 0 fo f f N T | NN µ m m µ σ 0 0 T | N N ∈ ∈ 0 0 N X X h f f 2 2 =   ]. The interaction of the Standard Model Higgs relevant for the direct mediated process, and the second factor is a rough estimate on the number 0 0 0 v v 2 N v v 47   m – W 2 − − 45     10 GeV, a non-negligible amount of mirror electrons and protons remain. Some , these states decay into mirror neutrons and mirror neutrinos through mirror v v . 2 2 ]. The relevant matrix elements of h h 0 √ √ 48 m 7.1.3 This matrix element, obtained from a lattice calculation, is consistent with the one extracted from = = 9 L the scattering cross section between hadron scattering data with the aid of chiral perturbation theory [ and matrix elements derived by a lattice calculation [ Here we assume thatUsing the the trace mass anomaly of formula for the Standard Model nucleon [ where we have takenons into [ accout the one-loop QCD correction to the coupling with glu- detection is given by 7.2 Direct detectionThe of above dark dark matter matter candidates, the exchange of theperiments Standard [ Model Higgs, and may be observed in direct detection ex- For of them laterand recombine into mirrorelectron atomic capture. states. Thusmatter. Unlike However, there there the is is cases a no constraint of constraint on sections the from ionized the component. self-interaction of atomic dark density of mirror protons and electrons. The freeze-out temperature is then given by If the freeze-out temperaturemirror is electrons much disappear smaller from than the thermal bath. This is the case for section of the JHEP11(2016)172 , = N eff 0 for = 3 N N (7.10) 6 /v m 0 may be v 0 N for ] by dashed m ] and Super- 4 57 − can be weaker. 59 0 -10 N level for 3 is estimated by the m − σ eff 10 . The regions depicted 120 N 0 N LZ . LUX m  level, if the mirror neutron and the lines in figure PandaX-II XENON1T σ 100 0 and a Standard Model nucleon neutrino floor d 0 N 80 . ], and the neutrino floor [ 0 without kinetic mixing. ] by solid lines. The higher mass region, and two N 56 at the 1(2) 1 / GeV as a function of 0 eff 54 m 60 requires the mirror neutron to be heavier 0 N' u eff N (1-30) GeV can explain the proximity of the e m eff N = 3 (5). We also show the sensitivity of the NN O σ ∼ – 24 – N 0 v'/v=3 v'/v=5 /v 40 = 0 N 0 v ], we obtain the bound µ is bounded from below. Let us consider the mirror N 0 58 m N level and 36 GeV (51 GeV) at the 1 20 m σ ], the LZ experiment [ -49 -45 -46 -47 -48

55

10 10 10 10 10

NN' / σ cm 2 ] will probe smaller kinetic mixing. 60 ] and the Panda-II experiment [ , we show the prediction on 8 53 (10) GeV is an interesting region. Note that, however, ∆ . The scattering cross section between dark matter 40 (110) GeV is excluded for O > = = 6 show that the constraint on ∆ Once kinetic mixing between hypercharge gauge bosons is introduced, In figure If kinetic mixing is absent, 0 0 0 N N f (1-10) GeV. Projected low-threshold experiments such as CRESST III [ Translating the bound derivedO in [ CDMS SNOLAB [ smaller and hence direct detectioncase, however, via direct Higgs detection exchange via maymatter the be kinetix interacts difficult mixing through to is its possible. observe.neutron magnetic In is The mirror this expected dipole neutron to moment. dark be of The order magnetic moment of mirror experiment [ m XENON1T experiment [ lines. We concluderegion that consistent dark with matter the can upper be bound on detected ∆ by near future experiments in the m ideal gas approximation of mirror quarks. The actual lowerby bound thin on (dotted) linesis are dark disfavored matter by and ∆ kinetic mixing is absent. We show the constraints from the LUX of mirror recombination.N It is madethan of 19 GeV one (26 GeV)(5). at the The 2 mirrorenergy neutron densities mass of of baryons and dark matter. Combined with the constraint from ∆ as a function ofby the mirror mass fermion of masses, andfermion dark the mass matter. matrix terms. element Here of we the assume trace that anomaly the is mass saturated of by the dark mirror matter isneutron, given which is free of constraints from self interactions as well as from the efficiency Figure 8 JHEP11(2016)172 0 . , . 0 3 3 L ν / / 4 4 − (8.1) (8.2) 0 ! ! (7.11) ) ) B d d (1) GeV) T T ( ( O 0 0 f f 4 2 g g = gauge field is 0 0 , N 4 + 2 + 2 L ) is excluded by 5 m 

− − 3 3 0 / / 0 4 4 10 B N   ) ) m d d 10 GeV T T  > . 80 80 ( ( ( 0  g g L 0 2 c    , or could be heavier than these − 0 5 0 e − gauge field. As we will see in the  < T 17 08 B . . 0 10 d L  and T vevs = 0 0 4 = 0 − 3 d 0  L / 3 , 4 0 / B 4 u ! − gauge field is absent, or the gauged α bind ) becomes ) ! 0 d α ) B L d T  ( – 25 – T 5.16 0 f − 2 ( 4 g 0 0 f 2 gauge field may alter dark matter phenomenology. g B cm gauge field is beneficial in identifying mirror baryons 0 0 ] 4 + L 34 L 2 + in eq. ( 61 −

− −

3 0 0 symmetry breaking beyond Minimal Mirror Twin Higgs. 10 / eff 3 4 / B B ]). 2 N 4 '  Z ) 62  0 17. 08. d ) . . 2 2 N d 75 T symmetry is also spontaneously broken by a large VEV breaking  75 breaking from . ( . T 2 m s 2 ( 2 ∗ 10 10 g Z g πα Z 4 bind  4  α 2 1. A kinetic mixing that allows 4 ' × × ' σ 7 4 4 7 is the fine-structure constant of the binding force that forms the atom. Here R = = The prediction on ∆ 0 0 bind f f 0 0 symmetry and a small/vanishing VEV breaking α can be as small as 0 can be as small as 0 10 ,γ ,γ L Next we consider the case with a massless Mirror atoms have a large radius and interact through screened charges of mirror eff eff eff eff In the supersymmetric Twin Higgs model, this is naturally achieved by a condensation of a mirror − N N N N 10 ∆ The mirror neutrino massescould now have arise a mass from of Diraclight order Yukawa states. the couplings lighter and states Also, of the a lightest, massless right-handed sneutrino at the TeV scale. ∆ ∆ 8.2 Additional dark matter candidates ∆ next sub-section, the massless with dark matter. The prediction for the amount of the dark radiation are given by heavy. B 8.1 Dark radiation Let us first consider thesymmetry case is where broken a at an intermediate mass scale and hence the 8 Non-minimal In this section weWhile consider the SM neutrinoscan be become much very heavier if lightcan the via be mirror achieved right-handed the if neutrinos seesaw the do mechanism, not obtain mirror large neutrinos masses. This where we assume various direct detection experiments.is Especially, excluded the by light CRESST mass II region [ ( baryons and electrons.nucleus The (per scattering nucleon) is cross given section by [ between atomic dark matter and a JHEP11(2016)172 0 B . neu- (8.3) (8.4) 2 0 − L 10 − 0 . Then the & asymmetry flat direction. B 0 0 0 asymmetries L L ¯ s 0 L 0 − − ¯ 0 d 0 L . 0 B − 7.1.1 B u 2 0 α . The asymmetry n − B = and 10 0 0 n 7.1.1 & B , with abundances that , charge and 0 L ] from the ¯ e − , n 0 0 B 64 . n , 0 . numbers are separately con- α 0 ) 7.1.2 n 63 0 L 4 3 e , e ¯ − d − symmetry is only approximate, a 0 u 0 = B and ( L 0 0 e 3 4 gauge interaction: mirror atom ,M asymmetries are given by gauge symmetry: mirror neutron d − n and disappears from the thermal bath. gauge boson: mirror atom and/or 0 0 − 0 0 , 0 0 0 ddd 0 L L = , and electrons L e B u = 0 L charge neutrality, can form an atom. The 0 p − 11 − 0 udd ,B 0 0 0 − and and L B = 0, only the mirror neutron remains, whereas 0 0 ) atoms. The constraints from self interactions B 0 B udd , n 0 0 ,L 0 0 0 – 26 – is unbroken during asymmetry genesis, only the − d B B e charge is conserved, the ) L n 0 0 , 0 n L 0 ,B + ¯ L B 4 3 L u − − 0 − uud − 0 0 gauge symmetry in the setup of section ddd disappear as discussed in section B 0 ( 0 = ¯ B B d , neutrons B 1 4 ,B 0 0 L u L − M = 0 − uud 0 uuu 0 0 0 B are absent, so that B B B B n 0 and with no light with unbroken with unbroken 3 4 asymmetries. If W 0 0 0 0 − e gauge symmetry in the setup in section ν 0 ν ddd L gauge field. However, a mirror neutron and a mirror neutrino, whose 0 , , = , B 0 0 0 0 L 0 , d d L e L − and states, leaving ( 0 , , , n − 0 0 0 0 0 0 uuu 0 B is exact, so that u d u ν B = B 0 symmetry is broken at a sufficiently high scale, a non-zero B 0 0 L e L n = 0, as would occur if − − 0 0 0 asymmetry may be produced before the mirror sphaleron process freezes out. neutron and neutrino asymmetries. On the other hand, if the L B 0 B 0 L − is sufficiently heavy, it decays to L If 0 0 These cases require asymmetry generation after the mirror sphaleron freeze out, as can be achieved in − ν B 0 11 out. The resultant number densities of mirror protons, neutrons and electrons are given by the supersymmetric Twin Higgs model, with Affleck-Dine baryogenesis [ mirror sphaleron process freezes out, the Here we assume thatmodel the plus matter three content of right-handed the neutrinos mirror just sector before is the identical mirror to sphaleron the process standard freezes depend on the if mirror proton and the mirror electron remain. may be generated above the mirror electroweak phase transition temperature. After the served. The following mirror particles are stable, Among them, is stored in mirror protons 8.2.3 Light If Reactions mediated by 8.2.2 Light With unbroken trality remove and the ionized fraction of atomic dark matter are evaded if Let us assume an unbroken mirror neutron itself is nothe longer unbroken a dark matternumber candidate, densities as it are feels identicalbound a due long-range from to force self by interactions as well as the ionized fraction is evaded for must be produced afterand the mirror sphaleronB process freezes out to avoid washout8.2.1 of Light When JHEP11(2016)172 , the (9.1) (9.2) 0 f N symmetry in the 2 Z depends on 0 f m 28) GeV. The upper bound . − 2 / , 1 2 λ / 1 2 /  1  SM 10 to break the mirror ∆ TH 10  ∆  × . It contains a complete mirror sector, so that × 4 and 10 respectively. breaking arises from Yukawa couplings and the – 27 – TH 3. The allowed range for 2 necessary . 0 4 TeV Z . /v < shows that, with Higgs mixing alone, there must be 1 0 7 TeV in the range of about (4 & . v 4 0 ' f We have constructed a completely realistic effective field eff phase transition temperature, necessary for a solution of = 5 m N 0 SM Λ TH , it is non-trivial that Higgs mixing can lead to a decoupling Λ 4 . Irrespective of the mirror fermions masses, the high-luminosity running breaking Higgs mass term necessary for the Twin Higgs mechanism and 2 2 Z breaking Yukawa couplings induce a sizable invisible branching ratio of the 2 Z gives a lower bound of ∆ 0 f As illustrated in figure The Mirror Twin Higgs models, however, predict the existence of extremely light particles, breaking Yukawa couplings not only suppress dark radiation to acceptable levels, but m 2 temperature less than thethe QCD dark radiation problem.light Figure mirror fermions with mass on SM-like Higgs boson throughreduction its of the mixing Higgs with couplingfrom to the unity Standard mirror of Model Higgs. the particles, leads Higgsshown This, to in signal-strengths a figure correlated together universal with with deviation theof the masses the of LHC mirror and fermions, the as ILC can probe only communication between theand sectors kinetic is mixing from ofZ Higgs hypercharge mixing, fields, required allowed by bygenerate Twin gauge the invariance. Higgs, Furthermore,raise the the mirror baryon mass as required for realistic dark matter. Minimal mirror twin Higgs. theory of Twin Higgs belowa the UV cut completion, off Λ Minimal which Mirror we Twin did Higgs, not the study, only can restore spacetime parity symmetry. In mixing; but the larger cut off may raise themirror masses photons of and new mirror particles neutrinos,leading that above to contribute the to constraints LHC the on reach. dark aactions radiation realistic that of theory. couple the the We universe, two haveYukawa sectors, found couplings. it that, is independent of the inter- The minimal theory has nopossibility new of colored discovery states modes to at be the produced LHC, at such as the production LHC. of It the does mirror offer Higgs the via Higgs while that of the Twin Higgs theory is constraints from self interactions and ionized fraction of atomic dark9 matter are relaxed. Conclusions The Twin Higgs mechanismallows significantly for relaxes a fine-tuning larger of cut the off electroweak scale. scale, The and cut off of the Standard Model is Dark matter is composed of comparable numbers of mirror atoms and mirror neutrons. The JHEP11(2016)172 . 0 . 5 eff f 6 N N ) for the , are less . For low eff 4.5 ) 8 ν m ), ( P could be extended 4.4 breaking does not 0 2 f Z m shows that this branching 5 limit by the XENON1T (LZ) σ remains about 0.3, but the enlarged imply lower bounds on the invisible increases, which is shown in figure eff 6 eff N inv) and the mirror baryon dark matter N and → 4 h – 28 – , Γ( is large enough to affect the thermal history of the  eff N = 3, the left panel of figure from figures we showed that the upper bound on /v 0 0 6 f becomes tighter, ∆ v phase transition, the allowed range of the light mirror fermion 0 m 0 f from scattering via a dipole moment. Mirror atomic dark matter m 4 − than the observed neutrinos, leading to important effects in both CMB = 5, the invisible branching ratio is reduced; the right panel of figure 10 2 ) − /v 0 3 v /v 0 − v 10  < In the Minimal Mirror Twin Higgs the seesaw mechanism yields light neutrino masses Mirror baryons and leptons are natural candidates for dark matter. Dark matter can If the kinetic mixing parameter The lower bounds on , PandaX and LUX have recently excluded large values for the dark matter mass. In /v 0 masses of mirror neutrinos,substantially affect however, either rely the on neutrinoand Yukawa the couplings therefore or assumption our right-handed that predictions neutrinorobust masses, for the than effective the sum predictionsdirect of detection neutrino for cross masses, ∆ section. ( and LSS. The effective2.3 sum times on neutrino greater massesstandard than relevant and in for mirror cosmological the neutrinos datasterile Standard could neutrinos. is Model. lead at Alternatively to least thestates mirror Small seesaw as neutrinos mixing could right-handed being lead between neutrinos observed toto degenerate these as Dirac with only Majorana neutrinos, massive the very with observed small the left-handed effects mirror states, on leading CMB and LSS. The predictions of eqs. ( with a sizable kinetic mixing such that the thermal history isfor affected both is sectors. excluded. Theseby neutrinos a can factor be ( Majorana with those in the mirror sector heavier on dark radiation canexperiment. be Adding kinetic fully mixing explored, tois the consistent up thermal with cosmological to history, lighter the themirror dark neutron limit 1(2) masses dark on matter ∆ that masses XENON1T inbound the and (1-10) GeV LZ region, are present direct unable detection to limits probe. For be composed of mirrormatter neutrons, can mirror be atoms, or directlyv even detected a via mixture Higgsthe of absence exchange, the of two. as kinetic illustrated Such mixing, dark in the region figure of dark matter masses allowed by present limits masses is enlarged. In section as high as 50be GeV, while lighter mirror than leptonsincluding 2 GeV could kinetic be provided mixing, as the therange light lightest lower of as bound mirror light 2 on mirror GeV. meson fermion ∆ A is masses mirror is heavier quark important than for can 2 mirror GeV. dark Even matter. the LHC. For shows that in someILC. cases Essentially the the entire signal parameter is space as can small be as probed. 0.002 -two sectors 0.01, near that the could QCD be probed by As the upper bound on branching ratio of theHiggs SM-like signal-strengths. Higgs boson For andratio the is universal typically in deviation the from range unity of of 0.05-0.15 that the can be probed by high luminosity running of number of light mirror fermion states, and narrows considerably for larger values of JHEP11(2016)172 ] is 75 – eff (A.1) N 69 ; constraints gauge boson, gauge boson, is a function. ]. (See [ 0 0 f vev can lead to 0 udd L L 0 68 B , L − − 0 0 65 , − , and ) B B 0 | c 0 vev can implement the B φ | ( L f − A -SU(3) 0 c φ B gauge field. Furthermore, there ∝ . The determinants of the mass 0 0 φ L M 0 c − 0 A large B SU(3) symmetry breaking in the Yukawa couplings . Without a massless det α 2 atoms and mirror neutrons Z symmetry is broken by VEVs of FN symmetry vevs. , 0 & – 29 – ) e 2 | L atoms, and constraints from self interactions and L Z φ | − 0 0 uud ( , ruining the Peccei-Quinn solution to the strong CP ¯ e − 0 B ¯ θ f , the theory still solves the strong CP problem. Due B α ¯ θ A 0 ddd B 6= φ 0 = B . We note that this mechanism is not peculiar to Twin ¯ θ ¯ 0 θ ∝ ¯ symmetry breaking in Froggatt Nielsen sector θ or charged fermions are proportional to = 2 0 M 0 c c 0 ν Z ¯ θ 0 udd SU(3) B det and SU(3) breaking from c 2 Z ]. If the Standard Model and mirror sectors have a common Peccei-Quinn ]. Here we show that if the 68 76 – is the anomaly coefficient of (FN symmetry)-SU(3) 65 A Let us denote the FN symmetry breaking field as Is it not clear whether this idea can be incorporated into models with Mirror Twin matrix of SU(3) where Note that the phases of the mass matrices do not depend on the absolute value of the is mandatory. This may leadproblem to [ breaking fields we regain Higgs models, but is generically applicable to Mirror World scenarios. to the small Peccei-Quinn symmetryQuinn symmetry breaking can scale be and easilyfor the understood models large as an of axion accidental an mass, symmetry invisible the [ QCD Peccei- axion with anHiggs. accidental As Peccei-Quinn we symmetry.) have shown in this paper, the One of theaxion advantages [ of thesymmetry, Mirror the World axion is massThe the is constraint possibility given from by of colliderssymmetry the a breaking and mirror scale heavy astrophsyics QCD is visible canQCD dynamics around and QCD be and the QCD evaded TeV becomes are scale. even identical, heavier. if If the the Peccei-Quinn physical theta angles of mirror the Institute for Theoreticalsupported Studies by ETH National and Science at Foundation grant the PHY-1066293. Aspen Center forA Physics, which is Heavy axion and The work of L. J. HallOffice and of K. Science, Harigaya was Office supported of inand High part Energy by by the Physics, the Department under of contract National Energy, R. No. Science DE-AC02-05CH11231, Barbieri Foundation wants under toETH grants thank Zurich PHY-1316783 Foundation Dr. and for PHY-1521446. support. Max R¨ossler,the The Walter work Haefner of Foundation L. and J. the Hall was performed in part at the ionized fraction aredark weekend matter if could befrom self a interactions mixture and of the ionized fraction are relaxed. Acknowledgments seesaw mechanism for the knownlarge neutrinos, Dirac while mirror a neutrino small masses.lowered to or Without zero 0.08 light (0.17) mirrorare neutrinos, without the new (with) minimal possibilities a ∆ massless fordark mirror matter dark could matter. be If there is a massless Additional JHEP11(2016)172 eff N (A.2) (A.3) . . If the ¯ θ 100 GeV, 1) = ]. − . Phys. Rev. 0 ¯ , and non-zero a θ . (1966) 1154] f ,N 3 eff SPIRE a N ··· IN f , 1 ][ , 10 TeV (1) TeV. 2 O = 0 ! Yad. Fiz. 0 VEVs, giving charged fermions may be as heavy 0 QCD k, k c φ ( Λ / 10 10 TeV, however, the QCD axion  and 0 = 0), the determinants of the mass On the possibility of experimental & hep-ph/0506256 (1966) 837 [ 0 QCD k N φ [ a A Λ 3 f ], constraints from colliders and ∆ πi

2 (100) GeV, which is excluded by hadron 68 charged pseudo Nambu-Goldstone bosons. If the  O × c are determined. For example, suppose that exp 0 φ – 30 – 1 MeV i| × (2006) 231802 0 The twin Higgs: natural electroweak breaking from φ and ' 100 GeV. For |h 96 a φ π charged particles can be as heavy as . f f = Sov. J. Nucl. Phys. c 2 a i , are determined by explicit breaking of the continuous FN 0 f ), which permits any use, distribution and reproduction in ! 0 φ h φ Question of parity conservation in weak interactions subgroup, so that vacua are given by , ]. 0 QCD QCD  Λ Λ and N Z k particles with masses of

N φ 0 c 10 TeV or SPIRE Phys. Rev. Lett. η (10), all SU(3) CC-BY 4.0 πi , IN O 2 m & [  This article is distributed under the terms of the Creative Commons . The difference of the FN symmetry breaking scales itself does not ruin a 0 ' f φ a exp m ]. and i| × φ (1956) 254 φ 12 is an integer, the theta angles remain identical in any vacua. If not, the theta |h SPIRE IN = mirror symmetry 104 observation of mirror particles [ Assuming that mirror quark masses are larger than the dynamical scale of mirror QCD, If the FN symmetry has no color anomaly (i.e. (1) TeV. However, it is difficult to avoid light SU(3) Z. Chacko, H.-S. Goh and R. Harnik, T.D. Lee and C.-N. Yang, I. Yu. Kobzarev, L.B. Okun and I. Ya. Pomeranchuk, If the Peccei-Quinn symmetry is broken by strong dynamics, SU(3) i A/N O φ 12 [3] [1] [2] h as domain wall number is Open Access. Attribution License ( any medium, provided the original author(s) and source areReferences credited. such a possibility isas excluded well as or a not,neutrino calculation masses, a of are calculation the required, of CMB whichthere is spectrum BBN should beyond including be with the SU(3) the decaying scopecolliders. effect of QCD of this axions, paper. ∆ For Comparing this formula withare figures satisfied 1 for and 2decays of into photons [ and mirror photons afterthe BBN baryon-to-photon begins, ratio giving a during slight difference BBN between and during recombination. To conclude whether angles remain identical in specific vacua. the mass of the QCD axion is given by the phase directions of symmetry to a discrete If the axion solution to the strong CP problem. matrices have the same phasesFN symmetry for has any a phases color anomaly, ofon the the theta how angles the may or phases may not of be the identical, depending VEVs of VEVs of JHEP11(2016)172 B ]. , , JHEP JHEP , (1979) , , SPIRE (2007) IN (1977) 421 , Conf. Proc. ][ (1974) 461 , Nucl. Phys. ]. D 75 (1998) 1531 (2015) 191801 , B 67 B 72 (2009) 095012 C 7902131 114 SPIRE D 57 ]. IN ][ D 79 Phys. Rev. , Phys. Lett. SPIRE , ]. Nucl. Phys. IN arXiv:1606.02266 Conf. Proc. , Phys. Rev. [ ][ , , Phys. Rev. Phys. Rev. Lett. SPIRE results. XIII. Cosmological , , IN ]. ][ , CERN, Geneva Switzerland (2014). ]. ]. The composite twin Higgs scenario ]. Naturalness in the dark at the LHC (2016) 045 muon decays? 2015 arXiv:1502.01589 9 [ SPIRE 08 10 IN SPIRE SPIRE ]. ]. 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