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Title Dark matter, dark radiation and gravitational waves from mirror Higgs parity

Permalink https://escholarship.org/uc/item/40s904z8

Journal The Journal of High Energy Physics, 2020(2)

ISSN 1029-8479

Authors Dunsky, David HALL, Lawrence J Harigaya, Keisuke

Publication Date 2020-02-01

DOI 10.1007/JHEP02(2020)078

Peer reviewed

eScholarship.org Powered by the California Digital Library University of California JHEP02(2020)078 – 8 (10 ∼ 0 Springer v dark matter 0 e October 17, 2019 January 24, 2020 February 12, 2020 : : : Received Accepted Published c and a Standard Model neutrino σ [email protected] Dedicated to the memory of Ann Nelson. , Published for SISSA by . Remarkably, this requires https://doi.org/10.1007/JHEP02(2020)078 `H , yielding the Standard Model Higgs as a i 3 GeV at 1 → H 0  ν  h 0 and Keisuke Harigaya v a,b [email protected] = , i 0 H h dark matter abundance is determined by freeze-out followed by 0 e . 3 4. With a low reheat temperature after inflation, the . Mirror electromagnetism is unbroken and dark matter is composed . 3 ) eV, consistent with observed neutrino masses. The mirror QCD sector − 1 Lawrence J. Hall 1908.02756 − 10 03–0 The Authors. . a,b Beyond Standard Model, Cosmology of Theories beyond the SM, Higgs 0 ∼ c –10

2 ) An exact parity replicates the Standard Model giving a Mirror Standard Model, , 0 ∼ . This “Higgs Parity” and the mirror electroweak symmetry are spontaneously − . Direct detection may be possible via the kinetic mixing portal, and in unified v 0 0 [email protected] ( e eff SM N SM λ and ¯ ) GeV, predicting a Higgs mass of 123 ↔ 0 e 10 School of Natural Sciences, InstituteNew for Jersey, Advanced 08540, Study, U.S.A. Princeton, E-mail: Department of Physics, UniversityBerkeley, of California California, 94720, U.S.A. Theoretical Physics Group, Lawrence BerkeleyBerkeley, National California Laboratory, 94720, U.S.A. b c a Open Access Article funded by SCOAP Keywords: Physics ArXiv ePrint: by future observations.with Mirror ∆ glueballs decayabundance to is mirror determined photons bymixing giving freeze-in portal. dark from radiation the SM sector by either the Higgs or kinetic theories this rate isture correlated after with inflation, the the protondilution decay from rate. decays of With10 mirror a neutrinos, high reheatmass tempera- of (10 exhibits a first order phase transition producing gravitational waves that may be detected SM broken by the mirrorPseudo-Nambu-Goldstone Higgs, Boson of an approximatepling SU(4) symmetry, with aof quartic cou- Abstract: David Dunsky, Dark matter, dark radiationfrom and mirror gravitational Higgs waves parity JHEP02(2020)078 6 15 ]. 5 – 2 21 9 8 9 17 25 ] or of supersymmetry [ 13 1 3 19 28 abundances 0 13 0 u ν symmetry 8 – 1 – 2 decay Z and 0 0 ν e 23 4 4 4 0 6 v 25 1 26 6.1.1 One6.1.2 generation of long-lived Universal coupling strength of neutrino portal A.1 Freeze-out A.2 Freeze-in 6.2 Freeze-in from Higgs portal and kinetic mixing 6.1 Freeze-out and dilution from 4.1 Direct detection4.2 by nuclear recoils Correlation between proton decay and direct detection 3.1 The Lagrangian 3.2 The mirror3.3 spectrum Prediction for 3.4 Kinetic mixing this new physics could be the breaking of PQ symmetry [ sector of the Standardbeen Model no discovery (SM) of any have physicsIf been that the would pursued SM lead Effective for to Field a Theory decades,finally natural is break valid explanation but well down? of above so the the A weak far weak possible scale. scale,enters answer there at at has what has the been mass scale provided scale where by will the it the LHC: SM perhaps Higgs new quartic physics coupling passes through zero. For example, 1 Introduction At high energy colliders, precision measurements of the electroweak symmetry breaking A Boltzmann equations for the B Energy densities of the mirror QCD bath 7 Gravitational waves from the mirror8 QCD phase Conclusions transition and discussions 6 Cosmological abundance of mirror dark matter 4 Direct detection and the correlation with proton decay 5 High and low reheat scenarios; BBN and dark radiation 3 The mirror standard model Contents 1 Introduction 2 Vanishing Higgs quartic from a JHEP02(2020)078 0 ]. 13 ]. However, under which 11 ]: parity forces R , 6 6 ]. Furthermore, since the 10 – 7 ). We have recently explored 0 . The constraint from long-lived , e 4 0 , we compute the relic abundance , l 6 0 are again stable and DM candidates; 0 , d ]. There are many implementations of 0 u occurs at the scale where the SM Higgs 6 is identified as the SU(2) [ , u 0 L 0 0 q − and , a doublet under the weak SU(2), with a B 0 production mechanism must be non-thermal e H 0 . In section (1) – 2 – e . The final section is devoted to conclusions and 5 U 7 ]. This theory solves the strong CP problem, with × ]. 12 R 17 – cancelling contributions from the ordinary quarks [ introduces the mirror copy of the SM with Higgs Parity 14 quark are also stable, and since the bounds on such heavy ¯ θ dark matter that is within reach of direct detection. How- 3 SU(2) 0 0 u e . In this theory × 0 L SM ↔ SU(2) × , a doublet under some SU(2) 0 does not couple to QCD, it is much less constrained by direct detection, 0 we review how Higgs Parity predicts the vanishing Higgs quartic coupling H u 2 transition. In the case of Freeze-Out DM, once the neutrino portal parameters ), into mirror quarks and leptons, ( 0 dark matter and dark radiation. The spectrum of the GWs from the mirror QCD 0 In section This Mirror Higgs Parity theory is highly constrained: the parameters in the SM In this paper we study a complete mirror sector where Higgs Parity doubles the entire In another class of theories, Higgs parity transforms SM quarks and leptons, Another possibility for this new physics is the breaking of a discrete symmetry, “Higgs /u 0 e to vanish and the quark Yukawa matrices to be Hermitian [ q, u, d, l, e its relation to the proton decaymirror rate glueballs is discussed is in investigated section inof section phase transition is estimateddiscussions. in section are chosen to give thein observed terms DM of abundance, measured the(DR) SM GW and parameters. GW signal signals can This and be their paper computed relation. entirely is devotedat to a the high DM, energy darkand scale. the radiation mass Section spectrum of the mirror sector. Direct detection of DM and, in unified theories, Lagrangian are the samethe as ones describing in portal the interactions: SMseveral one for Lagrangian, for kinetic the so mixing, neutrino that onecan portal. for the no the only Although longer Higgs the new solve portal the and doubling parametersthe strong of QCD are CP QCD problem, implies there that is Higgs now Parity a gravity wave (GW) signal from but since now allowing successful DM productionago, a via mirror Freeze-Out copy with ofspace-time dilution the inversion or symmetry SM [ with via an Freeze-In. unbroken parity Long was introduced as a way to restore ever, hadrons containing the hadron dark matter arerather very than strong, thermal. the Standard Model: SM such a theorynary where and the mirror electroweak quarks groupmirror are colored is quark [ contributions doubled, to Although but there QCD is is nothe immediate not, interesting path so possibility to of both gauge coupling ordi- unification, the theory does have θ breaking of SU(3) quartic vanishes, a remarkably successfulthe unification theory of needs couplings extending results to [ incorporate dark matter (DM). ( Parity”, that interchanges thepartner Higgs, SM Higgs, this idea. Onethe elegant right-handed possibility quarks is and that leptonsinclude transform SU(2) spacetime as parity doublets. and lead In to this a case solution Higgs of Parity the may strong CP problem [ JHEP02(2020)078 . 2 0 1) v (or m , 0 ]. In (2.1) (2.2) (2 H H 19 and the , . 18 H 0 H , the potential 0† 0 v is 0 HH  † H v ) representation. We at tree-level, the Low 0 H to a very small value 0 0 . 0 λ 2 and ) λ H + H H 2 SU(2) † ) , 0 H ( H acquires a large vacuum expec- 0†  0 0 λ H , is then extremely small. H λ 2 symmetry that exchanges the SU(2) 1, necessary for + SM 2 λ Z H 1 + † , |  obtains a vacuum expectation value, the 0  H 0 H 0 symmetry λ | ( λ is H 2 2 λ − Z gauge interaction, and the Higgs field H ] that yields the near vanishing of the SM Higgs – 3 – 6 0 . After integrating out H ) + 0 † , it is necessary to tune /λ 0 H H 2 v 0† 2 m 0 is much larger than the electroweak scale. With v H ) is calculated in the next section.  0 = 0 , the vacuum alignment in the SU(4) space is determined v m + λ v 0 2 ( 0 λ v H = † SM ) = λ i H H ( H ( is much smaller than the Planck scale and the typical unification 2 h that is explicitly broken, while Higgs Parity is exact. 0 , with 0 0 m v LE is introduced to solve the little hierarchy problem. The global SU(4) , quantum corrections from SM particles renormalize the quartic v 0 2), where the brackets show the (SU(2) V 0 − H , v = H (1 i 0 0 ) = 0 and H H h H ; the quartic coupling of the Higgs H,H 2 symmetry as Higgs Parity. The scalar potential for ( 0 ) becomes SU(4) symmetric. After 2 V /v Z 2 2.1 v ) is in a fundamental representation. For . A further difference is that the Twin Higgs mechanism requires a discrete symmetry 0 0 v , which is physically equivalent). (The SU(4) symmetry implies that the Higgs boson Although the scale We note a connection of the mechanism with the Twin Higgs mechanism [ Below the scale The vanishing quartic can be understood by an accidental SU(4) symmetry under which H ∼ −  0 H,H arise from the dynamics ofv the theory, but ratherinterchanging accidentally arises from the requirement scale, the theory is no more fine-tuned than the SM because of Higgs Parity. The required the Twin Higgs mechanism,mirror a (twin) global Higgs SU(4) symmetrysymmetry involving results the from SM the Higgs UVstrong dynamics of dynamics. the theory, In such as our the mechanism, global symmetry the of approximate UV global SU(4) symmetry does not in contribution to the Coleman-Weinberg potential does not affect thecoupling, vacuum orientation.) and itenergies, becomes the positive. scale atThe From which threshold the the correction SM to perspective Higgs of quartic running coupling vanishes from is low identified to with high in eq. ( SM Higgs is understood as ain Nambu-Goldstone this boson limit with of a vanishing extremely potential.by small the Note that Coleman-Weinberg potential.so that The the top true contribution vacuum beats is the the asymmetric gauge one, contribution where the entire condensate lies in To obtain the hierarchy λ ( positive, the Higgs parity istation spontaneously value of broken and Energy potential in the effective theory for We assume that the mass scale In this section we review thequartic framework coupling of [ at a highweak energy scale. gauge interaction Consider a withwith a its new partner SU(2) call the 2 Vanishing Higgs quartic from a JHEP02(2020)078 (3.1) (3.2) (2.3) much 0 µν B m , so that 0 µν v B = to obtain the , and may be  2 2 gauge group is 0 µ Λ λ Y / ) + mirror. The next 2 0 v (1) 0 2 H U Z 0† 1 (1) ) describe the neutrino × H U its c symmetry can be extended to describes mixing between )( 3.2 0 × 0 2 at the scale H 0 λ Z † SM f from the vacuum expectation y + L H ( 0 ¯ e λ 00 symmetry which maps = v , 0 3 dimensionless flavor matrices. SU(2) . λ 0 0 2 f c = f . y × , ` × Z y 0 00 0 ¯ , ) + d λ 0 = 2 , 2 + h 0 v Λ 0 0 ¯ u . 0 f ,H are 3 , D 0 0 = ¯ SU(3) e ]. ξ m H q kinetic mixing between ordinary and mirror M , HH 0 2 2  ) 21 0 v , l m , ↔ ↔ ↔ 0 gauge group. The theory solves the strong CP – 4 – ¯ and d 20 , × Y η 0 ¯ e ` ξ ` ¯ H u 2 2 symmetry sets (1) , (1) 0 Λ m + ( 2 U ¯ q d, `, U ( Z 2 0 × 0 M ¯ u, ) is not unique. For example, the H SM M q, ) L 0 3.1 ] considers the case where the SU(3) . The 0 SU(2) η ` ) + v 0 11 ` × . , . The total tuning is the same as in the SM, 6 1 m ¯ e, H + ( ] replicates the 2 M SU(3) ¯ d, l, H 12 M ) link the SM and mirror sectors: ¯ ) u, 3.2 are large mass scales and q, ( ` η ` is the SM Lagrangian up to dimension 4 and SM mapping described in ( M,D L + ( 2 SM M Z L = , as shown in figure L The 1 /v 0 value of the mirror Higgs, mirror fermion masses are largerv than their SM counterparts by a factor of approximately spacetime parity if space is inverted and SM fields are mapped to their Hermitian conjugated mirrors. hypercharge. The dimension 5sector. operators in the second line of3.2 ( The mirrorThe spectrum charged mirror fermions acquire a mass where two terms of ( the ordinary and mirror Higgs doublets and 3.1 The Lagrangian The most general gauge and Higgs Parity invariant Lagrangian up to dimension 5 is where matter is described by 2-component, left-handed, Weyl fields. SM gauge group. Refs.not [ replicated. The theory solvesunification. the strong Ref. CP problem [ andproblem can be and embedded has into SO(10) anwhere interesting the dark SM matter gauge group candidate. is In entirely replicated this by paper a we study a theory explained by environment requirements [ 3 The mirror standardThe model phenomenology of the theory crucially depends on the action of Higgs Parity on the where the first factorsmaller in than the the left cut handelectroweak off scale side scale from is Λ, the and the fine-tuning second to one obtain is the the scale fine-tuning in fine-tuning of the theory is JHEP02(2020)078 is M 0 MS and DM M S (3.3) (3.4) (3.5) 0 ), and u 0  0 EM nonper- v ( 0 0 D (1) S and M α U 0 . In the e 12 ) = 0 10 with 0 QCD v ν ( S α and 0 ν 2 

may come to dominate the

.

=

.1eV 0.01 , and if heavy enough, sub- 0 11 ν

0 m 11 ν γ 10 / 0 4 0 GeV , equating γ 0 v .  v 10 .

10 5

QCD

= 01 eV 0.10

ν 0 GeV m Λ 10 eV. v .   10  10 10 , acquire mass chiefly from SU(3) . Long-lived 01–0 0 0 . ν 10  S v'[GeV] , d m 0 QCD 03 eV 0 . Mirror neutrinos are unstable and decay to , is not a free parameter, but is determined by . – 5 – ν 0 , u 8Λ 0 Mirror Spectrum . m  e restoration scale 6 0 QCD 2 ' 190 GeV Z GeV ] 0 S 5 ' 9 m 23 10 , 10 22 ' 0 QCD 0 Λ ν , giving e' u' d' S' ν' 2 m ) /v for two values of 0 1181 up to the . The latter are visible decays which may occur during BBN if v . † 8 ( 1 down to lower scales until it diverges at the scale Λ 4 3 7 6 5 ). We will be interested in small mixing between 10 ' 0

'

10 10 10 10 10

S ) ] [ GeV Mass HH 3.2 ν α Z m . /m ( 0 S ν 6 . Mass spectrum of key mirror particles. The purple band shows the range of mirror α m , respectively, and thus stable and viable DM candidates. We explore 0 Standard and mirror neutrinos obtain mass from the dimension 5 operators on the Mirror electrons and up quarks are the lightest fermions charged under Unlike mirror quarks, mirror glueballs, or if heavy enough, beta decay to as shown in figure `H Mirror glueballs are unstabledominantly to and dominantly decaylong-lived. to We investigate such constraints in section second line of ( so that then running scheme the dynamical scale is given by turbative effects, with mass [ The mirror QCD confinementrunning scale, Λ SU(3) in section Figure 1 neutrino masses for SM neutrino masses betwen 0 JHEP02(2020)078 0 ]. = u 12 h as a (3.6) 0 m v 3σ and 4) GeV, 0 . , e 0 174.0 2 2σ  √ e 0 . g/ ] and show the 173.5 ln GeV is possible. 24 4 8 g

) = 0.1203 2 Z = (173 no symmetry forbids

(m is shown in the right ). However, as shown S 0.1192 π [GeV] t

α 3 0011), and Higgs mass t 173.0 2 . 0.1181 0 m 64 m 0 v

0.1170 3.2

0.1159  + 1σ1σ 2 172.5 / ) 1181 2 . 0 2σ g is not exactly zero because of the 0 e + 172.0 v 2 3σ as small as few 10 ) = (0 g ( 0 Z 9 8 v 12 11 10 m 10 10 q

with an accuracy of few tens percent [ 10 10 10

( ] [ GeV v' 0 , the running of the Higgs quartic coupling S ln 0 v 2 α v ) 2 0 – 6 – 18 g 10 17 + 10 16 2 10 g ) = 0.1181 15 ( Z for a range of top quark mass 2 10 (m S 14 2 boson mass π 10 3 13 Z 10 128 12 10 + 11 = 125.18 GeV,α =3σ t 10 t h e 10 y 0 ], 10 v 9 ) =3σ ),Δm μ[GeV] ln Z Z 12 10 4 t (m (m 8 S S y 10 . =3σ 2 7 , Δα , Δα 16) GeV. . h h h = 173.0 GeV,m 12 . 3 . For each top quark mass and QCD coupling constant, the range of π t 10 − 0 m Δm Δm Δm 6 8 2 10 10 6.1  5 from being order unity in the effective Lagrangian ( . 10 t 16) GeV. Within the uncertainties,   4 . ' − 18 MS scheme is assumed. The prediction for the scale m  . 0 10 ) 0 3 . (Left) Running of the SM quartic coupling. (Right) Predictions for the scale v 10  ( 2 10 18 is exactly the same as in the SM. We follow the computation in [ SM . The value of the SM quartic coupling at the scale 0.05 0.00 0.10 = (125 Diagrams contributing to kinetic mixing via the Higgs portal only occur beyond four loops, likely λ λ 2 h SM 3.4 Kinetic mixing Even though quantum correctionsa to tree-level the kinetic mixing are small, inducing an panel of figure the prediction corresponds to(125 the 1-sigma uncertainty inFuture the measurements measured can Higgs pin mass, down the scale threshold correction [ where the λ running in the left panelQCD of coupling figure constant at the m energy density of theupon universe decaying. and We release investigateDM significant the in entropy effect section into of the such SM entropy dilution thermal3.3 on bath freeze-out Prediction for Between the electroweak scale and the scale Figure 2 function of JHEP02(2020)078 0 0 ' the  (1) (3.8) U 9 G v × 10 0 ) induce . ) 3.7 6 Pl SU(2) ). As shown in × is non-zero, the /M 0 . Above 6 G 6 8 must vanish above c v 3.8 4 ( ) (3.7)  10 O 6 Pl SU(3) + MAJORANA × /M ] and references therein.) 4  (1 (1) 25 O U 7 × GeV G 10 ) + 0 v . 16 F G 2 0 v SU(2) 10 ×  = 8 )(Σ i the Higgs fields. The first term is absent is strongly constrained by nuclear and 0 c 0 F Σ 6 8 h 2 10 Σ [GeV] − , − 10 = (Σ e'

i 10

-

II CDMS

10 XENON1T 4 Freeze-In m Pl Σ 8 – 7 – h c × the higher dimensional operators in ( & , M

0

or charged under some symmetry. When Σ and Σ

⨯ LZ, 15years ton , shown qualitatively in figure 0 3 v  1 2 . is that SU(3) , G G  G v v 5 + 6

) + ope rudrecombination around Coupled 0 2 10 F  0 )(Σ at scale as DM via the induced kinetic mixing of ( GeV G F 0 0 v e G 16 (Σ 4 by gauge invariance. 10 2 × Pl 6 10 G c  v M G 6 c 1 2 5 . Constraints on kinetic mixing if DM is composed of mirror electrons.  − symmetry is unbroken above 10 2 3 are the gauge field strengths and Σ Z × 0 , mirror electron DM with 5 1 10 , essentially independent of DM mass. If the dim-6 coefficient -6 -7 -8 -9 -1 -2 -3 -4 -5 . -10 -11 3 3 Figure 3 , the correct DM abundance can be produced for a kinetic mixing parameter 11 10 10 10 10 10 10 10 10 10 F,F 10 10 − ' 4 ϵ  For example, consider a theory where the SM gauge group and the mirror gauge group 10 Since the 3 × It is possible tofigure freeze-in 4 if Σ is not an adjointacquire representation a of vacuum expectation value a kinetic mixing operators that induce kinetic mixing between the standard and mirror sectors are: where A natural explanation for such a small unifies into a larger gauge groupthe with no unification abelian scale factors. Consequently, separately unify to in figure electron recoil experiments, ionization signals, and cosmology ([ JHEP02(2020)078 0 e (4.4) (4.1) (4.2) (4.3) GeV. 2 / 1 , a total − th , 6 c E 0 Y is given by t 13 year (1) tar 10 rel . × v . The gauge groups U × G M 0 EM G 0 EM v 1 DM abundance can v ⇥ ton 0 , . 0 ' s (1) dark matter to scatter e ) (1)  / 0 3 G U th e 3 U v 0 (2) − . , and 0 E 0 ⇥ ( ⇥ 0 f 0 dqq SU is v, v 3. . 0 (3) A ⇥ (3) . max th = 220 km q 0 Mirror Model th ' E , q rel 0 2 Z v SU ) 10 keV | SU (3) )  , as well as a velocity distribution of N 2 th q 3 / , v (  E ) m 1  ( 2 0 SU F 3 | , with relative velocity f A − 2 /v 131 = 2 Z  q 2 2 2 3  v | Y Y q ) is the nuclear form factor. The number of ) 2 Z − is non-zero, the correct max q q 2 2  rel ( ( – 8 – 3 GeV/cm (1) (1) 8 v . F F c , q πα Z 54 | U U exp( 8 EM ], we find th  , and atomic weight 2 3 dq 0 GeV. E ⇥ ⇥ = v v T 27 (1) 4 N , / π 0 max 1 4 U e q m th dq (2) (2) dσ 26 GeV √ − q 2 GG m 8 7 Z ⇥ c dv  p SU SU 10 15 2 = (3) and atomic number ⇥ ⇥  ) = ) = 10 th 8 v N th q ( Standard Model × − SU  (3) (3) E m 5 ( 10 dvf f  ' SU SU G × , an exposure time v 6 . tar and below, or be induced by higher dimensional operators at µ x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ------M G = 1 v ) takes into account the suppression of the scattering by the form factor, 0 G DM abundance can be produced for the unification scale th do not contain any abelian factors so that kinetic mixing can only be radiatively generated v v v is the momentum transfer and . Qualitative picture of the effective field theory at scales 0 0 E e event q ( G f vanishes, and the dim-8 coefficient N 6 c and Assuming the Helm form factor [ where we assume a local DM density of 0 Here where expected events in atarget mass direct detection experiment with an energy threshold electromagnetically with a nucleus. Theand Rutherford a cross nucleus section of for mass scattering between 4 Direct detection and4.1 the correlation with Direct proton detectionKinetic decay by mixing nuclear induced recoils from higher dimensional operators allows at the scale correct If be produced for Figure 4 G JHEP02(2020)078 0 e so for G (5.1) (4.7) (4.5) (4.6)  GeV, v 9 , 10 2 show the / 1 5 ≤ gauge group. 0  , v 2 Y  2 (1) GeV 0 U 9 v 10 0 W × = 0. The direct detection 2 103 GeV 6 .  c 0 8 1  c . 2 4 2 / 3 / 1  / 1 1    GeV 0 G e v GeV year 0 16 GeV m 2 v = 10. 10 × 9 10 8 10 c 10 ) and we assume   ton  ], requiring – 9 – t 3.8 10 3 25 − − tar years event 10 10 M N 35 × '  10 1 encodes the relevant hadronic matrix element extracted (1) factor which eventually joins the 0 × 2 U dec v 3 ]. We also assume that below the heavy gauge boson mass T  < 1, the kinetic mixing parameter is stronger for fixed is close to this bound, future experiments may detect years ' 29  35 > ) + 8 10 e c ) and the proton decay rate are correlated with each other, 041 GeV 0 . × π 0 . If 4.2 3  3 → . The blue region shows that if XENON1T were to detect a nuclear recoil ' ]. The bound obtained there can be interpreted as an upper bound of ( p 5 ) ( t 28 103 + 1 . e − tar 0 Γ GeV. For example, the dashed blue and orange contours of figure π = 0 /M 9 GeV comes from XENON1T [ | 0 → 2 10 p W ( | 10 event = 1. The orange region shows the analgous signal region for LZ. For & 1 N 0 8 − At sufficiently high temperatures, the SM and mirror sectors are kept in thermal XENON1T searches for a recoil between DM and Xenon with a threshold energy > c v Γ 0 e is whether the two sectors were brought into thermalequilibrium equilibrium by after the inflation. Higgs portal; the sectors then decouple at a temperature 5 High and lowSince reheat all scenarios; the BBN parametersaffect and of the dark the cosmology radiation SM of haveinflation the and been the Mirror determined, portal parameters Higgs the that Parity only connect the theory free SM are parameters and mirror the that sectors. reheat A temperature key question after Hyper-Kamiokande and LZ bothsignals, can respectively. detect If correlatingthat proton nuclear recoil decay experiments and andfor proton nuclear detect recoil experiments may findreach correlating of signals XENON1T and LZ, respectively, for as shown in figure signal, the proton lifetime would generallyfor be longer than Hyper-Kamiokande could detect, scale the gauge group contains a (This case excludes, for example,scale.) the The Pati-Salam kinetic gauge mixing group israte breaking given at by eq. an ( intermediate with heavy gauge bosons mediating proton decay. The proton decay rate is where from a lattice computation [ as shown indark figure matter. 4.2 Correlation betweenLet proton us decay consider and a direct case detection where the SM gauge group is embedded into a unified gauge group around 10 keV [ of 16 onm the expected number of the events. Currently, the strongest bound on JHEP02(2020)078 . 3 ) c and (5.2) 0 /T γ 0 c T decay; for via higher- 0 G 75)(

ν

.

v G ] [ GeV v , whose energy 0 106 S 15 / 16 10 4⨯10 12 10 = (16 r , the two sectors were initially abundances are given by freeze- dec 11 is about 0 T e 0 10 c T . 75). On the other hand, if the reheat 0 . c and 0 u 106 / A r T Hyper-Kamiokande Super-Kamiokande ) 10 3 4 dec 10 , the – 10 – T = v'[GeV] ( confine to form mirror glueballs 0 0 ∗ , the two sectors were never in thermal equilibrium dec 0 = 1 corresponds to the limit where a massless ideal S g s g ρ dec A ]. At this point, the mirror bath contains only

9)( =

10 > T T .

8

/ 22 c [ B is generally much less than one. RH 9 , and lead to very different mechanisms for the abundance

T LZ 10 = (8 dec 0 QCD /T r 0 T T 26 Λ

.

= 10

8

c XENON1T = 1 8 0 c 10 T 35 34

10 10

] [ yr Lifetime Proton takes into account the non-trivial dynamics before and after the phase transi- dark matter. For . The rates are related as they both depend on the unification scale 0 we assume that only the SM sector is reheated, so that DM arises from freeze-in. 0 A v is above or below u scheme and identify regions of parameter space that give signals of dark radiation . Correlation between the proton decay rate and the DM-nuclear scattering rate as a dec RH RH T and T < T After the mirror QCD transition, The mirror QCD confinement transition occurs when the mirror thermal bath cools In both high and low reheating cosmologies, long-lived mirror glueballs are produced 0 e so that the ratio of entropies of the two sectors at RH 0 The factor tion and is estimated in appendix in thermal equilibrium and temperature after inflation is below and ratio of temperatures density normalized by the entropy is given by high and perturbed light element abundances. to a temperature g If the reheat temperature after inflation is greater than These two schemes for DM production are discussed inwhose the decay next products section. may yieldlight substantial elements. dark In radiation this or section alter weof the study mirror relic the general abundances glueballs. constraints of on the These maximum results production will be used in the next section to place limits on the Our two cosmological scenarios correspondtion, to whether the reheatof temperature after infla- out as the temperatureT drops below their masses, followed by dilution from Figure 5 function of dimensional operators. JHEP02(2020)078 † H H (5.4) (5.5) (5.6) (5.3) is the 0 ,ν MD T 0 and the mirror as shown in the h 0 0 c T Q decays during BBN. , where 0 0 0 (right). 0 q q S † ,ν . ]. Since the amplitude is 0 RH S 31 [ H,H 0 /T 3, so that the mirror glueball µν 0 GeV 0 0 q / . , into the SM hadronic sector and charge . 3 S ,ν 0 0 0 F 0 8 u 14 4 9 S q vis 5 S 0 m 0 µν − ρ = 2 v 7 MD m m m m . 0 F T 2 10 2 2 Q  (left) and  = 0 0 0 ++ . 0 2 π α g g i ' S 0 7 γ π 7 . 0 0 D c . F 0 2 γ 2 T is given by 270 ++ 16 ] 0 S giving 0 4 q † , exceeds about 1 s, |  0 r  α D 32 1 m u µν a π π − 1 A ) 1 8 π 0 0 = HH – 11 – G 2 † 16 0 3 4 γ γ 0 a µν q Q ' † 240 HH ' G , mirror neutrinos are a natural candidate to provide † 2 s → 0 0 0 HH g = γ S dec 0 S 2 → HH Γ γ 0 0 / γ S → 0 1 → 0 0 0 | > T γ + Γ Γ S q S 0 , we take 0 h → 0 Γ Γ γ 0 q = 0 RH S γ = T m L → vis s 0 ∆ 0 0 ρ 0 . Mirror glueball decay to S ++ q q S 0 . The decay rate to (Γ F . After confinement this becomes a dimension-5 operator connecting 6 ' 6 may inject substantial energy density, 1 0 0 0 − S q Figure 6 S ] 30 [ , generated by a loop of mirror quarks of mass 0 0 is a generic dilution factor which may arise if there exists a particle which comes γ G 0 0 0 0 D γ g g G In the cosmology with Mirror glueballs are typically long-lived. The lifetime of the mirror glueball is dom- 0 0 to F S 0 0 to dominate the energyentropy density into of the the SM universe thermal and bath. decays before BBN,such thereby dilution injecting since theyrelativistic, are and abundantly are produced, long-lived. decouple In from this the scenario, mirror bath while altering the neutron to protonimmediately ratio after, before leading nucleosynthesis to or the disassociating constraint light [ elements Here, If its lifetime, Γ If this occurs, The mirror glueball can alsoright decay panel to of the figure SM sector via the Higgs portal as shown by the with matrix element dominated by the smallest decay rate to mirror photons is F left panel of figure S gas of mirror gluonsglueball suddenly number becomes density pressureless is mirror conserved glueballs afterward. at inantly set by its decay rate to mirror photons, described by the dimension-8 operator JHEP02(2020)078 0 . S , is eff are 12 0 N (5.7) /T when 10 0 γ 0 0 T γ depends ,ν is known is known in orange BBN . In the left r RH → eff † D 7 , T 0 N , 11 S 10 dec dec ,HH , ∆ 0 γ D > T 0 < T γ in figure

1.0 0 RH 0.3 10 → 0.1 v RH T = 1. 10 0 = 0.03 v'[GeV] T

eff S A ΔN decays before MD decays during MD decays after MD. 9 0 0 0 10 ]. When S S S . . When 7 32 7 3 / 3 1 1 matter-dominated era / A 8 1 0 D 1 10 ν -4 -1 -2 -3  Pl 3 3 4 4 / / 10 10 10 10 / / 1 1 Pl M 0 1 1 0 c ) ) T 0 T' ) ) 0 0 ,ν T S 0 0 M S S into mirror photons is constrained, even if 0 S S Γ Γ Γ 0 S Γ Γ 12 RH T T – 12 – √ Γ T T S 10 ( ( T ( ( S S √ ∗ ∗ r D ∗ ∗ g g g g induced matter-domination begins and  3 decay leading to a dilution factor BBN / 0 4 3 4 0 4 / / / ν are produced, so they do not give a matter dominated ν 1 1 1  11 0    10 ν After MD 4 2 2 2 43 10 10 10 π π π 

  

4 7 1.0

0 0.3                     

γ 0.1 uigMD During 10 0

0 γ

×

10 = 0.03 eff S v'[GeV]

→ , there is no particle in the mirror standard model to provide such Δ N 0 Γ S dec Γ cosmology, with cosmology few . ' 9 contours (purple) and BBN constraints (orange) from < T 10 = 1. We show the BBN constraints as a function of 7 decays before, during, or after the RH eff S' Decays Before MD RH eff = 4 MeV 0 T D RH N T S N = 1. Contours of the dark radiation abundance produced from T RH,ν' ∆ T . ∆ D 8 4 3 2 8 7 6 5 9 1 10 11 10 10

10 10 10 10 10 10 10 10 is constrained, as shown in the left panel of figure

10 10 Dilution is constrained, as shown in the right panel of figure In addition, the energy deposited by D r For the low era and shown in figure For the high on whether so so does not decay duringdensity BBN. is The conventionally expressed mirror as photons an behave excess as in the dark effective radiation, number whose of energy neutrinos ∆ temperature of the SMit bath ends. when If dilution and using the precise energy yield constraints calculated in [ Figure 7 (right) panel the twoproduced sectors via were freeze-out (were and not) dilutionevaluated initially (freeze-in). at thermally the The coupled mirror temperature confinement so ratio temperature. of that the For DM two clarity, is sectors, we thermally take JHEP02(2020)078 0 4 . e 0 ν Sector Decoupling ν' Decoupling s' Freeze-Out d' Freeze-Out u' Freeze-Out e' Freeze-Out QCD' Confinement s' Decays d' Decays 12 10 ] for a recent estimation of the 34 , decoupling, as well as the mirror 0 33 ν 11 10 . In this case, the relic abundances of mirror dec decay – 13 – 10 0 ν 10 > T v'[GeV] RH T 9 10 shows the temperatures around which each particle freezes-out 8 8 6 5 4 3 2 9 8 7 1 10 11 10 -2 -1 10

10 10 10 10 10 10 10 10

10 10

10 10

] [ GeV T' dark matter are set by freeze-out followed by dilution from the late decays of . Temperatures of the mirror bath around which each mirror fermion freezes-out (solid) 0 annihilate and freeze-out. Although heavier mirror charged leptons and quarks are u 0 u Furthermore, the maximum temperature of the universe after inflation is taken less than the mirror 4 and 0 (solid lines) and decays (dashed lines). Here we ignore the effects causedelectroweak scale by to avoid late domain wall decays problemsthe of from maximal the temperature spontaneous breaking is of highermaximal Higgs than temperature. Parity. the Generically, reheat temperature. See [ e As the temperature ofand the universe drops, unstableunstable, mirror their particles decay widths decay, are while muchelectroweak stable smaller scale. than their Figure masses because of the large mirror 6 Cosmological abundance of mirror6.1 dark matter Freeze-out and dilution from In this section, we takeat the which reheat the temperature two of sectors the decouple, universe larger than the temperature Figure 8 and decays (dashed). Mirror temperaturesQCD of phase sector transition, decoupling, are shown as dotted lines. JHEP02(2020)078 0 0 . c 2 µ T ) , 0 S 0 c GeV, α . For 0 ) GeV, q quickly a small 2 10 11 0 . For all m below pair, and 0 0 0− QCD ( d 0 u 10 0 10 u ¯ q 0 e 0 π/ u q − < 0 8 u 0 and v 0 is comparable to . u 0 0 u , 0 0 d experimental values. . s u 0 decay producing σ u . The solid lines con- A or 0 c 0 partially decay producing that produce 9 does not change in either , generally forms a tightly 0 u 0 0 0 µ s d 0 e and s 0 0 s or b and 0 , but they thermalize quickly. 0 in the range of (10 0 s phase transition make s decay into GeV the Coulomb binding energy . 0 d 0 c 0 0 v ]. Nevertheless, these tightly bound T d 10 37 and 10 [ is the dominant component of DM. On 0 0 and 1 u > e 0 − , 0 s ) 0 is too large to be DM. This is problematic as v e 0 S 0 are shown in figure e α 0 0 that q – 14 – u 9 m ( ]. For ) to lie beyond their current 3 and ∼ Z 36 0 , , and so most of the mirror quarks initially form tightly m 0 e c ). To the left of the vertical dotted line, the QCD’ phase ( 35 T [ S 8 annihilates efficiently so that its beta decay contributions drops when annihilations of mirror hadrons continue after 0 α 0 0 γ s e freezes-out, which is why its abundance dramatically increases and 0 u t m does not change even if the cross-section is as large as Λ . Still, (1) fraction of these mirror quarks form loosely bound states with 0 0 . With such a large cross-section, these mirror hadrons scatter among v O e freeze-out. During these annihilations, . freeze-out. During these annihilations, 0 0 0 is short-lived and does not affect the above processes. A set of Boltzmann s d 0 QCD e 0 Λ τ requires abundances are determined by the following processes in chronological order: 0 GeV, the thermal abundance of 0 . The annihilations also produce ∼ and and 0 v and u 7 s 0 0 0 µ u u 10 freezes-out. , , , 0 0 0 0 and ], and so an × d e annihilate. Mirror hadrons composed of b c and are small. 0 6 QCD’ phase transition occurs. Mirror hadrons composed of We see from the solid lines of figure The thermal abundances of We elaborate on the fourth process. After the mirror QCD phase transition, mirror e 35 0 [ 3. 4. 1. 2. e > 0 c 0 the other hand, efficientcomponent annihilations of DM, after which thev exists QCD today insuch the a form low of mirror hadrons like confinement. Even thoughcase, the the annihilation relic cross-section abundance of the of QCD’ phase transitionare since effectively absent any (see beta figure transition decays occurs from before if hadronic annihilations are assumed to cease below to servatively assume that theThe annihilation abundance of cross-section ofcomparison, mirror the dashed hadrons line is assumes mirror hadrons completely cease annihilating after of mirror hadrons is largerbound than states with astates smaller still radius have aThe relatively mirror large baryon containing radius onlybound and mirror state scatter strange for and quarks, all annihilate relatively efficiently. the Coulomb bindingT energy of mirrorlarge hadrons radii containing a themselves efficiently, rearranging their quark constituentsubsequently until they annihilate contain a into We note that equations describing the freeze-out dynamics is shown in appendix quarks are tied with each other by strings and form bound states. For the mirror neutrinos, and include them momentarily. For JHEP02(2020)078 (6.2) (6.1) dilution 0 e 12 required to , the mirror 0 10 0 v v via the neutrino `H , with a decay rate 0 ¯ are sufficiently long-lived d 0 ) is sufficiently large. The 11 ν 10 3.2 and decays to 0 0 ν of ( , u . 0 0 decouple from the mirror thermal e 4 decay rapidly and study 5 ν . Consequently, the D 0 0 0 0 . v u m ν M ν 2 2 D 3 0 , ), and for sufficiently large v π M 1 and dec 0 and 10 ν 3.5 π T 0 0 192 8 e d 10 m 3 8 v'[GeV] m e' FO with mirror hadrone' annihilations FO without mirror hadronu' annihilations FO with mirror hadronu' annihilations FO without mirror hadron annihilations 0 = decouple while relativistic with an initial yield = – 15 – ν + 0 0 ¯ lh 0 d ν 0 u → u , 0 0 0 m e ν ν Γ → + 0 m ν 0 e Γ 9  e' and u'-Out Freeze Abundances 10 < m 0 dec 004. With this initial abundance, if . 0 T ν with a decay rate m = 5 ) ) = 0 dec 3.2 dec T is reduced, mirror neutrinos can decay well after they becoming non- T ( 8 = 1. D /s 4 3 2 9 8 7 6 5 η 1 10 ) -1 -2 -3 10 M 10 10 10 10 10 10 10 10 =

10 10 10

dec

DM shifts to higher scales. Ω ξ

h . Since 2 T 0 ( e . The cosmological abundance of mirror electrons and up quarks from freeze-out and from 8 0 ν n Shortly after the two sectors decouple at Nevertheless, in the above discussion, we have ignored mirror neutrinos which are ' We take 5 0 ν The mass of the mirror neutrinoneutrino is is given by massive eq. enough ( that it can beta decay into For our first example, wefrom assume decays that of two the flavors single of portal long-lived operator flavor. of The ( long-lived bath as the mirror weakin interaction figure rate drops below theY Hubble expansion rate, asthey shown dominate the energy density of the6.1.1 universe prior to decaying. One generation of long-lived electromagnetic charge conservation, andHowever, the as latter suppressesrelativistic decays to to SM the particles, SMproduce thereby sector. diluting decays of heavier charged mirror fermions. Dilution from mirror neutrino decays is not included. cosmologically stable if former prevents decays to the mirror sector, due to mirror fermion number and mirror Figure 9 JHEP02(2020)078 DM (6.5) (6.3) (6.4) 0 e can be 0 ν . m

. 10 M ] [ GeV M . Vertical gray 0 σ by a factor γ γ 15 16 14 n to 10 10 10 12 ). 0 10 D S 1 6.3 M . ν 4 /

1 m

v

) '  2 Pl 0 relative to

2

) plane in figure v

decays. An approximation for + + e e' M 0 11 π ν

0 Light u 10

√

ν

DM 10 ν →

' 6

RH , m

0 are large enough that Ω

> ∗

− ' e

Heavy v g

Ω and

ν ( ' D 10 0 

e

. 2 M σ 0 × / 2

2

0

1

5 eff Δ N = 0.3 v ) v 10

0 0 is determined to yield the correct 0.2 ≈ Pl 10 ν ν resulting from decays of

Y

D 0 0.1 M m 1 D 0 ν

DM ') eff ν vʹ[GeV] 0.06 2 M

– 16 – = N S

>Ω dark matter arises from freeze-out and dilution from / m

e' 0.05 ∗ 1 ∗ = 4 MeV σ 1 0

Ω ν g g e 2(Γ . (ν'→ e'+u'+d m 1 BBN 2 31(23) Pl 9 1

) deviate from their central values by 0 to 3

10

0 2

0 ν

0

2 e is the lifetime of the mirror neutrino. The numerical factor σ > 0.3 eV 2 ) when m ,ν RH,ν' ,ν m ( ν ν 1 T m πα

S

2 ν, Heavy − 0.06 RH ) MD α Σm m , m 0 0 T 35 ¯ T

d

v

0

Z S t σ > ) ( Δα Δ 3 m , m u ≈ 0 8 = and . Here e 0 ), the parameter ]. We solve the Boltzmann equation for the abundance of mirror , including freeze-out, the change of the expansion rate during the 10 t

0 ν ν

→ D nufcetDr Matter Dark Insufficient 0 FO 38 m , A ν s 0 , m e 0 ρ v +Γ 1 . Purple contours show ∆ -3 -2 -1 0

when lh

yield from freeze-out and dilution is ν

10 10 10

by the scaling ν ] [ eV m 0 0 → v 0 ν e ν m . Constraints on ( is the dilution factor provided by mirror neutrino decays ( dominantly decay into the SM sector, the decay products heat up the SM thermal = (Γ 0 D ν 1 − 2 is taken from [ For a given ( ) . 0 ν Further constraints on this scenario are shown in the ( abundance. Furthermore, themapped resulting to values of the resulting where (Γ of 1 fermions in appendix mirror neutrino matter-dominated era, and dilution from contours show When bath, thereby diluting the frozen-out abundance of Figure 10 one long-lived species of of the long-lived JHEP02(2020)078 0 , σ is 0 e for a . To 0 2 31(23) 0 ν ¯ e m 0 e should be ) GeV. In 2 , 0 t 1 23 is one of the ν 0 m is in the range ν –10 is kinematically → 0 to reproduce the eff 18 0 3 ¯ d 0 ν N Pl u decay, and too little 0 ], excluding the pink- e M 0 0 ¯ d ν 41 0 ) and is proportional to → Γ u 0 0 give comparable dilutions; is also excluded since the e √ 6.1 ν 1 , , with the vertical axis inter- decays, we take the strength 0 dark matter, the purple con- 2 ≈ 10 → ν 0 0 0 4 MeV [ 0 e ν 10 ,ν ), ν 2 > ν RH 0 T ,ν ) for all three generations of < m RH T 1 6.1 ν m are excluded in this case. The upper bound must lie within the range (10 eff  all give comparable dilutions. ]. The gray-shaded region is excluded at the 3 3 N D – 17 – ν 1 , 42 M 2 m , 0 3 abundance. In the orange-shaded region the sum of 1 eV. ν must be light enough that beta decay is forbidden. . ) of SM neutrinos, the mirror neutrino responsible for ) is dominated by the heaviest mirror neutrino. For a 0 and ∆ from decays of mirror glueballs, produced at the QCD 3 0 cannot occur. However, if the long-lived 0 u ν ν for an inverted hierarchy. In either case, too much 0 v 6.3 eff ¯ e 0 0 ] and within range of the sensitivities of CMB Stage IV ¯ e e N 0 0 and < m e and the Higgs and top masses. If the long-lived species is the 42 freeze-out and dilution from ν 0 3 0 2 abundance is smaller than the dark matter abundance without 0 S ν e ν 0 e ]. We adopt the bound α e m decays to be given by ( → 40 , 1 , all three  , . The corresponding SM neutrino mass should be above ∆ 0 0 0 2 lH e 1 39 e ν ν m → 0 ]. ν 43 creates ; for an inverted hierarchy ( 0 0 3 ν ν then beta decay to 0 ν 4, allowed by Planck [ . In addition to these bounds, there is a constraint from the decay The bounds from BBN, too much dark matter from In the resulting allowed region of parameter space for In the allowed white region, we find . Consequently, the dilution ( 0 ν 03–0 . on the heaviest neutrinois from 0.1 eV. the cosmological limit on the sumnormal of hierarchy the or neutrino masses produced. Regardless of whether the SM neutrinos obey a normal or inverted hierarchy, and for a quasi-degenerate spectrum dark matter from freeze-out arepreted approximately as as the in heaviest figure neutrino,0.05 eV. which is Thus the constrained larger by values oscillation of data to be at or above avoid overproducing Thus the total decay ratem of each mirror neutrinonormal hierarchy is ( given by ( dilution is 6.1.2 Universal coupling strength ofAs a neutrino second portal illustrationof of the neutrino portal couplingbasis, to we be take independent of generation. Thus, in a neutrino mass tours show our prediction forconfining ∆ transition, to mirror photons.0 Throughout the entire region ∆ experiments [ long-lived excluding the lightly yellow-shaded region.above its The present allowed central region value and,of is remarkably, 10 not the of large: neutrino its mass present must upper be within bound a of factor 0 allowed, creating too much the SM neutrino massesCosmic are Microwave Background above (CMB) 0.3 eV, [ level which from is measurement of disfavored bylightest the observations of the heavier states, then the lightly green-shaded region of figure the red-shaded region, the dilution. For too smalldark a matter neutrino abundance mass, the isnumber required of below neutrinos the [ MeVshaded scale region. and In affects the blue-shaded BBN region, the as mirror well beta decay as the effective JHEP02(2020)078 ]. and 46 – (6.6) t eV. m 3 44 − is always 10 eff , where the when

N

>

0 M ] [ GeV M v ν 11 is also the mass neutrino. Purple m 15 16 14 2 31 10 10 10 12 m 10 ) appears in green. If lightest

6.6

) ' .

SM neutrino: + + e e' 11

.

10 Light

0 10

DM v v ν →

' ].

e Ω >

' e Heavy

43

Ω ν ( ' m

lightest

−

σ 0 0 is the mass of the v v . Vertical gray contours show 10 ν . In the allowed white region, ∆ 0 γ e σ 2 31 10 m

m m

to

eff Δ 4 N = DM 0.1 0 dark matter are shown in figure ') ∆ vʹ[GeV] – 18 –

>Ω S dark matter arises from freeze-out and dilution from

e' 0

e σ Ω 1 0 0.06 > e

(ν'→ e'+u'+d GeV, the lightest neutrino mass is predicted to be 9 0.05 9 10 10

lightest 0.05 is the atmospheric neutrino mass difference squared and

× σ > 0.3 eV 2 ν, ) when 2 ν ν m

Σm , m 0.06 , but decays before the onset of the BBN for SM neutrino mass. The bound of ( 0

v

D

Z S t σ > ) ( Δα Δ 3 m , m 8 05 eV) . M 10

(0 nufcetDr Matter Dark Insufficient lightest resulting from decays of | ' 1 2 1 -3 -2 -1 eff

10 10 10

m ν N ] [ eV m Lightest 03, which will be probed by CMB Stage IV [ − . 2 3 . Constraints on ( m ) deviate from their central values by 0 to 3 ≡ | Z The sum of the masses of the three neutrinos can be constrained through its imprint on The constraints on this scheme for 2 31 is the electron mass. We have made the good approximation that ∆ m turns out to be larger than 4 ( with universal neutrino portal couplings. Here e m S 0 0 neutrino for a universal the structure of the universe.are Future measurements expected of to the CMB, determineOne BAO, the and can sum 21 check cm of the emission consistency the of masses the with the an measurements uncertainty and of the bounds 10 we meV have [ obtained. This bound is shown in the yellow hatched regionvertical of figure axis is the v in a narrow range. The lightest mirror neutrino is longer-lived than the heaviest mirror ∆ m squared difference between the lightest and heaviest SM neutrino in an inverted hierarchy. α greater than 0 this constraint can be translated to a bound on the Figure 11 ν contours show ∆ JHEP02(2020)078 6 7 . 0 e (6.7) (6.8) m ≈ freeze-in 0 T 0 0 e , g , g 0 0 does not grow γ γ 0 ). Thermalization e 6.7 0 is the freeze-in cross- f . Reheat temperatures i σv RH h T 0 0 . f f when the reheat temperature i ) 0 and a freeze-in number density e RH T RH 0 ( . T h 2 2 e 0 σv to be DM since the small y v h 0 ]. We will discuss the implication of the e π 2 1 ) ) is Hubble, and 8 47 , like the growth of a weakly interacting by scatterings involving highly energetic particles 0 e 0 , any pre-thermalized contribution is typically small. RH RH = e H as production almost ceases below T T , mirror DM is produced via freeze-in through i , cosmic perturbations of massive components ( ( RH † ) 0 6 T ) H H ν – 19 – dec H H RH n T . Although the mirror fermion and gauge boson RH T . 4 9 ( . The thermal evolution of the mirror electrons is < T < m . A set of Boltzmann equations describing the freeze- A 12 /T 0 0 σv e e A h RH may thermalize during reheating, altering ( ) = m T m 0 γ & RH and only the SM sector is reheated. Since the SM and 0 0 ¯ T f f ( and RH n dec 0 T e T , ], which we find is not efficient enough to reproduce the DM abundance. 0 h RH 49 , T 48 tightly couples to mirror photons, the perturbation of 0 e and grow the perturbation of 0 e and high † 0 , the mirror electrons carry a typical energy production still occurs for v H H . Freeze-in production of mirror fermions (left) and mirror gauge bosons (right) through 0 e RH T At During the matter dominated era by is the SM Higgs thermal number density, Some For low 6 7 H produced by inflatons [ cools the mirror bathportal. so Since that freeze-in production mirrorEven is particles so, maximized freeze-out at we consider instantly this but effect are in then appendix replenished by the Higgs n section given by in dynamics isas shown follows. in appendix production rates are UV-dominated, theUV production entropy production so that during the reheatingbelow dominant negates production the far- occurs mirror around electronabundance mass is yield further insufficient dilutedConsequently, by we ( focus on 6.2 Freeze-in from HiggsIn portal this section, and we kinetic considerof mixing the the relic abundances universe ofmirror mirror is sectors below are weakly coupledthe below Higgs portal, as shown in figure by itself. The perturbationscatter of with mirror glueballsmassive grows, particle during decays a into matter mirrorgrowth dominated to photons, era [ the which future searches for ultra compact mini halo elsewhere. Figure 12 the Higgs portal. can grow. Since JHEP02(2020)078 . & as A 0 e RH of the T RH and T 0 γ 12 10 increases, mirror fermions

and BBN constraints from

- - 8 5

- 11 10 10 (Higgs Portal) (6.9) .

eff

RH

= ϵ 10 T as they are much weaker than ). N 11 13 10

13 DM 6.1 . ∆

and the reheat temperature

Ω <

0

' e

v Ω exceeds that of dark matter for 6.1 3 processes as discussed in appendix RH ). On one hand, the freeze-in abundance →

T

10

) ( - iei Mixing Kinetic In Freeze are ineffective during subsequent freeze-out. Pl DM 3.8 10 0 DM can also be frozen-in via kinetic mixing are effective during subsequent freeze-out and . However, as at v'[GeV] e M 0 >Ω 0 0 e e e' – 20 – e 13 RH Ω , 2 and 2 T 0 3 e v y →

3.4

Freeze-In(Higgs Portal) LZ S , which transfer much of their abundance to

DM.

9 as DM via the kinetic mixing portal (shown by the dotted XENON1T ∗ 0 g 10 0 RH e 2 e / T 1 1 is overproduced via freeze-in from the Higgs portal. In the red ) 0 ∗ e g

(

etr Coupled Sectors

- Out Freeze from the Higgs portal is 01 . 0 8 0 e GeV, annihilations of 3 2 7 6 5 4 10 to freeze-in

GeV, annihilations of

, the two sectors were once in thermal equilibrium and the situation

8 ≈ 10 10 10 10 10 10

RH ] [ GeV T  8 10 dec FI rises, as shown in figure , 10 0 s T × e are produced at × ρ 4 0 RH ≥ , the frozen-in abundance of 4 e 0 T & v . 0 . Constraints on the mirror electroweak scale RH 0 v T v Finally, as mentioned in section For For For all . For 0 e reverts to traditional freeze-out discussedfrozen-in in mirror section glueball decaysthe are bound not on shown in overproduction of figure induced from higher dimensional operators ( reproduces the correct DM abundance, as shown in figure the allowed heavier than they annihilate and thermalize via 2 In this regime, a reheat temperature approximately equal to the mirror electron mass reheat temperature is highfreeze-in enough regime that reduces the to two the sectors freeze-out were regime originally (see thermally section coupled and the m The freeze-in yield of Figure 13 universe. In theregion, blue the region, required counters) is large enough to produce nuclear recoil signals in XENON1T. In the orange region, the JHEP02(2020)078 0  ], e 53 (7.1) (7.2) is the directly . On the 2 0 . e h 0 QCD 6 / , LZ has the col 1 0 , 3 Λ v .  < m necessary for , which is a free 1 5 . 0  GW .  0 b e 0 RH ' is the total energy T  0 , c 2 25. / T tot / 1 0 ρ SM e is the latent heat of the  is given by 0 . 3 m /ρ g 1 f show the lat is the DM since the required 2 − tot ∼ ρ /ρ / 0 (Kinetic Mixing) (6.10) ρ 13 H e 3 tot 2 ρ   lat kin 3 ρ ), and whose value can be chosen to ρ 2  /  1 of figure G 0  0 g v 0 lat e  e ρ ρ 0 RH c ) parametrizes the duration of the phase m 10 T ]. T dec 2 β/H tot 100 g 3 < m − 52  / 100 GeV from kinetic mixing is /ρ β/H  0 2 0 dec to be produced as DM for 0 60 4 g – 21 – g RH e ) 0 through kinetic mixing is set by ρ p e T 0    3 exp  e ) dark matter abundance is set by freeze-out followed 2 3 3 p 0 / / f/f  e 4 1 10 0 β/H − − e f/f RH (  D D m 3 + ( T . × × 0  Hz 8 5 Pl − − M 10 10 2 decays. The abundance of gravitational waves Ω decay. The ratio (  0 0 2 DM for reheat temperatures as low as × × ν ν 0 2 2 e πα , the freeze-in yield of consistent with the observed top quark mass, mirror quark masses are ' ' 0 e 0 02 DM via kinetic mixing is large enough to already produce recoil signals A similar calculation for the proposed LZ experiment, which can probe . p v 2 from in comparison with the Hubble time scale 0 0 f 8 h e 1 < m ≈ D ]. The phase transition proceeds by nucleation of bubbles, which collide with is the energy density of the mirror gluon bath, − f col , β 0 model is not necessarily excluded. FI ln g RH , 51 0 0 s , ρ d T e GW is not the DM, or is produced in a non-thermal way, the red region is not applicable and the ρ 50 0 Ω SM e through the Higgs portal is dominantly set by its yukawa coupling, which is fixed d We consider the case where the For 0 × If is close to the frequency at the peak of the distribution and e 8 to freeze-in p the dilution transition density, SM f temperature of theassuming mirror that QCD the phase velocity of transition. the bubble Here wall we is use the speed the of results light, of and take ref. into [ account by dilution from late produced by the bubble collisions as a function of a frequency 7 Gravitational waves from theIn mirror the range QCD of phasemuch transition larger than theorder mirror [ QCD scale.each The other and mirror produce QCD gravitational phase waves transition [ is then first  at XENON1T. an order of magnitudepotential smaller, to probe produces nearly the all green reheat contour temperatures ‘LZ’. capable For of low freezing-in The black dotted contours into the be region frozen-in as DM. The shaded red region is excluded if and whose smallness preventsother sufficient hand, the freeze-in abundanceparameter of (indirectly set bysufficiently the produce unification scale of JHEP02(2020)078 = -19 -20 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 3 10 10 10 10 10 10 10 10 10 10 10 10 10 (7.3) (7.4) 10 β/H 2 10 ) >3σ . Z 6 . The ratio m 10

( / S 4 1

t, Δα  2 1 Δm σ 5 0 . 0 2 QCD b is -1 0 T 1σ  10 0 2  b h is the energy density DECIGO -2

BBO 2 SM / 10 ]. Likewise, we expect 10 = 1

0σ tub β/H , -3 SM parametrizes the energy 3 /ρ 0 54  ') > 0.3 ρ 10 g 0  f[Hz] 0

LISA eff b g ρ tot GW -4 3 ΔN DM ρ / 2 10 /ρ >Ω / 2 11 e' -5 / 3 Ω tot 3 10 8) (ν'→ e'+u'+d . ρ  -6  10 lat kin + 0  3 -7 ρ ρ / p 10 1 0 = 100 g 3 lat -8  0 ) ρ c β H ρ f/f 10 p T )( dec away from their current central values. tot 100 g 3 100 GeV σ f/f / 2 /ρ are the degrees of freedom of the SM and the 0 2 ] finds a spectrum of gravitational waves produced by 0 dec 9( H/β 10 60 g g ) >3σ – 22 – ρ Z 55   02 0

(m dec  . S 10  g , Δα

t 3 3 2

σ ] assuming that the bubble walls expand at the speed / / 1 Δm 10 + 0 4 1 β/H . p − − and -1

1σ 57  , B 10 D D f/f -2 55

DECIGO dec × ×

BBO ( g 10 ) lie more than 2

0σ Hz 9 4 Z -3 ]. − − 10 m f[Hz]

> 0.3 as is the case for a weakly coupled, relativistic transition, and LISA (

eff 55 -4 10 10 0 ΔN S ') g 10 α × × ρ -5 DM is the kinetic energy of the bubble wall and 4 1 10 = 1 [ >Ω and e' -6 ' ' kin (1), and so we approximate this ratio as unity [ Ω t lat ρ 10 = 100 (right). Future gravitational wave detectors such as LISA and BBO may (ν'→ e'+u'+d p 2 O m -7 ]. The abundance of such gravitational waves Ω f /ρ h 10 is = 10 56 kin -8 0 β/H f β H tub g ρ , 10 ln . Gravitational wave spectrum generated by the mirror QCD phase transition for Numerically, this contribution is smaller than the one from the bubble collision. -9 /ρ is estimated in appendix d GW 9 10 -8 -9 0 -16 -17 -18 -19 -20 -11 -12 -13 -14 -15 -10 lat

g Ω 10 10

ρ

10 10 10 10 10 10 10 10 10 10 10 lnf d is comparable to d Gravitational waves are also produced by the turbulent motion of fluids induced by

/ρ

Since the mirror QCD bath couples to the standard model particles very weakly, bubbles only induce

GW

Ω h d 9 2 SM kin Here we use the resultsof of refs. light. [ turbulent motion of mirrortransition glueballs. generating In magnetic particular, a turbulence, turbulent ref. magnetic [ field is not induced. For a phase the bubbles [ so we take mirror sector at the decouplingdensity of of the the two sectors, mirrorρ respectively. gluons just before the phase transition, phase transition, of the SM bath,show all of whichρ are evaluated at the phase transition. Lattice calculations Figure 14 10 (left) and detect a signal if JHEP02(2020)078 ). S and 0 , α c = 4 is t T m n ]. are shown via  61 0 ), leading to , v the prediction 60 3.7 fails as is shown 14 0 of (10,100) in the ν ). The entire SM La- . The scaling depends β/H 1 3.2 , taking t ]. If the top quark mass is large m 58 into a group such as SU(5), the proton , is dark matter, with an abundance set 0 e dark matter can occur via kinetic mixing (100) [ G v 0 O e , which sets the scale at which the SM Higgs 0 – 23 – . The correlation of these two rates for v ). matter-dominated era. This condition is satisifed G 0 Z v ν m ( S α . Furthermore, since kinetic mixing vanishes in the unified ) for the gravity wave spectrum depends on ) is likely to be 4 and G t 7.3 /v 1 m β/H ), ( ∝ , are not well-understood because of the non-perturbative nature. p determined by the top quark mass, we show in figure 7.1 lat 0 v /ρ . A large fraction of the allowed parameter space of the theory will be level for is a model-dependent, positive integer. Thus proton decay excludes small ]. kin 5 σ ρ n 57 3 ]. We note that prediction for the gravitational wave spectrum assumes that . With . The ratio ( − 59 . D ) and 10 3 , where n G If the SM gauge group is unified at scale There are several interesting theories containing the Higgs Parity mechanism for the We also note that many aspects of the phase transition in QCD-like theories, such The prediction ( v β/H and direct detection excludes large ∝ G v shown in figure probed by a combination of Hyper-Kamiokandeturbulent and magnetic fields LZ. similar to thatprovided from in turbulent ref. motion [ of fluids, and hence we simply use the fitting in figure decay rate scales astheory, Γ it may arise from a higher dimensional operators, such as in eq. ( vanishing of the Higgs quarticwhere at the high Higgs energies. Parity Mirror partnerby Higgs of thermal Parity the mechanisms. is electron, the Directand simplest detection theory leads of tofrom a XENON1T recoil and the spectrum future characteristic reach of of LZ photon on exchange. the kinetic mixing The parameter present bound and the only unknown parameterstals are that those connect of the the two kineticof sectors. mixing, the Higgs SM The and spectrum, spectrum neutrino as ofonly shown por- the for on mirror the light sector the mirror is Higgs particlesquartic a in vanishes Parity scaled figure and breaking up will version become scale better determined by precision measurements of ( 8 Conclusions and discussions We have introduced the Mirror Higgsgrangian, Parity including theory, dimension described 5 by operators ( for neutrino masses, is replicated by Higgs parity as ( Once the phase transitiontency is of well-understood, future it measurements will ofFor the become a top possible survey quark of to mass gauge check and theories the the exhibiting consis- gravitational first wave order spectrum. phase transitions, see [ in figure enough, gravitational waves can be detectedand by BBO future experiments [ such asthe LISA, phase DECIGO transition occurs beforein the the region whereis, future at experiments the may 2 detect the gravitational wave spectrum, that especially for the spectrum of(left, the gravitational right) waves panel. for various collision The and dashed the and turbulent motionIn dotted respectively, the lines and blue show the shaded solid the region, lines the contribution show freeze-out from the followed the sum by of the bubble them. dilution from JHEP02(2020)078 0 , v eff and N (8.1) 0 . Part v between 14 . t eff m . ) GeV, is first N  ]. The mass is 12 relic abundance 70 0 ]. If Higgs Parity – e –10 , , for the case that 8 and ∆ 65 67 – t , the SM and mirror . Fixing the neutrino 10 m 62 GeV DM can arise via freeze- a RH 0 f = (10 T `H e 10 0 , . The constraints on this v 0 10 ν RH m 11 T / ) GeV, predicting a Higgs mass , by requiring the Higgs quartic 8 σ 10  decay to and hence does not solve the strong 0 0 , we predict the contribution to dark ]. The mass is correlated with –10 , for a suitable value of ν 8 0 0 e GeV 66 ) plane in figure v 10 ν 9 m ]. In this case, it is even possible to have a 10 , m 70 0 3 GeV at 1  v (40–1000) GeV for  – 24 – ∼ which determine 0 ν T 4 keV . . In the rest of the unshaded region this occurs via m decay dilution factor, and is shown in figure . Remarkably, the corresponding neutrino is required 0 0 = 0 RH ν ν T ), and 123 ) at their measured central values, we predict S a , and S 2 f 0 0 QCD e , α t GeV, since the large mass prevents the production of axions in Λ , α m h 9 6 m . m 10 must be in the range of (10 in ( = 0 . 0  t 0 v σ a m m . On the edge of the blue shaded region this occurs via the Higgs portal, ). An advantage of such a heavy axion is that it is easier to understand 13 . Fixing ( 0 8.1 v 5) GeV. . 6 GeV at 3  , which determines 0 2–175 . Alternatively, if the QCD axion is neutral under Higgs Parity it couples to QCD and Mirror Higgs Parity exchanges SU(3) with SU(3) For low values of the reheat temperature after inflation, Within this allowed unshaded region of figure Since all the mirror quarks are much heavier than the mirror confining scale, the mirror For large values of the reheat temperature after inflation, v mirror QCD with the sametheta decay angles constant. in Since the Higgs twogiven sectors, Parity by ensures the eq. the strong ( equality CPthe of problem PQ the is symmetry as still an solvedsmall accidental [ decay symmetry constant [ where the topological susceptibility ishence taken with from the [ top quark mass. Both axions may contribute to the dark matter density. kinetic mixing, dominated at temperatures near CP problem. One possibletransforms solution the is QCD to axionlike-particle introduce into with a a a QCD mirror mass axion QCD [ axion, the mirror QCD axion is an axion- gravity wave signal in these experiments would be correlatedin with production. The observedregion DM of abundance figure may bewhich obtained is anywhere UV in dominated the around unshaded QCD phase transition, which occurs at order and produces gravitational waves from bubblethe dynamics transition. and turbulent The fluid spectral motion energy at is density then today, normalized obtained to by the includingof critical energy the the density, allowed region of the theory can be probed by LISA, DECIGO and BBO, and a radiation arising from decaysshown of by mirror purple glueballs contours, toand varies mirror from therefore photons. with about The 0.04 resulting to ∆ 0.4, and is highly correlated with dilution is dominated byto a have single a massdata. larger Furthermore, than 0.01of eV, 123 in the rangeto vanish of at masses determined(173 from oscillation sectors reach thermal equilibrium viaarises the first Higgs portal from interaction. freeze-outportal The and parameters to is obtain thenare the diluted observed by abundance,scheme the for remaining relevant dark parameters matter are shown in the ( JHEP02(2020)078 q are s (A.4) (A.5) (A.6) (A.1) (A.2) (A.3) ,  and , µ 1 6 q,IR 3 = rel α v 8 8 , , b , c πc c n n 2 b n c c Γ  except for the titles of 2 f | 0 cb + 3Γ UV + 5Γ V , the decays of b | q, b s , , n 2 3 n ) b b )) is negligible and we solve the bound state, namely the soft Γ q eq 2 q Γ πα + 11 2 b ¯ q b, ) and | 2 m | c q m n ( cb n µ T Nuclear Physics, of the US stellar objects and meson decays. We will discuss the phenomenology of axion dark matter JHEP02(2020)078 , ) e (A.7) (A.8) (A.9) m (A.16) (A.10) (A.11) (A.12) (A.13) . − .  0 f  q T is negligible. ) (A.14) ) (A.17) = 0) by the mirror f ) (A.15) d f = 0) )Θ( e e 2 γ m m g , , µ , m . n 0 s , µ γ s n s 0 − − T n n n − 0 n − ( 0 s T s and s 0 ( 2 e T Γ T Γ eq Γ T 2 u n 2 eq 2 | g, ( | | γ, n i )Θ( us )Θ( us with a charge us n )Θ( 2 g 2 f 2 g V V , rel 2 e V | 2 f ` n n | | n v , , the decay of n ,  n ) e ) 0 − − 0 − − + 7 − + 3 + 3 2 g T rel 2 g T πα 2 f 2 f 2 γ µ v n µ n 2 µ n therm n − ( n n and − ( n n σ ( µ + i  f µ i h − µ e u i i , rel m rel m  rel v d rel . v f v ) ) + v are 4 ) + Γ = 0) 2 f f ) + Γ ) + Γ 2 ` 3 )Θ( v )Θ( ` 3 eq 2 f 2 e are set by freeze-in. During the reheating eq m m eq → m 5 f g πα , µ therm π n u, n 2 d, therm e, 0 u therm n σ − − σ n m σ n n ! T h σ − h 0 0 − h 2 ( f h ) − ) − T T q , − 1536 γ eq γ s and 2 u – 26 – 2 d 2 e ) + ) + g, ) + n n ) + n n 2 f e 2 γ = , µ 2 g , µ s )Θ( )Θ( n f<` ( X ( 2 e 0 ( 0 n n n 2 γ 2 f Γ i f 2 f i i , n 2 n n µ n | Γ /T − /T − − rel rel rel − e 1 + n f v v v us − − − 2 H µ 2 H 2 H d u e

m 2 H V m 2 f 2 γ n | 2 f n n ( σ σ σ ( n ( ( ( n n 4 4Γ n = ( is transferred into the abundance of ( ( eq eq

i  i i − h − h − − h − 2 γ, 2 γ, d rel i rel rel rel During the freeze-out of n rel n rel rel v v v = = = = = v ( ( v ` rel v v ¯ f i γ g i s e ¯ e d u v µ σ 2 2 f e 3 rel rel → → → → v v † † † → Hn Hn † Hn Hn Hn therm 2 therm f e σ σ σ σ σ h h h h h HH HH HH HH + 3 + 3 + 3 + 3 + 3 σ σ σ σ s e d u + + + + + µ

h ˙ ˙ ˙ n n ˙ ˙ n n n , the relic abundances of = = = = freeze-out. e g γ f 0 dec e Hn Hn Hn Hn < T is defined by f and + 3 + 3 + 3 + 3 RH 0 e g γ f T ˙ ˙ u n ˙ ˙ n n n , 0 A.2 Freeze-in For era, the Boltzmann equations are given by The freeze-out abundance of beta decay. The Boltzmann equation is given by where the summation is taken for mirror fermionsd lighter than The annihilation cross section of a mirror lepton Here Γ JHEP02(2020)078 , g or EM (A.26) (A.22) (A.23) (A.24) (A.25) (A.18) (A.19) (A.20) (A.21) α n  0 m T equals , and gluons, i is the SM Higgs. − γ α  H exp ) bremsstrahlung cross- 2 2 / ,   3 , 2 ggg 2 f  3  ! 0 Q ] i → 1 6 3 m T 0 n i f 73  T q X Pl α , , the thermalization cross-section and subscript 8 X π 4   T  M 72 .

¯ T 4 2 r 2 i RH fg, gg 0 ln ∼ 4 2 2 2 RH 0 0 T v T f 1 ) = ) T v T 2 πα T − is the total number density of the mirror 2 T T 4 2  →  ( (  /  0 i 1 ¯ π f 0 tot µ α 0 S T π 0 tot 4 0 tot n  n i i ≈ 4 α 2 ρ T n ∗ 2 f 0  α + g y v 1 3 rel  π 2 0 v  – 27 – ¯ 1 π π fγ, f 10 π 1 1 m 3 T i = 8 16 2 f 2 , and 0 α  ' ' ' − T T → therm 5

18  is taken for mirror fermions and quarks with masses ¯ σ f i ≈ h rel rel rel f q = v v v rel ¯ exp g v γ f 2 2 f H 2 3 depending on whether the exchange involves mirror photons / → → → 3 and → † f † † 2  n f σ 0 h HH . HH HH π or 3 σ σ 0 σ 2

mT e

T n 2  g 2 π g =  . Initially possessing a typical energy equals i T ) is the total energy density of the mirror sector frozen in via the Higgs portal , µ n = 0) = 0 T ( m T , µ 0  0 tot ρ T ) and eq ( 0 n Soft-scattering keeps the mirror bath in kinetic equilibrium (but not necessarily chem- eq T ( n is the mirror fermion with the largest mass below S the equilbrium number densities are where when the universe is atsector a temperature determined from the Boltzmann equations. For mirror photons, α or gluons. ical equilibrium), establishing an effective temperature and is the Hubble scale during the reheating matter-dominated era. Here, while the soft, number-changing ( sections are given by greater than among mirror charged fermions is given by where the summation on The production cross sections from the SM Higgs are [ f JHEP02(2020)078 0 ) is e , e ) 2 f , and A.21 n g (A.29) (A.27)  < m ) 2 ( − , n 0 e ) T g → , n = 0) f )) (A.30) , µ ) (A.31) 0 n γ g g , µ 0 n /T , µ T , µ 0 f 0 ( ) and 2 m eq /T ( /T g, f f eq . n ) (A.32) , m 2 e 2 g, m ) 2 g 4 A.22 ( ( e 2 n n n . This thermalization acts  0 ( eq n eq 3 ( are in the thermal bath; the i − T − 2 γ, 2 g, 0 − RH T ) n 2 g n rel γ γ T → γ 2 n v − n −  f ) if a lepton. Nevertheless, these ( , µ + 2 f  0 2 f σ ). g i h n n n ( ( , and A.8 rel via 2 /T 0 . i i and any pre-thermalized contribution v e ) (A.28) + ) g ) = 0) ) + = 0) γ A.32 2 f e rel , γ rel m 2 f 0 n v ( v n m RH s n RH , µ γ g )–( , µ f 0 f determine the evolution of 0 T eq , T − therm 0 n n + − σ and − ( σ and decreasing T 2 γ, T ) σ h d 0 h e ( ) ( h n e , 0 tot RH γ n T 0 ( A.27 eq eq 0 tot /T ρ i u ) + f γ, g, – 28 – ) + n + ) + , µ , 2 γ 2 g 0 0 n n rel )Θ( e 2 < T f m n n v 2 γ µ 2 f 2 f ( n 0 n e n /T , n n − 0 eq − 1 3 f σ − e h − − − 2 H, ) if a quark, and ( 2 H 2 H m 2 H < T ' 2 e n ( 2 f 2 f n = n n ( ( ( n e n n ( e eq ( A.2

0 tot 0 tot i   m i 2 γ, ρ n i i rel rel rel n rel Hn rel v ( drops below its mass. The Boltzmann equation for v v 1 3 v rel rel v ¯ i 0 f g γ ¯ e v v 2 f 2 e = T + 3 , thermalization of 3 3 rel , discussed below ( 0 → → → → v e † † † → → † ˙ T n therm 2 f 2 RH RH given by ( σ σ σ σ T T h h h h HH HH HH HH i σ σ σ σ + + + + rel

h , the universe is radiation dominated. The mirror bath remains in kinetic v f = = = = RH σ e g γ f h and high Hn Hn Hn Hn 0 v T < T freezes-out when ). Around this temperature, + 3 + 3 + 3 + 3 0 e e g γ f ˙ 5.1 ˙ The SM and mirror sectors decouple from each other at the temperature shown in For n ˙ ˙ n n n , and are given by γ which is used to estimatearound the the dark mirror QCD radiation phase abundance.in transition. We enhancement Entropy assume of production entropy the via conservation signals. super-cooling will result eq. ( B Energy densities ofIn the this mirror appendix QCDthe bath we energy estimate density the atgravitational energy the waves, density and phase of the transition, the energy which mirror density is QCD of used bath. the to mirror estimate We glueballs the derive after magnitude the of transition, Last, The Boltzmann equations for n is typically small, theoccurs most at and important below contributions to the present-dayequilibrium abundance (not of necessarily chemical equilibrium), establishing an effective temperature processes are effective, thereby increasing to cool the mirrorcross-section bath so that mirrorfrozen-out particles particles are freeze-out then instantly continuallytal. with replenished Since by an fresh freeze-in annihilation particles production from is the maximized Higgs at por- For low JHEP02(2020)078 3 2 ' ) f 0 S → until (B.3) (B.4) (B.1) (B.2) /a c m 1 a − ( ) as shown 3 a 0 c c T . (ln /a - an excellent 45 1 f 3 is the would-be / 0 f a − − 2 c is determined by T a  0 π a g c f = 1. A . s ∝ a a x B 60. After decoupling, = 32 ln . = 0 ' 3  g / s 4 W ∝ ∼ 0 dec  is determined by the former 1 g W e f − as function of 0 dec , 60 a , 0 g ! . We take 3 6 5 S is then numerically solved for as A f / 1 0 1  c a m 2 reactions among mirror glueballs 75 c . f  6 dec a GeV, /T a 5 0 g . →  f b 9 0 0 106 0 T , the ratio of the actual mirror glueball c π S 3 10 4 T  A m 0 dec and energy density as 3 60  ] and equating it with > g /  0 1 3 a 2 22 ) v c  dec π g  ln B 5 g 2 . b ) to determine is the scale factor of the universe. The 3 – 29 – (4 0 ∝ π  a 45 3 B.4 32 , the ratio of the temperatures of the two sectors is . 75 . 4 c ) from [ i ' 0 0  c g 2 v 3 = 0 106 2 ) /T b T 0 f 0 2 π 2 annihilations keep warm the mirror glueballs so that →  g = T 3 SM 5 (2 ]. Here, ( T 0 σ . , the non-trivial dynamics around the mirror QCD phase T 0 g h → g

76 . ρ ρ 5 ] – = 0 W 15 76 74 0 0 . 3 , the energy density of the mirror glueball bath deviates from that S 0 g 3 0 c if the mirror gluons remain an ideal gas until the phase transition, c SM ) is invalid. In this strongly interacting regime, ρ − T ρ m 0 T c a 2 T and inserted into ( B.4 . , the energy and the entropy density of the mirror QCD bath is well- 4 3 0 c c 0 ∝ = T (1) number whose value weakly affects T 0 = /a 7 . T 0 f O 0 c f 0 . a T T 0 c 3 4 . T ) is the product-log function, which is a solution of 0 7 . is the effective number of degrees of freedom of the SM bath at the mirror QCD x = is an ]. Entropy conservation within this decoupled mirror bath implies its entropy T ( c is the scale factor of the universe when the 3 A 22 g B [ W f 0 a For 0 As discussed in section For c T 3 . taking the lattice result for approximation since the glueball gas isis nearly the pressureless. entropy Here, density ofa the function mirror of glueball bath. for both regimes in figure freeze-out, or the mirror glueballsand decay. otherwise For by the latter. of a weakly-interacting ideal gassecond equality composed of of ( the lightest mirror glueballs, and hence the Here, scale factor at and where transition are encoded in theenergy density modification to factor that derived by anumber non-interacting conservation, ideal gas approximation and the glueball they decouple or decaycross-section [ is given by [ approximated by that of5 the ideal gas ofdensity the scales lightest mirror as glueballstheir with temperature a falls mass approximately as where phase transition. The ratio of the energy densities is the entropies of thetransition, two sectors the are mirror gluon separatelyof bath conserved. the is Around mirror nearly the gluon pressureless. mirror bath Parametrizing QCD by the phase energy density effective number of degrees of freedom of the mirror sector is JHEP02(2020)078 ] , ]. JHEP ]. , Phys. Rev. , Phys. Rev. , SPIRE SPIRE 30 IN IN noninvariance ' ][ c ][ t arXiv:1208.6013 ]. [ is defined as the energy and p A SPIRE 25 . IN ]. c ][ /a (2012) 103 f a 11 SPIRE arXiv:1312.7108 arXiv:1405.3692 IN ' T'≲ 0.7T [ 20 c [ ][ JHEP , T'≳ T 0.7

Approximation f c a a (2014) 214 arXiv:1403.8138 15 ]. Flat Higgs potential from Planck scale

Gas [ as a function of noninvariances in a superweak theory – 30 – (2014) 075006 A T Ideal , Split Dirac supersymmetry: an ultraviolet completion of SPIRE B 732 Natural suppression of strong Grand Unification, axion and inflation in intermediate P IN arXiv:1803.08119 D 90 CP violation in seesaw models of quark masses ][ 10 acting [ (2014) 137 ]. Inter ), which permits any use, distribution and reproduction in ]. 06 Strong . Implications of Higgs discovery for the strong CP problem and Grand unification and intermediate scale supersymmetry Axion-Higgs unification

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