Jhep05(2021)160

JHEP05(2021)160 Springer May 3, 2021 1 May 19, 2021 : March 4, 2021 : : , Accepted Received Published , Zhao-Huan Yu 1 Published for SISSA by https://doi.org/10.1007/JHEP05(2021)160 [email protected] , [email protected] [email protected] , , . 1 3 2102.01588 The Authors. Beyond Standard Model, Cosmology of Theories beyond the SM, Higgs c

We investigate the potential stochastic gravitational waves from first-order , [email protected] Corresponding author. 1 School of Physics, SunGuangzhou Yat-Sen University, 510275, China E-mail: [email protected] [email protected] Open Access Article funded by SCOAP ArXiv ePrint: space-based interferometer experiments LISA, Taiji, and TianQin. Keywords: Physics, Thermal Field Theory scalar fields in theat model the obtain early Universe. vacuum expectation Wetransitions, search values, taking for related into the account to parameteritational the phase points wave existed that transitions spectra experimental can are cause constraints.strong first-order further The phase evaluated. gravitational resulting wave Some grav- signals, parameter points which are have found the to opportunity induce to be detected in future Abstract: electroweak phase transitions in atwo model Higgs with doublets. pseudo-Nambu-Goldstonebounds, dark matter and The and can achieve dark the matter observed relic candidate abundance can via the naturally thermal evade mechanism. direct Three detection Zhao Zhang, Chengfeng Cai,and Xue-Min Hong-Hao Jiang, Zhang Yi-Lei Tang, Phase transition gravitational wavespseudo-Nambu-Goldstone from dark matter anddoublets two Higgs JHEP05(2021)160 ]. Therefore, the 6 – 4 ], where the DM candidate is a pNGB protected – 1 – 20 – 7 14 ]. Therefore, other experimental approaches are crucial 19 , 5 9 , ]. Thus, the DM relic abundance would be determined by the 8 16 3 6 – 10 1 1 2 ]. Although loop corrections give rise to a nonzero scattering cross section, 21 7 There are various ways to experimentally test DM models, including direct and in- Such a situation can be circumvented if one can effectively suppress DM-nucleon scat- one-loop calculations show that nearto future direct probe detection the experiments would pNGBfor not DM be exploring [ able these models. direct DM detection, collider searches, etc. The discovery of gravitational waves (GWs) tering at zeroepoch. momentum An transfer appealing without approachGoldstone to reducing boson achieve (pNGB) DM this DM is models annihilation provided [ by at by a Higgs-portal the global pseudo-Nambu- freeze-out symmetryture which makes the is tree-level softly DM-nucleontransfer broken scattering limit by [ amplitude vanish quadratic in mass the terms. zero momentum The pNGB na- of various direct detection experimentsscattering. searching Nonetheless, for no nuclear DM recoil signalto signals is robustly stringent induced found by constraints in DM on thesethermal experiments the so production DM-nucleon far, paradigm leading scattering faces cross a section serious [ challenge. It is conventionally believedat that the dark early matter Universeannihilation (DM) [ cross originates section at from the freeze-out thermalthat epoch. production the The relic natural abundance strength observationbe suggests of close the to DM the couplings weak to interaction standard strength. model (SM) This particles motivates should the worldwide establishment 1 Introduction 6 Gravitational wave spectra 7 Numerical analyses 8 Summary 3 Experimental bounds 4 Effective potential 5 Phase transitions Contents 1 Introduction 2 The model JHEP05(2021)160 ], S 54 U(1) – give → − 52 7 ]. These S becomes a 45 S , both carrying 2 . Section Φ ]. The singlet and ], TianQin [ 6 symmetry 7 [ which is only softly 2 51 ], we assume that the 2 and 2 Z Z 1 64 Φ Φ into → − 2 Φ U(1) . The effective potential at finite or 3 1 ], are able to probe sub-Hz bands and Φ Higgs doublets L 60 , → − 59 conservation is assumed in the scalar sector, 1 symmetry is further spontaneously broken after explicitly violated into a which is a SM gauge singlet. The Lagrangian – 2 – Φ SU(2) . We analyze key properties of the EWPT in S S ]. Such stochastic GWs typically peak around the CP ], or the most general breaking terms [ iα 4 49 U(1) e 36 – → 22 S ], and BBO [ symmetry , we briefly introduce the model and the particle masses. 2 2 58 Z , 57 ], to which ground-based laser interferometers are not sensitive. ], which is expected to involve more phase transition patterns. The 50 symmetry 15 . U(1) , and a complex scalar 8 ]. This model involves two 2 / 15 1 ] provides a new path. DM fields could be relevant to strong first-order elec- ], DECIGO [ 21 ], we study the possibility of extending the minimal pNGB DM setup with an addi- ] showed that such phase transitions can only be of second order and impossible to 56 , which are relevant to the GW spectra discussed in section , 63 5 12 55 – symmetry and a quadratic term that softly breaks As in the simplified versions of the two-Higgs-doublet models [ In the following section It would be rather interesting if we can find out a pNGB DM setup that allows both The minimal setup of the pNGB DM involves a complex scalar singlet with a global 61 develops a vacuum expectation value (VEV). Then the imaginary part of pNGB, acting as a DM candidate. scalar potential respects a broken by quadratic terms.leading to Moreover, only real coefficients. The potential satisfying these two assumptions and the found in ref. [ hypercharge respects a global by soft breaking quadratic terms. The S paper in section 2 The model In this section, we briefly describe the model we are interested in. More details can be Existed experimental bounds aretemperature described is in constructed section section in section numerical analyses of GW signals based on random parameter scans. We summarize the the vanishing DM-nucleon scatteringthe and notable detectable GW stochastic signalsels GW from [ signals. the strong Inspired FOPTstional by obtained Higgs in doublet [ the two-Higgs-doubletcorresponding mod- phase transitions and stochastic GWs will be studied in this paper in detail. symmetry, e.g., the softefforts cubic successfully terms achieved [ first-order phase transitionsthe (FOPTs) essential and merit stochastic of GWs, the but vanishingtransfer tree-level limit DM-nucleon scattering is in sacrificed. the zero momentum look for the stochastic GW signals. U(1) the SM Higgs doubletstudy together [ could induceproduce two-step stochastic phase GWs. transitions. Further studies Nevertheless, tried a to introduce extra terms to break the troweak phase transitions (EWPTs) thatproposed produce future detectable GW stochastic experiments GW [ mHz signals frequency in the band [ Nonetheless, future space-based GW interferometer plans,Taiji e.g., [ LISA [ by LIGO [ JHEP05(2021)160 . soft (2.9) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (2.1) (2.2) V . For (2.10) ∗ + S ]. develops 7 ,S i 4 2 → Φ | 2 Φ V S Φ | = or 2 2 1 λ V symmetry remain Φ + ] 4 | 2 ) 1 CP 1 symmetry , , Φ ) | ) Φ 2 s 2 s 1 † 2 . v 2 v 2 λ 1 | CP 2 κ κ S | + (Φ 2 , , , ) + + + | ) 1 2 2 2 real and positive through a phase 2 2 2 1 ! ! ) , respectively. Thus, at the zero 0 S Φ v Φ v 2 s 2 2 2 . | 0 S † 2 m v c . 2 Φ . √ √ 5 345 m † 1 345 κ / / − λ λ λ ) ) , + Φ 2 s 1 2 + 2 1 + + H [(Φ 2 + v + acts as a stable DM candidate [ 2 v v iη iη 5 4 | 2 S , and 2 1 Φ 2 2 2 . λ 2 + 1 + 2 λ S λ † 1 v S S v + + | v 1 φ φ 2 2 iχ 2 0 S 1 2 + + + λ | , (Φ λ 4 ρ ρ ( 1 1 ( 3 2 2 2 m + | 2 12 – 3 – v 2 1 2 gains a nonzero VEV, the photon would become 1 2 v arctan λ Φ 2 + + s | − 2 , leading to three stationary point conditions, m √ 2 1 Φ 1 2 − ) κ ≡ ≡ − + κ † 1 v v Φ s = − ( ( β s β β + Φ 2 + | , v v

| 345 or 2 1 4 2 4 2 soft λ v | 1 λ cot tan = = = V 1 , v Φ S | Φ 1 κ | 1 2 + 12 2 12 2 S ( v 2 22 S Φ Φ 2 ( 2 1 2 | m m λ m 2 = = = Φ + + developed must be real, and the dark | 2 2 S 2 2 | 22 2 11 2 would be spontaneously broken. Detailed discussions on vacuum | | 1 S 1 m S m m | Φ Φ | 2 S | 3 CP λ m 2 11 develop nonzero VEVs m + − . Consequently, the potential respects a dark S = S ,S symmetry reads i Φ soft breaking terms , and ]. Here we are particularly interested in the case that only the neutral real parts V , the VEV of 2 0 65 Φ U(1) , U(1) > 1 If the charged component of We can always make the soft breaking parameter Φ 2 0 S where The potential is minimized at temperature, these scalar fields can be expanded as massive, and the theoryan is imaginary unacceptable. VEV, Ifconfigurations the and neutral parameter component relations of in in ref. general [ two-Higgs-doublet modelsof can be found are further introduced in the potential. Thus, the total scalarredefinition potential of is m unbroken, ensuring that the imaginary part of The global JHEP05(2021)160 ]. 15 , (2.14) (2.15) (2.16) (2.11) (2.12) (2.13) 7 -nucleon scat- . The Yukawa χ 3 ., ., , c c . . 2 , -even neutral scalars ]. In this paper, we + H + H = 1 2 1 . CP gauge couplings, respec- are transformed into the 67 are transformed into the , Φ Φ . We remark that due to must behave as a SM-like 2 2 i Y + 2 R R i, j η i v i i d with a mass of φ 1 ` ` 66 h , v L L , χ i i ) 2 U(1) 0 5 64 ¯ ¯ and L L , and g i i and λ , and ` ` 1 i ∗ 2 ˜ ˜ y y η 2 + + 1 + u and v Φ 2 φ 4 − − 1 2 g L λ v of this matrix are the masses squared 2 2 ( 5 iσ q ˜ ˜ Φ Φ 3 1 2 λ . 2 2 h v R R ≡ 0 S 0 j 0 j − − m u u SU(2) = 2 m L L 2 12 2 12 ˜ i i Φ Z = ¯ ¯ are the Nambu-Goldstone bosons eaten by the m , m Q Q – 4 – χ T ij ij u u is the Fermi constant. , and ± ) , satisfying the experimental observations. After ˜ ˜ y y 2 m β β L F G , respectively. One of 2 h , m 0 i are extra Higgs bosons, whose masses squared are 3 − − G m 2 mass-squared matrix for the h , d ± 2 1 gv cos cos 1 1 , L denote the 3 -odd neutral scalars 0 i Φ Φ and β β 1 H is a massless Nambu-Goldstone boson due to the global = 0 is equivalent to the Higgs VEV in the SM and can be u 2 h R R gauge bosons are given by 0 × ( g 0 j 0 j χ 125 GeV v m CP sin sin d d G 3 W , and Z L L ≡ 2 and i i . , where can be diagonalized by unitary matrices, which transform the ∼ m = = into the mass eigenstates h 2 ¯ ¯ and + L Q Q a / i , 2 a 0 i ij u ± 1 , while the charged scalars and ij ij 1 g , the d d ˜ d H y Q a − ˜ ˜ 2 H m y y h β ) , − − F W m T ) and and G and and = = L , and 2 i 0 i I ij 2 d 2 + 0 , II u ˜ √ , y v , ` Y G G Y L L i + L = ( ν 2 1 ( v v . The eigenvalues gauge bosons. , the neutral boson s only appears in pairs in the interaction terms, guaranteeing its stability to ≡ q ± χ L ≡ i = 0 W L v 2 0 S , and symmetry. The soft breaking terms endow the pNGB 2 For the two Higgs doublets, four types of Yukawa couplings without tree-level flavor- The masses of the The VEVs contribute to a m ρ and , 1 coupling matrices gauge eigenstates the similarity of thein Yukawa couplings the in leptonic the sectors, quark manycast sectors of to and the the following the lepton analyses smallness specific for of (flipped) the the case. type-I ones (type-II) case can be given by where tively. Thus, weexpressed observe as that changing neutral currents (FCNCs)only focus can on be the type-I constructed and type-II [ Yukawa couplings, whose Lagrangians are respectively where Besides, become a DM candidate.tering The amplitude pNGB in the feature zero also momentumThus, eliminates transfer limit this the without model tree-level any is parameter tuning hardly [ constrained by DM direct detection experiments. For U(1) mass eigenstates Z given by ρ for the mass eigenstates Higgs boson with arotations mass with the of angle mass eigenstates JHEP05(2021)160 ] d i 74 v [ and (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) − ann µ σ h + µ . channel from → 0 . b s ¯ b ≥ B 2 12 ]. The numerical significantly con- ( FeynRules 2 23 . The analysis by 70 a − ± [ β ] should be satisfied: µ , m 13 H s in the + a 69 i , µ tan , 0 12 0 v , v a , 68 → 2 0 χ ≥ ann d 16 GeV and q 2 . σ ≥ h B 0 λ ± should be positive to guarantee + 1 is the ratio of the DM energy 23 ± H λ 1 β, m a + λ is the Hubble constant in unit of m 2 H p 13 , 18 DM a . p 0 tan m h + 2 Ω ] to calculate the prediction of the DM 12 | κ ≥ a 5 , 76 2 2 S λ + [ λ = 125 √ 2 , and 2 2 a − | λ h λ , κ + 4 1 m p 1 m p λ ], where 1 λ – 5 – ), κ + + 3 77 , κ acting as the SM-like Higgs boson with a mass p [ , MadDM 3 2 3 + 2 S 2 i κ λ κ , S h λ ≡ ≡ , which is corresponding to DM annihilation processes + , λ 5 0012 5 p = 1 2 . 23 0 12 ] are employed to test the parameter points. ) − , λ i 0 | ( 0 ] plugin 5 78 10 p , λ i λ ± 1 4 2 h 75 ≥ confidence level (C.L.) are abandoned. , a [ × , a κ 0 0 − | m S 2 + 4 , λ ≥ ≥ 1200 3 λ S . 95% -meson decays, depending on 2 S λ , λ + B λ λ 0 ] is used to test whether the SM-like Higgs boson is consistent with ] shows that the strongest constraint on the type-I (type-II) Yukawa 1 1 p 3 , λ is also used to compute the DM annihilation cross section = 0 3 λ λ 2 λ ≥ 73 λ 72 2 . Only the parameter points predicting the observed relic abundance are , p p 2 h 1 + + ( , λ 71 + + − 1 [ S S range of the measured value 3 1 DM λ λ λ , λ λ κ 2 2 σ Ω 0 MadDM 3 Mpc 3 λ λ ≡ ≡ 1 1 1 ≥ ). We further reject the parameter points that are excluded at 95% C.L. by these λ λ − γ 1 MadGraph5_aMC@NLO 12 13 s a a p p λ -ray observations of dwarf galaxies by the Fermi-LAT satellite experiment and the X Lilith 2 Below, we study the effective potential, cosmological phase transitions, and gravita- Then we impose the constraints from DM phenomenology. We utilize Although FCNCs have been forbidden at tree level, they can arise from loop correc- Furthermore, we require one of Firstly, we require that γ → at dwarf spheroidal galaxies. Thethe 95% C.L. upper limitsMAGIC on Cherenkov telescopes [ tional waves for the parameter points surviving from all the experimental bounds above. relic abundance. The observedis value given of by thedensity relic to abundance from the the critical100 Planck km density s experiment of thepreserved. Universe and with an average velocity of couplings comes from the measurementB of the FCNC decay flavor physics constraints. and the excluded by the data at tions. In particular,tribute the to loops the involving FCNC the the Gfitter charged Group [ Higgs boson within the tool LHC run 1 and run 2 Higgs measurements from ATLAS and CMS. Parameter points Each parameter point in the scans should be testedphysical by scalar existed masses. experimental bounds. Moreover,from in below, order the to following ensure conditions that from the copositivity criteria scalar [ potential is bounded In our analyses,parameters we are carry adopted out as the random free scans parameters: in the parameter space. The following 12 3 Experimental bounds JHEP05(2021)160 ) 2.1 (4.9) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.1) (4.2) , and (4.10) 2 are the ρ , 2 i 1 ˜ m . ρ 2 ˜ s 2 2 ), respectively. ˜ ρ ˜ s. 2 2 4 2.8 κ ˜ ρ 2 + κ 2 is the renormalization ˜ s = 2 , 2 1 ˜ µ ρ , ˜ ), and ( ρ 1 ! 23 i 1 ˜ ρ 2 , 4 h, κ C 2.7 = 1 2 12 ˜ , M L ) − . + m = 2 γ receives the Coleman-Weinberg 2 ), ( ˜ s , the elements of the symmetric 2 2 2 i ˜ n s 2 ˜ s 1 , Z, γ, t, b. ± ˜ − ρ L S ), its longitudinal mode can con- ˜ µ ˜ s, m 2 1 V 2.6 ± 2 = W λ 1 ˜ ρ , , ˜ s ˜ 3 ρ n ) ) L 4.2 ln 2 1 2 2 0 S , and -even neutral scalar fields Z , 4 ˜ ˜ s s

, and 345 2 − κ = n 1 2 2 m ˜ ρ λ 4 i 2 2 ˜ 12 ρ T κ κ ˜ = , ρ CP ˜ γ = + m ], 4 , , χ, W 1 2 − + i 1 n ˜ + + ρ ± χ 2 S 13 κ 4 ˜ n ρ 80 2 2 2 2 = 2 ˜ n s , while for longitudinal gauge bosons, scalar h, = ˜ ˜ m ρ ρ − 2 i b ˜ S 2 2 8 ,G coupling to the classical fields, and / M = X 2 1 T n λ – 6 – λ 0 ˜ 345 ρ Z i 0 2 − 1 λ n = G + π , 2 2 κ 1 = 1 t + 3 2 ,G n ˜ 4 2 ρ + = i ˜ 64 ρ ˜ 2 1 ρ − ± 2 1 -even neutral scalar bosons are derived as 1 ˜ C 2 22 = ρ 2 ± ˜ 2 2 ˜ ρ ρ 8 0 S count the degrees of freedom of the particles, given by λ characterize the feature of fermion loops. The subscripts a G 1 , n m ) = i 345 m b λ n n CP , which will be discussed below. 345 ˜ s ( + n λ n + , λ D (3 ( 2 1 = = = 4 4 1 . 2 , a, H 2 1 V 1 2 1 2 ˜ ρ ˜ i 3 2 ˜ ρ ρ + ± ± T − h 1 , / and 8 1 + + H n λ 2 11 W 2 S 2 12 t 2 , h ρ n for the n 2 (˜ n m 2 11 2 22 should be expressed as in eqs. ( = 3 m m + 1 would not contribute to eq. ( 2 h i V m m − − 2 S ˜ , h γ C M renormalization scheme [ 1 m ) = = = = = h ˜ s , 11 22 33 12 2 MS ˜ 2 2 2 2 ρ h, h, h, h, , ˜ ˜ ˜ ˜ , and 1 M M M M ρ 2 22 (˜ 0 m V , ] in the ) is 2 11 79 m 2.2 In terms of the classical background fields At zero temperature, the one-loop effective potential The tree-level effective potential in terms of the classical fields derived from eqs. ( can develop VEVs in the cosmological history. The effective potential is then expressed where the minus signs for L and T denote the longitudinal and transverse polarizationsmass-squared of matrix the gauge bosons. Although the photon tribute to the daisyscale. potential For transverse gaugebosons bosons, and fermions, where the sum runs overcorresponding all particle masses the squared particles inwe terms only of take the into classical account the fields.much top For smaller and the bottom Yukawa SM couplings. quark fermions, contributions, Hence, and all neglect the all particles the other we include in the calculations are Here, terms [ and ( In order to investigate the cosmologicalthe phase transitions effective in potential. the model, We we assumes need that to construct only the as a function of the classical background fields 4 Effective potential JHEP05(2021)160 (4.23) (4.18) (4.19) (4.20) (4.21) (4.22) (4.11) (4.12) (4.13) (4.14) (4.15) (4.16) (4.17) , . 4 ! ˜ s 2 ! s / 2 ) / 2 δλ ) ˜ s 2 2 2 / + ˜ s κ 2 2 4 2 ˜ ρ ˜ 2 κ ρ 1 + ˜ 2 ρ . ρ 2 1 1 ) + )˜ ˜ ˜ 2 ρ ρ δλ 5 , 2 1 ˜ s 5 ˜ . , ρ λ . , o S λ . 2 1 + 2 2 3 345 ) 3 2 ˜ λ ˜ ρ ρ 2 2 + λ ˆ 4 1 ! λ 2 h . ˜ s + 0 ˜ ρ ρ . 4 2 2 ˜ β + β 2 o m 1 ) + + ˜ 0 ρ , λ 2 12 gg 2 2 2 2 2 2 + ˜ ± s g 2 2 2 2 2 2 , b b ˜ ρ o 2 ρ δλ ˜ − ˜ 2 2 1 ρ m ρ y y 2 H ], 2 2 a are defined the same as in the SM. 2 h 2 + ( 2 ρ 0 ˜ + ˜ − κ + δλ m ˜ λ λ ˜ m 2 sin 2 cos )(˜ 2 m 2 1 2 12 2 82 gg boson and the photon are 2 , /v g , + + , 0 ρ ˜ s , b = (˜ 0 ± − = m + ( 2 s g + ( 2 1 1 2 Z 81 2 ˜ 2 h 2 G 2 G ˜ ρ s 2 b m

2 b − 4 g 1 2 1 + 2 22 2 22 2 ) ˜ ˜ ˜ ˜ ˜ δm m m m ˜ m ρ m -even neutral scalar bosons. To keep them κ 2 2 2 m s √ m n n n = ρ g is obtained as 1 + gauge ﬁelds is ( + 2 2 , , . 2 2 ± = CP T – 7 – + ˜ 1 4 3 χ 2 / / 2 2 δλ 2 2 ˜ 0 S ρ ) ) ) = ) = ) = ˜ 2 1 b ˜ W 2 ρ ρ 2 2 2 2 2 2 2 2 h 0 W y m ρ = + + / ˜ = 0 s ˜ s ( (˜ m β β ˜ ˜ 2 ˜ 1 2 2 1 T M M ˜ L δm 1 2 1 4 M 2 ρ -odd neutral scalar bosons is 2 ˜ ρ κ κ 2 t 2 2 Z 2 γ t = ˜ 1 2 1 y and y and + 2 ρ ˜ + = ρ + m m + ± )˜ ˜ L ρ 2 1 CP 5 2 S 2 2 2 sin 2 2 1 2 sin 12 B ˜ ρ 2 W = ˜ = ˜ ˜ ,B ˜ ˜ /v ρ λ ρ ρ 2 1 t 3 m 3 ˜ 5 δλ m L = = T boson is given by + λ 2 λ − W 2 γ m 2 Z 345 2 t 2 t δm + 4 2 ˜ ˆ ± ˜ λ ˜ + m + ˜ m M ˜ = λ m m √ 2 1 eigenvalues( eigenvalues( eigenvalues( + W 2 χ , while the mass-squared matrix for the charged scalar bosons is ˜ 2 12 ρ ) = 5 2 1 1 = + ( ˜ ˜ s m ˜ m ρ λ , λ t 1 2 2 12 − y λ ˜ − ρ m , + ( 4 1 λ − + ( ρ 2 11 (˜ + 2 11 m 3 CT

m λ V

= ≡ 2 = + ˜ 2 0 345 M ˜ ˆ λ M Notice that loop corrections generally shift the values of the VEVs as well as the renormalized mass-squared matrix ofintact, the we introduce the following counterterms [ where the couplings For the type-II Yukawa couplings, the masses squared become For the type-I Yukawa couplings, the masses squared of the top and bottom quarks are After diagonalization, the masses squared of the The mass squared of the The mass-squared matrix of the The masses squared of the DM candidate The eigenvalues of these matrices give the masses squared of the scalar bosons, i.e., with The mass-squared matrix for the JHEP05(2021)160 ) ) s in ,v with 2 2 IR 4.26 (4.34) (4.28) (4.29) (4.30) (4.31) (4.32) (4.33) (4.24) (4.25) (4.26) (4.27) ,v 1 v = Λ = 0 ± 2 ) and ( p 2 G , ) = ( ˜ s m .   , , 2 , 2 4.25 ! 1 ˜ , ˜ ρ 1 = ˜ 1 ρ 2 i 2 2 2 2 V , 1 V V ˜ ˜ ˜ ρ 2 s s 0 1 ˜ T ˜ 2 2 s∂ m ∂ ∂ ∂ ∂ ρ 2 G , ∂ ∂V ∂ ∂ (˜

, 1 ˜ 2 m 2 ˜ − − − 2 ρ 1 F v 1 1 ˜ ρ V ∂ J 8 V i 2 2 = = = 2 ˜ ˜ ρ n s∂ ∂ , , − 1 ∂ 2 ∂ ˜ 1 2 ∂ s , ˜ 1 1 ρ 2 t,b ˜ ˜ CT CT 1 2 ρ ρ 2 to be the mass of the SM-like ∂ 2 s ˜ s. ˜ s 1 V V ∂V ˜ ˜ = ρ V V X ρ v v vanish at , i 2 2 ˜ s∂ ∂ V 2 2 ˜ ˜ 1 2 s∂ s∂ ∂ 1 1 ∂V 2 IR 2 ∂ ∂ ∂ v v ∂ ∂ ˜ ± ˜ s∂ ρ ∂ ∂ + 3 2 ∂ Λ 4 4 , ∂ v 1 2 1 G 1 s s ! v v 8 s ], i.e., to set − − , v v ρ 2 2 i 4 4 2 v 1 , v 1 4 81 ˜ , ˜ = = = ρ 1 T = ˜ m and 2 2 V + + 1 2 V s 2 i ˜ 2 ρ 0 ] can be expressed as ˜ +

˜ ρ 2 2 s∂ 12 1 1 ∂ ∂ ˜ ˜ ∂ ∂V ∂ ∂ G ρ ρ 1 1 1 B 87 1 ˜ ρ V V ∂ ∂ J − − − V i 2 2 2 2 , φ ]. An approximate treatment is to give an IR 2 ˜ ˜ n ρ ρ ˜ s∂ ∂ ∂ 2 G 2 = = = ∂ ∂ ∂ ∂ – 8 – µ ˜ 86 m 1 2 1 s , 2 1 1 2 ˜ v v . Here, we take v ρ 2 2 CT CT CT v v ˜ v ρ ) ˜ 4 4 4 ln 85 ρ X V V s ), we obtain ∂ ˜ s∂ ∂ 2 2 =bosons ∂V 2 G j i + + ,, δλ δλ + ∂ , v ∂ ∂ ˜   1 1 2 m 1 1 1 2 1 2 2 1 2 2 4.26 ∂φ 2 V V ˜ ˜ 2 s ρ ∂ V V V ˜ ˜ 4 , v ˜ ρ ρ s , 2 2 2 2 2 π ∂ ∂ 1 1 ∂ ∂ ∂ ∂ ∂ i T 2 G )–( 2 ∂ ∂ ∂ v ˜ , δλ , ρ 1 2 2 1 s ˜ , m 1 4 1 4 1 1 4 1 V ∂ v v ∂φ 1 1 1 2 1 ˜ 1 ρ 1 2 2 ∂ V ˜ 8 8 ρ V ˜ ˜ 4.24 ρ 2 ρ ) = + + + ∂ 2 ∂ ) = ( ˜ s∂ ∂ ∂ ∂ ∂V − − ˜ 1 1 ∂ s 1 1 2 . ∂ ˜ s ) ˜ ˜ , ρ ρ − − − 1 1 1 ˜ s s, T 2 s ∂ ˜ s ∂ ∂ ∂V ∂V ∂V ˜ , ρ , ˜ ]. Similar problems exist in the effective potential with higher loops, v ρ ∂ ) = = = 2 , v 1 1 2 s , ∂ 1 ∂V ∂V s ˜ 2 ρ 1 v v v v 3 3 3 1 3 3 1 s 84 , 4 4 4 4 ρ , 1 ˜ , v ρ v v 1 1 , v 1 2 1 (˜ CT CT ˜ CT 2 ρ 1 8 8 ˜ ∂ ρ ρ − − − − V 83 V (˜ ∂ v 2 ∂ 2 , v 2 ∂V ˜ ρ ======1 ∂ ∂ 1T ∂ v s s 1 2 2 2 s 1 2 V 1 ) = ( to the Nambu-Goldstone boson masses [ δλ δλ ˜ s δm δm δm δλ , ) = ( IR 2 ˜ s ˜ ρ Λ , , 2 1 ˜ Thermal corrections to the effective potential are crucial for studying the EWPT. The The masses of the Nambu-Goldstone bosons ρ ρ , (˜ 1 ρ one-loop finite-temperature effective potential [ at Higgs boson. Solving eqs. ( massless Nambu-Goldstone modes, and onebosons can on fix shell it [ byand setting more the details momenta of can the becutoff Higgs found in refs. [ the logarithms at proportional to This problem is due to the ill-defined renormalized Higgs boson masses at in the Landau gauge, inducing logarithmic IR divergence terms in eqs. ( (˜ The nine counterterm coefficients are determined by the following nine equations at JHEP05(2021)160 ) ). T ( X 4.18 (4.47) (4.39) (4.40) (4.41) (4.42) (4.43) (4.44) (4.45) (4.46) (4.35) (4.36) (4.37) (4.38) Π , , in eq. ( 1 2 ,B y y 3 2 . The subleading W . + + , 2 ˜ X i 1 2 M )] 2 ˜ κ κ / M T 3 ( . ) ] 2 i , + 4 + 4 D X ) V 4 4 m 2 b ( ˜ 89 λ λ y , + β − dy, dy. ) + Π + . 88 2 + 4 + 4 2 1T ˜ s 2 t β / 2 t , 3 3 V are defined as x x 3 y 2 y 2 sin ) λ λ . + + ˜ ρ F 2 i + ]. The diagonal elements of 2 2 , 12 , J y y 12( sin 1 2 m + 8 + 8 = 0 ρ ( ¯ √ √ 90 CT T , (˜ h 1 2 = = 2 − − V T i 2 λ λ 1 X γ and e e g 2 2 88 n + ˜ B M − , 1 J 2 = 2 1 1 + = Π V , y + 12 + 12 T 3 L 2 2 β T 2 2 b X + 0 0 0 2 W Z y – 9 – g g =bosons g ln ln 0 i 2 2 V , y 12 cos y y π + 3 + 3 = Π = 2 = Π T . 2 2 ∞ ∞ 12 = = 0 L ) = ) ± ± 0 0 L T g g 2 B 1 1 − Z Z 9 9 W W κ y y Π ˜ s, T Π Π ≡ ≡ denotes the thermal corrections to 2 2 ) = eigenvalues[ + , can be neglected [ ) = ) ) 2 48 48 T T 1 ) ) ˜ x x ρ κ ( ( , T ˜ T s, T = = ( 1 F B ˜ ( s, T , + ρ J , J 2 X X (˜ 2 11 22 ˜ ρ S , , Type I: ˜ Π ρ are the corrections to the diagonal elements of , Π λ Type II: eff + + , 1 ( L V 1 ρ 2 B ρ (˜ represents the mass-squared matrices or masses squared in terms of 6 (˜ 2 i T Π = Π = Π ) D ¯ m ˜ s V = , 11 22 are the contributions from the Yukawa couplings. For the type-I and -II , , 2 0 0 χ and ˜ ρ 2 , 3 y ]. L 1 W 91 ρ = Π = Π = Π (˜ Π 2 and is the temperature and the functions X 33 11 22 ˜ h, h, h, 1 M T y Π Π Π Finally, we obtain the total effective potential We also consider the daisy diagrams, which can be significant. The corressponding are the field-dependent boson masses squared with thermal corrections in the high- Discussions on theoretical uncertainties in perturbative calculations of the effective potential can be 1 2 i ¯ Note that found in ref. [ The thermal corrections to the electroweak gauge bosons are Here, cases, they are given by off-diagonal elements of for the scalar bosons are derived as temperature limit and can be derived by where the classical fields, and contribution to the effective potential can be estimated by [ m where JHEP05(2021)160 ] 1 = ˜ ρ 92 [ ) (5.1) , and ) T ( T s develops ( 2 , v ˜ s v ) , T ) ( T 2 ( 1 , a second-order , v , we demonstrate v ) . 1 T ) at the local minima ( s ]. 1 ˜ s v , v 2 CosmoTransitions 94 , 460 GeV , v slightly decreases. At zero 93 1 are the Euclidean actions of ) v , and ' 3 2 T ˜ S ρ ( is the global one of the zero- becomes nonzero much earlier T = ( s , . In the plots, v ) ) 1 , the green minimum becomes a s ˜ 7 ρ c and T does not evolve synchronously with ( T , v (0) 4 s s 2 ) S v T . , v , v , ( , while 1 } s S v v (0) − 2 /T e 3 in section 4 . Below , v 1 ,S ) = ( 4 AT (0) ˜ s – 10 – -even neutral scalar fields would develop VEVs, S 1 , { ] ∼ v 2 (100) GeV ˜ , implying the restoration of the electroweak gauge ρ Γ O CP , , the blue minimum appears, accompanied with a 96 0) 1 , 119 GeV , ρ 0 (˜ 95 = min -symmetric bubbles, respectively. The three-dimensional , ' c S (3) T O 148 GeV ) = (0 to trace the temperature evolution of the local minima. ˜ s , ' 2 ˜ ρ - and , T 1 , the system stays at the red minimum with (4) ρ (˜ O constant and , probably due to the less couplings of the singlet field to other fields increase from zero to ) -dependent values of the classical fields (1) ) T T T O ( ( 460 GeV 2 2 v v & is an T then gain VEVs in an subsequent phase transition, which could be a strong FOPT CosmoTransitions , respecting the electroweak gauge symmetry. At and A are the and 2 0) ) ) ˜ ) ρ Below we discuss the dynamics of the FOPTs. The bubble nucleation rate per unit In our parameter scans, we usually find that The effective potential at the blue minimum is higher than at the green minimum At In this model, the three classical , 0 T T T ( ( ( , 1 1 s time and unit volume is given by [ where the scalar fields for v compared with the twothan Higgs the conventional doublets. EWPT epochand Typically, via a second-ordersimilar or to first-order those phase in transition. the conventional two-Higgs-doublet models [ quantum tunneling and turns intonucleates the blue bubbles, minimum, inside or which the thev “true system vacuum”. Such is a trapped FOPT attemperature, the the true true vacuum. vacuum satisfies In this FOPT, a nonzero VEV.barrier that At separates it fromtemperature. the green minimum. These two minima coexistuntil till the the zero critical temperature metastable state, i.e., a “false vacuum”. The system finally undergoes a FOPT through of the effective potential.local minima. The red, green, and blue(0 lines indicate the positionsphase of transition occurs three and the system turns into the green minimum, where typically leading to multi-step cosmologicalthe phase temperature transitions. evolution In of figure multipleters phases can for be a found in benchmarkv the point BP3 (BP), column whose of parame- table in a stochastic GW background.to We analyze utilize the the numerical phase package ify transitions. whether For or each parametertemperature not point effective the potential. in minimum The the parameterwe random points use scans, that fail we this ver- test are rejected. Then Based on the effective potentiallution constructed with in temperature. the At previousmized sufficiently section, at high we the temperatures, can origin the study effectivesymmetry. its potential As evo- is the mini- Universetwo coexisted cools minima down, separated extra by a minima high barrier, appear. strong FOPT In could particular, take place if and result there are 5 Phase transitions JHEP05(2021)160 700 (5.4) (5.2) (5.3) , and ) T ( 600 2 v , plane. ) ) 500 ) T T ( ( T 2 1 ( v

2 v s ) i v - x 400 V a e . T t p G a (

T a T 300 is given by the bounce m i , n i ) 700 ) M , and r n ( 200 ,T ˜ T s i c , , , T ) φ c 600 ( r T n ( 100 T false i 2 eﬀ ˜ φ p ρ V plane. i T (b) Minima in the , ) 500 eﬀ ) ) T 0 + ( r T s ∂φ 0 ( v ) = ( i ∂V 50

s -50

s 300 250 200 150 100

) 2 i v ) V e G ( ) ( T v ˜ dr ρ ∞ - x 400 V dφ = ( a e

T i i t i G a (

dr dφ dr a T dφ 300 ) = m 1 2 i , φ r 2 – 11 – r n i ( i 700 2 M + φ = 0 200 i 2 φ c dr r 2 =0 T 600 dr r d

∞ n i 100 0 T Z p dr plane. dφ T (c) Minima in the ) π 500 ) T 0 ( T 1 0 ( v 50

1 -50

= 4

s 300 250 200 150 100 ) i s v ) V e G ( ) ( v - T x 400 3 V a e T

S t G a ( a T 300 m i n i M 200 c is the field configuration of the false vacuum. T n can be simplified to 100 T . Temperature evolution of the positions of the minima in the axes is the radius of the bubble. 3 false i p T (a) Minima in the S r φ for BP3. The red, green, and blue lines denote three local minima. The vertical dashed lines 0 0 )

-50

200 150 100 300 250

T 1 ) V e G ( ) ( T v ( s with boundary conditions where where solution of the equations of motion v indicate the critical, nucleation, and percolation temperatures action Figure 1 JHEP05(2021)160 , 0 n t is n T T T , . As 120 c (5.7) (5.5) (5.6) c T T , as well 100 . Thus, 40 GeV . 2(a) Γ (1) ∼ 80 , where the nu- ) O n T n V is the smaller one T T e G ( 60 /T p ]. In order to evaluate T 3 T S , 98 Nucleation rate Nucleation and the temperature ) 40 0 t t, t , ( , the nucleation rate increases is reached at ] c (b) as functions of the temperature 3 (b) Nucleation rate 20 r T Γ = 1 ) 0 /T 100 t 3 , , ( T ) S 0 3 4 Γ 1 4 7 0 3 6 9 2 5 8 1 4 7 0 τ w 99 . We ﬁnd that a 4 4 4 5 5 5 5 6 6 6 7 7 7 8 ------( v

H ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0

a 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1

) V e G ( Γ t 4 dT dτ 2(b) Γ( c t 0 0 n t T denote the critical and nucleation times, re- T dt Z n Z t , an increasing number of bubbles thrive and c t – 12 – t n Z c = T ) = T 0 in ﬁgure π 3 is the comoving radius of a bubble growing from 120 3 4 and Γ ) Γ t, t H 0 , and then decreases. ( c T − 4 . The minimal of / t r S 3 dt t, t S 100 (a) and the nucleation rate ( n t r c 4 t S Z as functions of the temperature for BP3 in ﬁgure 40 GeV . 5 GeV 4 80 ) S ) = exp ∼ and ∼ t /T n V ( T 3 e T ] S G /T P , we need to estimate the fraction of space that still remains in the and ( , which can be computed by [ 3 60 Action p t p 97 t T S T /T 3 and in the radiation-dominated epoch. S 4 40 S dT (a) 1 − is the scale factor. ) 20 ) is the Hubble rate, and 0 t . The actions ( HT ( H a . 0 p − 1 5 4 3 2 8 7 6 Below the nucleation temperature The bubbles are actually nucleated at the nucleation temperature We present T

0 0 0 0 0 0 0 0 , given by

1 1 1 1 1 1 1 1 t = 4 3 S , T / S for BP3. The dashed lines denote the critical, nucleation, and percolation temperatures where to collide with eachstochastic other. GWs is expected The to bethe maximum reached percolation when of percolation time occurs bubble [ false collisions vacuum at that time remarkably produces where spectively. Note that the differentialdt relation between the time before the peak around cleation probability for acan single be bubble estimated within by [ a Hubble volume reaches as the corresponding nucleation rate until temperatures below the Universe cools down, below the critical temperature Figure 2 T and JHEP05(2021)160 , . . ] 0 w ]. T v 101 106 [ (5.8) (5.9) = 105 ]. For (5.13) (5.10) (5.11) (5.12) – T 29 . 110 0 , with the at 103 , , ] i 71 ∼ 4 . ) is positive at 98 S 0 is the effective 109 ,T ∗ ' ]. Hydrodynamic g ) true i p φ 106 and the cosmological t dS/dT ( ( 1 eff P − V β , − , increases to ] , we have ) ) 2 t p 0 ) ( t 0 T t ,T P = = − T 0 −

t . t false i 1 φ [( α ( or . ) O dS dT n ) eﬀ ) α is typically smaller than 0 + 2 t 0 V , 2 T h T ) + ( = 0 ( α HT t /T 0 3 vac β rad ∂ 3 H t ∂T ρ ρ 3(1 + − S √ t T ≡ = √ ( ≡ – 13 – 0 ) t 0 β − 1 + α = ) T t − (

= ˜ can be derived by requiring ) β ,T 0 t dt dS p ( CJ t v true i S φ ( ' characterizing GW production from FOPTs [ ≡ − . In addition, ) eff p around the time t β β V ( T S S , we define a dimensionless quantity − the radiation energy density in the plasma. 1 ) − 30 ,T H / 4 false i T ]. Thus, it is difficult to completely work out the bubble wall velocity ∗ φ ]. The density of the released vacuum energy is given by [ g ( 2 eff 108 π 108 V – is the field configuration of the true vacuum. It is useful to define a dimension- = is the velocity of the bubble wall. For randomly distributed spherical bubbles = 106 i true w , leading to positive v 0 vac rad φ ρ ρ T Expanding the action The expansion of the bubbles depends on the interactions between the bubble walls and FOPTs are able to release latent heat from the vacuum energy, which drives the ex- ]. Thus, the percolation time = T In order to conveniently compareexpansion the time phase scale transition time scale where can be roughly understoodthe as the electroweak inverse FOPTs time in duration which of we the phase are transition interested, [ the derivative This is a typical assumption when evaluating GW signals. bubble walls [ For Jouguet detonations, the Chapman-Jouguet condition leads to a wall velocity of [ relativistic degrees of freedom in the plasma. the plasma, analogous toanalyses chemical show combustion that in bubble a propagation haveweak relativistic diverse fluid detonations, modes, [ including subsonic Jouguet deflagrations, detonations, supersonic deflagrations (hybrid), and runway where less strength parameter with pansion of the bubbles andplasma also [ converts into the thermal and bulk kinetic energies of the with equal size inthe the fraction three-dimensional space, of the space102 percolation converted threshold to is the reachedcorresponding true when temperature vacuum, where JHEP05(2021)160 to (6.1) (6.2) (6.3) (6.4) (5.14) (5.15) GW ]. This Ω . 110 , 8 2 . The peak . ]. Numerical ∗ 3 . 8 t ) . ] w when GWs are 2 v ) col and = ∗ 117 t , , 111 1 col t 6 ]. Denoting f/f / 1 f/f 8( + 3 ln 116 ). The efficiency factor . , 115 0 ) 8( , t ∗ – . ] show that the resulting p ) g 3 T 100 110 n = ( 100 . T 1 + 2 112 t ˜ 2 β ( 100 118 3 ˜ , h β / 1 ln 4 ln ∗ 107 turb . T ∗ − − 2 w g , while the percolation temperature v 100 + Ω 100 GeV + 2 ∗ 2 ] n p h , as denoted in figures evaluated at the time w Hz T T ˜ v βh sw ˜ 5 1 β α 118 64 GeV . α − 100 GeV 100 GeV 62 0 φ . ' + Ω κ 0 10 1 + − and 2 60 GeV n – 14 – 4 ln 4 ln redshifted to today ( 8 × h . T α ∗ 2 ' 1 − − t col ˜ 65 β p . ) = ∗ ∗ = T 2 w 3 g g w t 100 100 v = Ω v = 1 col , the temperature corresponding to can be approximately determined by [ 2 ∗ f ), further calculations show that the nucleation and per- ) 11 h . T p ∗ 2 ln 2 ln t 0 T 5.6 42 + ( ) . = − − 0 GW t H (0 0 5 ( Ω ) . . T The nucleated bubbles expand and finally collide with each other. and ∗ a 5 t ( n − 132 141 a ) and ( T 10 ' ' = 5.5 is slightly lower, × ) ) ∗ p n h CJ n p T T 67 . ( ( v T T 3 3 S S = = 1 w 2 is evaluated at v h ∗ col g Ω Based on eqs. ( characterizes the fraction of the available vacuum energy converted into the gradient φ which is the inverse Hubble timeκ at energy of the scalar fields. The redshift of the frequency has been taken into account by the factor where frequency of the spectrum can be modeled as [ (a) Bubble collisions. Their collisions break the sphericalprocess symmetry and can generate be gravitational well wavessimulations [ described for by bubble the collisionsGW in envelope spectrum the approximation at thermal [ present plasma can [ be approximated by sound waves, and magnetohydrodynamic (MHD)be turbulence the present [ GW energy densitydensity, per we logarithmic separate frequency the interval contributions divided by from the the critical three sources as Electroweak FOPTs could induceWalker significant metric perturbations and of producerelevant the stochastic to Friedmann-Robertson- GWs the around relic theproduced. GW mHz spectrum band. There are are Two three key coexisting parameters GW sources at a FOPT, namely bubble collisions, For BP3, the nucleationassuming temperature is 6 Gravitational wave spectra colation temperatures JHEP05(2021)160 , ]. ]. ]. 104 115 (6.8) (6.9) (6.5) (6.6) (6.7) 126 (6.10) (6.11) 120 ]. Since , ]. Thus, , , 107 113 ) ] [ 122 2 ∗ , / v 7 κ 124 1 , . 113 , πf/h 2 sw 0 121 /f ' 3 108 2 ) 7 f (1+8 3 turb / turb κ 11 ) ]. This source lasts until the 4 + 3 f/f ( ] ] 3 turb 121 is the fraction of the available – , . ) v 108 α f/f . κ sw α 126 f , 119 α f ∗ v H κ . . (1+ 113 3 3 ∗ nl and ∗ 98 + / 3 . τ 4(1 + 1 / ] w 0 ∗ 1 ˜ ˜ 1 α βh βh t v w s 5 √ 2 v √ . ∗ 121 54 w 3 = . ∗ g 1 + 2 ]. It takes several Hubble times for the MHD v 100 0 g 3 t ∗ 100 √ = / – 15 – 1 = 125 2 135 + ) ] − ˜ 2 . ] βH π / 0 sw turb 3 α f f (8 124 α 123 = v α κ ∼ α Υ = 1 1 + CJ v nl κ turb τ 1+ can be approximated by [ κ w v v ˜ β κ Bubble collisions can stir up turbulence in the fluid, as the Υ w ˜ β v 6 − 4 The explosive bubble expansion in the plasma induces a sound shell − , and 10 is the Hubble rate at 10 CJ × v ) × ∗ t 17 = ( . 35 . w is typically negligible, except for runaway bubble walls [ H v = 1 φ = 3 ≡ κ 2 2 h ∗ h ]. Therefore, the duration of the sound wave source can be determined by the sw H Ω turb 123 In general, the contribution from the sound waves dominates in the GW spectrum [ In an expanding radiation-dominated Universe, the finite duration of the sound wave Ω – Based on the suggestion from simulations, we optimistically set Moreover, we omit the contribution from the bubble collisions in the following calculations. The corresponding GW spectrum can be fitted as [ with (c) MHD turbulence. energy injection to the plasmathe results plasma in is an extremely fullyconsidered, high leading ionized, Reynolds to the number MHD [ turbulence magnetic turbulence [ to field, decay, along and with the stochastic the GWs velocity arise field, continuously should during this be period [ where the peak frequency is estimated to be [ Thus, the GW spectrum contributed by the sound waves is given by [ source leads to a suppression factor [ where vacuum energy converted intodetonations, the kinetic energy of the fluid bulk motion. For Jouguet around the bubble wall. After theas bubble sound collisions, waves, which the become sound a shells significant propagatesound GW into source the waves [ fluid are disrupted121 by the development ofnonlinearity timescale nonlinear estimated shocks as and [ turbulence [ (b) Sound waves. JHEP05(2021)160 , , ) s 2 2 | v 5 Z Φ λ and 6.8 | , α | ), ( → − 4 . Larger λ 2 | β 6.2 . The relic Φ 1000] GeV , | , 02 . The purple 3 . or and λ plane assuming 0 3 | 1 [10 α 1 , Φ . 2 − ∈ ˜ 1 β λ s - − v , α ˜ β 1 → − λ 1 is equivalent to one with . 0 , 0 Φ 7 1 0 3 6 9 2 12 1 1 1 1 − − − − − 20] m 0 0 0 0 1 1 1 1 10 , 5 . has a VEV near the electroweak and 1 [0 - J and S can be either positive or negative. C 0 β v 1 3 2 symmetry ∈ . Thus, we can just take positive . = κ 0 w 2 2 12 v tan β

Z , . 2 m 2 2 h − α tan W 0 2 - , and G 1 plane, as presented in ﬁgure 0 α , 1 ˆ Ω . 1

2 κ 1 f 4 corresponds to a longer FOPT time duration. o , −

ensures that for the two types of Yukawa couplings. We − s 5 ˜ r β β – 16 – - λ u are required to satisfy the bounded-from-below 10 8] o , α t ] GeV 4 n 3 2 S and we can easily read oﬀ the GW signal strengths , - o λ 0 1 TeV λ C 3 1 , 01 3 . and the two Higgs doublets could be important. 500 to Type I Type II BP1 BP2 BP3 BP4 λ [0 , , as well as to cause a FOPT. , S lead to stronger GW signals, as implied in eqs. ( [1 would be totally equivalent to a positive one due to the 3 2 12 , and 1 s 2 4 - | ∈ m − v 0 λ 2 ˜ 1 2 3 4 5 6 7 1 β ------| ∈ 10 GeV κ ,

0 0 0 0 0 0 0

| to denote the peak amplitudes of the GW spectra. The contours

1 1 1 1 1 1 1 1 2 12 −

˜ 2 , 1 λ | m and h | 1 κ , α | GW . Besides, a parameter point with range of ˆ Ω , since the potential respects the S s 2 12 v . ). A negative m , and 11 − → − − 8] 3.2 800] GeV are demonstrated in ﬁgure S 10 , in the scans, while , ). For the surviving parameter points, we calculate the resulting values of . Contours of the peak amplitudes of the GW spectra in the 8 2 . β and ∼ h [58 [0 2 β 6.10 h tan We introduce The strength of the stochastic GW signals from the FOPT depend on Note that positive ∈ GW ∈ , and then project the points in the 1 ˆ Ω implies a stronger FOPT, while smaller tan χ GW − S ˜ ˆ Ω The parameter points lie in theGW ranges spectra of for the parameter points are further evaluated, assuming Jouguetof detonations. of the parameter points from the plot. The strongest GW signal we ﬁnd reaches up to α Consequently, larger and ( β and green points are corresponding to type-I and type-II Yukawa couplings, respectively. − expect for the softand breaking quadratic terms with In addition, the scale, and the interplay between in the logarithmic scale.constraints described The in parameter section points are required to passconditions ( all the experimental symmetry We perform random scans with them model parameters in theλ ranges of assume that the prior probabilities for the random parameters follow uniform distributions Figure 3 Jouguet detonations. Purple andYukawa couplings, green respectively. points Four denote BPs the are parameter also points indicated. for type-I7 and type-II Numerical analyses

JHEP05(2021)160

A S I L R N S . thr can and (7.1) 50 40 30 20 10 100 90 80 70 60 ) . , and ], and ]. We , with f a 10 ( 10 60 4 SNR , 1 113 3 0 < , > [ 1 J 2 GW , C v 1 Ω 0 = BBO h 0 LISA w 1 LISA v , ], BBO [ s e 1 56 - d SNR 0 u t 1 SNR i = 10 (50) l p ) LISA z 2 m - ]. The DECIGO curve H thr a 0

( 1 k f ], Taiji [ a ]. Some of the curves are e 127 3 - p 54

0 SNR 58 1 W . The next-generation plans for the parameter points. For TianQin G

Ultimate DECIGO , BP3 BP4 4 I f 10 - I

0 df, e 1 p ) ) Taiji < y (b) Type-II Yukawa couplings. f f T 5 - ( ( 0 ], Tianqin [ 0 1 2 3 4 5 6 7 8 9 0 1 8 9 1 - - 1 1 1 1 1 1 1 1 1 1 2 2 ------0 0

0 0 0 0 0 0 0 0 0 0 0 0

LISA

1 1 2 sens

2 GW 51

. The gray points yield 1 1 1 1 1 1 1 1 1 1 1 1

W G Ω h

ˆ 2 Ω Ω . The signal-to-noise ratio can be deﬁned 10 for them as the color axes in ﬁgure T ]. SNR > max ], and DECIGO [ ], Taiji, and TianQin. The signal f min

A S I L LISA R N S 60 – 17 – Z 114 LISA T 50 40 30 20 10 100 90 80 70 60 SNR s is larger than a signal-to-noise ratio threshold SNR ≡ 1 0 ], BBO [ 1 J C versus the peak frequency SNR 56 v . BP1 and BP2 (BP3 and BP4) correspond to the type-I 0 2 BBO = SNR 0 1 w h 1 v , s e 1 GW - d 0 u , like BBO and DECIGO, may probe much more parameter points. ˆ 1 Ω t i ], Taiji [ l ) p LISA z 2 54 ] are also plotted. The color axes denote the LISA signal-to-noise ratio - m s (3 years) for LISA [ H 0 a (

1 1) Hz 58 7 k . f a e 3 10 - (0 p

0 is the sensitivity of the experiment. Below we take the practical observation 1 × W ) TianQin G Ultimate DECIGO

illustrates f BP1 BP2 , 4 ∼ O 46 ] - I (

. 0 4 e 1 f p . Peak amplitudes of the total GW spectra versus frequency for the parameter points ], TianQin [ Taiji for the parameter points with (a) Type-I Yukawa couplings. y sens T = 9 128 5 - , 51 Ω 0 3 4 5 6 7 8 9 0 1 0 1 2 8 9 1 - - T 1 1 1 1 1 1 1 2 2 1 1 1 ------are all below 1 TeV, while the mass of the DM candidate is less than 140 GeV. The 0 0 LISA For a closer look at the results, we choose four benchmark points, whose detailed The GWs produced by FOPTs become an isotropic and stochastic background in the Figure

0 0 0 0 0 0 0 0 0 0 0 0

1 1 113

1 1 1 1 1 1 1 1 1 1 1 1

W G Ω h

ˆ ± 2 H the remaining gray pointsaiming at corresponding to information is listed in(type-II) table Yukawa couplings. In these BPs, the masses of the Higgs bosons where time be detected if the corresponding For the six (four) linkfind configuration that of some LISA, parameter thecould threshold points be is yield probed by the LISA. LISA We signal-to-noise denote ratio increases with the practicalas observation [ time converted from the sensitivity onconversions of amplitude the spectral related density quantitieswe or adopt can characteristic here be strain. is found the The be in, ultimate sensitivity regarded e.g., that as ref. is an [ only observational limited limitation by [ quantum noises, andpresent it Universe. can The detectability of the GW signals in the space-based interferometers SNR comparison, we also plotLISA the [ sensitivity curves for the future space-based interferometers Figure 4 with type-I (a) and type-IIfor (b) the Yukawa couplings future assuming space-based Jouguetultimate detonations. GW DECIGO Sensitivity interferometers curves LISA [ [ JHEP05(2021)160 , , α s 1 / 3 cm 5 2 , with 27 26 − − 10 − 10 2 66 04 08 56 87 82 . In table 10 × ...... 10 20 678 5755 α 346 . . 14591 38075 for the LISA, . × . The BPs are 8644 4643 8002 1732 . × . . 80222 87839 . . . . 7 × . . 0 0 75 GeV 5 15 − 2 82 5042 . − − . . . 2 6 1 p . Assuming Jouguet 2 T TianQin − 5 3 26 . − 10 − 10 SNR 10 3 155 1 120 2 47 60 631 87 874 98 582 83 307 38 124 03 76 × 10 × ...... 07 9 987 138 8556 181 0 . . 2 83745 . × 0689 4 3967 4 5741 2 2074 0 5297 0 . . × . 80378 91655 1 . . . . . 7 . . 2 0 71 − 47 GeV 72 7696 . − , and . . to 7 1 1 3 − Taiji 2 3 27 10 − in the rest BPs. The DM annihilation − 10 × SNR ]. 1 10 9 42 7 60 9 60 94 655 49 911 83 650 77 158 26 64 10 × 4 ...... , h 39 3 459 125 191 134 2142 0567 160 0 . . . 78 . . × 0295 0 5502 6 4715 5 2628 1 1882 1 2654 0 . × 85479 1 ...... 1 4 . 02 − − LISA 68 5876 . . . – 18 – 4 3 1 SNR 4 2 26 − − 10 5 641 ranging from . 10 3 23 6 37 3 74 96 280 75 496 02 124 11 91 88 78 40 384 10 × ...... 1 50 2 4852 1 2187 240 0 . . We find that the GW signal strengths decrease according . . − × 1727 4048 2 0027 1 3925 2 1496 2 8616 3 . × 80887 0 5 83 96 ...... BP1 BP2 BP3 BP4 ˜ 1 6 β 0 . 4 1 3 3 2 2 2 0 33 − − 30 0210 . in BP2, while it is . . 1 . Detailed information for four benchmark points. 2 and h 3 at dwarf galaxies predicted by the BPs are below s) 1 d / i ) 2 3 v 2 Table 1 ann β 1 (cm Taiji LISA 5 4 3 2 1 2 1 S σ (GeV) 402 − (GeV) 1014 (GeV) 282 (GeV) 125 α h TianQin (GeV) 117 κ κ (GeV) 664 λ λ λ λ λ (GeV λ d (GeV) 55 ˜ (GeV) 542 β i ± 3 2 1 tan Type I I II II a χ p s h h h v 2 12 H SNR SNR v T m m m m m SNR m m ann σ h For the four BPs, percolation of the FOPT occurs in also indicated in figures to the order of BP4,we BP1, also BP3, list and BP2, theTaiji, reflecting and signal-to-noise the TianQin ratios descending experiments, order respectively.BPs, of LISA while and TianQin may Taiji look probe promising BP4 to with detect a all sightly longer observation time. cross sections beyond the reach of Fermi-LAT and MAGIC [ ranging from 0.16 to 0.35detonations, and we derive the GW spectra for these BPs, as presented in figure SM-like Higgs boson is

JHEP05(2021)160

W G Ω h ] .

ˆ 2 3 1 2 3 4 5 6 7 8 9 0 1 2 1 1 1 1 1 1 1 1 1 2 2 2 133 . ------0 0 0 0 0 0 0 0 0 0 0 0 – 1 1 1 1 1 1 1 1 1 1 1 1 β , we plot 131 [ tan 3 for Jouguet γ s J 2 C v h X 1 = 0 1 w v → GW , s ˆ Ω g B . Thus, the FCNC n i l − 0 p β u µ 0 o 1 + n c a

µ t a → plane, with the color axis w s a B BBO k J , 750 GeV ± H u γ C 1 TianQin . s Y - , the parameter points for the v

m 2 I 0 X I 0 - . - = 1 6 h 0 → e β 1 w p B v ± y , T (b) Type-II Yukawa couplings. ], excluding a region with GW H a r tan ˆ Ω t 2 m - c 0 e 130 plane for type-I (a) and type-II (b) Yukawa 1

800 600 400 ,

)

1400 1200 1000 s ± H ) V e G ( z m

Ultimate DECIGO ± H e for type-I and type-II Yukawa couplings, H m 129 v ( [ - a f ± β 3 w - −

W G µ Ω a h 1 ˆ 2 – 19 – n tan + o i 1 2 3 4 5 6 7 8 9 0 1 2 µ t 1 1 1 1 1 1 1 1 1 2 2 2 ------a , the most stringent FCNC bound comes from the 0 0 0 0 0 0 0 0 0 0 0 0 t 1 1 1 1 1 1 1 1 1 1 1 1 i → 4 v - a 0 d r

BP4 BP3 BP2 BP1 B G 6(a) , the bounds from the observations of and light LISA J β C v 1 Taiji 5 6(b) - = 0 ] exclude a region with 1 0 w tan v 3 4 5 6 7 8 9 0 1 2 6 7 8 9 1 - - - - 1 1 1 1 1 1 1 1 1 1 , ------0 0 0 0 s

0 0 0 0 0 0 0 0 0 0

1 1 1 1

1 1 1 1 1 1 1 1 1 1 130 W G

Ω h n

i , l 2 p β u ] to reject parameter points. In ﬁgure o n 129 c a [

t 73 a w . GW spectra for four benchmark points assuming Jouguet detonations. − a k µ − u µ Y +

+ I 0 - µ µ 0 e 1 p → d y . Parameter points projected in the (a) Type-I Yukawa couplings. T → B Figure 5 s ]. The color axes denote the peak amplitude of the GW spectrum B For the type-I case in ﬁgure In order to show the most important ﬂavor constraints mentioned in section 73

400 200 800 600

1000 1400 1200 ± H ) V e G ( m LHCb and CMS measurements of For the type-II case in figure and constraints remove small our parameter points confronting theGfitter FCNC global bounds. fit [ We havetwo adopted types the of data Yukawa fromindicating couplings the the are peak projected amplitude in of the the GW spectrum, Figure 6 couplings. Yellow regions arefit excluded [ at 95% C.L.detonations. by the FCNC bounds from the Gfitter global

JHEP05(2021)160

W G Ω h

ˆ d 2 ), i v 1 2 3 4 5 6 7 8 9 0 1 2 . In CJ 1 1 1 1 1 1 1 1 1 2 2 2 ------2 (7.2) (7.3) (7.4) . For 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 h 3 ann can be < v σ √ h α / GW w 1 ˆ Ω . There are J ]. , due to the C < v and v ± s s 78 / = H c 500 w 3 w v v is mainly related m plane. Although , s cm g d on n ) i i l V p v 26 v e u − κ o G c (

ann 78 GeV a . 10 χ σ Fermi-Magic w 5 h m a / - . . ∼ × k 2 χ u ) α α Y χ 3

I . Below, we study the eﬀect of α m I 100 - + + m e CJ p plane for type-I (a) and type-II (b) α α y v 5 T (b) Type-II Yukawa couplings. d / 5 √ √ i 6 / w = 997 + 2 v 50 . 4 5 6 7 8 2 2 2 2 2 α w α - - - - - 083 037 αv

0 0 0 0 0 . .

ann h i 9

1 1 1 1 1 d n n

a 0 ) s m c ( / v σ .

σ 3 6 h ), supersonic deﬂagrations ( - − s χ in the type-I case leading to large m 73 + 0 36 017 + (0

. . .

< c

W G

Ω

h 0 1 0 ˆ . 2 – 20 – w − v 1 2 3 4 5 6 7 8 9 0 1 2 = = = 1 1 1 1 1 1 1 1 1 2 2 2 in the relativistic plasma is very close to ------W 0 0 0 0 0 0 0 0 0 0 0 0 B v s A v v D 1 1 1 1 1 1 1 1 1 1 1 1 region is basically excluded by the FCNC constraints, + 600 GeV c κ κ κ . W : : ± s s → c J H c C . One reason is that some annihilation channels kinematically , the vacuum energy fraction converted into the bulk kinetic v i m 600 GeV = = = 1 : χχ v 500 w v . w w w = 1 ]. The pile-up of points around , ann v v v s ± g w σ ) n v i h H l 134 has the following analytic approximations, based on ﬁt. V p e u v m G o κ ( c

χ a Fermi-Magic ]. The sound speed , and -ray observations of dwarf galaxies by Fermi-LAT and MAGIC [ w m s a γ , we project the parameter points in the c k u 108 Y 7

= I 100 - e w p y v . Parameter points projected in the (a) Type-I Yukawa couplings. T , s c 50 6 7 8 4 5 9 2 2 2 2 2 2 ------In the above numerical analyses, we have assumed the bubble propagation mode to In ﬁgure

0 0 0 0 0 0

i 1 1 1 1 1 1

d n n a c ( ) s m / v σ 3 w Furthermore, for subsonic deflagrations ( v energy of the fluid to the annihilation channel be Jouguet detonations withvarious bubble bubble wall propagation velocity modes.found In in general, ref. [ the dependence of all these parameter pointscan are required deviate to from predictvelocity the dependence the of canonical observed relic annihilation abundance, forbidden cross at section low velocities couldthe be resonance opened effects at [ the freeze-out epoch, and another reason is respectively. We remarkmany that parameter strong points with GW signalsthe type-II typically case, favor the small and hence it is more difficult to achieve strong GW signals. Figure 7 Yukawa couplings. Dashed linessection denote from the the 95% C.L. upper limits on the DM annihilation cross JHEP05(2021)160 (7.6) (7.7) (7.5) , along , 8 i δκ ) s c . On the other 8 . − . w v CJ v , ( = 15 0 D v lead to a stronger GW 0 κ 1 − 7 . Compared to Jouguet 3 8 . 72 B v 0 . 87 1) κ = . 5 2 J BBO 0 2 7 . . . C TianQin − − 0 0 0 v 1 1 = 0 TianQin - 4 = = = = = w 0 = 0 D v P w w w w w w 1 v CJ v κ v v v v v B v κ , SNR CJ h a CJ v + ( r v 3 t κ c 2 give much weaker GW signals. e 2 , = CJ v / - s p s 5 0 κ 2 c A v s ) c 1 .

w 2 ) κ e 0 w / v z − Ultimate DECIGO 5 v 5 − CJ / H a v /v 6 s ( ] w w CJ c

3 v f l v and CJ 3 w - a v 1) 0 v n ( 1 o – 21 – 05 3 B v i − . t + + κ a 1) t . These spectra are demonstrated in ﬁgure w B v i A v 1 v v δκ κ κ = 0 − ( a ) ) 4 5 r - s 5 / w 0 c G / − CJ 1 v . 11 s 3 v 11 w c − )] ( , and LISA is roughly given by v 1) α w v 72 v − . √ − κ Taiji 5 0 - 5 0 / , ), + ( , supersonic deﬂagrations with 3 4 5 6 7 8 9 0 1 2 6 7 8 9 1 CJ - - - - 2 1 1 1 1 1 1 1 1 1 1 11 s ------. v 0 0 0 0 B v

c . GW spectra for BP4 with several assumptions of 0 0 0 0 0 0 0 0 0 0 CJ

(1 +

1 1 1 1 CJ 0

( [( 1 1 1 1 1 1 1 1 1 1

κ v W G

Ω h v , 2 α/ & = 05 . √ ) = ) = ) = w s w v v CJ CJ = 0 v 9 ln[ Figure 8 < c . w 0 < v & w v − v w w ( v v = ( κ v < v κ δκ s c ( According to these expressions, we derive the GW spectra for BP4 assuming the bubble v κ experiments. The three scalar fieldswhich in should the be model have developedstrongly nonzero first VEVs from at order, EWPTs zero stochastic temperature, athas GWs could the been be early carefully effectively Universe. produced. constructedthermal corrections, with The If allowing effective one-loop us such potential corrections to EWPTs carry at are out zero of accurate temperature analyses on as EWPTs. well as 8 Summary In this paper, wemodel have studied comprising the the stochastic pNGBor GW dark type-II signals matter Yukawa from couplings. framework electroweak and Thenucleons FOPTs DM two vanishes in at candidate Higgs the zero is doublets momentum a transfer, with evading pNGB the type-I whose constraints tree-level from scattering direct detection off with the previously obtaineddetonations BP4 with spectrum for signal, which could behand, properly subsonic tested deflagrations by with TianQin with where wall velocity and detonations ( JHEP05(2021)160 β , we tan Ann. CJ v , = w v Front. Phys. ]. , ]. ]. SPIRE IN SPIRE SPIRE IN IN ][ ][ -meson decays, the Planck observation of (2017) 121202] [ B for type-II Yukawa couplings at large 12 ± hep-ph/0404175 arXiv:1003.0904 H [ – 22 – [ m Particle dark matter: Evidence, candidates and . For the benchmark point BP4, supersonic deflagra- -ray observations of dwarf galaxies by Fermi-LAT and w γ v in this model is more probable to induce a strong GW , which could be well detected by the future space-based (2005) 279 ), which permits any use, distribution and reproduction in Erratum ibid. ± (2010) 495 11 [ − H 405 48 10 – lead to a slightly weaker GW signal, while subsonic deflagrations 13 result in much weaker signals. − can induce a stronger GW signal than Jouguet detonations with = 1 CC-BY 4.0 10 05 . 72 w . 0 A survey of dark matter and related topics in cosmology This article is distributed under the terms of the Creative Commons v (2017) 121201 Phys. Rept. , Dark Matter Candidates from Particle Physics and Methods of Detection = 0 12 and 2 w . v . In this optimistic case, BP4 could be well tested by LISA, Taiji, and TianQin. = 0 CJ w constraints Rev. Astron. Astrophys. (Beijing) v v J.L. Feng, B.-L. Young, G. Bertone, D. Hooper and J. Silk, We have also investigated the effects of different bubble propagation modes with several Assuming that the bubble propagation mode is Jouguet detonations with We have performed random scans in the 12-dimensional parameter space, taking into = [2] [3] [1] w Attribution License ( any medium, provided the original author(s) and source areReferences credited. in part by the NationalNo. Natural 11875327, Science Foundation No. ofdation 11905300, China under under and Grants Grant No. No. No. 11805288, Universities, 12005312, 2018M643282, and the the the Fundamental China Sun Research Yat-Sen Postdoctoral Funds University Science for ScienceOpen Foundation. the Foun- Central Access. with Acknowledgments We thank Fa Peng Huang and Ligong Bian for helpful discussions. This work is supported values of the bubble walltions velocity with v Detonations with DECIGO are capable oflighter probing charged much Higgs more boson signal. parameter points. Since the We FCNCare have constraints more noticed on stringent that than thoseto a for stronger type-I GW Yukawa signals. couplings, the type-I case typically leads and the characteristic time duration ofrelic the GW FOPT. spectra Based on from such sound information, waves the and resulting MHD turbulencehave have found been that the evaluated. FOPTs ofGW some spectra parameter around points couldGW induce interferometers peak LISA amplitudes of and the Taiji. The next-generation GW interferometers BBO and account the constraints from boundedsurements of from the below 125 conditions, GeVthe Higgs LHC boson, DM run relic FCNC 1 abundance, andMAGIC. and run The the 2 surviving mea- parameter pointsWe are also have required further to induce analyzed an the electroweak FOPT. characteristic temperatures, the phase transition strength, JHEP05(2021)160 ] ]. , ]. ]. , ] (2020) ] SPIRE SPIRE L ] IN SPIRE 12 IN Phys. Rev. − IN ][ ]. ]. ]. , ][ B ][ One-loop ]. JHEP SPIRE SPIRE ]. , SPIRE arXiv:1907.09684 Electroweak IN [ IN Direct and indirect IN ][ ][ SPIRE ][ IN ]. arXiv:1812.05996 SPIRE [ ][ arXiv:1602.03837 IN [ Probing pseudo-Goldstone ][ arXiv:1912.04008 Global fit of ]. Pseudo-Nambu-Goldstone dark [ SPIRE arXiv:2001.05910 arXiv:1812.05952 [ Scalar Singlet Model IN [ arXiv:1901.09751 ]. (2019) 075011 [ ][ SPIRE IN U(1) 100 ]. ][ SPIRE (2019) 075028 ]. (2020) 015 arXiv:1708.06917 arXiv:1805.12562 arXiv:1708.02253 IN (2016) 061102 [ [ [ 99 ][ 04 Observation of Gravitational Waves from a arXiv:1906.02175 SPIRE (2021) 055024 116 SPIRE [ (2019) 015009 IN arXiv:2001.03954 (2019) 095036 IN [ ][ ][ JHEP 103 – 23 – 100 99 , Phys. Rev. D arXiv:1608.07648 Cancellation Mechanism for Dark-Matter-Nucleon , Non-thermal Production of PNGB Dark Matter and Pseudo-Nambu-Goldstone dark matter from gauged [ ]. (2017) 181302 (2018) 111302 Pseudo-Goldstone dark matter in a gauged (2017) 191801 Phys. Rev. D arXiv:2012.10286 , collaborations, (2020) 057 [ (2019) 035023 119 121 119 Phase Transitions and Gravitational Wave Tests of arXiv:1810.08139 SPIRE 05 Phys. Rev. Lett. Dark Matter Results From 54-Ton-Day Exposure of PandaX-II Probing pseudo Nambu-Goldstone boson dark matter at loop [ , IN Pseudo-Goldstone dark matter confronts cosmic ray and collider Phys. Rev. D 100 Phys. Rev. D Phys. Rev. D Dark Matter Search Results from a One Ton-Year Exposure of , ][ , Virgo , arXiv:1810.06105 (2017) 021303 arXiv:1901.03333 [ JHEP Results from a search for dark matter in the complete LUX exposure (2021) 130 [ and , 03 118 (2018) 089 Pseudo Nambu-Goldstone Dark Matter: Examples of Vanishing Direct Phys. Rev. Lett. 12 Phys. Rev. Lett. collaboration, Phys. Rev. Lett. , Phys. Rev. D , , (2019) 138 JHEP collaboration, , symmetry ]. ]. ]. ]. , L 01 − collaboration, (2019) 115010 JHEP arXiv:2008.12985 B , [ 99 SPIRE SPIRE SPIRE SPIRE IN IN IN IN Inflation LIGO Scientific Binary Black Hole Merger [ U(1) extended standard model Corrections in a Pseudo-Nambu034 Goldstone Dark Matter Model Revisited [ pseudo-Nambu-Goldstone Dark Matter [ Detection Cross Section anomalies matter and two-Higgs-doublet models [ Pseudo-Goldstone Dark Matter inD the Softly Broken level dark matter at the LHC probes of Goldstone dark matter Interaction contribution to dark-matter-nucleon scattering inJHEP the pseudo-scalar dark matter model Phys. Rev. Lett. PandaX-II Experiment XENON XENON1T LUX Y. Abe, T. Toma and K. Yoshioka, N. Okada, D. Raut and Q. Shafi, S. Glaus, M. Mühlleitner, J. Müller, S. Patel, T. Römer and R. Santos, C. Arina, A. Beniwal, C. Degrande, J. Heisig and A. Scaffidi, Y. Abe, T. Toma and K. Tsumura, J.M. Cline and T. Toma, X.-M. Jiang, C. Cai, Z.-H. Yu, Y.-P. Zeng and H.-H. Zhang, K. Kannike and M. Raidal, D. Karamitros, K. Huitu, N. Koivunen, O. Lebedev, S. Mondal and T.T. Toma, Alanne, M. Heikinheimo, V. Keus, N. Koivunen and K. Tuominen, C. Gross, O. Lebedev and T. Toma, D. Azevedo, M. Duch, B. Grzadkowski, D. Huang, M. Iglicki and R. Santos, K. Ishiwata and T. Toma, [7] [8] [9] [5] [6] [4] [20] [21] [18] [19] [16] [17] [14] [15] [12] [13] [10] [11] JHEP05(2021)160 ]. ]. 08 (2016) (2019) ] 99 ]. SPIRE SPIRE ] IN 94 79 IN ][ ][ JHEP SPIRE , (2019) 183 IN ][ 02 ]. (2018) 006 Phys. Rev. D 12 , Phys. Rev. D SPIRE Gravitational wave, , JHEP arXiv:1907.08899 , IN Eur. Phys. J. C (2017) 095028 [ , arXiv:1502.07574 ][ JHEP [ , 96 arXiv:1801.06184 (2019) 048 arXiv:1702.02698 [ (2020) 035001 [ ]. 02 Gravitational waves and electroweak arXiv:1908.00829 101 [ (2020) 197 SPIRE JHEP IN 80 , ][ (2017) 009 (2016) 065032 Phys. Rev. D (2018) 015032 , 09 ]. 93 arXiv:1704.04201 (2019) 062 [ 98 Impact of a complex singlet: Electroweak Modifying dark matter indirect detection signals by ]. Gravitational wave signatures from an extended Phys. Rev. D 10 – 24 – , SPIRE Unified explanation for dark matter and electroweak JCAP ]. Gravitational wave signals of pseudo-Goldstone dark , IN SPIRE ][ ]. Gravitational Wave Signals of Electroweak Phase Eur. Phys. J. C IN JCAP , , SPIRE ][ Phys. Rev. D Strong gravitational radiation from a simple dark matter ]. ]. ]. ]. ]. IN arXiv:1809.01198 , Electroweak Phase Transition, Gravitational Waves and Dark (2018) 095022 Phys. Rev. D Leptophilic dark matter from gauged lepton number: [ SPIRE ][ , Exploring inert dark matter blind spots with gravitational wave IN 98 Thermally modified sterile neutrino portal dark matter and Probing the baryogenesis and dark matter relaxed in phase ][ SPIRE SPIRE SPIRE SPIRE SPIRE IN IN IN IN IN Two-step strongly first-order electroweak phase transition modified Gravitational waves from scale-invariant vector dark matter model: ][ ][ ][ ][ ][ (2019) 190 complex singlet model 3 arXiv:1811.03279 05 Z [ arXiv:1702.06124 Phys. Rev. D [ , ]. ]. arXiv:1605.08663 JHEP [ , arXiv:1811.09807 [ SPIRE SPIRE arXiv:1907.13136 IN arXiv:1810.03172 arXiv:1809.09110 arXiv:1810.02380 arXiv:1709.09691 IN matter in the [ inert doublet dark matter model Matter in Two Scalar360 Singlet Extension of The Standard Model Probing below the neutrino-floor [ gravitational waves from phase[ transition: The Freeze-in case FIMP dark matter, gravitational(2019) wave 055003 signals, and the neutrino mass [ baryogenesis in a global study[ of the extended scalar singlet model thermal effects at freeze-out model Phenomenology and gravitational wave signatures signatures transition by gravitational waves[ and colliders Transition Triggered by Dark Matter collider and dark matter(2017) signals 108 from a scalar singlet electroweak baryogenesis baryogenesis and dark matter [ baryogenesis with direct detection and055006 gravitational wave signatures K. Kannike, K. Loos and M. Raidal, A. Paul, B. Banerjee and D. Majumdar, V.R. Shajiee and A. Tofighi, A. Mohamadnejad, L. Bian and X. Liu, A. Beniwal, M. Lewicki, M. White and A.G. Williams, L. Bian and Y.-L. Tang, I. Baldes and C. Garcia-Cely, E. Madge and P. Schwaller, F.P. Huang and C.S. Li, A. Hektor, K. Kannike and V. Vaskonen, A. Beniwal, M. Lewicki, J.D. Wells, M. White and A.G. Williams, F.P. Huang and J.-H. Yu, M. Chala, G. Nardini and I. Sobolev, W. Chao, H.-K. Guo and J. Shu, M. Jiang, L. Bian, W. Huang and J. Shu, [36] [37] [34] [35] [33] [31] [32] [29] [30] [27] [28] [25] [26] [23] [24] [22] JHEP05(2021)160 ]. ]. , ]. Phys. , Phys. SPIRE Phys. , SPIRE , IN SU(2) IN SPIRE ][ ]. ][ IN ]. [ ]. Class. Quant. Phys. Rev. D , , colliders and SPIRE SPIRE ]. − IN SPIRE e IN Class. Quant. Grav. [ ]. IN ][ + , e ][ Observable Gravitational SPIRE ]. IN SPIRE [ ]. IN arXiv:1811.01948 [ [ arXiv:2003.02276 [ arXiv:1702.00786 Testing the Dark Confined ]. ]. ]. SPIRE , SPIRE ]. IN ]. IN Taiji program: Gravitational-wave [ ][ SPIRE SPIRE SPIRE arXiv:1807.09495 arXiv:2012.11614 ]. SPIRE IN IN IN [ , arXiv:1912.12634 ]. SPIRE (2020) 046 IN ][ ][ ][ [ IN (2019) 076901 [ 08 ][ SPIRE Electroweak Phase Transition with an arXiv:2003.08828 82 Singlet-doublet fermionic dark matter and ]. SPIRE , IN arXiv:2012.03920 IN – 25 – , ][ ][ JCAP Complementarity of the future (2020) 082 , Dark matter, electroweak phase transition, and SPIRE (2017) 685 Complementary probe of dark matter blind spots by arXiv:2008.09605 IN (2020) 2050075 Filtered pseudo-scalar dark matter and gravitational waves arXiv:2012.15113 4 [ 07 [ , 35 Review of cosmic phase transitions: their significance and First-order electroweak phase transition in a complex singlet ]. arXiv:2010.03730 arXiv:2003.02465 arXiv:1911.05579 [ [ [ JHEP The Taiji Program in Space for gravitational wave physics and the Gravitational Wave Emissions from First Order Phase Transitions arXiv:1512.02076 Rept. Prog. Phys. , The TianQin project: current progress on science and technology TianQin: a space-borne gravitational wave detector [ , SPIRE (2020) 080 Scale-genesis by Dark Matter and Its Gravitational Wave Signal Laser Interferometer Space Antenna IN arXiv:1912.12899 [ 10 arXiv:1803.03368 [ Natl. Sci. Rev. [ Pseudo-Goldstone dark matter: gravitational waves and direct-detection , Fundamentals of the orbit and response for TianQin symmetry arXiv:2012.09758 3 (2021) 035012 (2020) 053011 (2020) 075047 , JHEP Z , collaboration, collaboration, (2016) 035010 Int. J. Mod. Phys. A 103 102 101 , collaboration, 33 (2020) 055028 (2018) 095008 arXiv:2008.10332 nature of gravity sources TianQin Grav. 35 TianQin from first order phase transition experimental signatures LISA Dark Sector Landscape: From Lattice to Gravitational Waves Rev. D blind spots lepton colliders and gravitational waves Rev. D with Two Component FIMP Dark Matter gravitational waves in the type-II two-Higgs-doublet model with a singlet scalar field model with Waves in Minimal Scotogenic Model gravitational waves in theRev. probe D of complex singlet extension to the standard model gravitational waves in a101 two-Higgs-doublet extension of the Standard Model W.-R. Hu and Y.-L. Wu, W.-H. Ruan, Z.-K. Guo, R.-G. Cai and Y.-Z. Zhang, X.-C. Hu et al., A. Mazumdar and G. White, T. Ghosh, H.-K. Guo, T. Han and H. Liu, W.-C. Huang, M. Reichert, F. Sannino and Z.-W. Wang, W. Chao, X.-F. Li and L. Wang, T. Alanne et al., Y. Wang, C.S. Li and F.P. Huang, M. Pandey and A. Paul, X.-F. Han, L. Wang and Y. Zhang, C.-W. Chiang and B.-Q. Lu, D. Borah, A. Dasgupta, K. Fujikura, S.K. Kang and D. Mahanta, Z. Kang and J. Zhu, B. Barman, A. Dutta Banik and A. Paul, N. Chen, T. Li, Y. Wu and L. Bian, [55] [56] [52] [53] [54] [50] [51] [47] [48] [49] [45] [46] [43] [44] [40] [41] [42] [39] [38] JHEP05(2021)160 , ] 72 (2018) ]. 98 Phys. ]. (2018) 675 , Phys. Rev. Theory and , 78 ]. SPIRE Phys. Rev. D Class. Quant. IN SPIRE , , (2019) 052 IN ][ (2006) 064006 arXiv:1106.0034 [ 7 SPIRE Eur. Phys. J. C [ Update of the global 73 IN , Phys. Rev. D ][ , ]. (2012) 1 (2017) 052 Eur. Phys. J. C (1977) 1966 ]. , SPIRE ]. Gravitational waves, inflation and 05 516 IN SciPost Phys. 15 Detecting a gravitational-wave [ , Phys. Rev. D A Second Higgs Doublet in the Early , astro-ph/0108011 SPIRE ]. [ SPIRE IN JCAP IN [ Constraining new physics from Higgs ][ ]. , arXiv:1502.04138 ]. (1985) 58 [ SPIRE Phys. Rept. IN , SPIRE 62 Phys. Rev. D Possibility of direct measurement of the ][ ]. IN , SPIRE Review of Particle Physics – 26 – ][ IN (2001) 221103 Gravitational wave and collider signals in complex ][ (2015) 440 87 SPIRE Symmetries of two Higgs doublet model and CP-violation IN hep-ph/0408011 ]. 75 Natural Conservation Laws for Neutral Currents [ ][ ]. ]. ]. arXiv:2001.01237 , Lilith: a tool for constraining new physics from Higgs ]. SPIRE collaboration, BBO and the neutron-star-binary subtraction problem IN SPIRE SPIRE SPIRE astro-ph/0504294 [ Theor. Math. Phys. Baryon asymmetry and detectable Gravitational Waves from IN IN IN [ gr-qc/0511092 , SPIRE arXiv:1909.02978 [ ][ ][ ][ [ IN Phys. Rev. Lett. (2005) 115013 , On Necessary and Sufficient Conditions for Some Higgs Potentials to Be ]. ][ Eur. Phys. J. C Diagonal Neutral Currents 72 , arXiv:1205.3781 Vacuum Stability Conditions From Copositivity Criteria [ (1977) 1958 SPIRE (2005) S955 ]. IN 15 [ 22 (2006) 042001 (2020) 015015 73 SPIRE arXiv:1908.03952 arXiv:1803.01853 IN arXiv:1611.05874 gr-qc/0511145 measurements with Lilith: update[ to LHC Run 2 results electroweak fit and constraints on[ two-Higgs-doublet models Particle Data Group 030001 measurements Bounded From Below (2012) 2093 Phys. Rev. D Rev. D Electroweak phase transition phenomenology of two-Higgs-doublet models [ Universe: Baryogenesis and Gravitational[ Waves two-Higgs doublet model with101 dynamical CP-violation at finite temperature the cosmic microwave background:Grav. Towards testing the slow-roll paradigm D acceleration of the universe usingantenna 0.1-Hz in band space laser interferometer gravitational wave background with next-generation space interferometers [ S. Kraml, T.Q. Loc, D.T. Nhung and L.D. Ninh, J. Haller, A. Hoecker, R. Kogler, K. Mönig, T. Peiffer and J. Stelzer, J. Bernon and B. Dumont, E.A. Paschos, K.G. Klimenko, K. Kannike, I.F. Ginzburg and M. Krawczyk, S.L. Glashow and S. Weinberg, R. Zhou and L. Bian, G.C. Branco, P.M. Ferreira, L. Lavoura, M.N. Rebelo, M. Sher and J.P. Silva, G.C. Dorsch, S.J. Huber, T. Konstandin and J.M. No, X. Wang, F.P. Huang and X. Zhang, C. Ungarelli, P. Corasaniti, R.A. Mercer and A. Vecchio, C. Cutler and J. Harms, H. Kudoh, A. Taruya, T. Hiramatsu and Y. Himemoto, N. Seto, S. Kawamura and T. Nakamura, [72] [73] [70] [71] [67] [68] [69] [65] [66] [63] [64] [61] [62] [59] [60] [58] [57] JHEP05(2021)160 90 , ] Phys. 439 (1974) , 07 ]. 9 Phys. , (2017) 121 (2014) 2250 02 SPIRE JHEP ]. IN , Phys. Rev. D Astron. , ][ , 185 Erratum ibid. (1994) 6662] ]. JHEP ]. Strong First Order [ Phys. Rev. D , SPIRE ICTP Summer School arXiv:2009.10080 , ]. 50 IN [ FeynRules 2.0 — A Theoretical uncertainties ][ SPIRE , in SPIRE IN IN (1995) 3 SPIRE ][ ]. ][ IN ]. ]. ][ 436 ]. (2021) 055 arXiv:1107.3559 [ SPIRE The Lightest Higgs boson mass in the Erratum ibid. 04 SPIRE IN [ SPIRE SPIRE IN ][ IN Taming Infrared Divergences in the [ IN hep-ph/9901312 ][ Comput. Phys. Commun. ][ JHEP , , ]. (2011) 089 Limits to Dark Matter Annihilation Electroweak Baryogenesis in Two Higgs Doublet arXiv:1406.2652 Nucl. Phys. B The Electroweak Phase Transition in the Inert arXiv:1601.06590 [ , ]. (1993) 3546 11 [ ]. – 27 – , (1999) [ arXiv:1504.05949 SPIRE (1973) 1888 [ 47 IN 7 Radiative Corrections as the Origin of Spontaneous SPIRE ][ SPIRE JHEP ]. IN , Electroweak phase transition in two Higgs doublet models IN arXiv:1804.00044 The Effective potential and first order phase transitions: hep-ph/9609240 (2014) 034 ]. ][ ][ [ (2016) 039 [ ]. ]. collaborations, arXiv:1807.06209 08 (2015) 028 SPIRE [ 02 Symmetry Behavior at Finite Temperature IN SPIRE Planck 2018 results. VI. Cosmological parameters [ 07 Phys. Rev. D SPIRE IN SPIRE , Phys. Rev. D IN ][ JHEP , IN JCAP MadDM v.3.0: a Comprehensive Tool for Dark Matter Studies ][ , The Effective potential at finite temperature in the Standard Model (1997) 3873 ][ , The automated computation of tree-level and next-to-leading order (2020) A6 JCAP arXiv:1406.2355 (2019) 100249 , ]. 55 [ Fermi-LAT Taming the Goldstone contributions to the effective potential hep-ph/9407389 arXiv:1405.0301 24 Finite temperature field theory and phase transitions (1992) 2933 641 [ and collaboration, ]. SPIRE 45 IN [ SPIRE IN hep-ph/9212235 arXiv:1612.04086 arXiv:1310.1921 Doublet Model for cosmological first-order phase transitions [ 3320 Rev. D Beyond leading-order [ Effective Potential (2014) 016013 Phys. Rev. D minimal supersymmetric standard model (1995) 466] [ Models and B meson anomalies Electroweak Phase Transition in the[ CP-Conserving 2HDM Revisited Satellite Galaxies Symmetry Breaking in High-Energy Physics and Cosmology Dark Univ. Planck Astrophys. MAGIC Cross-Section from a Combined Analysis of MAGIC and Fermi-LAT Observations of Dwarf complete toolbox for tree-level phenomenology [ differential cross sections, and their(2014) matching 079 to parton shower simulations N. Blinov, S. Profumo and T. Stefaniak, D. Croon, O. Gould, P. Schicho, T.V.I. Tenkanen and G. White, M.E. Carrington, P.B. Arnold and O. Espinosa, J. Elias-Miro, J.R. Espinosa and T. Konstandin, S.P. Martin, L. Dolan and R. Jackiw, J.M. Cline and P.-A. Lemieux, J.A. Casas, J.R. Espinosa, M. Quirós and A. Riotto, J.M. Cline, K. Kainulainen and M. Trott, P. Basler, M. Krause, M. Muhlleitner, J. Wittbrodt and A. Wlotzka, S.R. Coleman and E.J. Weinberg, M. Quirós, J. Alwall et al., F. Ambrogi et al., A. Alloul, N.D. Christensen, C. Degrande, C. Duhr and B. Fuks, [90] [91] [88] [89] [85] [86] [87] [83] [84] [81] [82] [79] [80] [77] [78] [75] [76] [74] JHEP05(2021)160 , , Nucl. ]. , 20 ]. (1983) ]. 183 . SPIRE (2020) 045 216 ]. SPIRE IN IN 05 ][ SPIRE Eur. Phys. J. C [ (2017) 086 , IN Adv. Phys. (1992) 3415 , SPIRE (1980) 963] ][ 12 IN 45 JCAP 44 ][ (1997) L585 , 30 (1981) 876 Nucl. Phys. B JHEP (2019) 003 , , ]. 23 04 Nucleation and bubble growth in arXiv:1712.08430 Energy Budget of Cosmological Comput. Phys. Commun. [ SPIRE Phys. Rev. D , Erratum ibid. , IN arXiv:1004.4187 JCAP J. Phys. A [ ]. Gravitational waves from a supercooled [ The Higgs Vacuum Uplifted: Revisiting , , ][ Gravitational radiation from first order astro-ph/9310044 [ Phys. Rev. D SPIRE , ]. (2018) 151 ]. IN Gravitational waves from the electroweak phase ][ (1980) 631 (2010) 028 05 ]. – 28 – Bubbles in the supersymmetric standard model ]. 44 SPIRE SPIRE 06 ]. Phase transition dynamics and gravitational wave IN IN (1994) 2837 An introduction to percolation theory SPIRE ][ JHEP On the Maximal Strength of a First-Order Electroweak A new insight into the phase transition in the early arXiv:1205.3070 IN , SPIRE 49 [ Precise determination of the critical threshold and exponents SPIRE Cosmological Consequences of a First Order Phase ][ JCAP IN ]. ]. ]. IN , [ Phase Transitions and Magnetic Monopole Production in the [ hep-ph/9801272 Grand Unified Model [ (1983) 544] [ SPIRE SPIRE SPIRE IN IN IN Phys. Rev. Lett. (2012) 024 , ][ ][ ][ 223 (1981) 37 Phys. Rev. D CosmoTransitions: Computing Cosmological Phase Transition 10 SU(5) (1982) 2074 , Relativistic Detonation Waves and Bubble Growth in False Vacuum Decay arXiv:1703.06552 100 25 [ (1998) 489 arXiv:1109.4189 Fate of the False Vacuum at Finite Temperature: Theory andDecay Applications of the False Vacuum at Finite Temperature [ JCAP . , ]. ]. 526 Erratum ibid. [ (2017) 570 SPIRE SPIRE IN arXiv:2003.08892 arXiv:1809.08242 IN arXiv:1705.09186 First-order Phase Transitions a first order cosmological electroweak[ phase transition spectra of strong first-order phase[ transition in supercooled universe Phys. Rev. D phase transitions 77 Phase Transition and its Gravitational[ Wave Signal (1971) 325 in a three-dimensional continuum percolation model electroweak phase transition and their detection with pulsar timing arrays Very Early Universe [ Transition in the Phys. B transition Universe with two Higgs doublets Phys. Lett. B 421 Temperatures and Bubble Profiles with(2012) Multiple 2006 Fields the Electroweak Phase Transition with[ a Second Higgs Doublet J.R. Espinosa, T. Konstandin, J.M. No and G. Servant, K. Enqvist, J. Ignatius, K. Kajantie and K. Rummukainen, P.J. Steinhardt, M. Kamionkowski, A. Kosowsky and M.S. Turner, J. Ellis, M. Lewicki and J.M. No, X. Wang, F.P. Huang and X. Zhang, M.D. Rintoul and S. Torquato, A. Kobakhidze, C. Lagger, A. Manning and J. Yue, A.H. Guth and E.J. Weinberg, V.K.S. Shante and S. Kirkpatrick, J.M. Moreno, M. Quirós and M. Seco, L. Leitao, A. Megevand and A.D. Sanchez, A.H. Guth and S.H.H. Tye, A.D. Linde, A.D. Linde, G.C. Dorsch, S.J. Huber, K. Mimasu and J.M. No, J. Bernon, L. Bian and Y. Jiang, C.L. Wainwright, [97] [98] [99] [95] [96] [93] [94] [92] [108] [109] [106] [107] [104] [105] [102] [103] [100] [101] JHEP05(2021)160 , 05 92 ]. ] (2009) 96 (2021) JCAP SPIRE ]. 12 , 01 IN ]. ]. ][ SPIRE Phys. Rev. D JCAP ]. JCAP IN , SPIRE , ]. SPIRE , IN ][ IN Phys. Rev. D ]. ][ arXiv:2003.07360 , ][ (1993) 4372 [ SPIRE SPIRE IN 47 IN ][ SPIRE [ Numerical simulations of Shape of the acoustic Gravitational waves from the IN (2014) 041301 ][ ]. arXiv:1512.06239 [ (2020) 050 112 Phase transitions in the early 07 SPIRE arXiv:1903.09642 Cosmological Backgrounds of (1992) 2026 IN [ ]. Phys. Rev. D arXiv:1704.05871 arXiv:2008.09136 ][ , 69 [ Gravitational wave energy budget in strongly ]. (2016) 001 JCAP Phase Transitions in an Expanding Universe: , ]. Gravitational radiation from cosmological astro-ph/0111483 SPIRE Gravitational radiation from colliding vacuum Gravitational waves from first order 04 [ IN SPIRE (2019) 024 ][ arXiv:1910.13125 ]. (2021) 1 – 29 – Phys. Rev. Lett. The stochastic gravitational wave background from IN [ SPIRE , [ 06 JCAP IN 24 , ][ (2020) 089902] [ ]. Gravitational waves from first-order cosmological phase SPIRE ]. IN Production of gravitational waves in the NMSSM Gravitational radiation from colliding vacuum bubbles: Gravitational Wave Production by Collisions: More Bubbles Phys. Rev. Lett. arXiv:1201.0983 ]. JCAP ][ , [ 101 (2002) 024030 ]. , (2020) 024 SPIRE SPIRE IN (1992) 4514 66 03 IN ][ SPIRE ][ SPIRE 45 arXiv:0806.1828 IN [ IN ][ (2012) 027 ][ JCAP Detecting gravitational waves from cosmological phase transitions with Science with the space-based interferometer eLISA. II: Gravitational waves , 06 arXiv:1504.03291 Erratum ibid. [ [ arXiv:0709.2091 Phys. Rev. D [ (2008) 022 , SciPost Phys. Lect. Notes JCAP ]. Phys. Rev. D , , 09 , arXiv:2007.08537 arXiv:0909.0622 [ [ SPIRE IN arXiv:1304.2433 astro-ph/9211004 turbulence turbulence and magnetic fields024 generated by a first-order phase transition transitions: lifetime of the[ sound wave source Stochastic Gravitational Waves in Standard001 and Non-Standard Histories gravitational wave power spectrum from(2017) a 103520 first order phase transition supercooled phase transitions sound of a first[ order phase transition acoustically generated gravitational waves at(2015) a 123009 first order phase transition envelope approximation to many[ bubble collisions JCAP universe cosmological phase transitions Sources from cosmological phase transitions LISA: an update bubbles (2008) 017 Gravitational Waves and eLISA/NGO: Phase Transitions, Cosmic Strings and Other A. Kosowsky, A. Mack and T. Kahniashvili, C. Caprini, R. Durrer and G. Servant, J. Ellis, M. Lewicki and J.M. No, H.-K. Guo, K. Sinha, D. Vagie and G. White, M. Hindmarsh, S.J. Huber, K. Rummukainen and D.J. Weir, J. Ellis, M. Lewicki, J.M. No and V. Vaskonen, M. Hindmarsh, S.J. Huber, K. Rummukainen and D.J. Weir, S.J. Huber and T. Konstandin, M. Hindmarsh, S.J. Huber, K. Rummukainen and D.J. Weir, M.B. Hindmarsh, M. Lüben, J. Lumma and M. Pauly, A. Kosowsky, M.S. Turner and R. Watkins, A. Kosowsky and M.S. Turner, C. Caprini et al., C. Caprini et al., S.J. Huber and T. Konstandin, P. Binetruy, A. Bohe, C. Caprini and J.-F. Dufaux, A. Kosowsky, M.S. Turner and R. Watkins, [125] [126] [123] [124] [121] [122] [120] [118] [119] [115] [116] [117] [113] [114] [111] [112] [110] JHEP05(2021)160 , ] Phys. Rev. Class. , , ]. ]. (2007) 022002 98 SPIRE decay from the IN SPIRE − arXiv:1411.4413 ]. ][ IN µ [ (2017) 191801 + ][ µ -lepton properties as of ]. 118 τ → SPIRE 0 s IN B (2015) 68 ][ branching fraction and effective SPIRE Phys. Rev. Lett. IN − , 522 ) µ ][ 2 s + α µ ( arXiv:1612.07233 -hadron, and O arXiv:1310.5300 c [ → [ Phys. Rev. Lett. Nature 0 s at , , B ) Gravitational-wave sensitivity curves γ – 30 – s X -hadron, arXiv:1503.01789 decays b [ (2017) 895 → − Observation of the rare arXiv:1408.0740 ¯ µ B 77 [ ( + (2013) 124032 ]. Sensitivity curves for searches for gravitational-wave µ B ]. 88 ]. → 0 Three exceptions in the calculation of relic abundances SPIRE Averages of B IN (2015) 221801 SPIRE Measurement of the SPIRE ][ IN IN [ ][ (2015) 015014 Updated NNLO QCD predictions for the weak radiative B-meson decays Estimate of 114 Eur. Phys. J. C 32 , Phys. Rev. D , collaboration, ]. and LHCb collaborations, (1991) 3191 43 SPIRE hep-ph/0609232 arXiv:1703.05747 IN D [ Phys. Rev. Lett. lifetime and search for [ HFLAV summer 2016 Quant. Grav. backgrounds CMS combined analysis of CMS[ and LHCb data M. Misiak et al., M. Misiak et al., K. Griest and D. Seckel, LHCb collaboration, E. Thrane and J.D. Romano, C.J. Moore, R.H. Cole and C.P.L. Berry, [132] [133] [134] [130] [131] [128] [129] [127]