Probing the gauge symmetry breaking of the early universe in 3-3-1 models and beyond by gravitational waves
PProbing the gauge symmetry breaking of the early universe in 3-3-1 models andbeyond by gravitational waves
Fa Peng Huang and Xinmin Zhang
Theoretical Physics Division, Institute of High Energy Physics,Chinese Academy of Sciences, P.O.Box 918-4, Beijing 100049, P.R.ChinaSchool of Physics Sciences, University of Chinese Academy of Sciences, Beijing 100039, China
Taking the 3-3-1 models (with SU (3) c ⊗ SU (3) L ⊗ U (1) Y gauge group) as examples, we study that aclass of new physics models with extended gauge group could undergo one or several first-order phasetransitions associated with the spontaneously symmetry breaking processes during the evolutionof the universe, which can produce detectable phase transition gravitational wave (GW) signals atfuture GW experiments, such as LISA, BBO, DECIGO, SKA and aLIGO. These GW signals canprovide new sources of GWs with different peak frequencies, and can be used to probe the evolutionhistory of the universe. PACS numbers:
I. INTRODUCTION
The observation of gravitational waves (GWs) by Ad-vanced Laser Interferometer Gravitational Wave Obser-vatory (aLIGO) [1] has initiated a new era of exploringthe cosmology, the nature of gravity as well as the fun-damental particle physics by the GW detectors [2–9].Especially, due to the limitation of the colliders’ energy,GW detectors can be used as new or complementary tech-niques to probe the existence of the new physics (NP)by detecting the symmetry breaking patterns or phasetransition history for large classes of NP models withan extended gauge group, which are motivated by themysterious experimental results in our understanding ofparticle cosmology (such as the dark matter problem orthe puzzling observed baryon asymmetry of the universe),and the absence of NP signals at current collider exper-iments. The increasingly attractive NP models with anextended gauge group have many new particles withoutleaving obvious observable imprints at current particlecolliders. However, the GW experiments may provide apossible approach to test their existence. For example, toexplain the baryon asymmetry of the universe via elec-troweak (EW) baryogenesis, a strong first-order phasetransition (FOPT) is needed to realize the departure fromthermal equilibrium by extensions of the standard model(SM) [10–12]. And during the FOPT, detectable GWswill be produced through three mechanisms: collisions ofexpanding bubbles, sounds waves, and magnetohydrody-namic turbulence of bubbles in the hot plasma [13–20].Phase transitions in particle physics and cosmology areusually associated with the symmetry breaking, i.e. wherethe universe transits from a symmetric phase to a sym-metry broken phase when the temperature drops belowthe corresponding critical temperature.For the first time, we have a realistic chance to exploreNP with gauge symmetry breaking processes throughphase transition GW signals after the discovery of theGWs by aLIGO, which is particularly exciting. In thispaper, we study the possibility to probe the gauge symme- try breaking patterns and the phase transition history ofthe early universe by the phase transition GW signals. Inparticular, we focalize our analysis to GW detection of theNP models with an extended non-Abelian gauge group,where the symmetry breaking at each energy scale mayassociate with a FOPT, as shown in Fig.1. The group G Hidden can spontaneously break into the SM gauge groupvia one or several steps and strong FOPT can take placein each step, which can produce detectable phase transi-tion GWs. For example, the gauge group G Hidden can bethe non-Abelian gauge group SU (3) c ⊗ SU (3) L ⊗ U (1) Y ,which is called 3-3-1 model [21, 22]. We show that manyversions of the 3-3-1 model can produce at least one strongFOPT at TeV scale in some parameter spaces, which canproduce detectable GW spectrum by the recently provedLaser Interferometer Space Antenna (LISA) [23, 24], BigBang Observer (BBO) [25], Deci-hertz InterferometerGravitational wave Observatory (DECIGO) [26, 27], andUltimate-DECIGO [28]. In general, there can exist severalspontaneous symmetry breaking processes in NP mod-els, which may also accompany several FOPTs with theevolution of the universe as shown in Fig.1. If the scaleof the FOPT associated with the symmetry breaking isabout 10 − GeV, the phase transition GW spectrummay be within the sensitivity of future aLIGO. If thestrong FOPT occurs at the QCD phase transition scale insome hidden QCD models [2], the produced GW signalsmay be tested by pulsar time array (PTA) at the SquareKilometre Array (SKA) [29] or the Five-hundred-meterAperture Spherical Telescope (FAST) [30].This paper is organized as follows: In Section II, weschematically discuss the GW detection of the gaugegroup symmetry breaking and show how to calculate thephase transition GWs during the FOPT. In Section III,we will study the phase transition GW spectra in someconcrete models with extended gauge group. In SectionIV, we show our final discussions and conclusions. a r X i v : . [ h e p - ph ] N ov FIG. 1: Symmetry breaking (phase transition) patterns in theNP models with extended gauge group during the evolutionof the early universe, where FOPT may occur.
II. FIRST-ORDER PHASE TRANSITIONGRAVITATIONAL WAVE SPECTRUM
In a generic classes of NP models, one or severalcosmological phase transitions can occur during eachstep’s symmetry breaking at different energy scalewith the evolution of the universe as shown in Fig. 1.For example, the symmetry breaking pattern may be G (HiddenN) · · · → G(Hidden1) → G(SU(3) C ⊗ SU(3) L ⊗ U(1) Y ) → G(SU(3) C ⊗ SU(2) L ⊗ U(1) X ) → G(SU(3) C ⊗ U(1) EM ). With the evolution of our universe, symmetrybreaking will happen at corresponding energy scale, wherethe strong FOPT may take place. Detailed models aregiven in Section III.With the evolution of the universe, the universe tran-sits from a ‘false’ vacuum to a ‘true’ vacuum, and strongFOPT occurs if there exists a sufficient potential barrierbetween them. These processes can produce observablestochastic GW signals, which can be detected in someGW detectors, such as aLIGO, LISA, BBO, DECIGO,Ultimate-DICIGO, SKA, FAST and so on. Their sensi-tivity range for some critical temperatures depends onthe energy scale of the FOPT for different gauge groupextended models, as shown in Section III. To discuss theGW spectra from FOPT, it is necessary to begin with theone-loop finite temperature effective potential V eff (Φ , T ): V eff (Φ , T ) = V tree (Φ)+ V cw (Φ)+ V ther (Φ , T )+ V daisy (Φ , T ) , (1)where Φ represents the order parameter field for the phasetransition, V cw is the one-loop Coleman-Weinberg poten-tial at T = 0, and V ther + V daisy is the thermal contributionincluding the daisy resummation [50]. During each stepof symmetry breaking in the NP models with extendedgauge group, strong FOPT may occur. During a strongFOPT, bubbles are nucleated via quantum tunneling orthermally fluctuating the potential barrier with the nu-cleation rate per unit volume Γ = Γ ( T )e − S E ( T ) andΓ ( T ) ∝ T [32], where S E ( T ) (cid:39) S ( T ) /T is Euclideanaction [33, 34] defined as S E ( T ) = (cid:90) dτ d x (cid:34) (cid:18) d Φ dτ (cid:19) + 12 ( ∇ Φ) + V eff (Φ , T ) (cid:35) . Then, Γ = Γ e − S /T [32] and S ( T ) = (cid:90) d x (cid:20)
12 ( ∇ Φ) + V eff (Φ , T ) (cid:21) . (2)From the above equations, in order to obtain the nucle-ation rate, the profile of the scalar field Φ needs to becalculated by solving the following bounce equation: d Φ dr + 2 r d Φ dr − ∂V eff (Φ , T ) ∂ Φ = 0 , (3)with the boundary conditions d Φ dr ( r = 0) = 0 andΦ( r = ∞ ) = Φ false . The bounce equation can be solvednumerically using the overshoot/undershoot method. TheFOPT terminates when nucleation probability of one bub-ble per horizon volume is of O (1), i.e., Γ( T ∗ ) (cid:39) H ∗ . Thatis to say, it should satisfy S ( T ∗ ) /T ∗ = 4 ln( T ∗ / . (4)It is known that there exist three sources for producingGWs during the FOPT, which are collisions of the vacuumbubbles [16], sound waves [17] and turbulence [18, 19] inthe plasma after collisions, respectively.The most well-known source is the bubbles collisions,and the corresponding phase transition GW spectrumdepends on four parameters. The first parameter is theratio α of the vacuum energy density released in the phasetransition to that of thermal bath, defined as α ≡ ∆ V eff ( T ∗ ) − T ∂ ∆ V eff ( T ∗ ) ∂T ρ rad ( T ∗ ) , (5)where ∗ specifies that the quantity is evaluated at T ∗ determined by Eq.(4). The parameter α measures thestrength of the phase transition GWs, namely, largervalues for α correspond to stronger phase transition GWs.The second one is the time duration of the phase transition β − with β ≡ − dS E dt | t = t ∗ (cid:39) d Γ dt | t = t ∗ , and one has βH ∗ = T d ( S /T ) dT (cid:12)(cid:12)(cid:12)(cid:12) T = T ∗ (6)since β = ˙Γ / Γ during the phase transition from its def-inition. In other words, β − corresponds to the typicaltime scale of the phase transition. The third one is theefficiency factor λ co , which characterizes the fraction ofthe energy density converted into the motion of the col-liding bubble walls. And the last one is the bubble wallvelocity v b . Here, for simplification, we choose the de-fault value v b = 0 .
7. Explicit calculations on the bubblewall velocity are beyond the scope of this work. Theenergy released into the GWs of peak frequency [35] is ρ GW,co ρ tot ∼ θ co (cid:16) H ∗ β (cid:17) λ co α (1+ α ) v b . The second and third sources are the GWs from thematter fluid effects, which can further contribute to thetotal energy released in gravitational radiation duringthe phase transition. Here, we just use the formulaegiven in Ref [36]. The second source is from the soundwaves in the fluid, where a certain fraction λ sw of thebubble wall energy (after the collision) is converted intomotion of the fluid (and is only later dissipated) [36] with ρ GW,sw ρ tot ∼ θ sw (cid:16) H ∗ β (cid:17) λ sw (cid:16) α (1+ α ) (cid:17) . The third source isfrom turbulence in the fluid, where a certain fraction λ tu of the walls energy is converted into turbulence [36] with ρ GW,tu ρ tot ∼ θ tu (cid:16) H ∗ β (cid:17) λ / tu (cid:16) α / (1+ α ) / (cid:17) . It is worth noticingthat these two contributions from the matter fluid effectsdepend on H ∗ /β linearly, and they are not fully under-stood. In some cases, these two effects may be larger thanthe one from bubble collisions.The peak frequency produced from bubble collisionsat T ∗ during the FOPT is given by [37, 38]: f ∗ co =0 . β/ (1 . − . v b + v b ). Considering the adiabatic ex-pansion of our universe from the early universe to thepresent universe, the ratio of scale factors at the time ofFOPT and today can be written as a ∗ a = 1 . × − Hz × H ∗ (cid:16) T ∗ GeV (cid:17)(cid:16) g t ∗ (cid:17) / . Thus, the peak frequency todayis f co = f ∗ co a ∗ /a , and the corresponding GW intensity isgiven by [37]Ω co ( f ) h (cid:39) . × − (cid:16) H ∗ β (cid:17) (cid:16) λ co α α (cid:17) (cid:16) g t ∗ (cid:17) × (cid:16) . v b .
42 + v b (cid:17)(cid:104) . f /f co ) . . f /f co ) . (cid:105) . The peak frequency of the GW signals from sound waveeffects is about f sw = 2 β/ ( √ v b ) a ∗ /a [17, 36] with theGW intensity [17, 36, 39]Ω sw ( f ) h (cid:39) . × − (cid:16) H ∗ β (cid:17)(cid:16) λ sw α α (cid:17) (cid:16) g t ∗ (cid:17) v b × (cid:104) f /f sw ) / f /f sw ) (cid:105) / , in which λ sw (cid:39) α (0 .
73 + 0 . √ α + α ) − [39] for rela-tivistic bubbles.The GW signals from the turbulence have the peakfrequency at about f tu = 1 . β/v b a ∗ /a [36] and theintensity [19, 40]:Ω tu ( f ) h (cid:39) . × − (cid:16) H ∗ β (cid:17)(cid:16) λ tu α α (cid:17) / (cid:16) g t ∗ (cid:17) v b × ( f /f tu ) (1 + f /f tu ) / (1 + 8 πf a / ( a ∗ H ∗ )) . The final phase transition spectra consist of the threecontributions above.
III. PHASE TRANSITION GRAVITATIONALWAVES FROM NON-ABELIAN GAUGE GROUPEXTENDED MODELS
In this section, we discuss the phase transition GWs insome NP models with extended non-Abelian gauge group, where one or several strong FOPTs may occur with theevolution of our universe at certain critical temperature.Firstly, the GW spectra in the gauge group extendedmodels based on the SU (3) c ⊗ SU (3) L ⊗ U (1) Y gaugesymmetry, commonly known as the 3-3-1 models [21, 22]are investigated. The 3-3-1 models can naturally explainthe electric charge quantization and three generationsof fermions [21, 22]. The collider phenomenology of the3-3-1 models have been extensively studied, such as therecent Ref. [41] and references therein, and the phasetransitions in some versions of 3-3-1 models have beenstudied in Refs. [42–44]. So far, no obvious NP signals arediscovered at the LHC, including the 3-3-1 models. Here,we use the GW signals to explore the NP models and theirphase transition patterns in three versions of the 3-3-1models (We discuss the minimal and the economical 3-3-1model in details, and only show the main results of thereduced minimal 3-3-1 models.) [42–44], where the scalarsfields are accommodated in different representations ofthe SU (3) L gauge group in each version.For simplicity, we limit our discussions of the FOPT tothe thermal barrier case, where the potential barrier inthe finite temperature effective potential origins fromthermal effects. In this case, the bosonic fields con-tribute to the thermal effective potential of the form V eff (cid:51) ( − T / π ) (cid:0) m ( X, T ) (cid:1) / in the limit of high-temperature expansion. For qualitative sketch of this typeof FOPT, we show the general effective potential near thephase transition temperature, which can be approximatedby V eff ( X, T ) ∼ (cid:0) − µ + c T (cid:1) X − e T ( X ) / π + λ X . (7)Here, X represents the order parameter field for thephase transition. For the EW phase transition in theSM, X field is just the Higgs field. The parameter e quantify the interactions between X field and thelight bosons, and can be schematically written as e ∼ (cid:80) light boson (degrees of freedom) × (coupling to X) / .And, the parameters c depends on interaction between X and light particles. For the heavy fields whose massesare much larger than the critical temperature, their con-tribution can be omitted from Boltzmann suppression.This can help to simplify our discussions when the modelshave many new fields at different energy scales. Thus,in this case of qualitative analysis, the wash out param-eter can be obtained as (cid:104) X (cid:105) ( T c ) T c ≈ e πλ , where the anglebracket <> means the vacuum expectation value (VEV)of the field X at T c . From the above qualitative analysis,we know that introducing new light bosonic fields (com-pared to the corresponding critical temperature) helpsto produce or enhance the FOPT. The 3-3-1 models justintroduce enough bosonic fields to produce detectablephase transition GWs. A. Gravitational wave spectrum in the minimal3-3-1 model
We firstly consider the phase transition GW spectrumin the so-called minimal 3-3-1 model [42], which corre-sponds to the electric charge operator Q = T −√ T + x I.Here, x represents the U (1) charge, and T and T are the generators. The gauge bosons, associated withthe gauge symmetry SU (3) L ⊗ U (1) Y , consist of anoctet W iµ ( i = 1 , · · · ,
8) and a singlet B µ . In thismodel, three SU (3) L triplets scalars ( η = (cid:0) η η − η +2 (cid:1) T , ρ = (cid:0) ρ + ρ ρ ++ (cid:1) T , χ = (cid:0) χ − χ −− χ (cid:1) T ) are needed tobreak the gauge symmetry, and generate the masses of thegauge bosons and the exotic quarks. The scalar potentialin terms of ρ , η and χ is given as [42, 45] V ( ρ, η, χ ) = µ η † η + λ (cid:0) η † η (cid:1) + µ ρ † ρ + λ (cid:0) ρ † ρ (cid:1) + µ χ † χ + λ (cid:0) χ † χ (cid:1) + (cid:2) λ (cid:0) ρ † ρ (cid:1) + λ (cid:0) χ † χ (cid:1)(cid:3) (cid:0) η † η (cid:1) + λ (cid:0) ρ † ρ (cid:1) (cid:0) χ † χ (cid:1) + λ (cid:0) ρ † η (cid:1) (cid:0) η † ρ (cid:1) + λ (cid:0) χ † η (cid:1) (cid:0) η † χ (cid:1) + λ (cid:0) ρ † χ (cid:1) (cid:0) χ † ρ (cid:1) + 12 (cid:0) f (cid:15) ijk η i ρ j χ k + H. c. (cid:1) . (8)The new gauge bosons acquire masses at several TeVscale when the SU (3) L × U (1) Y group breaks down to SU (2) L × U (1) X triggered by the SU (3) L triplet scalar χ ,while the ordinary quarks and SM gauge bosons obtaintheir masses during the last step symmetry breakingtrigged by the triplet scalar fields η and ρ . There existthree CP-even neutral scalars including the lightest onewhich corresponds to the SM Higgs boson h and theother heavier scalar bosons H and H . There is also amassive Z (cid:48) gauge boson, which has been constrained bythe current LHC data.Numerically, we find that there are parameter spaces al-lowed by the collider constraints [46] that can give a strongFOPT, when the gauge group spontaneously breaks from SU (3) L × U (1) Y to SU (2) L × U (1) X [42]. During thisphase transition, the order parameter field X here is justthe H field. Then, the phase transition GW spectrumcan be obtained from the above GW spectrum formulae.Since this model has so many free parameters, whichmakes it very complicated to study the whole parameterregions allowed, we only show some sets of benchmarkpoints, which are favored by the collider data and theconditions of a strong FOPT. Since from the collider con-straints (especially the constraints from new gauge boson Z (cid:48) at LHC) favor the parameter spaces with (cid:104) χ (cid:105) (cid:38) Z (cid:48) > LISA BBO - - - - - - - f [ Hz ] h Ω G W FIG. 2: The GWs spectra in the minimal 3-3-1 model. Thecolored regions correspond to the expected sensitivities ofGWs interferometers LISA and BBO, respectively. The redline, green line and black line depict the phase transition GWspectrum for the benchmark sets I, II, III in Tab. I, respectively,during SU (3) L ⊗ U (1) Y → SU (2) L ⊗ U (1) X . signals are given in the recent Ref. [41] and referencestherein. Taking the benchmark set III as an example,when (cid:104) χ (cid:105) = 5 . m Z (cid:48) = 5 . Benchmark set (cid:104) χ (cid:105) [TeV] m Z (cid:48) [TeV] ( T ∗ , α, βH ∗ )I.Red line in Fig.2 6.1 6.2 (1.21 TeV, 0,70, 534)II.Green line in Fig.2 5.4 5.5 (0.82 TeV, 0.69, 619)III.Black line in Fig.2 5.1 5.2 (0.71 TeV, 0.78, 582)TABLE I: The benchmark sets in the minimal 3-3-1 modelfor the strong FOPT after considering the constraints fromcurrent experimental data. It is worth simply discussing what makes the GW signalfrom the TeV FOPT available for LISA and significantlylarger than that of the EW phase transition. It is becausethe FOPT discussed here comes from the the potentialbarrier in the finite temperature effective potential bythermal effects. Thus, the phase transition strength isproportional to e ∼ (cid:80) light boson (degrees of freedom) × (coupling to X) / , namely, the summation of the effectivecouplings between the order parameter field and the ther-mal particles (To make efficient thermal contributions,the particle masses should be much less than 3 T c .). Forthe EW phase transition, the order parameter field isthe Higgs field and only the particles whose masses aresmaller than 1 TeV can make thermal contributions tothe EW phase transition. Thus, some new heavy bosonin the minimal 3-3-1 model can not make sufficient con-tributions to the EW phase transition when their massesare much heavier than the critical temperature (The crit-ical temperature of EW phase transition is about 100GeV). Further, the couplings between the Higgs bosonand other particles are greatly constrained by currentdata, especially the diphoton decay data, see the detaileddiscussions on the tensions between strong EW FOPTand LHC data in Ref. [47]. That is why the EW phasetransition is rather weak. For the TeV phase transition,the critical temperature is around 1 TeV, and most of thebosons can make efficient thermal contributions to thephase transition. And the order parameter field in TeVphase transition is the new scalar field H , whose col-lider constraints on the couplings between H and otherparticles are not as strong as the Higgs boson case. B. Phase transition gravitational wave spectra inthe economical 3-3-1 model and the reducedminimal 3-3-1 model
In the economical 3-3-1 model [43], one chooses thesimplest SU (3) L representations for the scalar fields withspontaneously symmetry breaking, namely, two complexscalar triplets ( χ = (cid:0) χ , χ − , χ (cid:1) T ∼ (cid:0) , − (cid:1) and φ = (cid:0) φ +1 , φ , φ +3 (cid:1) T ∼ (cid:0) , (cid:1) ) with different hypercharge areneeded. The scalar potential is written as V ( χ, φ ) = µ χ † χ + λ ( χ † χ ) + µ φ † φ + λ ( φ † φ ) + λ ( χ † χ )( φ † φ ) + λ ( χ † φ )( φ † χ ) . (9)The SU (3) L ⊗ U (1) Y gauge group is broken sponta-neously via two steps. In the first step, the symme-try breaking SU (3) L ⊗ U (1) Y → SU (2) L ⊗ U (1) X hap-pens when the triplet scalar χ acquired the VEV givenby (cid:104) χ (cid:105) = √ ( u, , ω ) T with ω (cid:29) v (cid:29) u . In thelast step, to break into the SM U (1) EM gauge group SU (2) L ⊗ U (1) X → U (1) EM , another triplet scalar φ isneeded to acquire the VEV as (cid:104) φ (cid:105) = √ (0 , v, T . In thisversion of 3-3-1 model, there exist two neutral scalars, oneis the SM Higgs boson h , the other is the heavy scalar H .It also contains singly charged Higgs boson H ± . Thereare also two new heavy neutral gauge bosons Z and X ,and the singly charged gauge boson Y ± . In this work,the modified package ‘CosmoTransitions’ [48] is used tonumerically calculate the FOPT using full one-loop ther-mal potential. During the first time symmetry breaking,the order parameter field for the phase transition is the H scalar field, namely, the X = H . Strong FOPT atthe TeV scale can be induced by the new bosons andexotic quarks if the masses of these new particle are from100 GeV to several TeV. During the last time symmetrybreaking SU (2) L ⊗ U (1) X → U (1) EM , the order parame-ter field for the phase transition is just the Higgs bosonfield, namely, X = h if it is compared to Eq.(7). FOPTat the EW scale can be triggered by the new bosons.Considering the current constraints from collider data,we have u < . < ω < ω larger than 3 TeV here, with some reduced parameterspaces shown in Tab. II. The corresponding ( T ∗ , α, βH ∗ )is shown in Tab. III. There is a different aspect from theview point of GW signal that in the economical 3-3-1model both the two step symmetry breaking can be thestrong FOPT, which will produce two copies of GWs. Benchmark set ω [TeV] m H [TeV] T ∗ [TeV] m H ± TeV]I. Black lines in Fig.3 3.0 0.80 0.94, 0.16 1.4II. Green lines in Fig.3 4.0 1.30 1.26, 0.11 2.5TABLE II: The benchmark sets in the economical 3-3-1 modelfor two FOPTs after considering the constraints of currentexperimental data. The T ∗ represents the corresponding nu-cleation temperature for the first step FOPT and the secondstep FOPT, respectively.Benchmark set ( T ∗ , α, βH ∗ ) ( T ∗ , α, βH ∗ )I.Black Fig.4 (0.94 TeV, 0.59, 305) (0.16 TeV, 0.14, 612)II.Green Fig.4 (1.26 TeV, 0.68, 413) (0.11 TeV, 0.19, 710)TABLE III: The corresponding nucleation temperature T ∗ , α and βH ∗ of each FOPT for the different benchmark set in theeconomical 3-3-1 model. For the set II of benchmark points with ω = 4 TeV, m H = 1 . m H ± = 2 . h , H ,doubly charged scalar h ++ , two SM like bosons Z , W ± , - - - - - - f [ Hz ] h Ω G W �� ( � ) � ⊗ � ( � ) � ⟶ � ( � ) �� �� ( � ) � ⊗ � ( � ) � ⟶ �� ( � ) � ⊗ � ( � ) � ���� ��� FIG. 3: The phase transition GW spectra h Ω GW for thebenchmark sets in the economical 3-3-1 model. The coloredregions correspond to the expected sensitivities of GW in-terferometers LISA and BBO, respectively. The black linesdepict the GW spectra of the benchmark set I for the twoFOPTs during SU (3) L ⊗ U (1) Y = ⇒ SU (2) L ⊗ U (1) X at theTeV (right line) scale and SU (2) L ⊗ U (1) X = ⇒ U (1) EM at theEW scale (left line), respectively. The green lines representthe corresponding GW spectra for the benchmark set II. the new heavy neutral boson Z , the singly and doublycharged boson U ±± and V ± . These new particles and ex-otic quarks can be triggers for the strong FOPT [44]. Weshow some benchmark sets for the strong FOPT allowedby the collider constraints in Tab. IV. The corresponding( T ∗ , α, βH ∗ ) is shown in Tab. V. Two copies GW spectrafrom two FOPTs are produced at different energy scales,as shown in Fig. 4, which can be detected by LISA andBBO. DECIGO and Ultimate-DECIGO can also detectthese GW signals. Since from the perspective of the GWsignals, the reduced minimal 3-3-1 model has no obviousdifferences with the economical 3-3-1 model, we will notdiscuss this model in detail. Benchmark set (cid:104) χ (cid:105) [TeV] m H [TeV] T ∗ [TeV] m h ++ [TeV]I.Black Fig.4 3.0 1.0 0.84, 0.080 1.9II.Green Fig.4 4.0 1.3 1.23, 0.082 3.3TABLE IV: The benchmark sets in the reduced minimal 3-3-1 model for the two strong FOPTs after considering theconstraints of current experimental data. The T ∗ representsthe corresponding nucleation temperature for the first stepFOPT and the second step FOPT, respectively. C. Discussions on gravitational wave spectra innew physics models with hidden gauge group
In general, if the SM is extended by non-Abelian gaugegroup, FOPT may occur associated with each step’s spon-
Benchmark set ( T ∗ , α, βH ∗ ) ( T ∗ , α, βH ∗ )I.Black Fig.4 (0.84 TeV,0.51,330) (0.08 TeV,0.12,659)II.Green Fig.4 (1.23 TeV,0.7,490) (0.082 TeV,0.16,719)TABLE V: The corresponding nucleation temperature T ∗ , α and βH ∗ of each FOPT for the different benchmark set in thereduced minimal 3-3-1 model. �� ( � ) � ⊗ � ( � ) � ⟶ � ( � ) �� �� ( � ) � ⊗ � ( � ) � ⟶ �� ( � ) � ⊗ � ( � ) � ���� ��� - - - - - - f [ Hz ] h Ω G W FIG. 4: The phase transition GW spectra h Ω GW for thebenchmark sets in the reduced minimal 3-3-1 model. Thecolored regions correspond to the expected sensitivities of GWinterferometers LISA and BBO, respectively. The black linesdepict the GW spectra of the benchmark set I for the twoFOPTs during SU (3) L ⊗ U (1) Y = ⇒ SU (2) L ⊗ U (1) X at theTeV scale (right line) and SU (2) L ⊗ U (1) X = ⇒ U (1) EM at theEW scale (left line), respectively. The green lines representthe corresponding GW spectra for the benchmark set II. taneously symmetry breaking processes, where the phasetransition GWs may be produced and can be used to testthe hidden NP models with gauge symmetry breaking.Thus, the phase transition GWs can be used to test thehidden symmetry breaking during the evolution of theuniverse. One class of well-motivated models is the phasetransition GW signals in dark matter models with SU(N)hidden gauge group, which are discussed in Ref. [2]. If thehidden QCD phase transition scale is about O (100) MeV,the FOPT can produce phase transition GWs with thepeak frequency in the 10 − − − Hz range [2], whichcan be probed by the PTA GW experiments, such as theSKA or FAST. A schematic GW spectrum for the hiddenQCD phase transition is shown in Fig. 5 with the red line.The study in Ref. [2] may applies to other cases of darkQCD models, such as the case of dark QCD in the famousrelaxion mechanism [31] , and another novel mechanismcalled “ N naturalness” [49] . The relaxion mechanism can technically relax the EW hierarchand the light Higgs mass comes from the dynamical cosmologi-
On the other hand, if a FOPT takes place at a criticaltemperature of O (10 –10 ) GeV [5, 53], such as someversions of grand unified models, this could potentiallyproduce detectable GWs spectrum in the future aLIGOor aLIGO-like experiments, and provide us with a uniqueprobe of the hidden NP models at very high energy scale,which is not directly accessible by particle colliders. Theschematic GW signal is shown as the purple line in Fig. 5. SKA LISA BBO aLIGOO5aLIGODark
QCD
MeV High scale10000
TeVTeV scale - - - - - - - - - - f [ Hz ] h Ω G W FIG. 5: Schematic phase transition GW spectra during theevolution of our universe. The colored regions represent theexpected sensitivities of GW detectors aLIGO, LISA, BBOand SKA, respectively. The red line depicts the possible GWspectrum if the FOPT occurs at the scale of O (100) MeV insome hidden QCD models [2]. The black line represents theGW spectrum for the FOPT at TeV scale in some models withextended gauge group. The purple line corresponds to the GWspectrum when the FOPT occurs at the scale of O (10000) TeVin some NP models with hidden symmetry breaking process. IV. CONCLUSIONS
In Fig. 5, the schematic FOPT GW spectra h Ω GW during the evolution of our universe are shown for a genericclasses of NP models with non-Abelian symmetry breakingat different energy scales. The colored regions representthe expected sensitivities of the GW detectors SKA, BBO,LISA and aLIGO, respectively. The red line depictsthe possible GW spectrum in a class of hidden QCDmodels [2], where the FOPT occurs at the scale of O (100)MeV and the associated GWs can be detected by the PTAexperiments, such as SKA or the FAST built in China. The black line represents the GW spectrum for the FOPTat TeV scale in a large classes of NP models with gaugesymmetry breaking. As examples, we have shown thatthree versions of the 3-3-1 models discussed above couldproduce detectable GWs at TeV scale when the gaugesymmetry SU (3) L ⊗ U (1) Y breaks to SU (2) L ⊗ U (1) X .Especially, in the economical and reduced minimal 3-3-1models, two FOPTs can take place, which will producetwo copies GW spectra with different characteristic peakfrequencies. In general, large classes of NP models withFOPT at the scale from O (100) GeV to several TeV canbe tested at future laser interferometer GW detectorsin space, such as the recently proved LISA [24], BBO,DECIGO, Ultimate-DECIGO, Taiji and TianQin [54].The purple line corresponds to the GW spectrum in somehidden NP models where the FOPT takes place at thescale of O (10000) TeV. The GW signals are within thesensitivity of the future aLIGO and provide us with aunique detection of the hidden gauge symmetry breakingat high energy scales beyond the abilities of LHC.It is worth noticing that this is the first study on thatthe universe could produce more than one copies of phasetransition GW signal with both solid calculation in realis-tic NP models (such as two GW signatures with differentcharacteristics result from two FOPT in the economicaland reduced minimal 3-3-1 models) and generic discus-sions. Our study also includes the detailed study on phasetransition GWs produced at TeV scale in realistic particlephysics models.To conclude, GW signals become a new and realistic ap-proach to explore the the symmetry breaking patterns inparticle cosmology after the discovery of GWs at aLIGO.For cosmology, we can only hear the non-trivial cosmolog-ical phase transitions using GWs to explore the evolutionof the universe. For particle physics, this GW approachcan compensate for the colliders, and provide a novelapproach to probe the symmetry breaking or phase tran-sition patterns. We are in an exciting expedition towardsthe revolutionary discovery of the NP models and cosmo-logical phase transitions at GW detectors. More detailedstudy will be discussed in our future work [52]. Acknowledgements.
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