Probing the seesaw scale with gravitational waves
aa r X i v : . [ h e p - ph ] S e p EPHOU-18-007
Probing the seesaw scale with gravitational waves
Nobuchika Okada ∗ Department of Physics and Astronomy,University of Alabama, Tuscaloosa, Alabama 35487, USA
Osamu Seto † Institute for International Collaboration,Hokkaido University, Sapporo 060-0815, Japan andDepartment of Physics, Hokkaido University, Sapporo 060-0810, Japan
Abstract
The U (1) B − L gauge symmetry is a promising extension of the standard model of particle physics,which is supposed to be broken at some high energy scale. Associated with the U (1) B − L gaugesymmetry breaking, right-handed neutrinos acquire their Majorana masses and then tiny lightneutrino masses are generated through the seesaw mechanism. In this paper, we demonstrate thatthe first-order phase transition of the U (1) B − L gauge symmetry breaking can generate a largeamplitude of stochastic gravitational wave (GW) radiation for some parameter space of the model,which is detectable in future experiments. Therefore, the detection of GWs is an interesting strategyto probe the seesaw scale which can be much higher than the energy scale of collider experiments. ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION The nonvanishing neutrino masses have been established through various neutrino os-cillation phenomena. The most attractive idea to explain the tiny neutrino masses is theso-called seesaw mechanism with heavy Majorana right-handed (RH) neutrinos [1]. Then,the origin of neutrino masses is ultimately reduced to questions on the origin of RH neutrinomasses. It is natural to suppose that masses of RH neutrinos are also generated associatedwith developing the vacuum expectation value (VEV) of a Higgs field which breaks a certain(gauge) symmetry at a high energy scale.As a promising and minimal extension of the standard model (SM), we may considermodels based on the gauge group SU (3) C × SU (2) L × U (1) Y × U (1) B − L [2] where the U (1) B − L (baryon number minus lepton number) gauge symmetry is supposed to be brokenat a high energy scale. In this class of models with a natural/conventional U (1) B − L chargeassignment, the gauge and gravitational anomaly cancellations require us to introduce threeRH neutrinos whose Majorana masses are generated by the spontaneous breakdown of the U (1) B − L gauge symmetry. In the case that the U (1) B − L symmetry breaking takes place atan energy scale higher than the TeV scale, it is very difficult for any collider experiments toaddress the mechanism of the symmetry breaking and the RH neutrino mass generation.Detection of gravitational waves brings information about the evolution of the very earlyUniverse. Cosmological GWs could originate from, for instance, quantum fluctuations dur-ing inflationary expansion [3] and phase transitions [4, 5]. If a first-order phase transitionoccurs in the early Universe, the dynamics of bubble collision [6–10] and subsequent tur-bulence of the plasma [11–15] and sonic waves generate GWs [16–18]. These might bewithin a sensitivity of future space interferometer experiments such as eLISA [19]; the BigBang Observer (BBO) [20] and DECi-hertz Interferometer Observatory (DECIGO) [21];or even ground-based detectors such as Advanced LIGO (aLIGO) [22], KAGRA [23] andVIRGO [24].The spectrum of stochastic GWs produced by the first-order phase transition in the earlyUniverse, in particular, by the SM Higgs doublet field, has been investigated in the literature.Here, the phase transition occurs at the weak scale. See, for instance, Ref. [25] for a recentreview.In this paper, we focus on GWs from the first-order phase transition associated with the2pontaneous U (1) B − L gauge symmetry breaking at a scale higher than the TeV scale. GWsgenerated by a U (1) B − L extended model with the classical conformal invariance [26, 27],where its phase transition takes place around the weak scale, have been studied in Ref. [28].GWs from a second-order B − L phase transition during reheating have been studied inRef. [29]. In this paper, we consider a slightly extended Higgs sector from the minimalmodel and introduce an additional U (1) B − L charged Higgs field with its charge +1. Thisis one of the key ingredients in this paper. GWs generated by a phase transition in thisextended scalar potential, but at TeV scale, have been studied in Ref. [30]. As we will showbelow, the new Higgs field plays a crucial role in causing the first-order phase transitionof the U (1) B − L symmetry breaking and the amplitude of resultant GWs generated by thephase transition can be much larger than the one we naively expect. II. GW GENERATION BY A COSMOLOGICAL FIRST-ORDER PHASE TRAN-SITION
In this section, we briefly summarize the properties of GWs produced by a first-orderphase transition in the early Universe. There are three main GW production processes andmechanisms: bubble collisions, turbulence [11] and sound waves after bubble collisions [16].The GW spectrum generated by a first-order phase transition is mainly characterized bytwo quantities: the ratio of the latent heat energy to the radiation energy density, which isexpressed by a parameter α and the transition speed β defined below. In this section, weintroduce those parameters and the fitting formula of the GW spectrum. A. Scalar potential parameters related to the GW spectrum
We consider the system composed of radiation and a scalar field φ at temperature T .The energy density of radiation is given by ρ rad = π g ∗ T , (1)with g ∗ being the number of relativistic degrees of freedom in the thermal plasma. At themoment of a first-order phase transition, the potential energy of the scalar field includes the3atent energy density given by ǫ = (cid:18) V − T ∂V∂T (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) { φ high ,T ⋆ } − (cid:18) V − T ∂V∂T (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) { φ low ,T ⋆ } , (2)where φ high(low) denotes the field value of φ at the high (low) vacuum. Here and hereafter,quantities with the subscript ⋆ stand for those at the time when the phase transition takesplace [32]. Then, a parameter α is defined as α ≡ ǫρ rad . (3)The bubble nucleation rate per unit volume at a finite temperature is given byΓ( T ) = Γ e − S ( T ) ≃ Γ e − S E ( T ) /T , (4)where Γ is a coefficient of the order of the transition energy scale, S is the action in thefour-dimensional Minkowski space, and S E is the three-dimensional Euclidean action [9].The inverse of the transition timescale can be defined as β ≡ − dSdt (cid:12)(cid:12)(cid:12)(cid:12) t ⋆ . (5)Its dimensionless parameter β/H ⋆ can be expressed as βH ⋆ ≃ T dSdT (cid:12)(cid:12)(cid:12)(cid:12) T ⋆ = T d ( S E /T ) dT (cid:12)(cid:12)(cid:12)(cid:12) T ⋆ . (6) B. GW spectrum
1. Bubble collisions
Under the envelope approximation and for β/H ⋆ ≫ f peak ≃ (cid:18) f ⋆ β (cid:19) (cid:18) βH ⋆ (cid:19) (cid:18) T ⋆ GeV (cid:19) (cid:16) g ∗ (cid:17) / Hz , (7) h Ω GW ( f peak ) ≃ . × − κ ∆ (cid:18) βH ⋆ (cid:19) − (cid:18) α α (cid:19) (cid:16) g ∗ (cid:17) − / , (8) For a recent development beyond the envelope approximation, see Ref. [31]. . v b .
42 + v b , (9) f ⋆ β = 0 . . − . v b + v b , (10)where v b denotes the bubble wall velocity. The efficiency factor ( κ ) is given by [11] κ = 11 + Aα Aα + 427 r α ! , (11)with A = 0 . GW ( f ) = Ω GW ( f peak ) ( a + b ) f b peak f a bf a + b peak + af a + b , (12)with numerical factors a ∈ [2 . , .
82] and b ∈ [0 . , . a, b, v b ) =(2 . , . , .
6) in our analysis.
2. Sound waves
The peak frequency and the peak amplitude of GWs generated by sound waves are givenby [16, 17] f peak ≃
19 1 v b (cid:18) βH ⋆ (cid:19) (cid:18) T ⋆ GeV (cid:19) (cid:16) g ∗ (cid:17) / Hz , (13) h Ω GW ( f peak ) ≃ . × − κ v v b (cid:18) βH ⋆ (cid:19) − (cid:18) α α (cid:19) (cid:16) g ∗ (cid:17) − / . (14)The efficiency factor ( κ v ) is given by [35] κ v ≃ v / b . α . − . √ α + α for v b ≪ c sα . . √ α + α for v b ≃ , (15)with c s being the sonic speed. The spectrum shape is expressed as [34] (cid:18) ff peak (cid:19)
74 + 3 (cid:16) ff peak (cid:17) / . (16)5 . Turbulence The peak frequency and amplitude of GWs generated by turbulence are given by [11] f peak ≃
27 1 v b (cid:18) βH ⋆ (cid:19) (cid:18) T ⋆ GeV (cid:19) (cid:16) g ∗ (cid:17) / Hz , (17) h Ω GW ( f peak ) ≃ . × − v b (cid:18) βH ⋆ (cid:19) − (cid:18) κ turb α α (cid:19) / (cid:16) g ∗ (cid:17) − / . (18)In our analysis, we conservatively set the efficiency factor for turbulence to be κ turb ≃ . κ v as in Ref. [34]. The spectrum shape is given by [15, 33, 34] (cid:16) ff peak (cid:17) (1 + ff peak ) / (1 + πfh ⋆ ) , (19)with h ⋆ = 17 (cid:18) T ⋆ GeV (cid:19) (cid:16) g ∗ (cid:17) / Hz . (20) III. GWS GENERATED BY SEESAW PHASE TRANSITIONA. B − L seesaw model Our model is based on the gauge group SU (3) C × SU (2) L × U (1) Y × U (1) B − L , wherethree RH neutrinos ( N iR with i running 1 , ,
3) and two SM singlet B − L Higgs fields (Φ and Φ ) are introduced. Under these gauge groups, three generations of RH neutrinos haveto be introduced for the anomaly cancellation. The particle content is listed in Table I. TheYukawa sector of the SM is extended to have L Y ukawa ⊃ − X i =1 3 X j =1 Y ijD ℓ iL HN jR − X k =1 Y N k Φ N k CR N kR + H . c ., (21)where the first term is the neutrino Dirac Yukawa coupling, and the second is the MajoranaYukawa couplings. Once the U (1) B − L Higgs field Φ develops a nonzero VEV, the U (1) B − L gauge symmetry is broken and the Majorana mass terms of the RH neutrinos are gener-ated. Then, the seesaw mechanism is automatically implemented in the model after theelectroweak symmetry breaking.We consider the following scalar potential: V (Φ , Φ ) = 12 λ (Φ Φ † ) + 12 λ (Φ Φ † ) + λ Φ Φ † (Φ Φ † )+ M Φ Φ † − M Φ Φ † − A (Φ Φ Φ † + Φ † Φ † Φ ) . (22)6 U(3) c SU(2) L U(1) Y U(1) B − L q iL / / u iR / / d iR − / / ℓ iL − / − e iR − − H − / N iR − U (1) B − L model. In addition to the SM particle content( i = 1 , , N iR ( i = 1 , , U (1) B − L Higgs fields (Φ and Φ ) areintroduced. Here, we omit the SM Higgs field ( H ) part and its interaction terms for not only simplicitybut also little importance in the following discussion, since we are interested in the casethat the VEVs of B − L Higgs fields are much larger than that of the SM Higgs field. Allparameters in the potential (22) are taken to be real and positive. At the U(1) B − L symmetrybreaking vacuum, the B − L Higgs fields are expanded around those VEVs v and v , asΦ = v + φ + iχ √ , (23)Φ = v + φ + iχ √ . (24)Here, φ and φ correspond to two real degrees of freedom as CP -even scalars, one linearcombination of χ and χ is the Nambu-Goldstone mode eaten by the U (1) B − L gauge boson( Z ′ boson) and the other is left as a physical CP -odd scalar. Mass terms of particles are For the case of a phase transition of the SM Higgs field interacting with new Higgs fields, see, for example,Ref. [36]. L mass = −
12 ( φ φ ) λ v + λ v − M v (cid:0) −√ A + λ v (cid:1) v (cid:0) −√ A + λ v (cid:1) λ v + λ v + M − √ Av φ φ −
12 ( χ χ ) λ v + λ v − M −√ Av −√ Av λ v + M + λ v + √ Av χ χ − g B − L (4 v + v ) Z ′ µ Z ′ µ − X i N ci Y N i v √ N i . (25)With the U (1) B − L symmetry breaking, the RH neutrinos N iR and the Z ′ boson acquire theirmasses, respectively, as m N iR = Y N i √ v , (26) m Z ′ = g B − L (4 v + v ) , (27)where g B − L is the U (1) B − L gauge coupling. The mass matrix of CP -even Higgs bosons ( φ and φ ) and the mass of the physical CP -odd scalar P can be, respectively, simplified as λ v + Av √ v v (cid:0) λ v − √ A (cid:1) v (cid:0) λ v − √ A (cid:1) λ v , (28)and m P = A √ v ( v + 4 v ) , (29)by eliminating M and M under the stationary conditions, λ v λ v v − M v − Av √ , (30) λ v λ v v + M v − √ Av v = 0 . (31)Let us here note the LEP constraint m Z ′ /g B − L = p v + v & m Z ′ & . , (32)for g B − L ≃ . B − L gaugesymmetry breaking by the Higgs fields Φ and Φ becomes of the first order in the early8 IG. 1: The three-dimensional plot of the one-loop corrected scalar potential of two B − L Higgsfields Φ and Φ at zero temperature, which induces a first-order phase transition in the earlyUniverse. Point g B − L v v A α β/H ⋆ T ⋆ A 0 . . .
086 109 1 . . .
69 104 16 .
18C 0 .
71 10 .
89 52 .
24 1515D 0 .
72 10 .
77 57 . Universe. In the following analysis, we set λ = 0 . , λ = 0 .
1, and λ = 0 . Y N i have been neglected, assuming Y N i ≪ g B − L , forsimplicity. We show in Fig. 1 the shape of the one-loop scalar potential (22).Implementing our model into the public code CosmoTransitions [43], we have evaluatedthe parameters α , β and T ⋆ at a renormalization scale Q = ( v + v ) /
2. We list our resultsfor four benchmark points in Table II. In Table III, we list the new particles’ mass spectrumfor point A, which can be tested by the future LHC experiment. Except for point A, onecan easily see the benchmark points are far beyond the reach of collider experiments.9 oint m Z ′ m P m H m H A 6 .
26 2 .
49 0 .
65 2 . Z ′ boson and new Higgs bosons for point A is shown inunits of TeV. In Fig. 2, we show predicted GW spectra for our benchmark points along with expectedsensitivities of future interferometer experiments. Here, the resultant spectra have beencalculated with a bubble wall speed of v b = 0 .
6. We have confirmed that the results arenot so significantly changed for other v b values of O (0 . gwplotter [44]. The sensitivities of DECIGO and BBOreach the results of points A and B. Point C is an example which is not marginally able tobe detected by DECIGO/BBO but its peak is within the reach of CE. IV. SUMMARY
The origin of heavy Majorana RH neutrino masses is one of the essential pieces to under-stand the origin of neutrino masses through the seesaw mechanism. Gauged B − L symmetryand its breakdown are a natural framework to introduce the RH neutrinos into the SM andto generate their Majorana masses. The seesaw scale is in general far beyond the reach offuture collider experiments. In this paper, we have investigated a possibility to probe theseesaw scale through the observation of stochastic GW radiation. We have shown in thecontext of a simple U (1) B − L extended SM that the first-order phase transition of the B − L Higgs potential can generate an amplitude of GWs large enough to be detected in futureexperiments. Such a detection is informative to estimate the seesaw scale. Grojean and Ser-vant have shown that GWs generated by phase transitions at T ⋆ ∼ GeV are in reach offuture experiments [49].(For recent studies, see e.g., Refs. [50, 51].) We have demonstratedthat the detection of GWs is indeed possible in our model context.At last, we should note a delicate and critical caveat about the issue of gauge dependence10
ECIGOELISA aLIGO CEBBO10 - - - - - - f @ Hz D W G W h v b = FIG. 2: The predicted GW spectra for the benchmark points with v b = 0 . of the effective Higgs potential. See, for example, Ref. [52] for recent discussions. So far, wehave no clear resolution to this issue. According to Ref. [52], the resultant GW spectrum hasone order of magnitude uncertainties under a specific gauge choice. Thus, even for the worstcase, our benchmark points A and B can still be within the reach of future experiments.Once a better prescription has been developed, we will reevaluate the amplitude of GWs. Acknowledgments
We are grateful to C. L. Wainwright and T. Matsui for kind correspondences concern-ing use of
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