On Large Lepton Number Asymmetries of the Universe

FTUV-17-03-17 IFIC/17-19 On large lepton number asymmetries of the Universe

Gabriela Barenboim1∗ and Wan-Il Park2† 1 Departament de F´ısica Teòrica and IFIC, Universitat de València-CSIC,E-46100, Burjassot, Spain and 2 Department of Science Education (Physics), Chonbuk National University, Jeonju 561-756, Korea (Dated: July 15, 2018) A large lepton number asymmetry of O(0.1 − 1) at present universe might not only be allowed but also necessary for consistency among cosmological data. We show that, if a sizeable lepton number asymmetry were produced before the electroweak phase transition, the requirement for not producing too much baryon number asymmetry through sphalerons processes, forces the high scale lepton number asymmetry to be larger than about 30. Therefore a mild entropy release causing O(10 − 100) suppression of pre-existing particle density should take place, when the background temperature of the universe is around T = O(10−2 − 102)GeV for a large but experimentally consistent asymmetry to be present today. We also show that such a mild entropy production can be obtained by the late-time decays of the saxion, constraining the parameters of the Peccei-Quinn sector such as the mass and the vacuum expectation value of the saxion field to be mφ & O(10)TeV 14 and φ0 & O(10 )GeV, respectively.

INTRODUCTION baryon number asymmetry. However, breaking the electroweak symmetry requires |L| to be larger than about Extensive analysis of Big Bang Nucleosynthesis (BBN) 10 [10] which is already too large to be consistent with and cosmic microwave background (CMB) data showed CMB data [2]. Besides, a large enough suppression of that the baryon number asymmetry (B) of the Universe the sphaleron processes requires even larger |L|. Hence, should be O(10−10) [1, 2]. On the other hand, BBN and in order for |Lα| ∼ O(1) at the present universe to be CMB data still allow O(1) lepton number asymmetries consistent with observations, a much larger |L| had to be generated above the electroweak scale temperature and of muon- and tau-neutrinos (Lµ,τ ) [3], even though the asymmetry of electron-neutrinos is tightly constrained to diluted to a safe level well before BBN. −3 For all these reasons, we will discuss how one can have be |Le| . O(10 ) by BBN [26]. Such large asymmetries might provide a better fit to various astrophysical and a fully consistent picture of a large lepton number asym- cosmological data [5]. metry, including the high temperature lower bound on |L| for symmetry-breaking, a large enough suppression of A large lepton number asymmetry (Lα ≡ ∆nα/nγ sphalerons, a plausible scenario of late-time mild entropy with α, ∆nα, and nγ being the neutrino flavor, the num- release, the generation of a large enough asymmetry, and ber density difference of να andν ¯α, and the photon number density, respectively) can be generated before or af- a brief discussion about dark matter. ter the electroweak phase transition (EWPT). If it were produced after (at temperatures lower than) the EWPT, the anomalous electroweak processes (sphalerons) [6, 7] SYMMETRY BREAKING DUE TO A LARGE which effectively could transform such a lepton num- TOTAL LEPTON NUMBER ASYMMETRY ber asymmetry into a baryon number asymmetry (or vice versa) do not take place, i.e., a pre-existing baryon number asymmetry would not be affected by the afore- Conventionally, the electroweak symmetry breaking of the standard model (SM) is supposed to take place said large lepton number asymmetry. However, if Lα when the temperature of the Universe drops down to arXiv:1703.08258v1 [hep-ph] 24 Mar 2017 were produced before (at temperatures higher than) the EWPT, the danger of an over-production of B due to Tew ∼ 100GeV. However, it is known that the presence sphaleron processes must be dealt with. of a large net lepton number asymmetry (coming, for example, from left-handed neutrinos) can cause a symmetry In this work, motivated by its potential cosmological breaking of some SM gauge groups [8, 9]. Specifically, in benefits, we consider the case of |L | ∼ O(0.1−1) which µ,τ the standard model the vacuum expectation value (VEV) in general requires even larger asymmetries at high tem- of the canonically normalized CP-even Higgs field is de- perature, well above the electroweak scale. Sphaleron termined by a finite temperature effective potential of the processes in this case should be sufficinetly suppressed form [10], in order to avoid baryon over-production. Interestingly enough, it has been known for a long time that, if P λ λ0 g2 n2 |L = Lα| is large enough, the electroweak symmetry 4 2 2 2 2 L α Veff = v + T v + C v + can be never restored [8, 9] and the sphaleron processes 4 2 8 T 2 can be exponentially suppressed, efficiently blocking the 4n2 3v2 + 12C2 + 14T 2 + L (1) conversion of the total lepton number asymmetry into a 54C2v2 + (87v2 + 96C2) T 2 + 112T 4 2

2.5 2.5

13.15≤Lini≤50 2.0 2.0

Lini=50 1.5 1.5 v(T)/T v(T)/T 1.0 1.0

Lini=13.15 0.5 0.5

0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 10 20 30 40 50

C(T)/T Lini

FIG. 1: Left: Vacuum expectation values of CP-even Higgs (v) and gauge ﬁeld (C) as a function of Lini. Right: v as a function of Lini.

where v is the VEV of the canonically normalized CP- MSSM flat-directions or stringy moduli are well moti- even neutral Higgs field, C the VEV of a gauge field, nL vated potential candidates for such a purpose. However, the net asymmetry of the lepton number density, generically, the former decays too early, and the latter produces too much entropy. On the other hand, the 0 1 2 2 2 3 02 2 Peccei-Quinn field responsible for the axion solution of λ = 6λ + yτ + 3yt + 3yb + g + 3g (2) 12 4 the strong CP problem [11] may fit well to our purpose, since the energy density and decay rate of the saxion, assuming that the Yukawa coulings (y s) are dominated i the scalar partner of the axion, can be adjusted by in- by the third generation contribution, and g0 is the gauge dependent parameters. For example, assuming that the coupling of the U(1) gauge group of the SM. From Y U(1) Peccei-Quinn symmetry was broken before or dur- Eq. (1), it is straightforward to see that v(T )/T appears ing inflation, one can regard the axion coupling constant to be nonzero for L 13.15 with L being the ini- ini & ini as a free parameter only lower bounded by astrophysical tial lepton number asymmetry at high energy, as shown constraints [12]. in Fig. 1. It is clear then that an effective suppression of the sphaleron rate in order to avoid over-production of baryon number asymmetry requires even larger Lini. Depending on the specifics of the theoretical model and Such a large Lini is difficult to be made consistent with the cosmological scenarios, it is possible that saxion field CMB data unless there is enough amount of dilution φ can start coherent oscillations with respect to its true caused by, for example, a late time entropy release taking minimum when the expansion rate (H) of the universe place below Tew but well before BBN. becomes similar to its mass scale at zero-temperature. The temperature at this epoch is

ENTROPY PRODUCTION FROM PECCEI-QUINN SECTOR

A late-time entropy release can be obtained, for ex- π2 −1/4 T ≈ g (T ) pm M (4) ample, when the dominant energy content of the early osc 90 ∗ osc φ P universe is no longer given by radiation but by non- relativistic particles which should decay before BBN. In this case, the pre-existing lepton number asymmetry will be diluted by the entropy production associated to the decay. Since we are interested in L ∼ O(1) after the where g∗(Tosc) is the number of relativistic degrees of entropy release, the dilution factor should be freedom, mφ is the zero-temperature mass of saxion, and M is the reduced Planck mass. The initial oscillation L P ∆ = ini O(10) (3) amplitude (φ ) is also model-dependent, but here we L & osc take it to be of the order of φ0, the zero-temperature VEV Now, the question is how we can realize such a mild late- of saxion, for simplicity. Then, saxion energy density time entropy generation. starts dominating the universe when temperature drops 3 to large lepton number asymmetry via sphaleron processes. 1/3 2 In this case, the density of the baryon number asymmetry 1 g∗s(Tosc) φosc T ≈ T (5) when sphaleron processes are practically terminated at tf ∗ 6 g (T ) M osc ∗s ∗ P (i.e., at T ∼ Tew) is given by 2 1/2 φosc MP Z tf ≈ c mφ (6) 1 3 MP mφ nB ≈ − 3 dtΓspha (t)nL (14) a ti 2 1/2 φosc mφ ' 27.4GeV × (7) where we take t to be the time at which the generation 1014GeV 10TeV i of L takes place, and ignore the back-reaction of nB to with nL (i.e., the conversion of a tiny nB to nL via sphalerons ). The sphaleron rate after the electroweak symmetry 1/3 2 −1/4 1 g∗s(Tosc) π c ≡ g∗(Tosc) ∼ O(0.1) (8) breaking is then given by [18] 6 g∗s(T∗) 90 10−2m7 W −Esph/T where g (T ) = 200 and g (T ) = 86.25 were used for Γsph(T ) ∼ e (15) ∗s osc ∗s ∗ α3 T 6 our numerical estimations. W The decay rate of the saxion depends on its couplings where mW = gv/2 is the W -boson mass with g and to SM particles. In simple hadronic axion models [13, 14], v being the gauge coupling of the SU(2)L gauge sym- saxions decay dominantly to light QCD-axions, bearing metry of the SM and the VEV of the CP-even canoni- 2 the potential danger of forming a standard thermal back- cal Higgs field, respectively. αW = g /4π, and Esph = ground which could jeopardize a successful BBN. Hence, E˜ ×2mW /αW is the sphaleron energy with E˜ ≈ 2 for the we should have direct interactions of saxion to SM par- standard model with a 125.5GeV Higgs mass [19]. Sim- ticles like, for example, in DFSZ axion scenarios [15, 16]. ilarly, when the SU(2)L is broken at high temperature, In this case, when kinematically allowed, the saxion can well above the electroweak scale, we can take decay dominantly to the SM higgs with a rate given by −1 7 10 v(T ) − 8πv(T ) [17] Γ (T T ) ∼ e gT (16) sph ew T 6 !1/2 m3 4 2 1 φ µ mh where we keep g being the low energy value [34]. Γs→hh = 2 1 − 2 (9) 2π φ0 mφ mφ It is important to stress that the matter-domination era starts at T Tew in the scenario we are considering. where µ and mh = 125.5GeV are the parameter of the Also, v(T )/T ≡ r is nearly constant for T Tew in Higgs bilinear in the minimal supersymmetric standard this set-up. Therefore, as T approaches v(T = 0), the model (MSSM) and the observed mass of Higgs particle, conversion of L to B is practically shutted down. Hence, respectively. Hence, for Γ ' Γ the decay tempera- s s→hh regarding r as a constant for T & Tf , from Eq. (14) one ture is found to be finds −1/4 π2 2 −1/2 p −1 π MP 7 − 8πr Td = g∗(Td) ΓsMP (10) B(Tf ) ∼ −10 g∗(Tf ) r e g Lini (17) 90 90 Tf µ 2 m 3 1014GeV 2 ' 2.86GeV × φ (11) At late-times, including the dilution caused by the en- mφ 10TeV φ0 tropy release, the contribution of lepton number asymmetry to the baryon number asymmetry at present be- where g∗(Td) = 75.75 and the instantaneous decay ap- proximation were used. From Eqs. (5) and (10), one finds comes

B0 = B(Tf Tew)/∆ ∆ ≈ T∗/Td (12) & 2 −1/2 2 5/2 4 π MP 8πr m 10TeV φ −1 7 − g ≈ 9.58 φ 0 (13) ∼ −10 g∗(Tf ) r e L0 (18) 14 90 Tf µ mφ 10 GeV

where L0 represents the net lepton number asymmetry where we used φosc = φ0 in the second line. Since we at present. need only ∆ = O(10) with Td < Tew, we may consider the case of T < T for simplicity. In Fig. 2, we show the expected present baryon number ∗ ew −2 asymmetry as a function of Lini for L0 = 10 , 0.1, and 1. From the ﬁgure, it can be clearly seen that L0 = 1 −10 LOWER BOUND OF L AT HIGH ENERGY and B0 . 10 requires

Lini 32.4 (19) Lacking another source, it is natural to assume that the & baryon number asymmetry is generated mainly from a which corresponds to v(T )/T & 1.57. 4

Dine ﬁeld Φ, is given by 105 2 1/2 φ2 ≈ mM 3 (22) AD (5λ2)1/4 P 10-5 0 and δCP is the CP-violating phase. Hence the lepton B number asymmetry at high temperature is found to be

10-15 π2 π2 3/4 L ≈ g (T ) sin δ ini ζ(3)(5λ2)1/4 90 ∗ CP 25 10- g (T )3/4 sin δ 10 20 30 40 50 ' 56 ∗ √ CP (23) 200 Lini λ where we used that H = m. It becomes apparent FIG. 2: Baryon number asymmetry at present as a function then that this scenario can easily satisfy the condition of the initial lepton number asymmetry at high energy for −2 in Eq. (19) for plausible ranges of δCP and λ. Tf = v = 246GeV and L0 = 10 , 0.1, 1 from left to right. −10 Dashed lines are indicating the value of Lini for B0 = 10 . It should be noted that, when the lepton number asymmetry is generated, for m ∼ TeV the temperature of the Universe is very high and gravitinos are likely to be over-produced if the mass of the gravitino is similar LEPTOGENESIS to the scale of the soft SUSY breaking masses of scalar fields. Hence, in this case, in the presence of only mild The lower bound for the initial lepton number asym- entropy production due to saxion decays it is necessary to metry in Eq. (19) is very large and therefore quite chal- have gravitinos much heavier than TeV scale, otherwise lenging from the model building point of view. However, late time decays of gravitinos would prevent a successful such a large lepton number asymmetry can be generated BBN. This is the reason we consider O(10)TeV as the in the early univrese, for example, by the Affleck-Dine typical mass scales of mφ and µ in the earlier discussion. mechanism [20–22]. Specifically, in Refs. [23, 24], it was Such a scale can arise naturally in the pure gravity me- shown that L ∼ O(10) can be achieved either along the diation scenario [25]. In such a scheme, the gravitino right-handed sneutrino direction or along the left-handed problem can be avoided and a right amount of relic dark sneutrino field involved in the MSSM flat-directions. Let matter at present can be provided by the wino playing us briefly describe the case of a MSSM flat-direction, fol- the role of the lightest supersymmetric particle. How- lowing Ref. [24]. ever, due to the late-time entropy release in our scenario, Let us consider a flat direction (Φ) lifted by a d = 6 su- wino dark matter would be sub-dominant in the end. 6 3 perpotential term, W = λΦ /MP with λ being a numer- The main component of dark matter can come from the ical coupling constant. In this case, through the Affleck- axion misalignment. Since the Peccei-Quinn symmetry Dine mechanism, a large lepton number asymmetry can is assumed to be broken before or during inflation, the be generated when the expansion rate H of the Universe misalignment is determined simply by the initial phase becomes similar to the soft SUSY-breaking mass m of the shift of the Peccei-Quinn field, and can be freely chosen flat direction Φ around the origin. We will assume that to provide the right amount of axion dark matter. the Universe was dominated by radiation when the lepton number asymmetry is generated. Then, at the generation of the asymmetry, the temperature of the Universe CONCLUSIONS is given by In this paper, we discuss a complete scenario able to −1/4 π2 accomodate a large lepton number asymmetry surviving T ≈ g (T ) (mM )1/2 90 ∗ P up to this date. −1/4 A good fit to astrophysical and cosmological data may g (T ) m 1/2 ' 7.2 × 1010GeV ∗ (20) require O(1) lepton number asymmetries L (normal- 200 10TeV α ized by photon number density) of neutrino flavors να and the corresponding number density of lepton asym- (α = µ, τ only). 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