Return of Grand Unified Theory Baryogenesis: Source of Helical Hypermagnetic Fields for the Baryon Asymmetry of the Universe

PHYSICAL REVIEW D 97, 103506 (2018)

Return of grand unified theory baryogenesis: Source of helical hypermagnetic fields for the baryon asymmetry of the universe

Kohei Kamada* Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS), Daejeon 34126, Korea and School of Earth and Space Exploration, Arizona State University, Tempe, Arizona 85287, USA

(Received 22 February 2018; published 14 May 2018)

It has been considered that baryogenesis models without a generation of B − L asymmetry such as the GUT baryogenesis do not work since the asymmetry is washed out by the electroweak sphalerons. Here, we point out that helical hypermagnetic fields can be generated through the chiral magnetic effect with a chiral asymmetry generated in such baryogenesis models. The helical hypermagnetic fields then produce baryon asymmetry mainly at the electroweak symmetry breaking, which remains until today. Therefore, the baryogenesis models without B − L asymmetry can still be the origin of the present baryon asymmetry. In particular, if it can produce chiral asymmetry mainly carried by right-handed electrons of order of 10−3 in terms of the chemical potential to temperature ratio, the resultant present-day baryon asymmetry can be consistent with our Universe, although simple realizations of the GUT baryogenesis are hard to satisfy the condition. We also argue the way to overcome the difficulty in the GUT baryogenesis. The intergalactic −16–17 −2–3 magnetic fields with B0 ∼ 10 G and λ0 ∼ 10 pc are the smoking gun of the baryogenesis scenario as discussed before.

DOI: 10.1103/PhysRevD.97.103506

I. INTRODUCTION at T ≳ 80 TeV [5,6]. Such primordial helical hyperMFs can have a strong impact on cosmology [7]. They can be the Baryon asymmetry of the Universe (BAU) is a long- seed for the galaxy and galaxy-cluster MFs and remain standing problem in particle physics and cosmology. One until today as the intergalactic MFs. Moreover, the baryon of the popular models is the GUT baryogenesis [1]. The asymmetry is (re)generated through the Standard Model baryon asymmetry is provided from heavy boson decays in (SM) chiral anomaly [8,9], which is not completely washed grand unified theories (GUTs). The most troublesome issue out by the EW sphalerons [9,10]. Therefore, we can is that only B (baryon)þL (lepton) but not B − L asym- imagine the following scenario: (i) the GUT baryogenesis metry is generated in the heavy boson decay in the SU(5) first generates the B þ L and chiral asymmetry, (ii) max- GUT. The electroweak (EW) sphalerons [2] wash out the imally helical hyperMFs are generated from the chiral B þ L asymmetry and, hence, no asymmetry is left in that asymmetry while B þ L asymmetry is eventually damped, scenario. and (iii) the hyperMFs (re)produce baryon asymmetry, However, an interesting feature in the GUT baryogen- especially at the EW symmetry breaking, which lasts esis, namely, the generation of chiral asymmetry, is still of against the EW sphalerons. Then, the asymmetry is interest. Since the first-generation electron Yukawa inter- responsible for the present Universe. In other words, the action, which is the weakest chirality flip interaction, is in GUT baryogenesis can be the indirect origin of the present equilibrium only at relatively low temperature, T ≲ 80 TeV BAU. In this article, we explore this scenario and clarify the [3], the chirality is a good conserved quantity at higher condition required for the successful Universe. Note that energy scales. In particular, it has been noticed that with the the essence of the scenario is the generation of chiral help of the chiral magnetic effect [4], maximally helical asymmetry carried mainly by the right-handed electrons hypermagnetic fields (hyperMFs) are generated if suffi- without B − L asymmetry and hence can be applied for ciently large chiral asymmetry exists in the thermal plasma other models beyond the SM than GUTs.

*[email protected] II. GENERATION AND EVOLUTION Published by the American Physical Society under the terms of OF HYPERMAGNETIC FIELDS the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to First, we give an analytic explanation of how hyperMFs the author(s) and the published article’s title, journal citation, are generated at high temperature above the EW scale, and DOI. Funded by SCOAP3. which is consistent with recent numerical studies [6].

2470-0010=2018=97(10)=103506(5) 103506-1 Published by the American Physical Society KOHEI KAMADA PHYS. REV. D 97, 103506 (2018) B 1 2α The basic equations to be solved are as follows. The d Y ¼ ∇2B þ Y μ ∇ B þ ∇ ðv B Þ E B Y 5;Y × Y × × Y ; evolution of the hypergauge fields ( Y, Y) in the comov- dτ σY π ing frame (with conformal time τ) is described by the ð4Þ Maxwell’s equations,

2 dB dμ5 6c1ye αY Y ¼ −∇ × E ; ∇ × B ¼ J : ð1Þ ;Y ¼ R B · ð∇ × B Þ dτ Y Y Y dτ πT2σ Y Y i Y 12 2 α2 E τ c1yeR Y 2 Here, we omit the displacement current d Y=d since it is − B þ Γ → μ5 : ð Þ π2 2σ Y h ee ;Y 5 suppressed by the order of the amplitude of the fluid Ti Y velocity v and large hyperelectric conductivity in the magnetohydrodynamic (MHD) approximations, which is From these equations, we can see that there is a conserved μ þ 3 2 α ðπ 2Þ appropriate to describe the dynamics of large-scale gauge quantity, 5;Y c1yeR Yh= Ti in the limit where the Γ ≡ fields we are interested in. The electric current consists of YukawaR interaction h→ee is negligible. Here, h the Ohm’s current and chiral magnetic current [4], ð1 Þ 3 Y B =V V d x · Y is the hypermagnetic helicity density Y 2α with being the hypercharge vector potential. Y Let us investigate the evolution of the system with initial JY ¼ σYðEY þ v × BYÞþ μ5;YBY: ð2Þ π μ ¼ μi B v conditions with a large 5;Y 5;Y and tiny Y and . When 2 the hyperMFs are small enough, μ5;Y is effectively constant Here, σY ≃ 10 Ti is the hyperelectric conductivity [11], α ¼ 02 4π μ ¼ and the last term in Eq. (4) is negligible. Then the equation PY g = is the hyper-fine structure constant, and 5;Y ξ 2 for the circular polarization modes of the hyperMFs is ð−1Þ i y μ is the comoving chiral hyper-chemical i i i potential. Ti is the temperature where we define the scale dBY k 2α ð Þ ξ k ¼ − ∓ Y μi Y ð Þ factor a T to be 1, i is assigned for 0 for right-handed k 5;Y Bk : 6 dτ σY π fermions and 1 for the left-handed fermions, and yi and μi are the hypercharge and comoving chemical potential of the If μi > 0ð< 0Þ, the þð−Þ mode feels instability for fermion i, respectively. The evolution of the chemical 5;Y −ðþÞ potential is determined by the anomaly equations. In the smaller k while the mode does not. The difference SM, since the sphalerons and Yukawa interactions except between two circular polarization mode is the hypermag- for the first-generation electron’s one are in equilibrium at netic helicity and hence the amplified hyperMFs are maximally helical. Since the resultant baryon asymmetry high temperature T ≳ 80 TeV [3], the evolution for μ5 is ;Y μi determined by the most weakly-coupled fermion, that is, is positive for positive 5;Y, we hereafter focus on the case μi 0 ¼ α μi π the right-handed electrons [5,8–10], with 5;Y > . The most unstable mode is at kc Y 5;Y= Yþ ∝ ½ 2τ σ and evolves as Bk exp kc = Y . Thus at the time μ 6 2 α 2 d 5;Y yeR Y τ ≃ σ =k ≡ τ , the instabilities start to grow. This corre- ¼ c1 E B − Γ ↔ μ5 ; ð Þ Y c c τ π 2 Y · Y h ee ;Y 3 d Ti sponds to the temperature ¼ μ μ 1 2 −1 where c1 5;Y= e is the ratio between the chiral and right- 6 α σ R ≃ 6 8 10 Y Y Tc . × GeV −2 2 handed electron chemical potential and Γ ↔ is the chirality 10 10 T h ee i flip rate of the first-generation electron Yukawa interaction. −1 2 i 2 g = μ5 =T The overline represents the volume average. The evolution of ;Y i ð Þ × 106 75 10−2 : 7 the fluid velocity v is, in principle, described by the Navier- . Stokes equation, which is hard to solve. However, numerical jμi j ≳ 10−3 studies showed that the velocity fields are emerged from If 5;Y =Ti , the hyperMFs are amplified at vanishing initial conditions due to the Lorentz force and T ≳ 80 TeV, before the electron Yukawa interaction gets μ reach at an equilibrium to the hyperMFs immediately [6,12]. effective and the 5;Y starts to decay. Thus, we assume here that the velocity fields obtain the As the hyperMFs grow, the Lorentz force drives the ∼ γ ≡ jvj21=2 comparable strength to the hyperMFs with a similar coherent velocity fields up to v BY in a short period (v , 1 2 −4 2 −2 −2 2 ≡ jB j2 = length, EM ¼ aðTÞ jBYj =2 ¼ γ EK ¼ γ ρjvj =2 with BY Y ). Since the velocity fields erase the small 2 4 ρ ≡ ð30=π ÞgT being the energy density of the scale structure, the effects of the fluid velocity on the Universe. The number of relativistic degrees of freedom evolution of hyperMFs are no longer negligible when the λ ¼ τ g is taken to be 106.75 for the SM. The ratio between the eddy turnover scale et v reaches at the instability scale λ ≡ 2π velocity and MF strength γ is found to be Oð0.01Þ − Oð1Þ c =kc. This happens when the hyperMFs evolve up ≃ ðπ2 30Þ1=22α μi 2 ðγσ Þ ≃ Oð1Þ [6,12]. From Eqs. (1), (2),and(3), we can remove the electric to BY g= Y 5;YTi = Y .Forc1 , the fields as hypermagnetic helicity is still smaller than the chiral

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μ ≫ 3 2 α ðπ 2Þ The MFs continue to evolve until today and exist in the chemical potential, 5;Y c1yeR Yh= Ti , and the hyperMFs still continue to be amplified, with the comovingpffiffiffi intergalactic void with the properties coherent length satisfying λY ≃ vτ ≃ γBYτ= ρ [6]. −1=3 −1=3 0 −16 −1=3 γ αY By estimating the amplification time scale as τ ∼ σYλY= ≃ 9 9 10 Bphys . × Gc1 −2 −2 4α μ5 , we can have the mean field strength and coherent 10 10 Y ;Y length at given time τ. μi =T 1=3 1=3 5;Y i g ð Þ The amplification of the hyperMFs terminates when the × −2 ; 13 2 2 10 106.75 hypermagnetic helicity is saturated, 3c1y α h=ðπT Þ≃ eR Y i μi 2 3 −1 3 5;Y. The resultant physical hyperMF strength and coherent −1 3 γ = α = λ0 ≃ 6.9 × 10−3 pcc = Y length at the saturation are evaluated as phys 1 10−2 10−2 μi 1=3 −2=3 γ −5 α 9 5 =Ti g phys 10 2 2 Y ;Y ð Þ B ðT Þ ≃ 1 4 10 c × −2 : 14 Y s . × GeV 1 10−2 10−2 10 106.75 −7 i 5 3=2 σ =T μ5 =Ti g × Y i ;Y ; ð8Þ These intergalactic MFs with positive helicities are the 102 10−2 106.75 smoking-gun of that scenario, as argued in Ref. [10]. γ 4 α −7 λphysðT Þ ≃ 0 48 −1c−2 Y III. (RE)GENERATION OF BARYON ASYMMETRY Y s . GeV 1 10−2 10−2 Equation (5) suggests that the chiral asymmetry (as well σ 5 μi −3 −3=2 Y=Ti 5;Y=Ti g as the baryon asymmetry) will not be completely washed × ; ð9Þ 102 10−2 106.75 out in the presence of hypermagnetic helicity but reach at the equilibration, dμ5;Y=dτ ≃ 0 [8–10], at the temperature 2 2 12πc1y α B =λ μ ≃ eR Y Y Y ð Þ γ −2 α 4 5;Y 24 2 α2 2 π þ 2σ Γ : 15 6 Y c1ye YBY= Ti Y h→ee T ≃ 2 4 10 c1 R s . × GeV 10−2 10−2 −3 i 2 1 2 Note that for the maximally helical fields with a positive σ =T μ5 =T g = Y i ;Y i : ð Þ B ð∇ BÞ ≃ 2π 2 λ × 102 10−2 106 75 10 helicity, it is approximated as Y · × BY= Y. . Moreover, when the electroweak symmetry breaking takes place, the hypermagnetic helicity is transferred to the After the saturation, the hyperMFs evolve according to the (electro)magnetic helicity, which gives a nonzero contri- ∝ τ−1=3 λ ∝ τ2=3 inverse cascade law, BY , Y [6,12] supported bution to the anomaly equation for the B þ L asymmetry. by the velocity fields, while the chiral asymmetry is erased This effect has been studied in detail in Ref. [10], which μ τ ≃ 0 so that Eq. (5) reaches to the equilibration, d 5;Y=d shows that in the SM crossover with the 125 GeV Higgs – [8 10]. At the temperature T during radiation domination boson the effect lasts for a while after the freezeout of the before the electroweak symmetry breaking, the physical EW sphalerons and hence the B þ L asymmetry is not properties of the hyperMFs are given by washed out completely. The resultant baryon and lepton asymmetry of the Universe today is evaluated as 7 9 −1 3 g ðTÞ = −1 3 γ = BphysðTÞ ≃ 0 82 2 s c = Y . GeV ð Þ 1 10−2 17 ðθ ÞS gs Ts η0 ≡ nB ≃ ð 2 þ 02Þ f W;T ð Þ B g g 16 −1=3 i 1=3 1=3 s 37 γ α μ5 =Ti g today w;sph T¼135 GeV × Y ;Y 10−2 10−2 106.75 with [13] 7=3 T ð Þ × 102 ; 11 θ GeV ðθ Þj ≡ − d W ð2θ Þ f W;T T¼135 GeV T sin W dT ð Þ 5=9 γ 2=3 ≃ ð5 10−4 0 3Þ ð Þ λphysð Þ ≃ 9 8 106 −1 gs Ts −1=3 × . ; 17 Y T . × GeV c1 −2 gsðTÞ 10 phys phys 2 1 3 λ ð Þ α −1=3 μi = −2=3 H Y BY Y 5;Y=Ti g S ≡ ; ð Þ × 16π3 18 10−2 10−2 106.75 sT T −5=3 T × : ð12Þ γ ¼ exp −147.7 þ 107.9 ; ð19Þ 102 GeV w;sph 130 GeV

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0 ð2Þ ð1Þ 5 where g and g are the SM SU L and U Y gauge be evaded if the GUT Higgs bosons in the representation couplings, s is the entropy density, nB is the baryon number of SU(5) are produced by the preheating [15]. In that case, density, θW is the temperature-dependent effective weak large chiral asymmetries can be generated. However, if we mixing angle, and H is the Hubble parameter. The uncer- identify the Higgs boson (5) is responsible for the EW tainty in fðθW;TÞ comes from the errors in the temperature symmetry breaking, it mainly decays into the third or dependence of the weak mixing angle in the EW crossover second generation fermions through the Yukawa interaction found in the one-loop calculation and lattice calculations and little right-handed electron asymmetry is generated. 1 [10,14]. Since the hyperMFs generated from the chiral As a result c1 ¼ μ5 =μ 1 is extremely large and hence the ;Y eR instability discussed in the above are maximally helical and hyperMFs as well as the resultant baryon asymmetry is the mechanism in Refs. [8–10] work. From Eqs. (11) and highly suppressed. (12), the resultant baryon asymmetry today in terms of the One way to overcome these difficulties is to consider initial chiral asymmetry is calculated as another Higgs field in the 5 representation of SU(5) GUT that is not related to the EW symmetry breaking but mainly α −1 μi 0 −5 −1 Y 5;Y=Ti couples to the first generation fermions. Imagine they once η ≃ 4.0 × 10 c1 fðθ ;TÞ; ð20Þ B 10−2 10−2 W dominate the energy density of the Universe through, e.g., 1 1 instant preheating [16] and eventually decay into eRuR and which is the main result of this article. Therefore, although ¯ 1 ¯ 1 QLQL pairs after they become nonrelativistic. Then a large precise evaluations of the temperature dependence of the chiral chemical potential mainly carried by right-handed weak mixing angle are needed for the quantitatively precise electrons can be generated. In that case, while the asym- estimate, the helical hyperMFs generated by the chiral metry of the right-handed electrons are unchanged for a η0 ∼ instability can be responsible for the present BAU B while, the asymmetries are rearranged to other fermions 10−10 μi ∼ 10−3 if the initial chiral asymmetry is 5;Y=Ti and through the Yukawa interaction and sphaleron processes −4 fðθ ;T ¼ 135 GeVÞ ∼ 10 with c1 ¼ Oð1Þ. Note that immediately so that we obtain c1 ¼ μ5 =μ 1 ¼ 553=481. W ;Y eR this predicts slightly large baryon asymmetry, but it is not The (physical) chemical potential at the time of decay is μi ∼ 10−3 problematic since for 5;Y=Ti the generation of evaluated as hypermagnetic fields might not be saturated before T ∼ 80 TeV when the chirality flip interaction becomes strong, phys i 2 μ5 μ5 π g T and hence the resultant magnetic fields properties can be ;Y ¼ ;Y ¼ ϵ dec ð Þ 5 c1 ; 21 slightly smaller so that they are appropriate for the present Tdec Ti mX i −3 BAU. On the other hand, for μ5 =Ti ≪ 10 , the chiral ;Y ϵ instability of hyperMFs do not occur at all. Consequently, where is the net right-handed electron asymmetry −4 produced by a single Higgs-anti Higgs pair, T is the even if fðθW;T ¼ 135 GeVÞ ≫ 10 the present BAU dec cannot be explained. It should be also noted that we suffer Higgs decay temperature, and mX is the mass of the Higgs ðθ ¼ 135 Þ ≫ field. Therefore, if e.g., the CP violation in GUT allows from baryon overproduction if f W;T GeV −3 −4 i −3 ϵ ≃ 10 and the decay temperature of the Higgs boson is 10 and μ =T ≳ 10 for c1 ¼ Oð1Þ. But if c1 is 5;Y i ∼ 10−2 appropriately large due to the nature of the chiral asymmetry tuned to be TR=mX , the chiral asymmetry ideal for −4 the present BAU can be generated. generation mechanism, fðθW;T ¼ 135 GeVÞ ≫ 10 and μi ≫ 10−3 5;Y=Ti can be accommodate to the present BAU. V. CONCLUSION IV. POSSIBLE SOURCE OF THE In this article, we discussed the possibility for baryo- CHIRAL ASYMMETRY genesis models without B − L generation such as GUT baryogenesis to be indirectly responsible for the present Finally, let us give a discussion on the possible origin of þ such large chiral asymmetry mainly carried by the right- BAU. It is usually considered that B L asymmetry is handed electrons. In the standard SU(5) GUT baryogenesis washed out completely by the EW sphalerons and no [1], thermally produced GUT bosons decay into quarks and asymmetry would remain. In the scenario discussed here, leptons with nonvanishing B þ L asymmetry. However, the the washout by the EW sphalerons are evaded by the thermal GUT symmetry breaking that would occur in that mechanism that the asymmetry is first transferred to the case suffers from the monopole problem. This problem can hypermagnetic helicity which in insensitive to the EW sphaleron. Baryon asymmetry is provided by the hypermagnetic helicity decay that is effective until shortly after 1 ðθ Þj Note that the uncertainty in f W;T T¼135 GeV is a the EW sphaleron freezeout [10]. conservative one, which includes the fitting function whose fit is not so good at T ≲ 150 GeV (Fitting function B of Fig. 2 in the There are several difficulties for the realistic model build- second article of Ref. [10]). If we omit the fitting function, we have ing. The efficiency is not as much as 100%, and relatively ðθ Þj ≃ ð0 04…0 3Þ μ ≃ 10−3 f W;T T¼135 GeV . . . large initial asymmetry is required, 5;Y=Ti .In

103506-4 RETURN OF GRAND UNIFIED THEORY BARYOGENESIS: … PHYS. REV. D 97, 103506 (2018) addition, the asymmetry should be carried mainly by the ACKNOWLEDGMENTS right-handed electrons. Nevertheless, our findings here K. K. is grateful to R. Jinno, T. Kuwahara, A. J. Long, T. opened a new direction in the study of the BAU. Further Vachaspati, M. Yamaguchi, and J. Yokoyama for useful studies on the realistic model building for thegeneration of the comments and discussions. The work of K. K. was sup- B þ L asymmetry as well as the determination of the ported by Institute for Basic Science (IBS) under the temperature dependence of the weak mixing angle are Project Code IBS-R018-D1, and by the Department of required. Energy under Grant No. DE-SC0013605.

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