Neutrino assisted GUT baryogenesis - revisited

Many GUT models conserve the difference between the baryon and lepton number, $B-L$. These models can create baryon and lepton asymmetries from heavy Higgs or gauge boson decays with $B+L \neq 0$ but with $B-L=0$. Since the sphaleron processes violate $B+L$, such GUT-generated asymmetries will finally be washed out completely, making GUT baryogenesis scenarios incapable of reproducing the observed baryon asymmetry of the Universe. In this work, we revisit the idea to revive GUT baryogenesis, proposed by Fukugita and Yanagida, where right-handed neutrinos erase the lepton asymmetry before the sphaleron processes can significantly wash out the original $B+L$ asymmetry, and in this way one can prevent a total washout of the initial baryon asymmetry. By solving the Boltzmann equations numerically for baryon and lepton asymmetries in a simplified 1+1 flavor scenario, we can confirm the results of the original work. We further generalize the analysis to a more realistic scenario of three active and two right-handed neutrinos to highlight flavor effects of the right-handed neutrinos. Large regions in the parameter space of the Yukawa coupling and the right-handed neutrino mass featuring successful baryogenesis are identified.


I. INTRODUCTION
The unaccounted baryon asymmetry in the Universe is one of the reasons why the Standard Model (SM) is imperfect and challenges physicists to come up with new ideas of asymmetry generation mechanisms. Grand Unified Theories (GUTs), on the other hand, elegantly unify all SM gauge couplings except for gravity and provide a seminal framework for baryogenesis model building. The simplest GUT based on the SU(5) gauge group proposed by Georgi and Glashow in 1974 [1], contains leptoquark gauge bosons which can mediate baryon number violating processes, making the proton unstable, while preserving the difference between the baryon and the lepton number B −L. A baryon asymmetry can be generated from heavy gauge or Higgs boson decays [2][3][4][5], which fulfill all Sakharov conditions [6], but is accompanied by a lepton asymmetry of equal amount. Thus the generated baryon and lepton asymmetries will be erased completely by the non-perturbative B + L violating sphaleron processes [7][8][9] which are effective when the temperature of the Universe falls below roughly 10 12 GeV (while B − L asymmetries are unaffected by the B − L conserving sphalerons).
The same problem also arises in other larger symmetry groups, such as SO(10), which contains U (1) B−L as a subgroup. In other words, as long as U (1) B−L is not broken when the baryon asymmetry is created, neither the baryon nor the lepton asymmetry will survive the sphaleron processes.
In principle there exist at least two remedies for GUT baryogenesis. For certain matter representations under SO (10) or larger groups it has been demonstrated in Refs [10][11][12][13][14][15], that a non-vanishing B − L asymmetry can still be realized. Alternatively, Fukugita and Yanagida [16] have proposed to involve right-handed neutrinos to revive GUT baryogenesis, where the right-handed neutrino N can be a singlet under SU (5) or be embedded into the 16 of SO (10). A Majorana mass of N , induced by a scalar vacuum expectation value (VEV) or simply imposed by hand, will explicitly violate the B − L symmetry, that can either be a subgroup of the original GUT symmetry or an accidental symmetry. The corresponding part of the Lagrangian reads This entire process can create a non-zero baryon asymmetry after the electroweak phase transition, if the N -induced washout processes are active before the onset of the sphaleron processes. It is important that the sphalerons and L washout processes do not coexist for too long since simultaneous Land (B + L)-violating interactions will erase both the B and L asymmetries. Note that the SM Yukawa couplings, that couple right-handed to left-handed leptons via the Higgs boson, are not in thermal equilibrium above a temperature of roughly 10 12 GeV for τ , 10 9 GeV for µ and 10 5 GeV for e. As a consequence, the washout processes in Eq. (2) may only erase an initial asymmetry stored in the SU (2) L lepton doublets but not in the right-handed charged leptons, depending on the temperature. Throughout this work, for simplicity we assume the L asymmetry is stored in the lepton doublet only, i.e., a change on the number of the lepton doublet is equivalent to that of the lepton number: For the ∆L = 2 washout processes, a change in the lepton number always comes with an equal amount of the H * number change, ∆L (= −∆(B − L)) = ∆H * , which can be easily seen from Eq. (2). In addition, when these processes are in chemical equilibrium, the chemical potentials of and H * are equal. Consequently, as shown in the center panel of Fig. 1 at most half of the initial L asymmetry is converted into that of H * without considering impacts of the Yukawa couplings 1 . In other words, the maximal induced B − L asymmetry by the ∆L = 2 interactions is one fourth of the initial B + L asymmetry from GUT baryogenesis. The SM Yukawa interactions later come into equilibrium and shuffle the asymmetry among quarks, leptons and Higgs bosons such that the final B and L asymmetries will be functions of the B − L asymmetry, as indicated in the right panel of Fig. 1. Note that as long as the B − L asymmetry is non-vanishing, the B + L asymmetry will not be completely erased by the sphalerons [17,18]. That is due to the fact the sphalerons couple only to left-handed particles, whereas the baryon and lepton numbers consist of both lefthanded and right-handed particles (those are connected by the Yukawa couplings) 2 . After the electroweak phase transition (EWPT), the H 0 asymmetry vanishes due to the Higgs VEV [18] H ± and the imaginary part of the electrically neutral component are eaten by the W ± and Z bosons.
In this work, we revisit the idea of N -assisted GUT baryogenesis by numerically solving the Boltzmann equations including the lepton number violating ( / L) processes as well as the sphaleron effects, which allows a quantitative study of the relevant parameter space. Special attention is paid to the investigation of the interplay between the washout and sphaleron processes, from which one can infer the condition for obtaining the maximal final baryon and lepton asymmetries. We start with the case of one lepton generation and one right-handed neutrino, and then generalize this scenario to the realistic case of three generations plus two right-handed neutrinos.
1 For temperatures of interest, the t (T 10 16 GeV) or b (T 10 12 GeV) Yukawa interactions are effective during the L washout. The lepton asymmetry can be further shifted into a quark asymmetry as discussed in Appendix A, leading to larger final (B − L) and B asymmetries. 2 To be more precise, if the sphalerons and all the SM Yukawa couplings are in equilibrium, all asymmetries can be expressed as functions of the lepton doublet asymmetry or its chemical potential, µ . If B − L = 0, it implies µ = 0 and thus B + L = 0.

II. BOLTZMANN EQUATIONS FOR L WASHOUT
In this Section, we briefly discuss the Boltzmann equations used for obtaining the time evolution of the particle density in question. More detailed discussions can be found in Refs. [19][20][21]. The Boltzmann equation for a particle in the presence of the interaction a 1 · · · a n ↔ f 1 · · · f m is, fm γ (f 1 · · · f m ↔ a 1 · · · a n ) − n n a 1 · · · n an n eq n eq a 1 · · · n eq an γ ( a 1 · · · a n ↔ f 1 · · · f m ) . ( Here Y (≡ n/s) denotes the particle number density normalized to the entropy density s and z = M N /T . The thermal equilibrium rate γ is defined as with the squared amplitude, |M | 2 , summing over initial and final spins. For a 2 → 2 process, γ can be further simplified if the corresponding amplitude only depends on the square of the center-of-mass energy s but not on the relative motion with respect to the thermal plasma: with s min = max{(m a 1 + m a 2 ) 2 , (m f 1 + m f 2 ) 2 } and the reduced cross-sectionσ(s) = 2sλ(1, m 2 a 1 /s, m 2 a 2 /s)σ(s). To simplify the analysis, we first focus on the 1 +1 scenario, one generation of SM leptons and one right-handed neutrino. Moreover, we assume that the scale of GUT baryogenesis is below the right-handed neutrino mass to avoid complications from finite-temperature effects such as N → HL being kinematically forbidden due to thermal masses when T M N [21].
This scenario with low injection scales ( M N ) can be realized when the heavy particles responsible for baryogenesis are non-thermally produced as proposed in Ref. [22]. Moreover, the B − L asymmetry can also arise even if the B + L injection scale is higher than M N as long as the Yukawa coupling y does not carry any CP phase 3 , and if lepton washout interactions and the sphalerons are not simultaneously effective for a long time. In fact, for certain regions of the parameter space, a higher injection scale leads to a larger B − L due to a longer washout period.
To compute the L washout, we include both ∆L = 1 and ∆L = 2 interactions. Following the same notation as used in Ref. [21], the relevant ∆L = 2 washout processes are H ↔¯ H * and ↔ H * H * (with thermal rates γ N s and γ N t , respectively). The ∆L = 1 As explained in Ref. [21], for ∆L = 2 processes one has to subtract contributions from onshell right-handed neutrinos N to avoid double counting, if the contributions of their decays and inverse decays are already taken into account, i.e., the processes H ↔ N ↔¯ H * have been included by successive decays. Alternatively, one can simply consider ∆L = 2 interactions with on-shell right-handed neutrinos without including the (inverse) decays as they are already incorporated in the the unsubtracted rate. In this work, we adopt the second method.
Assuming no CP violation sources in the N decay as mentioned above, the Boltzmann equation for the lepton asymmetry with with the Hubble parameter always refer to the equilibrium density: Y eq i ≡ Y eq particle i = Y eq anti-particle i . Here M P l is the Planck mass and g * is the number of relativistic degrees of freedom (106.75 for the SM). Y H is the L asymmetry change due to As we shall see in Appendix A, the impact of the heavy quark Yukawa couplings on the washout processes can be characterized by the Note that we do not consider the τ Yukawa coupling, which is in equilibrium for T 10 12 GeV. This is justified as the contribution is relatively small, compared to those of t and b Yukawa couplings in light of the color factor.
Finally, the Boltzmann equation for the number density of N is given by: As all reduced cross-sectionsσs for the ∆L = 1 interactions are given in Ref. [21], we here only provide the reduced cross-sectionsσ N s andσ N t for the ∆L = 2 processes, H ↔¯ H * and ↔ HH * respectively, as displayed in Fig. 2.
N -mediated lepton number violating processes. 4 The symbol "∆" is reserved for the change of the particle number due to L washouts, such as ∆L and ∆H. 5 In the absence of the Yukawa couplings, Y H = Y H since ∆L = ∆H * = −∆H as mentioned above.
The reduced cross section of H ↔¯ H * with the center-of-mass energy squared s is: where Here, θ is the angle between the incoming and the outgoing lepton. For ↔ H * H * , the reduced cross section reads:

III. SPHALERONS PROCESSES
We are now in the position to include the sphaleron processes into the Boltzmann equations to study the interplay between the / L and (B + L)-violating (denoted by B + L) interactions. The sphaleron effects can be expressed as [24,25] where Clearly the sphalerons erase the B and L asymmetries at the same rate so that the B − L asymmetry remains constant. Thus the sphalerons alter Y B+L but not Y B−L in the Boltzmann equations: After all SM Yukawa interactions reach thermal equilibrium, the final baryon and lepton asymmetries are directly related to the B − L asymmetry created by the washout processes: where c s = 28/79 [17,18] for non-supersymmetric models with one Higgs doublet as assumed here. where m is an arbitrary mass scale such that the product m · t is dimensionless. One can see that the L asymmetry is eliminated faster than the B asymmetry and thus consequently a net B − L asymmetry is created. At later times, the damped / L interaction becomes inefficient and the sphalerons yield Y B + Y L = 0, leading to a non-vanishing baryon asymmetry: The resonance enhancement appears when T ∼ M N -a smaller Yukawa coupling implies a narrower decay width of N , resulting in a larger enhancement.  [27], respectively. Therefore, to obtain the mass-squared difference corresponding to atmospheric neutrino oscillations from the type-I seesaw mechanism and to achieve Y final B−L /Y initial B+L 10 −1 require a lower bound on the righthanded neutrino mass: M N 10 13 GeV with y 0.1. Note that the ratio Y final B−L /Y initial B+L is independent of the value of Y initial B+L since the Boltzmann equations are linear in Y B and Y L . In Ref. [16] it was concluded that for M N ∼ 10 16 GeV and y ∼ 1 the observed baryon asymmetry can be reproduced. These values of M N and y actually fall into the region of maximal washout, i.e., In other words, our numerical results of the 1 + 1 scenario are consistent with those of Ref. [16].

V. GENERALIZATION TO A 3 + 2 CASE
Finally, we study a more general case including three generations of SM leptons and two right-handed neutrinos, required to reproduce two mass-squared differences inferred from the solar and atmospherical neutrino oscillations.
Before showing numerical results, we would like to comment on the properties of the 3 + 2 case. Here we mainly focus on the maximal washout region, which as indicated above occurs for M N 10 13 GeV. As a consequence, none of the charged lepton Yukawa couplings is efficient in the regions of interest (T 10 13 GeV) and so the three lepton generations are indistinguishable [28,29], implying that the neutrino mixing matrix (PMNS matrix) will not appear in the result. Since one linear combination of ( e , µ , τ ) will not couple to the right-handed neutrinos and hence remains massless, the lepton asymmetry stored along this direction will not be washed out. In contrast, the two massive light neutrinos which couple to N (1,2) will participate in the washout processes and the asymmetries stored therein will be partially converted into the asymmetry of H. In other words, in the 3 + 2 case only two linear combinations, denoted by 1 and 2 , will give rise to a non-zero (B/3 − L (1,2) ) and consequently also a final B, whereas the (B/3 + L 3 ) asymmetry along the 3 direction will be completely destroyed by the sphalerons 6 .
In the following, we will not explore the full parameter space which is quite large for the 3 + 2 case. Instead a simplified framework, where each massive light neutrino couples only 6 The situation is quite similar to cases in leptogenesis where heavy neutrino flavor effects are important; not only N 1 but also N 2 contributes the L asymmetry generation [28,[30][31][32][33], depending on the flavor structure of Yukawa couplings y αj¯ α HN i (α = (e, µ, τ ) and i = (1, 2)) and on how strong the corresponding L washout effects are.
to one of the heavy right-handed neutrinos but not to both of them, is chosen to highlight the main features. The relevant Lagrangian is where the neutrinos ν 1 and ν 2 in the SU (2) L doublets 1 and 2 , respectively, are the mass eigenstates of m 1 = y 2 1 v 2 /M 1 and m 2 = y 2 2 v 2 /M 2 while ν 3 in the doublet 3 remains massless (m 3 = 0). Due to the fact the PMNS matrix will not influence the washout calculation for T > 10 12 GeV, we only require m 1 and m 2 to reproduce the two mass-squared differences associated with solar and atmospheric neutrino oscillations.
With these assumptions the previous washout calculation is repeated, taking into account   When M 2 becomes much bigger than M 1 , the B + L injection scale is fixed to be M 1 /3.
In this region, L violation is created mainly by N 1 because of Γ 1 Γ 2 , leading to a sizable final B/3 − L 1 asymmetry but a tiny B/3 − L 2 asymmetry. That is the reason why the maximal Y final B−L /Y initial B+L becomes smaller than that of the previous case with M 2 ∼ M 1 as seen by comparing the red/middle cross with the orange/right one in the left panel.
To summarize, for regions with large M 1 and M 2 sizable L washouts can be obtained and the observed mass-squared differences can be realized in the simplified 3+2 case. For general 3 + 2 cases, each light neutrino will receive both N 1 and N 2 contributions. The conclusions drawn above, however, remain valid as long as there is no severe cancellation between the two contributions.

VI. CONCLUSIONS
In this work, we have revisited the idea proposed in Ref. [16] where lepton number violation, induced by the right-handed neutrinos, results in a nonzero B − L asymmetry. In other words, the lepton number violating processes can make part of an original baryon asymmetry We here investigate how the existence of the heavy quark Yukawa couplings modify the Boltzmann equations. In the following, it is always assumed that all asymmetries (quarks, leptons and Higgs bosons) are much smaller than the equilibrium density of the corresponding particle. That is, for f = Q 3 (third generation quark doublet), U 3 (right-handed t), D 3 (right-handed b), and H, where Q 3 , and H have two degrees of freedom from gauge multiplicity. To make the notation consistent with Y B and Y L which denote the baryon and lepton asymmetry densities, we drop '∆' in Y ∆f , i.e., In contrast, Y eq f = Y eq f always represents the equilibrium density of the (anti-)particle. The Boltzmann equation for the lepton asymmetry, including the Yukawa interactions with z = M N T . To simplify the calculation without including quarks in the Boltzmann equations, one can assume that the relevant Yukawa couplings quickly transfer the asymmetry from H to Q 3 and U 3 when T 10 16 GeV, and also to D 3 for T 10 12 GeV. In light of the chemical equilibrium of the Yukawa interactions and the correlations of the particle numbers among H, Q 3 and U 3 7 , it is straightforward to show that for 10 12 T 10 16 GeV where Y H is the total asymmetry obtained from the L washout, i.e., Y H (z) ≡ Y initial L −Y L (z).
Similarly, for T 10 12 GeV, one has