Bridging resonant leptogenesis and low-energy CP violation with an RGE-modified seesaw relation

We propose a special type-I seesaw scenario in which the Yukawa coupling matrix $Y^{}_\nu$ can be fully reconstructed by using the light Majorana neutrino masses $m^{}_i$, the heavy Majorana neutrino masses $M^{}_i$ and the PMNS lepton flavor mixing matrix $U$. It is the RGE-induced correction to the seesaw relation that helps interpret the observed baryon-antibaryon asymmetry of the Universe via flavored resonant thermal leptogenesis with $M^{}_1 \simeq M^{}_2 \ll M^{}_3$. We show that our idea works well in either the $\tau$-flavored regime with equilibrium temperature $T \simeq M^{}_1 \in (10^9, 10^{12}]$ GeV or the $(\mu+\tau)$-flavored regime with $T \simeq M^{}_1 \in (10^5, 10^9]$ GeV, provided the light neutrinos have a normal mass ordering. We find that the same idea is also viable for a {\it minimal} type-I seesaw model with two nearly degenerate heavy Majorana neutrinos.


Introduction
A special bonus of the canonical (type-I) seesaw mechanism [1][2][3][4][5] is the thermal leptogenesis mechanism [6], which provides an elegant way to interpret the mysterious matter-antimatter asymmetry of our Universe. The key points of these two correlated mechanisms can be summed up in one sentence: the tiny masses of three known neutrinos ν i are ascribed to the existence of three heavy Majorana neutrinos N i (for i = 1, 2, 3), whose lepton-number-violating and CP-violating decays result in a net lepton-antilepton number asymmetry Y L which is finally converted to a net baryon-antibaryon number asymmetry Y B as observed today.
In the standard model (SM) extended with three right-handed neutrinos and lepton number violation, it is the following seesaw formula that bridges the gap between the masses of ν i (denoted as m i ) and those of N i (denoted as M i ): where M ν represents the light (left-handed) Majorana neutrino mass matrix, v 174 GeV is the vacuum expectation value of the SM neutral Higgs field, M R stands for the heavy (righthanded) Majorana neutrino mass matrix, and Y ν is a dimensionless coupling matrix describing the strength of Yukawa interactions between the Higgs and neutrino fields. The eigenvalues of M ν (i.e., m i ) can be strongly suppressed by those of M R (i.e., M i ) as a consequence of M i v (for i = 1, 2, 3), and that is why m i v naturally holds. Although such a seesaw picture is qualitatively attractive, it cannot make any quantitative predictions unless the textures of M R and Y ν are fully determined [7]. Without loss of generality, one may always take the basis in which both the charged-lepton mass matrix M l and the heavy Majorana neutrino mass matrix M R are diagonal (i.e., M l = D l ≡ Diag{m e , m µ , m τ } and M R = D N ≡ {M 1 , M 2 , M 3 }). In this case the undetermined Yukawa coupling matrix Y ν can be parametrized as follows -the so-called Casas-Ibarra (CI) parametrization [8]: where U is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix [9][10][11] used to diagonalize M ν in the chosen basis (i.e., U † M ν U * = D ν ≡ Diag{m 1 , m 2 , m 3 }), and O is an arbitrary complex orthogonal matrix. This popular parametrization of Y ν is fully compatible with the seesaw formula in Eq. (1), but the arbitrariness of O remains unsolved. Note that it is the complex phases hidden in Y ν that govern the CP-violating asymmetries ε iα between the lepton-number-violating decays N i → α + H and N i → α + H (for i = 1, 2, 3 and α = e, µ, τ ) [6,[12][13][14]. In particular, the flavored asymmetries ε iα depend on both (Y [15][16][17][18][19][20][21][22]. Given the CI parametrization of Y ν in Eq. (2), one can immediately see that ε i have nothing to do with the PMNS matrix U [23][24][25], while ε iα will depend directly on U if O is assumed to be real [26][27][28][29][30][31].
Note also that both U and D ν in Eq. (2) are defined at the seesaw scale Λ SS v, which can be related to their counterparts at the Fermi scale Λ EW ∼ v via the one-loop renormalization-group equations (RGEs) [32][33][34][35][36][37]. In this connection the RGE-induced correction to the CI parametrization of Y ν has recently been taken into account [38] 1 : where T l = Diag{I e , I µ , I τ }, and the evolution functions I 0 and I α (for α = e, µ, τ ) are given by in the SM with g 2 , λ, y t and y α standing respectively for the SU(2) L gauge coupling, the Higgs self-coupling constant, the top-quark and charged-lepton Yukawa coupling eigenvalues [38]. Eq. (3) tells us that the unflavored CP-violating asymmetries ε i should also have something to do with the PMNS matrix U at low energies because of a slight departure of T l from the identity matrix. This new observation makes it possible to establish a direct link between unflavored thermal leptogenesis and low-energy CP violation under the assumption that O is a real matrix [38,39], but one may still frown on the uncertainties associated with O.
In this work we simply assume the unconstrained orthogonal matrix O to be the identity matrix (i.e., O = 1), so as to reconstruct the Yukawa coupling matrix Y ν in terms of not only M i at the seesaw scale but also m i and U at low energies. Considering the fact of y 2 e y 2 µ y 2 τ 1 in the SM, we find that I e I µ 1 and I τ 1 + ∆ τ are two excellent approximations, where denotes the small τ -flavored effect [38]. Then the expression of Y ν in Eq. (3) can be somewhat simplified and explicitly written as in which the scale indices Λ SS and Λ EW have been omitted for the sake of simplicity, but one should keep in mind that the values of m i and U αi (for i = 1, 2, 3 and α = e, µ, τ ) are subject to the Fermi scale Λ EW . With much less arbitrariness, we are going to show that such a special RGE-modified seesaw scenario allows us to account for the observed baryon-to-photon ratio η ≡ n B /n γ (6.12 ± 0.03) × 10 −10 7.04Y B in today's Universe [40] by means of flavored resonant thermal leptogenesis with M 1 M 2 M 3 [41][42][43][44] 2 . We find that our idea works in either the τ -flavored regime with equilibrium temperature T M 1 ∈ (10 9 , 10 12 ] GeV or the (µ + τ )-flavored regime with T M 1 ∈ (10 5 , 10 9 ] GeV, if the mass spectrum of three light Majorana neutrinos has a normal ordering. In addition, we show that the same idea is also viable for thermal leptogenesis in a minimal type-I seesaw model [50,51] with two nearly degenerate heavy Majorana neutrinos.

Resonant leptogenesis
In the type-I seesaw scenario the lepton-number-violating decays N i → α + H and N i → α + H are also CP-violating, thanks to the interference between their tree and one-loop (self-energy and vertex-correction) amplitudes [6,[12][13][14]. Given M 1 M 2 M 3 , however, the near degeneracy of M 1 and M 2 can make the one-loop self-energy contribution resonantly enhanced [41][42][43][44][45][46][47][48][49]. As a result, the flavor-dependent CP-violating asymmetries ε iα between N i → α + H and N i → α + H decays (for i = 1, 2 and α = e, µ, τ ) are dominated by the interference effect associated with the self-energy diagram [42,43]: where ξ ij ≡ M i /M j and ζ j ≡ Y † ν Y ν jj / (8π) with the Latin subscripts j = i running over 1 and 2. Taking account of the expression of Y ν in Eq. (6), we immediately arrive at together with The flavored CP-violating asymmetries in Eq. (7) turn out to be 2 For such a heavy Majorana neutrino mass spectrum, the role of N 3 in thermal leptogenesis is expected to be negligible because its contribution has essentially been washed out at T M 1 M 2 .
where α = e, µ, τ and j = i = 1, 2; and ζ j = . One can see that ε iα ∝ ∆ τ holds, and hence ε iα will be vanishing or vanishingly small if O = 1 is taken but the RGE-induced effect is neglected. Note that the first term in the square brackets of Eq. (10) depends only on a single combination of the two so-called Majorana phases ρ and σ of U [7], denoted here as φ ≡ ρ − σ; and the second term is only dependent on the Dirac phase δ of U . So a direct connection between the effects of leptonic CP violation at high-and low-energy scales has been established in our RGE-assisted seesaw-plus-leptogenesis scenario.
In the flavored resonant thermal leptogenesis scenario under consideration, the CP-violating asymmetries ε iα are linked to the baryon-to-photon ratio η as follows [52,53]: where κ 1α and κ 2α are the conversion efficiency factors, and the sum over the flavor index α depends on which region the lepton flavor(s) can take effect. To evaluate the sizes of κ iα , let us first of all figure out the effective light neutrino masses Then the so-called decay parameters K iα ≡ m iα /m * can be defined and calculated, where 1.08 × 10 −3 eV represents the equilibrium neutrino mass and H(M 1 ) = 8π 3 g * /90M 2 1 /M pl is the Hubble expansion parameter of the Universe at temperature T M 1 with g * = 106.75 being the total number of relativistic degrees of freedom in the SM and M pl = 1.22 × 10 19 GeV being the Planck mass.
• For M i 10 12 GeV (for i = 1, 2), all the leptonic Yukawa interactions are flavor-blind. In this case the unflavored leptogenesis depends on the overall CP-violating asymmetry ε i = ε ie + ε iµ + ε iτ 0 in our scenario, as one can easily see from Eq. (10).
• For 10 9 GeV M i 10 12 GeV, the τ -flavored Yukawa interaction is in thermal equilibrium and thus the τ flavor can be distinguished from e and µ flavors in the Boltzmann equations [52,53]. In this case one has to consider two classes of lepton flavors: the τ flavor and a combination of the indistinguishable e and µ flavors. We are then left with the flavored CP-violating asymmetries ε iτ and ε ie + ε iµ together with the flavored decay parameters K iτ and K ie + K iµ , and the latter can be used to determine the corresponding conversion efficiency factors. [ 21,54]. Given the initial thermal abundance of heavy Majorana neutrinos, the efficiency factor κ (K α ) can be approximately expressed as [21,55] where z B (K α ) 2 + 4K 0.13 α exp (−2.5/K α ). We proceed to numerically illustrate that our resonant leptogenesis scenario works well. First of all, the values of I 0 and ∆ τ at the seesaw scale are illustrated in Fig. 1 with Λ SS ∈ [10 5 , 10 12 ] GeV in the SM. Adopting the standard parametrization of U [7], we need to input the values of eleven parameters: two heavy neutrino masses M 1 and M 2 (or equivalently, M 1 and d); three light neutrino masses m i (for i = 1, 2, 3); three lepton flavor mixing angles θ 12 , θ 13 and θ 23 ; and three CP-violating phases δ, ρ and σ (but only δ and the combination φ ≡ ρ − σ contribute). For the sake of simplicity, here we only input the best-fit values of θ 12 , θ 13 , θ 23 , δ, ∆m 2 21 ≡ m 2 2 − m 2 1 and ∆m 2 31 ≡ m 2 3 − m 2 1 (or ∆m 2 32 ≡ m 2 3 − m 2 2 ) extracted from a recent global analysis of current neutrino oscillation data [56,57] Given the above inputs, we can estimate the size of K α with the help of Eq. (12). It is found that K e 2.4, K µ 2.9 and K τ 2.6 in the normal neutrino mass ordering case; or K e 44.9, K µ 20.8 and K τ 26.4 in the inverted mass ordering case. Now that K α > 1 holds in either case, any lepton-antilepton asymmetries generated by the lepton-number-violating and CP-violating decays of N 3 with M 3 M 1 M 2 can be efficiently washed out. It is therefore safe to only consider the asymmetries produced by the decays of N 1 and N 2 . Now let us use the observed value of η to constrain the parameter space of φ and d by allowing m 1 (or m 3 ) and M 1 to vary in some specific ranges; or to constrain the parameter space of m 1 (or m 3 ) and M 1 by allowing φ and d to vary in some specific ranges, and by taking account of both the τ -flavored regime with T M 1 ∈ (10 9 , 10 12 ] GeV and the (µ + τ )flavored regime with T M 1 ∈ (10 5 , 10 9 ] GeV. We find no parameter space in the inverted neutrino mass ordering case, in which the conversion efficiency factors are strongly suppressed. Our RGE-assisted resonant leptogenesis scenario is viable in the normal neutrino mass ordering case, and the numerical results for the τ -and (µ + τ )-flavored regimes are shown in Figs. 2 and 3, respectively. Some brief discussions are in order.
• The τ -flavored regime (i.e., T M 1 ∈ (10 9 , 10 12 ] GeV). As can be seen in the upper panels of Fig. 2, φ is mainly allowed to lie in two possible ranges: [0, 2π/5] and [π, 7π/5]; and the dimensionless parameter d satisfies d 4 × 10 −5 . These two ranges of φ differ from each other just by a shift or reflection; and they are symmetric about φ = π/5 and φ = 6π/5, respectively. Such a feature can easily be understood. Considering ε ie + ε iµ + ε iτ = 0 and M 1 M 2 , we have η ∝ ε 1τ + ε 2τ ∝ sin 2(φ − ϕ τ ) with ϕ τ ≡ arg U * τ 1 U τ 2 e iφ being dominated by the CP-violating phase δ whose value is around 19π/20. And thus if φ is replaced by π + φ (or 7π/5 − φ) and 2π/5 − φ (or 12π/5 − φ), the value of η will keep unchanged. Note that even if φ = 0 holds, there can still exist some parameter space for the four free parameters. In this special case the Dirac CP phase δ, which is sensitive to leptonic CP violation in neutrino oscillations, is the only source of CP violation in our flavored resonant leptogenesis scenario. As shown in the lower panels of Fig. 2, M 1 varies in the range (10 9 , 10 12 ] GeV and m 1 0.01 eV holds. But for a given value of d, the parameter space of M 1 is generally constrained to a specific range; and when d decreases, the allowed range of M 1 increases correspondingly. For the most part of the allowed range of φ, the smallest neutrino mass m 1 can approach zero with a given value of d, and the value of η is almost independent of m 1 when m 1 becomes small enough since η is dominated by the term containing m 2 m 1 in this case. When the value of φ approaches the edge of the allowed range of φ, there will be a lower limit on m 1 which can be seen from the orange band in the lower-left panel of Fig. 2. This feature is mainly a consequence of the reduction of ε iτ in magnitude, which is proportional to sin 2(φ − ϕ τ ).
• The (µ + τ )-flavored regime (i.e., T M 1 ∈ (10 5 , 10 9 ] GeV). It is obvious that in this case the parameter space is largely reduced as compared with that in the τ -flavored regime.
It is finally worth mentioning that the normal neutrino mass ordering is currently favored over the inverted one at the 3σ level, as indicated by a global analysis of today's available experimental data on various neutrino oscillation phenomena [56][57][58]. This indication is certainly consistent with our RGE-assisted resonant leptogenesis scenario.

On the minimal seesaw
Since we have focused on resonant leptogenesis with M 1 M 2 M 3 based on the type-I seesaw mechanism, it is natural to consider a minimized version of this scenario by switching off the heaviest Majorana neutrino N 3 . That is, we can simply invoke the minimal type-I seesaw model [50,51] with two nearly degenerate heavy Majorana neutrinos to realize resonant leptogenesis. In this case the Yukawa coupling matrix is a 3 × 2 matrix, and thus the arbitrary orthogonal matrix O in the CI parametrization of Y ν is also a 3 × 2 matrix. To remove the uncertainties associated with O, we may take corresponding to the normal (m 1 = 0) or inverted (m 3 = 0) neutrino mass ordering. Then the expression of Y ν in Eq. (6) can be simplified to with m 1 = 0, m 2 = ∆m 2 21 and m 3 = ∆m 2 31 ; or with m 3 = 0, m 2 = −∆m 2 32 and m 1 = −∆m 2 32 − ∆m 2 21 . In other words, the mass spectrum of three light neutrinos is fully fixed by current neutrino oscillation data in the minimal seesaw model, so the uncertainty associated with the absolute light neutrino mass scale disappears. Another bonus is that one of the Majorana phases of U (i.e., ρ) can always be removed thanks to the vanishing of m 1 or m 3 , and therefore we are left with only two low-energy CP-violating phases (i.e., δ and σ) which affect the flavored CP-violating asymmetries ε iα . In our numerical calculations we simply input the best-fit values of θ 12 , θ 13 , θ 23 , δ, ∆m 2 21 and ∆m 2 31 (or ∆m 2 32 ) as given below Eq. (13). Then the observed value of η can be used to constrain the parameter space of σ and d by allowing M 1 to vary in some specific ranges; or to constrain the parameter space of M 1 and d by allowing σ to vary in (0, 2π]. We find that in this minimal type-I seesaw model our RGE-assisted resonant leptogenesis scenario is viable only for the normal neutrino mass ordering with m 1 = 0 and only in the (µ + τ )-flavored regime. The numerical results are briefly illustrated in Fig. 4.
An immediate comparison between Fig. 3 and Fig. 4, which are both associated with the (µ+τ )-flavored regime for resonant leptogenesis, tells us that the parameter space in the minimal seesaw case is slightly larger. This observation is attributed to the smaller cancellation among the contributions of three flavors, since the efficiency factor for the e flavor [i.e., κ(K e )] is much larger than those for µ and τ flavors [i.e., κ(K µ ) and κ(K τ )] in the minimal seesaw scenario. Note that if κ(K e ) = κ(K µ ) = κ(K τ ) held, η would vanish due to ε ie + ε iµ + ε iτ = 0. As shown in Fig. 4, σ is mainly located in two disconnected intervals [0, 3π/10] and [π, 13π/10]. But these two intervals are different from each other only by a shift (σ → σ + π) or a reflection (about σ = 13π/20); and each of them has a symmetry axis (σ = 3π/20 or σ = 23π/20). We see that M 1 6.3 × 10 5 GeV holds, and d is allowed to vary in a wide range between 10 −11 and 10 −7 . When the value of M 1 deceases, the lower and upper bounds of d are both reduced; meanwhile, the allowed range of σ becomes smaller. That is why when M 1 is smaller than 10 7 GeV and σ is switched off (i.e., δ is the only source of CP violation), it will be very difficult (and even impossible) to make our RGE-assisted resonant leptogenesis scenario viable.
One may certainly extend the above ideas and discussions from the SM to the MSSM, in which the magnitude of ∆ τ is expected to be enhanced by taking a large value of tan β. In this case it should be easier to obtain more appreciable CP-violating asymmetries ε iα , simply because they are proportional to ∆ τ . So a successful RGE-assisted resonant leptogenesis can similarly be achieved in the MSSM case. In this connection the main concern is how to avoid the gravitino-overproduction problem [59][60][61][62][63], and a simple way out might just be to require M 1 10 9 GeV and focus on thermal leptogenesis in the (µ + τ )-flavored regime.

Summary
Based on the type-I seesaw mechanism, we have reconstructed the Yukawa coupling matrix Y ν in terms of the light Majorana neutrino masses m i , the heavy Majorana neutrino masses M i and the PMNS matrix U by assuming the arbitrary orthogonal matrix O in the CI parametrization of Y ν to be the identity matrix. To bridge the gap between m i and U at the seesaw scale Λ SS and their counterparts at the Fermi scale Λ EW , we have taken into account the RGE-induced correction to the light Majorana neutrino mass matrix. This RGE-modified seesaw formula allows us to establish a direct link between low-energy CP violation and flavored resonant leptogenesis with M 1 M 2 M 3 , so as to successfully interpret the observed baryon-antibaryon asymmetry of the Universe. We have shown that our idea does work in either the τ -flavored regime with equilibrium temperature T M 1 ∈ (10 9 , 10 12 ] GeV or the (µ + τ )-flavored regime with T M 1 ∈ (10 5 , 10 9 ] GeV, provided the mass spectrum of three light Majorana neutrinos is normal rather than inverted. We have also shown that the same idea is viable for a minimal type-I seesaw model with two nearly degenerate heavy Majorana neutrinos.