Leptogenesis in the $\mu\tau$ basis

We formulate three-flavor type-I leptogenesis in the $\mu\tau$ basis which is convenient because in the three-flavor regime, both $\mu$ and $\tau$ charged lepton Yukawa interactions are in thermal equilibrium and the thermal bath is symmetric under the exchange $\mu \leftrightarrow \tau$. We apply this formalism to models with $\mu\tau$-reflection $\mathsf{CP}^{\mu\tau}$ symmetry. We confirm the previous result that leptogenesis fails in the three-flavor regime with exact $\mathsf{CP}^{\mu\tau}$ symmetry. Allowing $\mathsf{CP}^{\mu\tau}$ symmetry to be broken to various degrees, we show that leptogenesis can be successful in the three-flavor regime only in certain tuned parameter space, which could further imply additional symmetry is at play. As a bonus, we derive analytical expressions which could be utilized whenever the branching ratios for the decays to $\mu$ and $\tau$ flavors are equal or approximately so.


I. INTRODUCTION
The leptogenesis mechanism for explaining the observed asymmetry between matter and antimatter in the universe is a beautiful and economical byproduct of the seesaw explanation for the smallness of neutrino masses [1]. The heavy degrees of freedom that suppress neutrino masses are also the ones that decay violating CP and induce the necessary lepton number asymmetry that are converted to the observed baryon asymmetry.
The typical mass scale for these heavy degrees of freedom is M ∼ 10 14 GeV if they contribute at tree level to the light neutrino masses and if their couplings to the SM leptons are order one. For the simplest type I seesaw, these heavy degrees of freedom are SM singlets, commonly called heavy right-handed neutrinos N i . For this simple case and for the high scale of 10 14 GeV or above, leptogenesis operates in a regime where the flavor content of the generated lepton asymmetry is not distinguishable, a regime known as the one-flavor regime.
Of course, without new states at intermediate scales, such a heavy right-handed neutrinos are not observable in terrestrial experiments and are very difficult to probe by other means. Lowering the scale of these right-handed neutrinos would be desirable to increase observability. However, the Davidson-Ibarra bound [2] constrains the maximum amount of total CP asymmetry that can be generated and consequently sets a lower limit of around 10 9 GeV for the leptogenesis scale induced by the total CP asymmetry [3]. The bound is not applicable if flavor effects are at play: the Yukawa interactions of the SM may be fast enough to distinguish some charged lepton flavors so that the asymmetry accummulated in some flavors may follow a different dynamics [4][5][6]. In particular, it becomes possible to go below 10 9 GeV where leptogenesis operates in the three flavor regime where all e, µ, τ flavors can be distinguished in the plasma [7,8]. Without resorting to resonant enhancement of CP violation [9,10] but only with flavor effects, some form of fine-tuning is still unavoidable if one were to go much below 10 9 GeV [11].
Although successful in explaining the lightness of neutrino masses, the seesaw mechanism by itself cannot explain the pattern of large mixing angles and the relative scale of neutrino masses that was uncovered in the last decades [12]. One approach to increase predictive power is to assume a flavor symmetry acting on the horizontal space of the three families of lepton fields; see Refs. [13] for a review. Unfortunately, if all the mixing angles are fixed by symmetry, one can show that there is no CP violation at low energy [14]. However, neutrino oscillation experiments are almost excluding CP conserving values for the Dirac CP phase [15,16], one of the remaining low energy parameters yet to be measured in the lepton sector.
One of the simplest symmetries capable of predicting all the CP phases at low energy, yet allowing for CP violation, is a symmetry called µτ -reflecion or CP µτ in which ν µ and ν τ interchange is combined with CP symmetry [17,18]; see also [19,20]. This symmetry predicts a maximal Dirac CP phase δ = ±90 • and trivial Majorana phases together with maximal atmospheric angle. The maximal values for the atmospheric angle and maximal δ CP = −π/2 agrees with current global fits [15] and it is strenghened by the recent T2K result [16].
Leptogenesis in the presence of the CP µτ symmetry has been studied in the past and it was shown that leptogenesis was not successful in the one [18] and three flavor [19] regimes. The first failure is due to the vanishing of the total CP asymmetry while the second one follows because the flavor projectors are µτ symmetric and the washout rates in the µ and τ flavors are the same. In the two flavor regime, the necessary lepton asymmetry can be easily generated [19], even in a highly predictive scenario where a texture zero is present in the heavy right-handed neutrino mass matrix [21].
Here we will analyze leptogenesis with CP µτ symmetry in various temperature regimes. Firstly, in Sec. II, we formulate Boltzmann equations for three-flavor leptogenesis in a basis we call the µτ basis where we explore the symmetry of the leptogenesis dynamics under µτ relabeling. If, in addition, the underlying theory is invariant by some form of µτ symmetry such as CP µτ , the Boltzmann equations decouple into two distinct pieces which evolves independently. In this case, we confirm in Sec. III the known result of the failure of three-flavor leptogenesis in the presence of CP µτ considering the flavor effects in full generality. The question that follows is then the amount of symmetry breaking necessary for successful three-flavor leptogenesis. The amount of symmetry breaking may not necessarily be small because CP µτ may be generalized to include nontrivial Majorana phases [22,23]. One example can be seen in Ref. [24] which is a more specific version of the Littlest seesaw model [25].
The rest of this paper is organized as follows. Sec. IV analyzes the necessary amount of CP µτ breaking that is necessary for successful leptogenesis. We conclude that three-flavor leptogenesis is barely possible with small breaking. So Sec. V considers large CP µτ breaking and demonstrates that some amount of fine-tuning is unavoidable. Sec. VI shows some examples of models where large CP µτ breaking may occur only in the CP asymmetries but not on the flavor projectors. The conclusions can be seen in Sec. VII and the appendices contain auxiliary material.

II. THREE-FLAVOR LEPTOGENESIS IN THE µτ BASIS
The type-I seesaw Lagrangian in the basis where charged lepton Yukawa y α , α = e, µ, τ , and Majorana mass M i , i = 1, 2, 3, for the right-handed neutrinos N i are real and diagonal is given by where α and H are respectively the SM lepton and Higgs doublets withH = iσ 2 H * , and e α are the right-handed charged leptons.
In the three-flavor regime, T 10 9 GeV, both µ and τ charged lepton Yukawa interactions are in thermal equilibrium, and all three lepton flavors can be distinguished in the plasma. Since µ and τ leptons are both massless and carry the same SM quantum numbers, the cosmic thermal bath is symmetric under the exchange µ ↔ τ , i.e., the "dynamics" with such a relabelling is the same. In this case, it is convenient to describe leptogenesis in the µτ basis that we will develop here.
In this regime, the Boltzmann equations (BEs) for ∆ α ≡ B 3 − L α charge produced from the decay and inverse decays of right-handed neutrinos N i ↔ α H, N i ↔¯ α H * are given by where z ≡ M 1 T with M i the mass of N i , Y eq = 15 8π 2 g with g the total relativistic degrees of freedom (g = 106.75 for the SM), Y a ≡ na s with n a the number density of particle a and s = 2π 2 45 g T 3 the cosmic entropic density, and Y eq N i = 45 2π 4 g a 2 i z 2 K 2 (a i z) with a i ≡ M i /M 1 and K n the modified Bessel function of the second kind of order n. The flavored CP parameters are defined as [26] iα ≡ is the N i tree-level total decay width and the one-loop function is given by The decay reaction is described by where H = 1.66 √ g T 2 M Pl is the Hubble rate with M Pl = 1.22 × 10 19 GeV. Finally, the flavor projectors are with α P iα = 1.
Since the system is symmetric under the exchange µ ↔ τ , the generic forms of the flavor They satisfy where Defining the flavor matrix we can rewrite eq. (2) in vector notation as where Next, let us define the projectors into µτ even and odd subspace with X 2 ± = X ± and X + X − = 0. We can recast eq. (13) in terms of µτ even and odd components Note that the µτ even component is two We will further rewrite P i as where Using the fact that X commutes with Σ i and anticommutes with δ i , i.e., [X, Σ i ] = 0 and {X, δ i } = 0, we have The above identities together with [X ± , F ] = 0, which follows from eq. (10), simplify eqs. (15) and (16) to Finally, the total B − L asymmetry is given by where p refers to the vector component of Y ± and we have used p (Y − ) p = 0. Although the µτ odd component Y − does not contribute directly to Y ∆ , it does contribute to Y + through the righthanded side of eq. (22). However, if P iµ = P iτ , as when µτ interchange or µτ -reflection symmetry is valid in the neutrino sector, it follows that δ i = 0 and the BEs for Y + and Y − in eqs. (22) and (23) decouple. In this case, we only have to solve for Y + in eq. (15) which is two-dimensional, i.e., essentially a two-flavor scenario. Analytical approximate solutions to eqs. (22) and (23) are derived in Appendix B for the case of δ i = 0 and |δ i | 1.
Eventually, as the ElectroWeak (EW) spharelons processes go out of equilibrium at T ∼ 130 GeV [27], the final baryon asymmetry is frozen to be [27][28][29] where we have assumed that the EW symmetry is already broken as suggested in [27] and do not include the contribution of top quark. In this work, we fix Y B | exp = 8.7 × 10 −11 as indicated by the Planck measurement [30].

III. EXACT CP µτ LIMIT
It was shown in Ref. [19] that leptogenesis with µτ -reflection or CP µτ can be successful only in the two-flavor regime where 10 9 GeV M 1 ∼ T 10 12 GeV. Here we review and confirm this result from a more precise calculation that includes off-diagonal flavor effects [4][5][6] which, in principle, potentially could (but do not) source flavor dependent lepton asymmetries in the threeflavor regime. These effects were not considered in Ref. [19]. Later, off-diagonal flavor effects were briefly considered in Ref. [31] but only a numerical example was given to illustrate the general case. Here, we treat these flavor effects in full analytical generality and, additionally, identify the features that preclude successful leptogenesis.
In the limit where µτ -reflection symmetry or CP µτ is exact in the neutrino sector, the flavored CP parameters in eq. (4) satisfy [19] ie = 0 and iµ = − iτ .
The symmetry also relates the N i Yukawa couplings as which implies If leptogenesis happens above 10 12 GeV where lepton flavor effect is not at play, it is clear that leptogenesis fails because [18,19] Interestingly, while the total CP parameter i is zero, the individual flavored CP parameters iα could be much larger than the Davidson-Ibarra bound [2] on i for hierarchical N i . This shows that one might be able to realize purely flavored leptogenesis [4,32] below the one-flavor regime T 10 12 GeV where interactions mediated by τ charged lepton Yukawa get into thermal equilibrium. 2 Indeed, this was shown to be the case in Ref. [19] in the two-flavor regime 10 9 GeV T 10 12 GeV. Focusing on diagonal flavor effects, Ref. [19] also demonstrated that in the three-flavor regime T 10 9 GeV, there is an exact cancellation resulting in vanishing final baryon asymmetry.
In the three-flavor regime, we can analyze the consequences of CP µτ on leptogenesis using the BEs in the µτ basis. Here, we keep the index i to demonstrate that our analysis holds for any N i in the three-flavor regime. Due to eq. (26), we can see that i is odd under µτ interchange and as a result As CP µτ symmetry further constrains P iµ = P iτ from eq. (28), the BEs for Y + and Y − in eqs. (15) and (16) decouple and we only need to solve for Y + . The solution for Y + is: where z 0 = M i /T 0 with T 0 the initial temperature. In the absence of preexisting asymmetry, Y + (z 0 ) = 0, and the final B − L asymmetry remains zero. The "preexisting" asymmetry can also come from two-flavor leptogenesis due to decays of heavier N i . In this case, one needs to make sure that the asymmetry survives the three-flavor washout from the lightest N 1 . For large K 1 and generic P 1τ , one needs large preexisting asymmetry to survive washout. For special parameters, such as for P 1τ ≈ 0 or P 1τ ≈ 0.5, N 1 washout can be very weak, even for large K 1 . In the absence of special conditions which allow the survival of a preexisting asymmetry, we have proven that three-flavor leptogenesis fails in the limit of exact CP µτ . In the context of CP µτ , a detailed study of the contribution from N 2 leptogenesis [33] is considered in Ref. [35] and will not be considered further in this work.
In between 10 9 GeV T 10 12 GeV where only the τ -lepton flavor can be distinguished, leptogenesis can proceed in the two-flavor regime that distinguishes τ from e + µ. In this case, the BEs are the same as (2) but now the flavor indices run through α = e + µ, τ , with So the flavor coefficients A, C have sizes 2 × 2 and 1 × 2 respectively. We can put the equations in matricial form as in (13) with the respective modifications that include Y ∆ = (Y ∆ e+µ , Y ∆τ ) and With CP µτ , we can still define an effective µτ interchange: Under this interchange, we still have odd CP parameter, but A, C are not symmetric by interchange and P e+µ = P τ as well. So the F matrix is also not µτ symmetric. Therefore, the BEs for the µτ even and odd components no longer decouple and an asymmetry in the total lepton number direction generically survives and may reach the necessary value [19,21].
Having confirmed that leptogenesis with CP µτ symmetry cannot proceed successfully in the oneor three-flavor regimes, we will analyze here how much breaking of CP µτ is necessary to account for successful leptogenesis in these regimes. We initially focus on small breakings of CP µτ such as induced by RGE running [20,34]. But we should keep in mind that models with large breakings solely in the Majorana phases can be constructed naturally [22][23][24]. In Ref. [22], CP µτ symmetry was denoted as µτ -R and the generalized version when Majorana phases were generic was called µτ -U. In other words, µτ -U ensures θ 23 = 45 • and δ = ±90 • but free Majorana phases. We will use the nomenclature µτ -R and µτ -U when the need to distinguish arises.
For simplicity, we consider the scenario where leptogenesis proceeds only through decays of N 1 and drop the corresponding subscript "1" unless it is required for clarity. This is realized for instance when the reheating temperature T R falls in the range M 1 < T R M 2 < M 3 . If M 1 and M 2 are of similar order but not quasi-degenerate (no resonant enhancement), both will contribute constructively, resulting in an enhancement of about a factor of two in the weak washout regime. In the strong washout regime, there is no enhancement since additional contribution in the source term is compensated by the additional washout.

A. One-flavor regime
Here we consider the scenario where 10 12 GeV M 1 < T R such that N 1 leptogenesis occurs in the one-flavor regime. We assume leptogenesis can be sourced by a small breaking of CP µτ leading to where Such a small breaking can be induced for example by RGE effects [20,34].
For hierarchical N i that we will consider here, = δ has to obey the Davidson-Ibarra bound [2]: where v = 174 GeV is the Higgs vacuum expectation value (vev), m l (m h ) is the lightest (heaviest) light neutrino masses and ∆m 2 hl ≡ m 2 h − m 2 l . In the following, we will fix ∆m 2 hl to the atmospheric mass squared splitting |∆m 2 atm | = 2.5 × 10 −3 eV 2 . The upper limit is the largest in the limit m l = 0 for which we have | | ≤ 9.9 × 10 −5 (M 1 /10 12 GeV). Having m l as large as 0.1 eV, we have instead | | ≤ 2.3 × 10 −5 (M 1 /10 12 GeV).
It is possible to divide the one-flavor regime of temperatures T 10 12 GeV, roughly into two ranges. The BEs within these ranges are different but the quantitative consequences are minor. If leptogenesis happens at T 2 × 10 12 GeV where the EW sphaleron interactions are out of equilibrium [36,37], we have where L is the total lepton number. If leptogenesis happens in this regime, lepton number is projected in ∆ ≡ B − L = α ∆ α once EW sphaleron interactions get into equilibrium at T 2 × 10 12 GeV and we have Y ∆ = −Y L .
If leptogensis happens at 4×10 11 GeV T 2×10 12 GeV where the EW sphaleron interactions are already in equilibrium [36,37], we should track the total abundance in Y ∆ by Notice that eq. (38) and eq. (39) differ by a small numerical factor in the washout term (second terms on the right-hand side). In either cases, the final baryon asymmetry Y B is related to Y ∆ through eq. (25).
Using the analytical approximate solution for eq. (38) or (39) shown in Appendix C, we obtain the necessary |δ | to achieve the experimental baryon asymmetry shown in Fig. 1 (black lines) as a function of the sole relevant variable with m * ≈ 1 meV andm In terms of K, eq. (6) can be written as with Y eq N i (0) = 45 π 4 g . The two solid (dashed) black lines are respectively the solutions of eq. (38) and eq. (39) assuming zero (thermal) initial N 1 abundance. In the K > 1 regime, the upper [lower] black line corresponds to the solutions of eq. (38) [eq. (39)].
To compare the amount of necessary breaking in δ with the flavored τ , we also show in Fig. 1 the possible values of | τ | in any model of type I seesaw with CP µτ for different cases. These include the different CP parities that are possible since Majorana phases are trivial [19]. In green and red lines we show | τ | for the different cases restricted to hierarchical N i and N 3 decoupled case, with M 1 = 10 12 GeV, in the low end of the one-flavor regime. For this case, we can see that leptogenesis cannot be successful solely with a small breaking of CP µτ . Masses for M 1 of the order 10 14 GeV are required to generate a sufficiently large | τ |, allowing an |δ | of 10% that is still sufficiently large.
Qualitatively, the same conclusion holds even if we allow mild hierarchies for N i and include the effects of N 3 in the loop. This can be seen from the scatter points in cyan and dark blue that are randomly generated for M 1 = 10 12 GeV and M 1 = 10 14 GeV, respectively, allowing the mild hierarchy 9M 1 ≤ 3M 2 ≤ M 3 . Only for the region 1 K 10 and | τ | 10 −5 , which contain very few cyan points, can leptogenesis be successful. We use the Casas-Ibarra parametrization [38] satisfying CP µτ and exclude large Yukawa couplings λ by requiring that |λ iα | ≤ 1 so that scatterings which violate lepton number by 2 units can be neglected during leptogenesis. 3 We use the and M 1 = 10 14 GeV (dark blue). 3σ ranges from Ref. [39] were used for the neutrino mixing parameters θ 12 , θ 13 not fixed by CP µτ and neutrino mass squared differences. We include the one-loop corrections as discussed in the Sec. V. restriction |ξ i | < 3 for parameters appearing in the Casas-Ibarra parametrization. The details of the parametrization are explained in Appendix D.

B. Three-flavor regime
Here we consider the scenario where M 1 < T R 10 9 GeV such that N 1 leptogenesis occurs in the three-flavor regime. We continue to assume N 1 dominated leptogenesis as discussed in the beginning of Sec. IV.
For the first possibility, = δ e +δ = 2δ should respect the Davidson-Ibarra bound in eq. (37). The largest upper bound is obtained for m l = 0 for which we have | | = |2δ | ≤ 9.9×10 −8 (M 1 /10 9 GeV). As we will see shortly, leptogenesis can barely be successful in this case. The second possibility is a purely flavored scenario where the total CP asymmetry is zero, = 0, which trivially satisfies the Davidson-Ibarra bound (37). In this case, there is in principle no bound on δ .
Applying the solution in eq. (B25), the required |δ | for the aligned and purely flavored case are plotted respectively in Fig. 2 and Fig. 3 on the plane of K − P τ . These are the relevant parameters in eq. (43) because D is proportional to K as in eq. (42) and P e = 1 − 2P τ .
For the aligned case in Fig. 2, we observe the following features. For zero initial N 1 abundance (left plot), in the weak washout regime K < 1, the final B −L asymmetry is suppressed by K 2 while in the strong washout regime K > 1, it is suppressed by 1/K. For thermal initial N 1 abundance (right plot), in the weak washout regime K < 1, the final B − L asymmetry saturate to maximal value while in the strong washout regime K > 1, it is also suppressed by 1/K. Since |δ | 10 −7 is required, leptogenesis can barely be successful due to the Davidson-Ibarra bound discussed below eq. (47).
For the purely flavored case in Fig. 3, we observe that the final B − L asymmetry is suppressed in the weak washout regime K < 1 for both zero (left plot) and thermal (right plot) initial N 1 abundance. This is a specific feature of purely flavored leptogenesis since in the absence of washout, leptogenesis fails because the total CP parameter is null. Since |δ | is not subjected to the Davidson-Ibarra bound, this scenario can be successful in a larger parameter space. Next, we will analyze the scenario where CP µτ is broken only in the flavor projectors P µ = P τ (1 − χ) with 0 < χ < 1 while we keep e = 0 and µ + τ = 0. 4 In this case, the small breaking is quantified by χ 1. From eqs. (22) and (23) with X + = 0, we have where X − is given in eq. (30) and Σ = diag (P e , P τ (1 − χ/2), P τ (1 − χ/2)) , In the above, we have P e = 1 − 2P τ (1 − χ/2). Although Y − does not contribute to the B − L asymmetry, it feeds into the BE for Y + through the second term in eq. (48).
For χ 1, the final asymmetry is proportional to χ τ as shown in eq. (B43). Applying this solution, we plot in Fig. 4 the required χ | τ | for zero (left plot) and thermal (right plot) initial N 1 abundance on the K − P τ plane for the case χ = 0.1. This case differs very slightly from the case with χ < 0.01 in which the contour of χ| τ | = 10 −6.5 slightly shrinks. For comparison, we also show the case with large CP µτ breaking in the projectors with χ = 0.8 in Fig. 5. In this case, eq. (B43) which holds for χ 1, is not a good approximation and we resort to solving eqs. (48) and (49) directly. Notice that leptogenesis is always inefficient in the weak washout regime K < 1, i.e., it requires large CP parameters. This is actually a feature of purely flavored leptogenesis as we discussed earlier. Here we can also understand this feature from eq. (48) in which the source term (the second term) is proportional to 1 2Y eq DδF Y − and reduces as K decreases. Finally, we can also have CP µτ breaking in both the CP parameters e = 0, µ + τ = 0 and the flavor projectors P µ = P τ . Nevertheless, the maximum amount of breaking required will be quantitatively similar and not worth showing. The reader can superimpose Fig. 2 or 3 with Fig. 4 or 5 to get a sense.

V. LARGE CP µτ BREAKING
Here we quantify large CP µτ breaking in the CP parameters by For small breaking as in eq. (36), we have seen in Sec. IV A that, for M 1 ≈ 10 12 GeV in the lower end of the one-flavor regime, leptogenesis was barely possible only in the uncommon region | τ | 10 −5 and 1 K 10; cf. Fig. 1. If we allow a large breaking in the form of large δ in eq. (35), leptogenesis is possible in the one-flavor regime for typical parameters as it approaches the generic case.
In an analogous manner, in the three-flavor regime analyzed in Sec. IV B, a large breaking in the CP asymmetries of at least |δ e | ∼ |δ | ∼ 10 −6 (cf. Figs. 2 and 3) in eq. (44) or for a small breaking in the flavor projectors, (P τ − P µ )/2 ∼ 0.1, a CP asymmetry of at least | τ | ∼ 10 −5 was required. These contrast with the typical value of | τ | ∼ 10 −8 shown in the scatter plots of Fig. 6 that show the possible ranges for the CP µτ symmetric model. 5 The dark (light) blue points assume |ξ i | ≤ 3 (|ξ i | ≤ 6) for parameters ξ i that appears in the orthogonal matrix R of the Casas-Ibarra parametrization; see explicit parametrization in Appendix D. The red points are a small subset of the light blue points with relatively large | τ | and moderate K, as can be seen in Fig. 6 (left). In contrast, Fig. 6 (right) shows | τ | against the ratio When this ratio is small, the second and third rows of R are very similar in magnitude and this feature may be regarded as indication of fine-tuning or additional (approximate) symmetry. Also, when this ratio is small, the first row of R is small in magnitude and this explains how the entries of R may be large but K is kept moderate . We can see that the red points are concentrated where the ratio is small which shows that fine-tuning is necessary to generate large τ but not too large K. We can then conclude that some fine-tuning (or additional symmetry) is required to achieve successful leptogenesis with small CP µτ breaking.
We can compare the tuning quantified in eq. (53) with other measures. Considering the type I seesaw formula in the basis ν L ν L , where m D = vλ andM R is the right-handed neutrino mass matrix which is real, positive and diagonal. The seesaw naturally explains the light neutrino mass of order 0.1 eV for λ ∼ 1 ifM R ∼ 10 14 GeV. Lighter right-handed neutrinos would require smaller λ unless some cancellation [40] between m D andM −1 R takes place in the seesaw formula in eq. (54). The degree of cancellation can be roughly quantified by where the double bars denote the matrix norm The larger the measure of eq. (55) the larger the degree of cancellation. A similar measure is given by some norm of the matrix R in the Casas-Ibarra parametrization [41].
We will also see that to get large | τ |, it is essential that we also consider the one-loop contribution to the neutrino mass matrix coming from Higgs and Z in the loop: where C eff (M R ) is a function on the diagonal entries for which [42] m H and m Z are the masses of the Higgs and the Z. The total neutrino mass matrix is then modified to KeepingM ν as the physical neutrino masses, one can include the one-loop contribution in the Casas-Ibarra parametrization by modifying the Yukawa coupling λ to an effective λ eff [43]. See Appendix D for the expression. To quantify the relative contribution from the one-loop contribution, we further use [11] T where SVD denotes the singular values of the matrix. The larger this measure, the larger is the one-loop contribution, and if it is much larger than unity, it means that the tree and one-loop contributions are large but they largely cancel each other in (59).
We illustrate the different fine-tunings needed in Fig. 7. The left plot shows how | τ | varies with the seesaw cancellation measure (55), using the effective λ eff that already takes the one-loop contribution into account. We can see that large | τ | 10 −6 (large |λ|), which includes our red points, is only possible if some cancellation takes place in the seesaw. Since a delicate cancellation between λ eff and M −1 R is necessary for large | τ |, the same cancellation is unlikely to happen between λ and M −1 R just for the tree-level contribution. This means that large cancellations between tree and one-loop contributions take place. We show this in Fig. 7 (right) where we plot | τ | against the measure of eq. (60). We also see that | τ | 10 −6 is only possible for large T loop . We note that for leptogenesis with (approximate) CP µτ a small value for the measure T R 2−3 (53) is a better indicator for successful leptogenesis than T SS or T loop because it allows large | τ | with moderate K. Instead, large T SS or T loop tend to lead to large K as well.
To conclude this section, successful leptogenesis in the three-flavor regime with approximate CP µτ symmetry is only barely possible in the upper end of mass value M 1 ∼ 10 9 GeV if we allow a large degree of cancellation in the neutrino mass matrix in the seesaw and consequently large cancellation between tree and one-loop contributions. The necessary fine-tuning 6 can be alleviated by allowing large breaking of CP µτ in the relevant parameters of leptogenesis, i.e., large difference between P µ and P τ or/and large deviation from the µτ odd property in eq. (26) for the CP asymmetries.

VI. EXAMPLES OF LARGE CP µτ BREAKING ONLY ON α
Here we show some examples where large deviations from the CP µτ symmetric case are possible in the CP asymmetries α but not on the flavor projectors P α . These models realize the scenario described in (43) where the flavor projectors still respect (28) but the CP asymmetries will not be µτ odd as in (26) but exhibit a large µτ even part (44). These examples are based on the generalization of CP µτ called µτ -U symmetry in Ref. [22] where PMNS matrix exhibit but Majorana phases are non-trivial. Another characterization is that the PMNS matrix obeys |(U ν ) iµ | = |(U ν ) iτ |, i = 1, 2, 3. These features may be achieved accidentally by symmetry [23]. But here we need full models with the type I seesaw embedded so that consequences on leptogenesis can be analyzed.
The first example is the Littlest mu-tau seesaw model (LSSµτ ) of Ref. [24]. The model is supersymmetric with only two right-handed neutrinos and requires a lot of auxiliary fields but at the leptogenesis scale it can be simply described by the simple Dirac mass matrix and the heavy singlet Majorana mass matrix where ω = e i2π/3 . The low energy neutrino mass matrix is with m s = −a 2 /(22M 0 ω). This model, with only one parameter at low energy, accommodates the low energy neutrino parameters within 3σ when RGE corrections are taken into account [24]. Without corrections, we indeed have θ 23 = 45 • and δ = −π/2. There is also one nontrivial Majorana phase K ν = diag(1, 1, e −iβ/2 ) with β ≈ −1.8.
After rephasing of the singlet neutrino fields, the Dirac mass matrix in the mass basis of N R is The phase ω will induce the necessary CP violation. It is clear from the first row that P e = 0 and P µ = P τ = 1/2, which respect the CP µτ property (for N 1 but not for N 2 ). The CP parameters defined in eq. (4) will, however, deviate from the CP µτ symmetric case: where g(x) is given in eq. (5). Using M 0 = 10 9 GeV and |m s | = 2.66 meV, we have with = α α = −4.36 × 10 −8 and K = 54.83. Although these CP parameters are too small for the three-flavor regime, it certainly may generate the right baryon asymmetry in the one-or two-flavor regimes.
The second example was proposed in Ref. [22] and is also based on constrained sequential dominance (see Ref. [44] for a review), a supersymmetric model where the Dirac mass matrix has special forms due to vev alignments from nonabelian flavor symmetry. We assume the charged lepton mass matrix squared M † l M l is invariant by a generator of groups such as A 4 or S 4 , so that in the flavor basis the charge leptons contribute the mixing matrix U † l having the form There are three right-handed neutrinos N iR with diagonal mass matrix where we allow for phases and the masses are not necessarily ordered from lighter to heavier. Special flavon vevs give the Dirac mass matrix the special structurē Note that u 1 ⊥ u 2 , u 3 and such a structure is easily obtained from vev alignments in indirect models [45] from, e.g., the A 4 symmetry in the CSD framework [46] in the real triplet basis of A 4 . The parameters a, b, c are chosen to be real by appropriate rephasing. In the mass basis for the charged leptons and right-handed neutrinos, this matrix becomes One can easily check that P iµ = P iτ . Moreover, m D obeys which generalizes the CP µτ symmetric case [19].
In the symmetry basis of (68), the neutrino mass matrix is given by where If η 2 −η 1 = 0, π, one can check that the real and imaginary parts of M ν commute so that it is diagonalized by a real orthogonal matrix, leading to a PMNS matrix obeying µτ -U [22]. Additionally, if we obtain NO spectrum with observables within 3σ and lightest mass m 1 ≈ 12 meV. This example was shown in Ref. [22]. The phase η can take any value and a nonzero value leads to nontrivial Majorana phases without disrupting the predictions of µτ -U mixing. In addition, the mixing obeys the TM1 form [47].
However, the additional Majorana phase does not induce a deviation of α from the CP µτ symmetric form in eq. (30). The reason is the following: because u 1 is orthogonal to u 2 , u 3 , the loop contribution from N 2 , N 3 to the CP asymmetries of N 1 vanish. 7 So N 2 or N 3 needs to be the lightest one and only the interference between N 2 and N 3 leads to nonzero CP asymmetries. Due to the property (73), one can show that µ = − τ from an analogous proof shown in Ref. [19].
To induce a deviation from the µτ odd form in (30), we need η 2 − η 1 = 0, π, which also leads to deviation from µτ -U in (61) but P µ = P τ is still valid. For example, we can take (M 2 , M 1 , M 3 ) = 10 9 (1, 10, 100) GeV and to obtain neutrino oscillation observables within 3σ and resulting in α ≈ 1.35 × 10 −8 withm α ≈ (12.24, 49, 49) meV. Although not sufficient for successful three-flavor leptogenesis, this is another example of large CP µτ breaking on α maintaining P µ = P τ . In order to maximize the CP parameters in eq. (77), one can show that |m c | ≤ |∆m 2 atm |/6 and hence we have setting |∆m 2 atm | = 2.5 × 10 −3 eV 2 . These CP parameters are clearly too small for the three-flavor regime while it could work naturally for one-or two-flavor regimes.

VII. CONCLUSIONS
We develop the BEs for leptogenesis in the three-flavor regime in the µτ basis where the description of approximate µτ reflection (CP µτ ) symmetric case is particularly simple. Effectively for leptogenesis, the symmetric case implies equality between the flavor projectors into µ and τ flavors and odd behavior of the CP asymmetries under µτ exchange: iτ = − iµ and ie = 0. The formalism is useful for all cases where these properties are (approximately) valid and makes use of the fact that the interactions relevant for leptogenesis in the primordial plasma are symmetric by exchange of µ ↔ τ . With the formalism, barring preexisting asymmetry, we confirm previous results that leptogenesis cannot be successful in the three-flavor regime in the exact symmetry limit, even if flavor effects are fully taken into account.
With small breaking of CP µτ , N 1 leptogenesis can be barely successful in the three-flavor regime only if we allow large fine-tunings that we quantify using some measures. The fine-tuning is necessary to push the CP asymmetry | τ | to higher values keeping the washout parameter K moderate. These values can be achieved in our case only if the second and third rows (associated to N 2 , N 3 ) of the orthogonal matrix of the Casas-Ibarra parametrization are very similar in magnitude.
Additionally, we confirm that this requires large cancellations in the seesaw formula, calculated using one-loop corrections, between the Dirac mass term and the heavy right-handed neutrino mass matrix. Barring some protective symmetry, large cancellations between tree and one-loop contribution to the neutrino mass matrix are generically required as well.
With large breaking of CP µτ , the parameter space for leptogenesis in the three-flavor regime widens and less fine-tuning is necessary. We end by discussing some examples in which the symmetry is broken only in the CP asymmetries but not in the flavor projectors. These examples can lead to successful leptogenesis in the one-or two-flavor regime but the CP asymmetries turn out to be insufficient for the three-flavor regime.
In the temperature regime 10 2 GeV T 10 4 GeV where all Yukawa interactions are in thermal equilibrium, we have In the µτ basis, the general Boltzmann equations for leptogenesis in 3-flavor regime are given by eqs. (22) and (23) in which we rewrite them here for convenience We can diagonalize the matrix Σ i F as follows where the eigenvalues are 8 with The matrix V i is given by where where with Assuming that leptogenesis is dominated by decays of particular generation of N i , it is useful to transform Y ∆ ± to the following basis 9 where we have defined Notice that the Y ∆α asymmetry can be recovered from the µτ even components in the new basis 9 The inverse of Vi is given by as follows In this new basis, we have where we have defined In the case where P iµ = P iτ =⇒ a i± , b i± = 0, we only need to solve The analytical approximate solution in eq. (C7) can be applied directly and we have If CP µτ is not broken in the CP parameters, i± = 0 and the final asymmetry is vanishing in the absence of preexisting asymmetry Y i± (0) = 0 in accordance to the result in Sec. III.
1. Perturbative diagonalization for |δP iµτ | 1 If |δP iµτ | 1, we can carry out the perturbative diagonalization and write down the solution at leading order in δP iµτ . In the basis While δR i is proportional to δP iµτ , R (0) i is already diagonal and therefore the perturbed matrix of eigenvectors for R will be I + δU . Denoting the perturbed eigenvalues matrix as δr i = diag (δr i+ , δr i− , δr io ), we have Keeping only the leading terms, we obtain Writing R Explicitly, we have (B33) Using the results above, the BEs with perturbative diagonalization up to order δP iµτ are given by The analytical approximate solution in eq. (C7) can be applied directly and we obtaiñ Let us consider the case of vanishing initial asymmetriesỸ i± (0) =Ỹ o (0) = 0. Transforming back to the basis of Y i± , we have If CP µτ is not broken in the CP parameters, i± = 0 and io = iτ , and the final asymmetry becomes which is proportional to δP iµτ iτ . We review here the Casas-Ibarra parametrization in the presence of CP µτ symmetry. It was used to produce the scatter plots in Figs. 1 and 6. Part of the formulas below were given in Ref. [19] with a slightly different notation.
Considering the seesaw formula in (54), the Casa-Ibarra parametrization can be written as where R is a complex orthogonal matrix that does not depend on low energy parameters. The hatted matrices are the diagonalized matrices and U ν is the PMNS matrix V .
The one-loop correction (57) to the neutrino mass matrix can be considered in the Casas-Ibarra parametrization by considering [43] m eff D = vλ eff = iM instead of (D1).
For example, the first option for K 2 ν means that K 2 ν = 1 3 . We use the convention that K R and K ν only contain 1 or i in the diagonal.