Addendum to “Leptogenesis in an SU ( 5 ) × A 5 golden ratio ﬂavour model” [Nucl. Phys. B 896 (2015) 311–329]

We derive and discuss the solution of the Boltzmann equations for leptogenesis in a phenomenologically viable SU ( 5 ) × A 5 golden ratio ﬂavour model proposed in [1,2]. The model employs, in particular, the seesaw mechanism of neutrino mass generation. We ﬁnd that the results on the baryon asymmetry of the Universe, obtained earlier in [2] using approximate analytic expressions for the relevant CP violating asymmetry and efﬁciency factors, are correct, as was expected, up to 20–30%. The phenomenological predictions for the low energy neutrino observables, derived using values of the parameters of the model for which we reproduce the observed value of the baryon asymmetry, change little with respect to those presented in [2]. Among the many predictions of the model we ﬁnd, for instance, that the neutrinoless double beta decay effective Majorana mass m ee lies between 3.3 meV and 14.3 meV.


Introduction
The origin of flavour is one of the most challenging unresolved fundamental problems in particle physics. The questions of why there are three generations (not more and not less), of the origin of the hierarchies of the fermion masses and of the very different quark and neutrino mixing patterns are still far from having received a satisfactory explanation.
In recent years an approach to the problem of flavour based on discrete flavour symmetries became widely used especially in treating the flavour problem in the lepton sector, for a recent review see, e.g., [3]. A large number of models employing discrete flavour symmetries have been proposed. However, many of these models focus only on leptons and only reproduce the observed neutrino mixing angles with possibly a few additional predictions for the leptonic CP violation phases and/or the absolute neutrino mass scale.
Here we will focus on a particular model [1] which reproduces all flavour information in the quark sector and in addition to reproducing the mixing angles in the neutrino sector, provides predictions for the absolute neutrino mass scale and the leptonic CP violation phases. In [2] we discussed a slight modification of the original model, which allowed us to accommodate successfully the generation of the baryon asymmetry of the Universe within the leptogenesis scenario [4]. In that previous publication [2] we used analytic approximations to calculate the baryon asymmetry, which can be expected to be correct only up to 20-30%. In the present article we go beyond these approximations and calculate the baryon asymmetry by solving the relevant system of Boltzmann equations numerically. We show that using this more precise method of calculation of the baryon asymmetry one can still generate successfully the observed value of it in the model considered. We discuss also the impact of the new results on the baryon asymmetry on the predictions of the low energy observables of the model -on the correlation between the angles θ 13 and θ 23 , on the values of the leptonic CP violation phases, on the value of the effective Majorana mass in neutrinoless double beta decay, m ee , etc.
The article is organised as follows. In Section 2 we review the model constructed in [1] and its modification proposed in [2]. In Section 3 we discuss the Boltzmann equations and the solutions for the baryon asymmetry we obtain. We update the results on the neutrino masses and mixing angles previously obtained in [1,2] in Section 4. Section 5 contains summary and conclusions.

The leptonic Yukawa and Majorana mass matrices
In this section we briefly recapitulate the Yukawa couplings and Majorana mass matrices in the lepton sector of the model of interest to fix notations. The structure of these matrices is justified by the flavour symmetries of the model and is discussed extensively in [1,2]. The interested reader is referred to these articles for details.
The right-handed neutrino Majorana mass matrix reads where v 2 and v 3 are complex (vevs) of a flavon breaking the A 5 family symmetry. This matrix is of the golden ratio pattern type A [5], i.e., it is diagonalised by is the golden ratio and P 0 is diagonal matrix containing the two CP violation Majorana phases α 1 and α 2 , P 0 = Diag(exp(− iα 1 2 ), exp(− iα 2 2 ), 1). The phases α 1 /2 and α 2 /2 are related to those in the convention used by the Particle Data Group [6], α 21 /2 and α 31 /2, as follows: The matrix of charged lepton Yukawa couplings has the form: where the a ij are complex parameters which are fixed by the quark sector (since it is a GUT model) and the charged lepton masses [1]. Note that we did not use here standard GUT relations but the relations proposed in [7] which are in good agreement with the current data on fermion masses and the Higgs mass results [8][9][10]. Since the a ij depend on tan β and to redo the fit is very time consuming we have fixed this parameter here to 30 which is in good agreement with the aforementioned GUT relations. The matrix of neutrino Yukawa couplings can be written as The matrix appeared in the original model [1], while was introduced in the second study [2]. The phases can be chosen such that y n 1 is real and then it turned out that the baryon asymmetry is generated only by the small correction δY ν (note that c 1). For more details and analytical estimates the reader is referred to [2].
It proves convenient for our further discussion to define the parameters which will play an important role in the analysis we will perform: where One of the Majorana phases which we choose to be φ 1 can be set to zero by applying a redefinition of the heavy Majorana fields. The remaining two phases φ 2 and φ 3 can be expressed in terms of ρ and φ using the complex mass sum rule We note that only normal ordering is viable in the model considered [1,2], and the Yukawa couplings are degenerate in LO, so that we have |M 3 | < |M 2 | < |M 1 |. The Majorana phases α 1 , α 2 are employed in the renormalisation group evolution package REAP [11] we are going to use in our numerical analysis, and the phases φ 2 and φ 3 are related, up to corrections of order c 2 , via

Boltzmann equations
In this section we discuss the Boltzmann equations for this model. Since we set the leptogenesis scale to be M S ≈ 10 13 GeV, we find with tan β = 30, 10 9 (1 + tan 2 β) GeV < M S < 10 12 (1 + tan 2 β) GeV. Values of M S in this interval correspond [12] to the two-flavour leptogenesis regime [13,14]. We will perform the analysis of the baryon asymmetry generation in this regime.
We use the set of Boltzmann equations in supersymmetric leptogenesis [15][16][17] which we briefly summarise below (for notational details and further explanations, see the original papers).
The baryon asymmetry generated in the two-flavour regime in leptogenesis is determined, in particular, by the evolution of the heavy Majorana neutrino and sneutrino number densities (abundances), Y N i and Y N i , and of the lepton charge and CP violating asymmetries in the charges , during the epoch of the evolution of the Universe when the abundances Y N i and Y N i start to deviate from their equilibrium values (the out-of-equilibrium Sakharov condition [18]). The evolution of the quantities of interest in the epoch of interest can be described by a system of coupled Boltzmann equations, which in the case of the MSSM and of the two-flavour regime we consider read [15][16][17] Due to the fact that leading order neutrino Yukawa coupling is unitary (except for an overall factor y n 1 2 ), we have 2 i ≡ e i + μ i = − τ i to leading order. We do not include here the thermal corrections to the CP asymmetries since they give only negligible contributions within the low temperature regime [19,20]. We neglect also the L = 2 process for the same reason as stated in [2], i.e., given the value of y n 1 , the L = 2 processes do not have a significant impact at the leptogenesis scale of interest M S ∼ = 10 13 GeV.
The final baryon asymmetry is As an example we show the result of the Boltzmann equations in Fig. 1 for a single set of parameters chosen from the numerical scan in the next section. The parameters for this plot are tan β = 30, M 3 = 8.51 × 10 12 GeV, c = 0.053, ρ = 6.29, γ = 3.33 .  Table 1 The best-fit values and the 3σ ranges for the parameters taken from [21]. The two minima for both θ 13  Note that at this point we had given two different example points in our previous publication [2]. But with the improved calculations none of them is in good agreement with the experimental result on Y B anymore and hence we have chosen here a different example point. This shows furthermore the importance of the current study.

Results
In this section we show the results for the masses and the mixing angles as well as the results for Y B in dependence on the parameters in our model. To obtain realistic values for Y B coming from the numerical solution of the Boltzmann equations given in Section 3 we had to slightly increase the value of the neutrino Yukawa coupling from y 1 = 0.1 to y 1 = 0.12 which is a 20% change as anticipated.

Masses and mixing angles
For our numerical scan we use the method as described in [1]. For the parameters which characterise the charged lepton and quark sector we use the fit results given there with tan β = 30 and M SUSY = 1 TeV. The renormalisation group evolution is done using the REAP package [11].
There are six free parameters left in our model, the moduli |X| and |Y |, the phases φ and δ e 12 in the leading order matrices and the modulus |c| and the phase γ , which come from the correction to the neutrino Yukawa matrix. We performed a random scan over these parameters and impose the experimental ranges for the mixing angles, mass squared differences (cf. Table 1) and Y B as constraints. For Y B we used [22,23] where the 3σ uncertainty is obtained, for simplicity, by multiplying the 1σ error by a factor of three. In order to calculate Y B we solved the Boltzmann equations given in Section 3 numerically. Before we present the results for the normal ordering of the neutrino masses we comment briefly on the case of inverted ordering. In the original model [1] the inverted ordering was not viable due to incompatible constraints for θ 12 coming from the mass sum rule on one hand and from the angle sum rule in our model on the other hand. Due to the correction for the neutrino Yukawa matrix the mixing angles are modified with corrections of order c which nevertheless have to be of the order of c ≈ 0.4 to save the inverted ordering as we have shown in an estimate given in [2]. Since c is associated with a higher dimensional operator such high values are not plausible.
Turning now to the normal ordering, we show the results for the masses and mixing angles in Fig. 2. We find all parameters in agreement with the experimental 1σ (3σ ) ranges for the masses, mixing angles and Y B .
The correlations between θ 13 and the Majorana phases are weaker than in [2] which is due to the constraints on the phase φ and the value of |Y | coming from the numerical solution of Y B . Furthermore we need here a larger value of c which washes the correlations out and enlarges the ranges for the phases. Nevertheless we find the phases to be in similar ranges as in [2], namely we obtain For the rephasing invariant J CP which determines the magnitude of CP violation effects in neutrino oscillations [26] we find values in the ranges J CP = ±(0.006, 0.036). Our predictions for neutrinoless double beta decay are shown in Fig. 3. For the lightest neutrino mass m 1 which is mostly determined by the mass sum rule we obtain values between 12 meV and 22 meV. For the observable in neutrinoless double beta decay m ee we obtain values between 3.3 meV and 14.3 meV. For the sum of the neutrino masses we predict m ν ∈ 0.077-0.099 eV , (4.5) which might be determined, e.g., from cosmology. So far there is only an upper bound [22] m ν < 0.23 eV, (4.6) which is well in agreement with our prediction. The second observable is the kinematic mass m β as measured in the KATRIN experiment [27] which is given as Here we predict m β ≈ 0.015-0.023 eV which is below the projected reach of m β > 0.2 eV of KATRIN.

Leptogenesis
Moving on to our predictions for Y B , we show the results of our parameter scan in Figs. 4 and 5.
We find parameter points which are in agreement with the 1σ (3σ ) range for Y B . The ranges for the φ and ρ are as in our previous study between 4.7 and 7.6 for ρ and between 0.66 and 1.38 or −1.38 and −0.66 for φ since they are mostly determined by the masses and mixing angles. However the constraints coming from the numerical solution of Y B now forbid values of φ between −0.66 and 0.66. Interesting is also the range for the mass of the lightest right-handed neutrino which is in the range from 0.778 · 10 13 GeV to 0.862 · 10 13 GeV which shows a clear correlation to c but no correlation to γ . The other correlations can be seen in Fig. 5. We furthermore want to note that the rather narrow range of M 3 is due to the fact that we fixed y n 1 to a certain value. As a rule of thumb the scale of light neutrino masses has to remain the same and hence a variation of 10% in y n 1 leads to a variation of 20% in M 3 .  [22] and the grey region in the upper part displays the upper bound on the effective mass from the EXO experiment [24]. The red, straight lines represent the sensitivity of GERDA phase I respectively GERDA phase II [25]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) The size of the correction c has to be in the range between 0.035 and 0.085 and γ is in narrow ranges around 0 or π . This also explains the disconnected regions for the mixing angles and phases which did not appear so in our previous studies. As in our previous study the sign of Y B is uniquely determined by γ . For the 1σ ranges of the parameters the values of γ have to be in the region around 0. This implies that we only find parameter points for the 1σ ranges compatible with a negative value of Y B . Note that this is mainly driven by the 1σ range of θ 23 which is not very well determined. Indeed, with the results from the Valencian fitting collaboration [28] where θ 23 is allowed to be in the second quadrant, even at the 1σ level positive values for Y B are possible.
As γ is in narrow ranges around the CP conserving values, the Majorana phases are the major sources for CP violation here, as it can be seen from Fig. 4.

Summary and conclusions
In this work we have revised predictions for leptogenesis in an SU(5) × A 5 golden ratio GUT flavour model from [2]. In that publication we had used approximations to calculate the baryon asymmetry which are known to be precise only up to 20-30%. Instead we have solved here the full set of Boltzmann equations numerically and could show that we can still successfully accommodate the experimental values of all mixing angles, fermion masses and the baryon asymmetry even at the 1σ level. Nevertheless, to do so we had to adjust some parameters. For instance, we have increased the neutrino Yukawa coupling by 20% from y 1 = 0.1 to y 1 = 0.12.
All of the main features of the original model are still valid. For instance, the neutrino mass sum rule is correct up to small corrections and the inverted ordering is still ruled out. Furthermore, the Yukawa coupling ratios y τ /y b ≈ −3/2 are unaffected by the modification of our model and hence imply non-trivial constraints on the spectrum of the supersymmetric partners of the Standard Model particles.
The phenomenological predictions are not changed significantly as well. For the lightest neutrino mass we find the range m 1 ∈ [0.012, 0.022] eV, but the strong correlation between θ 13 and the phases is somewhat weakened due to an increased value of c. For the phases we find δ ∈ [ We have proposed the first SU(5) × A 5 model to our knowledge which can successfully accommodate for all parameters in the neutrino sector and for the baryon asymmetry of the Universe as calculated from the Boltzmann equations and for the not yet measured quantities in the neutrino sector we make testable predictions. For this our model needs comparatively few parameters which makes it very appealing and testable in future experiments.