A possible relation between leptogenesis and PMNS phases

We propose a new scenario for baryogenesis through leptogenesis, where the CP phase relevant for leptogenesis is connected directly to the PMNS phase(s) in the light neutrino mixing matrix. The scenario is realized in case only one CP phase appears in the full theory, originating from the complex vacuum expextation value of a standard model singlet field. In order to realize this scheme, the electroweak symmetry is required to be broken during the leptogenesis era and a new loop diagram with an intermediate $W$ boson exchange including the low energy neutrino mixing matrix should play the dominant contribution to the CP violation for leptogenesis. In this letter, we discuss the new basic mechanism, which we call type-II leptogenesis, and give an estimate for maximally reachable baryon asymmetry depending on the PMNS phases.


I. INTRODUCTION
The origin of the baryon asymmetry of the Universe (BAU) has been a longstanding theoretical issue [1]. Among Sakharov's three conditions [2] for successful generation of an asymmetry from a symmetric initial state, the first, i.e. baryon number violation, and the second, i.e. C and CP violation, rely most strongly on model building beyond the Standard Model (SM) of particle physics. Along this line, there already exist plenty of theoretical models to generate the BAU [3][4][5][6][7], with different ways to depart from thermal equilibrium, e.g. from heavy particle decay outside equilibrium to first-order phase-transitions or to the dynamical Affleck-Dine (AD) mechanism [5].
In this letter, we would like to follow an alternative route, realizing instead the leptogenesis mechanism within a phase with broken electro-weak symmetry at high temperature and relying mostly on SM physics in the neutrino sector to achieve the necessary CP-violation. The only ingredients beyond the SM that we need is the presence of various species of SM singlets, with the quantum numbers of right-handed(RH) neutrinos, different Higgs doublets, in order to allow for a non-vanishing contribution to the CP asymmetry from a W -boson loop and to keep the electroweak symmetry broken.
Indeed, at the level of the SM of particle physics, CP violation is related to the charged-current interaction and determined by two CP phases, one in the quark sector, the Cabibbo-Kobayashi-Maskawa (CKM) phase δ CKM [8,9] and the other the Pontecorvo-Maki-Nakagawa-Sakada (PMNS) phase δ PMNS [10], while two more Majorana phases are appearing in the leptonic sector. In a family unified grand unified theory (GUT), these two phases can be related if only one single complex vacuum expectation value(VEV) appears in the full theory [11]. From the early time on, it has been an interesting issue to investigate a possibility of relating the baryon asymmetry with the SM phase(s) δ CKM or/and δ PMNS .
The first obvious possibility is to exploit δ CKM for the BAU, but it has been known long time that such phase, appearing always with small mixing angles, is not enough for the baryon number generation in GUT baryogenesis [12]. Even in other scenarios, relying on the quark sector, like the AD mechanism [5] through baryon number carrying scalars or the electroweak baryogenesis, additional CP violating phases are needed to provide a large enough baryon asymmetry [13]. Therefore, it is difficult to explain the BAU just considering the CP violation in the quark sector.
A more promising road is given by baryogenesis through leptogenesis [6], since the phases in the leptonic sector are unconstrained and the mixings large. This mechanism relies on the fact that sphaleron processes are effective before/during the electroweak phase transition and violate both baryon (B) and lepton (L) numbers, but conserve baryon minus lepton number (B − L). Therefore, baryogenesis or leptogenesis above the electroweak scale must generate a non-vanishing B − L number, that is then translated into a baryon asymmetry before or at the electroweak transition.
In this paper we consider a leptogenesis scenario, which allows us to relate the CP-violation during leptogenesis to the phase δ PMNS in the light neutrino mixing matrix. In order to be able to have a well-defined neutrino mixing matrix when the lepton asymmetry is cosmologically created, we require that the SM gauge group SU(2) L ×U(1) Y remains broken during the leptogenesis epoch. In fact, the Brout-Englert-Higgs(BEH) mechanism for SU(2) L ×U(1) Y breaking at high temperature is possible for some regions in the parameter space of BEH bosons h u and h d [14].

II. A NEW TYPE OF LEPTOGENESIS
In the leptogenesis scheme with one or two BEH doublets, the lepton asymmetry arises from the decay of the lightest heavy Majorana neutrino N 1 producing light leptons and antileptons and the Higgs particle by the decay where ℓ i (l i ) is the i-th lepton (antilepton) doublet and h u is the up-type Y = 1/2 BEH doublet. We follow here the supersymmetric notation, but the mechanism can work also without supersymmetry. In this case therefore the same mother particle N has two decaying channels with different lepton number and therefore the model satisfies the Nanopoulos-Weinberg theorem [15,16]. In classical leptogenesis, the CP violation in the decay arises from the interference of the tree-level with the one-loop diagrams involving the heavier RH neutrinos N j , j = 2, 3 (for the case of three generations) and the CP violation arises in general from the complex Yukawa couplings and has in general no direct relation to the low-energy CP-phases [17], apart in case of particular textures [18] or CP conservation in the heavy RH neutrino sector [18,19].
In this letter we would like to extend the model in order to have a large contribution to the CP violation from an electroweak loop involving explicitly the PMNS matrix. In order to do so, we introduce another copy of the Higgs doublet H u , heavier than the SM one h u and with vanishing vacuum expectation value (VEV), as well as another generation of RN neutrinos N 1 . All these particles can mix with h u , N 1 respectively and allow for the presence of the diagram (2b) in Fig. 2, where the virtual particles are all SM particles and one of the vertices include the PMNS matrix directly. We consider here for simplicity the case where the field H u is heavier than the right-handed neutrino, so that the decay of N into ℓ i + H u is negligible.
The CP phase in the PMNS matrix is required to descend down from high energy scale by a complex VEV. To relate different phases, we assume that only one SM singlet field X develops a CP phase δ X . Thus, all Yukawa couplings and the other VEVs are real and all CP violation parameters arise from δ X .
While the SM Higgs doublet(s) do not carry lepton number, we define the fields H d and H u to carry the lepton number L = +2 and −2, respectively, and N instead to have L = 1, while N will be defined to carry L = −1. We can then write for the Higgs doublets and the heavy neutrinos the Yukawa couplings: The Yukawa couplings (f 's) of the inert Higgs doublets H u,d to the lepton doublets are distinguished from those (f 's) of the BEH doublets h u,d and no mixing is allowed at this level due to the different lepton number assignments. We have as well other lepton number conserving interactions as where ∆m 0 is real. The first term of (3) gives directly a Majorana mass term between N 1 and N 1 without a phase because we defined it preserving the lepton number. The Dirac mass for the seesaw neutrino mass is via N 1 h u ℓ L which appears as in the 'Type-I leptogenesis' . The needed lepton number violating couplings are introduced by the couplings which allow for mixing also in the Higgs sector and for the see-saw mechanism. There are more L violating terms such as h d H u , h u H d and h * d H d , which are not relevant for leptogenesis. We will assume ∆m 0 ≫ m ′ 0 , m ′′ 0 , i.e. the L conserving mass parameter is much larger than the L violating mass parameters. In this case, (N 1 , N 1 ) are maximally mixed and we call N the lightest mass eigenstate obtained from the mixing of these two states. We can then define an effective Yukawa coupling for this lightest RH neutrino as by considering the large mixing angle θ N between the neutrinos and also the small mixing between H u and h u .

III. WITH ONE PHASE
The process (1) can include the phase δ X by the interference terms with the diagram with an intermediate W -boson. To relate the leptogenesis phase δ L to the SM phase(s), one needs a families-unified GUT toward a calculable theory of the physically measurable phases. In the anti-SU(7) [20], indeed δ PMNS and δ CKM are shown to be related [11]. In this paper, we attempt to relate the phases in leptogenesis and δ PMNS [21]. In other words, we attempt to express the lepton asymmetry ǫ L in terms of δ PMNS . For this, the W -boson loop must dominate over the other one-loop corrections.
To obtain a calculable theory for phases, we introduce a single Froggatt-Nielsen(FN) field [22] X developing a complex VEV, X = x e iδX [23,24]. The Yukawa coupling matrix of the doublet h u include powers of X such that some symmetry behind the Yukawa couplings is satisfied. The Yukawa couplings of the three RH neutrinos obtain then a complex phase from different powers of the Froggatt-Nielsen field X depending on the generation. To simplify the discussion, let us assume that the heavy Majorana neutrinos have a mass hierarchy and let the lightest heavy Majorana neutrino N dominate in the leptogenesis calculation.
For the tree ∆L = 0 decay mode corresponding to Fig. 1 (a), we show the relevant Feynman diagrams interfering in the N → ℓ j + h u decay in Fig. 1 (a) and (b) giving rise to a new contribution to leptogenesis. In Fig. 1 (a), (c), and (d), we also show the relevant diagrams for N → ℓ i + h u decay in the classical leptogenesis scenario, discussed in [25][26][27]. In the basis where the N s and the charged leptons masses are diagonal, possible phases appear at the vertices with the red bullets in Fig. 1. In models where a single complex VEV appears in all the Yukawa couplings, even if with different powers, the classical leptogenesis diagrams given in Fig. 1 (c) and (d) do not contribute to the CP asymmetry because the overall phases cancel out with that of Fig. 1 (a). Indeed, we see that in the diagrams the directions of X nj are opposite, indicating that the phases are equal and opposite. We are then left to compute the contribution from the diagram with an intermediate W . As we will see explicitly below, such contribution vanishes in the presence of a single Yukawa coupling, but gives a non-vanishing contribution in our model.
Let us calculate the effect of the vertex correction via the W boson of Figs. 1 (b), in the simple mass-insertion formalism. A new conclusion will be drawn from this calculation. With this set-up, it is a standard procedure to calculate the asymmetry ǫ L , i.e. the difference of N decays to the ℓ andl, where ℓ(l) is a (anti-)lepton. We have the following interference term from Figs. 1 (a) and (b), where I denotes the loop integral depending on the internal masses and the external momenta.  (7) is the mixing angle between h 0 u and H 0 u in the mass eigenbasis when µ ′ 2 ≪ m 2 H 0 , m 2 h 0 . Through such Higgs flavor change at the blue bullet in Fig. 2 (b), the lepton number is violated. As in the classical case, the loop integral is UV divergent, but its imaginary part is finite and quite simple in the limit of vanishing mass for the leptons and m W ≪ m 0 . Indeed we obtain where we have used 2P · p ℓ = m 2 0 − m 2 h + as set by the kinematical constraints. This expression is IR divergent for vanishing m W , but in that limit the PMNS matrix U ij becomes trivial and the CP violation vanishes automatically. Indeed considering also the neutrino final states in Figs. 1 (a) and (b), in which case the PMNS matrix is U instead of U † , and the analogous diagram with the W -loop attached to the tree-diagram in Fig. 1 (a), we obtain The asymmetry (9) in the limit of unbroken SU(2), but m h 0 ,H 0 = m h + ,H + , is given by the simple matrix multiplication f † 1 (U + U † )f 2 where f 1,2 are column vectors. The imaginary part has the form 1 is diagonal. This is consistent with the fact that in these diagrams the lepton violation is on the left side of the cut as discussed in [28]. Nevertheless in our case thanks to SU(2) breaking, the masses of the particles in the loop are different, so the loop factors are not exactly equal and an imaginary part is present. Indeed expanding for example in the Higgs mass difference If the SU(2)×U(1) Y is broken, by some choice of BEH boson couplingsà la Ref. [14], the mass splitting is ∆m 2 h ∝ v 2 (T ) and a substantial CP asymmetry is present at T ≤ m 0 . Moreover it is then possible to relate the lepton asymmetry directly to δ PMNS .
For concreteness, we use the PMNS matrix U ij , in the vertex diagram of Fig. 1 (b), presented in [11,29] together with Majorana phases δ a,b,c , where only two phases out of three phases e iδ a,b,c are independent. Out of three e iδ a,b,c , we choose one freely to match to physics of the problem. As mentioned before, Figs. 1 (c) and (d) do not have the interference term with (a). So, we choose the Majorana phase of the dominant SM lepton e iδ0 such that it does not have the phase dependence on e iδ0 in the interference with Fig. 1 (a). If we assume that the third generation Yukawa couplings dominate, Here we see that the CP asymmetry is directly related to the PNMS phases. The first factor of Eq. (10) is about 10 −3 , since the asymmetry is enhanced by the smallness of the W mass. This we will see is an advantage and not a problem since sphaleron transitions are suppressed during the EW symmetry breaking epoch and we can realize baryogenesis even if only a small fraction of the lepton number is converted into baryon number.

IV. RELATION OF THE PHASES
Now we can relate the phases in our plan of spontaneous CP violation [30] with one complex VEV, i.e. the phase of X . Following the argument of Ref. [11], we can conclude that there will be no observable lepton asymmetry if δ X = 0. Therefore, all the interference terms in Eq. (9) must have factors of the form sin(N ij δ X ) where N ij is an integer. For example, consider the imaginary part of a specific term in Eq. (9) before taking the sum with i and j. From the product of (b) and (a) * of Fig. 1, we read one convenient term, i.e. for i = 3 and j = 1, which has the overall phase e i[δPMNS+δa−n1δX+δ0]+i[n3δX−δ0] where δ PMNS and δ a are defined in Eq. (11). The Majorana phase δ 0 is the phase of the heavy lepton sector, which does not appear in this phase expression with i = 3 and j = 1 if the lightest neutral heavy lepton dominates in the lepton asymmetry. The imaginary part of this term is In Ref. [11], we argued that the observable phase δ PMNS in low energy experiments must be integer multiples of δ X since there will be no electroweak scale CP violation effects if δ X = 0 and π. Along this line, we argue that δ PMNS = n P δ X and δ a = n a δ X , which are sufficient for the physical requirement. In this case, Eq. (13) becomes sin[(n P + n a − n 1 + n 3 )δ X ]. Now, consider the sum with i and j. We observe that each term has the form of Ae i(±nP δX +δ ′ ) + Be iδ ′ where A and B are real numbers formed with real angles and δ ′ = n ′ δ X = n a δ X , n b δ X , or n c δ X , viz. Eq. (11). It is of the form which has the phase δ ij = arctan ({A sin[(±n P + n ′ )δ X ] + B sin[n ′ δ X ]}/{A cos[(±n P + n ′ )δ X ] + B cos[n ′ δ X ]}). Thus, every term has the vanishing phase if δ X = 0 and π. Thus, the sum in Eq. (13) gives 0 if δ X = 0 and π. Even at this stage, we have obtained an important conclusion: the phases in the heavy lepton sector does not appear. For further relations, we must use a specific model relating n P , n ′ , n i , and n j , as we used the flipped-SU(5) model in relating δ PMNS and δ CKM [11]. Thus, the asymmetry takes a form, where A ij are a ij times appropriate ratio of Yukawa couplings. Note that there are only two independent n ′ as commented before, below Eq. (11).

V. SPHALERON PROCESSES DURING THE EW BROKEN PHASE
Contrary to simple approximations, sphaleron transitions [4] are suppressed, but not vanishing when the electroweak symmetry is broken. For a relatively large range of Higgs VEVs, as long as v ≤ T , one can obtain at least a partial conversion of L into B. Indeed the sphaleron rate in the equilibrium broken phase is given by [31,32] Γ broken where κ is a constant, g W , α W are the electroweak coupling and coupling strength and E sph the energy of the sphaleron energy barrier, proportional to the Higgs VEV, E sph = 1.524πv/g W . So, as found in [32], in the SM with a Higgs mass of 125 GeV, the sphaleron processes remain in thermal equilibrium until one reaches temperatures of the order T * = (131.7 ± 2.3) GeV, where v T > 1. In case the Higgs VEVs remain non-vanishing, as we advocate here, such VEVs are proportional to the temperature in the high T regime, v(T ) = v(0) 2 + k 2 T 2 , as discussed in [14]. Therefore, for k ∼ 1 the sphaleron processes may enter equilibrium for low temperatures above the electroweak scale T ≥ v(0) as long as In case the sphaleron processes are active only for a very short range of temperatures, nevertheless a partial conversion of lepton number into baryon number may still be possible, giving rise to the observed baryon asymmetry if the lepton number is sufficiently large.

VI. CONCLUSION
By introducing only one CP phase by a complex VEV of a SM singlet field X and assuming leptogenesis via the lightest Majorana neutrino out of equilibrium decay during a phase where the electroweak symmetry is broken, we have that the dominant contribution to the CP asymmetry in the early Universe arises from a W -boson loop, directly containing the PMNS phase δ PMNS . In this way we are able to have a novel mechanism to relate high and low energy CP violation, in particular in the case when a single CP phase is introduced by spontaneous mechanism at a high energy scale along the Froggatt-Nielsen method.
Even in case more phases are present and a non-vanishing contribution from the classical loop with the heavier RH neutrinos states is present, the PMNS phase could still represent the dominant part and allow for a direct correlation of the baryon asymmetry to neutrino observables.