Flavor dependence of CP asymmetries and thermal leptogenesis with strong right-handed neutrino mass hierarchy

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1 PHYSICAL REVIEW D 73, (006) Flavor deendence of CP asymmetries and thermal letogenesis with strong right-handed neutrino mass hierarchy O. Vives Theory Division, CERN, CH-, Geneva 3, Switzerland (Received 3 January 006; ublished Aril 006) We rove that taking correctly into account the leton flavour deendence of the CP asymmetries and washout rocesses, it is ossible to obtain successful thermal letogenesis from the decays of the second right-handed neutrino. The asymmetries in the muon and tau-flavour channels are then not erased by the inverse decays of the lightest right-handed neutrino N. In this way, we reoen the ossibility of thermal letogenesis in models with a strong hierarchy in the right-handed Majorana masses that is tyically the case in models with u-quark neutrino-yukawa unification. DOI: 0.03/PhysRevD PACS numbers: 3.35.Hb, 4.60.St,.30.Hv, 4.60.Pq The measure of nonvanishing neutrino masses and mixings and the use of the seesaw mechanism to exlain the smallness of these masses has made letogenesis [] one of the most romising mechanisms to generate the observed value of the baryon asymmetry of the Universe. The seesaw mechanism requires the resence of heavy righthanded neutrinos with comlex Yukawa coulings that can generate a leton asymmetry through their out-ofequilibrium decays. This leton asymmetry is then transformed into baryon asymmetry by the standard model shaleron rocesses violating (B L) []. When the righthanded neutrinos are roduced through scatterings (mainly inverse decays) in the thermal lasma this scenario is known as thermal letogenesis. In the thermal letogenesis mechanism, it is usually assumed that the leton asymmetry is generated by the out-of-equilibrium decays of the lightest right-handed neutrino N. The asymmetry generated by the two heavier right-handed neutrinos N and N 3 that we take as hierarchically heavier than N, is usually assumed to be either negligible or efficiently washed out by the N inverse decays [3]. In these conditions only N generates a baryon asymmetry. The final value of the baryon asymmetry normalized to the entroy density B is then related to the CP asymmetry " in the decays of the lightest right-handed neutrino as B 8 n N 3 s " " ; () where the factor 8=3 is the fraction of the B L asymmetry converted into baryon asymmetry by shalerons. n N =s is the number density of right-handed neutrinos normalized to the entroy density that in equilibrium is aroximately equal to 0:=g with g 06 the number of roagating states in the standard model lasma. Finally is the efficiency factor describing the fraction of the CP asymmetry that survives the washout by inverse decays and scattering rocesses. Now, for hierarchical right-handed neutrinos, it is ossible to relate " to the mass of the lightest right-handed neutrino, the observed neutrino masses, and an effective letogenesis hase as " 3 8 M v m 3 m sin L 3 8 M m atm 0 6 M matm 0 0 ; () GeV 0:05 ev where the maximum value for this asymmetry corresonds to fully hierarchical neutrinos with m 3 m atm 0:05 ev, m 0, and a maximal hase L. The baryon asymmetry is recisely measured from the cosmic microwave background as CMB B 0:9 0 0 [4]. Thus, using Eqs. () and () with we obtain a lower bound on the mass of N [5]: M M min :5 0 9 CMB B 0:05 ev GeV 9 0 : m atm However this lower bound reresents a serious roblem for many flavour models, secially those incororating some grand unification symmetry. In these models we usually exect that neutrino Yukawas are closely related to the uquark Yukawa matrices. As show in the aendix of [6] (see also Ref. [7]) and exlained below, in these models it is ossible to obtain the mass of the lightest right-handed neutrino as jm j y v jwk m k j m u jw m W m sol W3 m atmj ; (4) with W ij the rotation from the basis of diagonal L masses to the basis where the (left-handed) neutrino-yukawa matrix Y Y y is diagonal, m k the left-handed neutrino masses that we take as hierarchical, and we have aroximated y v m u. From this exression it is straightforward to see that we can only obtain M 0 9 GeV if m u 0 9 GeV 4 m 06 u ev MeV v (3) * jw m W m sol W 3 m atmj: (5) =006=73(7)=073006(5)$ The American Physical Society

2 O. VIVES PHYSICAL REVIEW D 73, (006) This imlies that, barring accidental cancellations (or some secific textures as in [8]), having a large enough N is only ossible if we have simultaneously m & ev, W & 5 04 (for m sol 0:008 ev), and W3 & (for m atm 0:05 ev). These conditions are usually not satisfied in most flavour models where the tyical values for the lightest right-handed neutrino mass are close to GeV with W of the order of mixing angle in solar neutrinos [9 ]. Therefore, aarently, these models cannot roduce a sufficient baryon asymmetry through the thermal letogenesis mechanism. It is evident that this reresents a very serious roblem for all these flavour models, that are forced to abandon thermal letogenesis and use other mechanisms to roduce the observed baryon asymmetry []. In this aer we resent a solution to this roblem. We show that when the flavour deendence of the leton asymmetries and the erasure rocesses are correctly taken into account, it is still ossible to generate a large enough baryon asymmetry in the decays of the second right-handed neutrino N that then survives the washout due to the inverse decays of N. The basic idea behind this statement is that the asymmetries generated by N in the different flavour channels are washed out by different N Yukawa interactions. In fact, the decays of the second right-handed neutrino N create different asymmetries in the,, and e channels. As we will show these asymmetries do not mix through the thermal interactions with the lasma before the N inverse decays become effective. Therefore, it is evident that the different flavour channels will only be erased by the corresonding N! HL i interaction and not by the sum over all leton flavours as usually done [3,6,3 6]. Taking into account these effects, we rove here that in models with Y Y u, the leton asymmetry in the and channels is only mildly erased by N decays. Some of these ideas were already resent in the literature. For instance, the ossibility of generating the asymmetry in N decays was reviously discussed by P. di Bari in the one flavour aroximation [3]; an early discussion of flavour effects in letogenesis, mainly in N decays, can be found in [7], and flavour effects in the context of low scale letogenesis models were recently studied in [8]. In this aer we show that combining these elements it is ossible to obtain successful letogenesis in realistic models of fermion masses with u-quark neutrino-yukawa unification. Therefore, we reoen the ossibility of thermal letogenesis in these models. In the following we give a general roof of this statement. Although this mechanism is equally valid in suersymmetric and nonsuersymmetric models we only discuss the nonsuersymmetric examle. The suersymmetric case is comletely analogous [9]. The general exression for letonic Yukawa coulings and Majorana masses in a seesaw standard model is, L Yuk L T Y e e c R H L T Y c R H ct R Mc R, with L i, e Ri, Ri,(ie; ; ) the three generations of letons and Y e, Y, and M, 3 3 matrices. In the basis of diagonal Y e and M, the neutrino-yukawa matrix is Y V y L D Y V R, where D Y is a diagonal mass of neutrino-yukawa coulings, that in the class of models discussed here is D Y diagy ;y ;y 3 diagm u =v ;m c =v ;m t =v. The effective Majorana mass for the left-handed neutrinos is obtained through the seesaw mechanism as m v Y M Y T U D m U T ; (6) where the mixing matrix U is the Pontecorvo-Maki- Nakagawa-Sakata (PMNS) matrix that is sufficiently known from neutrino exeriments, and D m diagm ;m ;m 3 the three neutrino masses. At the moment, we know only two mass differences, m atm :7 0 3 ev and m atm 7:0 0 5 ev. Although in the following we assume hierarchical sectrum with normal hierarchy, other cases can be easily treated in the same way. Now we define the Hermitian matrix v 4 y4 My M [6] and, in the basis of diagonal Y y Y, from Eq. (6) we have v 4 y4 D Y V L my V y L D Y V L m V T L D Y D Y y D Y D Y ; (7) with V L m VL T WD m W T. From here the matrix can be written as 0 ij ~ y i ~ j X ~ i k ~ j k ; ~ i y i B k y C i A: i y y 3 3i (8) Given the strong hierarchy in the neutrino-yukawa eigenvalues (similar to the hierarchy in the u-quark masses) we have in ractice always that j ~ j j~ j, j ~ 3 j. For instance, to change this hierarchy in the simlified case where we take W 3 0 and y =y 3 0 would require that j j < j jy =y. In these conditions it is very easy to understand that the largest eigenvalue of the matrix, corresonding to v 4 y4 =jm j, will be given in very good aroximation by the (, ) element of this matrix. Therefore the lightest right-handed Majorana neutrino mass is given by: jm j y v j j y v j j m u jwk m k j : (9) From this exression we can estimate the mass of the lightest right-handed neutrino. If the observed neutrino mixings are not resent in the neutrino-yukawa matrices and come mainly from the seesaw mechanism itself we have that V L and W U PMNS. Then we have that jm j m u=m sol 0 6 GeV. Only in the

3 FLAVOR DEPENDENCE OF CP ASYMMETRIES AND... PHYSICAL REVIEW D 73, (006) case W we can have jm j m u=m ev=m GeV. Therefore, tyically N is too light to generate a sizeable baryon asymmetry. In the same way the associated eigenvector to this eigenvalue is given by ^ ~ =. Using this eigenvector we can also obtain the Yukawa coulings of N in the basis of diagonal Y e and M as 0 Y i V y L D Y ^ i y V y A 3 0 y U k m k W k U k m k W C k A; (0) U 3k m k W k M ImYa M Y ayk Y k : () Owing to the hierarchy in the neutrino-yukawa matrices and the right-handed Majorana neutrinos, the first term in Eq. () usually dominates and therefore we take it to estimate the size of the asymmetry. Now we have that, =v ImY a m ak Yk =M 3 ImYa Y a3yk Y k3 =M ImYa Y ayk Y k. Therefore, the numerator in a includes the M 3 contribution to the m ak element. To maximize the asymmetry we assume this element to be of order m 3 m atm and define the corresonding CP-violating hase as, then we have a 3 M m 3 jy a j 8 v jy 3 j ; () where we use that the hierarchy in the Yukawa elements is such that Y 3j >Y j, Y j. It is clear that asymmetry in the channel can be as large as 3 06 M =0 0 GeV. In the muon and electron channels the asymmetry will be further suressed by jy j=jy 3 j and jy j=jy 3 j, resectively, and thus we can exect a smaller asymmetry secially in the electron channel [0]. These asymmetries are artially converted, through shaleron rocesses, to a a 3 B L a asymmetry and, in art, washed away by N inverse decays. The final leton asymmetry after the decay of N will be given by a n N s a a ; (3) with n N =s the number density of N normalized to the entroy density []. The efficiency factor a is the fraction of the roduced asymmetry that survives after the end of N decays and inverse decays and it is aroximately given by where we used that P iv L ij i P ku jk m k W k. At this oint we can already discuss the behavior of a leton asymmetry created by the out-of-equilibrium decays of the next-to-lightest right-handed neutrino N. The asymmetry in the different flavour channels (a e; ; ), assuming M M M 3, is given by [3] a 3 M 8Y y ImYa Y M Y a3yk Y k3 3 a N! l a H Hj T M ~ma m ; (4) with m a v Y aya =M [4] and m is the equilibrium mass equal to 0 3 ev in the SM [4]. The final values of this efficiency factor are model deendent and therefore we kee this factor as a arameter in the following. At temeratures slightly below M we have three different asymmetries in the,, and e channels. These asymmetries remain basically unchanged from T M to T M and, in articular, the different flavours do not mix. In fact, at this temerature, both N and N 3 righthanded neutrinos have already decayed. Therefore the only ossible Yukawa interactions in the lasma are those with the right-handed charged letons and with N. Taking into account that the Y3;3 e Yukawa interactions are in thermal equilibrium at temeratures below 0 4 GeV, it is more convenient to use the basis of diagonal charged leton Yukawas to include these interactions in the thermal masses. Thus, only the Yukawa interactions with N can change flavour, although N are very rare in the lasma u to temeratures of the order of M when they reach the equilibrium density [5,6]. Therefore, we have to discus the washout of the different asymmetries by the N decays and inverse decays. First, it is easy to see that this washout will be exonential and not linear if the N are thermally roduced at temeratures close to M. From Ref. [5] and taking " 0, the Boltzman equation for this case would be r 0 4 z e z z ~m a 0:86 0: 0: A m ab b ; with AT 0 9 0:06 0:65 0:07 A; (5) 0:06 0:07 0:65 with ~m a v Y ay a =M takes into account that a given asymmetry a can only be erased by the corresonding leton flavour, z M =T and m 0 3 ev [4]. Now we can use Eqs. (9) and (0) and we see that

4 O. VIVES PHYSICAL REVIEW D 73, (006) U k m k W k j =j j jm W = m W = j =j j; ~m e jx k ~m jx k ~m jx k U k m k W k j =j j jm W = m W = m 3 W 3 = j =j j; U 3k m k W k j =j j jm W = m W = m 3 W 3 = j =j j; (6) where m 3 m atm and m 3 m sol and we have used bimaximal mixings in the PMNS matrix to make an estimate. The final values of ~m a deend on the matrix W. In case the mixings in the neutrino-yukawa matrix are small, we have that W U and so ~m e m m m sol ; ~m m m ~m m m m m m sol 4 ; m sol 4 : (7) Using these values in the case ini ini ini e, Eq. (5) can be easily integrated (taking z z, e z 0 and A A 0:63) and we obtain fin ; ini 0:75 ~m; ; e =m : (8) If now we take the central value from the solar mass difference as m sol 0:008 ev, we have that the muon and tau asymmetries are only erased by a factor ex:5 0: and hence they can survive the washout from N decays. This should be comared with the washout that we would obtain in the single flavour aroximation, that would be e 0:75P a ~ma =m e 0:75m sol=m : Now the final asymmetry in the channel would be 0: fin M m 3 v e 0:75m sol=4m & 6:5 0 0 M =0 0 GeV ; B 8 79 fin fin fin ; (9) where we have used that and e 0. Thus in this case, it could be ossible to generate a sufficient baryon asymmetry from the and channels even for efficiency factors and of order 0.5 and M 0 0 GeV. For smaller efficiency factors we could still have successful letogenesis with a somewhat heavier M. This situation is tyically found for instance in models with non-abelian flavour symmetries. In this aer we have only resented a simle estimate to show that this mechanism can generate a large enough baryon asymmetry. A full comutation in a model with an SU3 flavour symmetry and sontaneous CP violation is now in rogress [9,,5]. From Eq. (6) and taking into account that P kwk m k we can see the washout by N can only be stronger than the result obtained in Eq. (7) if the contribution from the atmosheric neutrino mass is sizable (assuming that m m sol ). This requires basically that W 3 m 3 W m. Clearly this ossibility cannot be excluded, but taking into account that m =m 3 =6, W V L U, and that U = and U3 0: it requires large left-handed mixings in the neutrino-yukawa matrix. This could be ossible in models based in Abelian flavour symmetries or discrete symmetries, although it is clear the final result is highly model deendent and the different models must be studied in detail. We have shown that taking correctly into account the leton flavour deendence of the CP asymmetries and washout rocesses, it is ossible to obtain successful thermal letogenesis from the decays of the second righthanded neutrino. The asymmetries in the muon and tauflavour channels are then not erased by the inverse decays of the lightest right-handed neutrino N. Therefore thermal letogenesis is still viable in models with a strong hierarchy in the right-handed Majorana masses that is tyically the case in models with u-quark neutrino- Yukawa unification. I acknowledge suort from the RTN Euroean Project No. MRTN-CT and from the Sanish MCYT FPA I thank L. Covi, S. Huber, A. Riotto, and secially S. Davidson for very helful discussions. [] M. Fukugita and T. Yanagida, Phys. Lett. B 74, 45 (986). [] V. A. Kuzmin, V. A. Rubakov, and M. E. Shaoshnikov, Phys. Lett. B 55, 36 (985). [3] W. Buchmuller, P. Di Bari, and M. Plumacher, Nucl. Phys. B665, 445 (003). [4] P. de Bernardis et al., Astrohys. J. 564, 559 (00). [5] S. Davidson and A. Ibarra, Phys. Lett. B 535, 5 (00)

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