Casas-Ibarra Parametrization and Unflavored Leptogenesis
aa r X i v : . [ h e p - ph ] J a n Casas-Ibarra Parametrization and Unflavored Leptogenesis
Zhi-zhong Xing ∗ Institute of High Energy Physics and Theoretical Physics Center for Science Facilities,Chinese Academy of Sciences, P.O. Box 918, Beijing 100049, China
Abstract
The Casas-Ibarra parametrization is a description of the Dirac neutrino massmatrix M D in terms of the neutrino mixing matrix V , an orthogonal matrix O and the diagonal mass matrices of light and heavy Majorana neutrinos inthe type-I seesaw mechanism. Because M † D M D is apparently independent of V but dependent on O in this parametrization, a number of authors haveclaimed that unflavored leptogenesis has nothing to do with CP violation atlow energies. Here we question this logic by clarifying the physical meaning of O . We establish a clear relationship between O and the observable quantities,and find that O does depend on V . We show that both unflavored leptogen-esis and flavored leptogenesis have no direct connection with low-energy CPviolation.PACS number(s): 14.60.Pq, 13.10.+q, 25.30.Pt Typeset using REVTEX ∗ E-mail: [email protected] Very compelling evidence for finite neutrino masses and large neutrino mixingangles has been achieved from solar [1], atmospheric [2], reactor [3] and accelerator [4]neutrino oscillation experiments. This exciting breakthrough opens a new window to physicsbeyond the standard electroweak model, because the standard model itself only containsthree massless neutrinos whose flavor states ν α (for α = e, µ, τ ) and mass states ν i (for i = 1 , ,
3) are identical. A very natural and elegant way of generating non-zero but tinymasses m i for ν i is to extend the standard model by introducing three right-handed neutrinosand allowing lepton number violation. In this case, the SU (2) L × U (1) Y gauge-invariant massterms of charged leptons and neutrinos are given by −L mass = l L Y l HE R + l L Y ν ˜ HN R + 12 N c R M R N R + h . c . , (1)where ˜ H ≡ iσ H ∗ , l L denotes the left-handed lepton doublet, and M R is the mass matrixof right-handed neutrinos. After spontaneous gauge symmetry breaking, we are left withthe charged-lepton mass matrix M l = Y l v and the Dirac neutrino mass matrix M D = Y ν v ,where v ≃
174 GeV is the vacuum expectation value of the neutral component of the Higgsdoublet H . The scale of M R can be much higher than v , as right-handed neutrinos belongto the SU (2) L singlet and are not subject to electroweak symmetry breaking. It is thereforenatural to obtain the effective mass matrix for three light neutrinos [5]: M ν ≈ − M D M − M T D . (2)Such a relation is commonly referred to as the type-I seesaw mechanism. Let us denote themass states of three right-handed neutrinos and their corresponding masses as N i and M i (for i = 1 , , m i ∼ v /M i as a naive result, which explainswhy m i is small but non-vanishing. Note that both light and heavy neutrinos are Majoranaparticles in this seesaw picture. Without loss of generality, one usually chooses the basiswith both Y l (or M l ) and M R being diagonal, real and positive (i.e., M l = Diag { m e , m µ , m τ } and M R = c M N ≡ Diag { M , M , M } ). In this basis, Casas and Ibarra (CI) proposed aninteresting parametrization of M D [6]: M D ≈ iV q c M ν O q c M N , (3)where V is the 3 × M ν (i.e., V † M ν V ∗ = c M ν ≡ Diag { m , m , m } ) , and O is a complex orthogonal matrix.Associated with the above seesaw mechanism, the leptogenesis mechanism [7] may natu-rally work to account for the cosmological matter-antimatter asymmetry via the CP-violatingand out-of-equilibrium decays of N i and the ( B − L )-conserving sphaleron processes [8]. TheCP-violating asymmetry between N i → l + H c and N i → l + H decays, denoted as ε i (for i = 1 , , Note that we have tentatively ignored tiny differences between the eigenvalues of M R (or M ν )and the physical masses M i (or m i ). See the next section for a detailed discussion. i = 18 πv ( M † D M D ) ii X j = i (cid:26) F ( x ij ) Im h ( M † D M D ) ij i (cid:27) , (4)where F ( x ij ) = √ x ij { (2 − x ij ) / (1 − x ij ) + (1 + x ij ) ln[ x ij / (1 + x ij )] } with x ij ≡ M j /M i is theloop function of self-energy and vertex corrections. In this unflavored leptogenesis scenario,a non-vanishing ε i depends on the imaginary part of M † D M D . Given the CI parametrizationin Eq. (3), it is straightforward to obtain M † D M D ≈ q c M N O † c M ν O q c M N , (5)which is apparently independent of V but dependent on O . Hence a number of authors havetaken it for granted that unflavored leptogenesis has nothing to do with CP violation at lowenergies (see, e.g., Refs. [10–15]). We find that this conclusion is questionable, because thephysical meaning of O has never been clarified in the literature.The main purpose of this note is to clarify the physical meaning of O in the CIparametrization by establishing a relationship between O and the observable quantities in ageneric type-I seesaw model without any special assumptions. Contrary to the naive obser-vation, we find that O depends not only on the neutrino mixing matrix V but also on thematrix responsible for the charged-current interactions of heavy neutrinos N i . The latter,which has clear physical meaning and is denoted as R , governs the strength of CP violationin V and that in leptogenesis. After a detailed analysis of the correlation between R and V , we draw a general conclusion that both unflavored leptogenesis and flavored leptogenesishave no direct connection with low-energy CP violation. After spontaneous SU (2) L × U (1) Y → U (1) em symmetry breaking, the mass termsin Eq. (1) turn out to be −L ′ mass = E L M l E R + 12 ( ν L N c R ) (cid:18) M D M T D M R (cid:19) (cid:18) ν c L N R (cid:19) + h . c . , (6)where E and ν L represent the column vectors of ( e, µ, τ ) and ( ν e , ν µ , ν τ ) L , respectively. Theoverall 6 × (cid:18) V RS U (cid:19) † (cid:18) M D M T D M R (cid:19) (cid:18) V RS U (cid:19) ∗ = (cid:18) c M ν c M N (cid:19) , (7)where c M ν = Diag { m , m , m } and c M N = Diag { M , M , M } have been defined before.After this diagonalization, the flavor states of light neutrinos ( ν α for α = e, µ, τ ) can beexpressed in terms of the mass states of light and heavy neutrinos ( ν i and N i for i = 1 , , ν α and α (for α = e, µ, τ ) canbe written as −L cc = g √ e µ τ ) L γ µ V ν ν ν L + R N N N L W − µ + h . c . (8)in the basis of mass states. So V is just the neutrino mixing matrix responsible for neutrinooscillations, while R describes the strength of charged-current interactions between ( e, µ, τ )3nd ( N , N , N ). V and R are correlated with each other through V V † + RR † = . Hence V itself is not exactly unitary in the type-I seesaw mechanism and its deviation from unitarityis simply characterized by non-vanishing R .Because both V and R are well-defined in Eq. (8), they can be used to understandthe physical meaning of O in the CI parametrization. To do so, we first derive the seesawrelation from Eq. (7). The latter yields V c M ν V T + R c M N R T = , (9)and S c M ν S T + U c M N U T = M R . (10)If Eq. (7) is rewritten as (cid:18) M D M T D M R (cid:19) (cid:18) V RS U (cid:19) ∗ = (cid:18) V RS U (cid:19) (cid:18) c M ν c M N (cid:19) , (11)we can directly obtain the exact results R = M D U ∗ c M − N , (12)and S ∗ = M − V c M ν . (13)Let us substitute Eqs. (12) and (13) into Eqs. (9) and (10), respectively. Then we arrive at V c M ν V T = − M D (cid:16) U ∗ c M − N U † (cid:17) M T D , (14)and M R = U c M N U T + (cid:16) M − (cid:17) ∗ V ∗ c M ν V † (cid:16) M − (cid:17) † ≈ U c M N U T . (15)The excellent approximation made in Eq. (15) implies that U is essentially unitary. Taking U to be unitary and combining Eqs. (14) and (15), we obtain M ν ≡ V c M ν V T ≈ − M D M − M T D , (16)where V is also unitary in this approximation. Eq. (16) reproduces the seesaw formula givenin Eq. (2). It is obvious that R ∼ S ∼ O ( M D /M R ) holds, and thus the seesaw relationactually holds up to the accuracy of O ( R ) [16].Now we look at the orthogonal matrix O in the CI parametrization. Given the basiswhere M R is diagonal, real and positive, Eq. (15) implies that M R ≈ c M N and U ≈ arevery good approximations. In this case, we get M D ≈ R c M N from Eq. (12). Substitutingthis relation into Eq. (3), we obtain O ≈ − i q c M − ν V † M D q c M − N ≈ − i q c M − ν V † R q c M N , (17)4hich shows that O is definitely dependent on V . It is worth remarking that both V and R ,which are respectively associated with the charged-current interactions of light and heavyMajorana neutrinos, have clear physical meaning. Hence it seems improper to draw theconclusion from Eq. (5) that unflavored leptogenesis is independent of low-energy neutrinomixing and CP violation described by V . If Eq. (17) is substituted into Eq. (5), however,we shall arrive at a much simpler expression M † D M D ≈ c M N R † R c M N . (18)This result is actually straightforward, just because of M D ≈ R c M N . It apparently hasnothing to do with V . So the question becomes whether unflavored leptogenesis dependson V through R . We have known that V is correlated with R via the exact seesaw relationin Eq. (9) and the normalization condition V V † + RR † = . To see this correlation moreclearly, one has to adopt an explicit and self-consistent parametrization of V and R . Here we make use of the parametrization of V ≡ AV and R advocated in Ref. [17]: V = c c ˆ s ∗ c ˆ s ∗ − ˆ s c − c ˆ s ˆ s ∗ c c − ˆ s ∗ ˆ s ˆ s ∗ c ˆ s ∗ ˆ s ˆ s − c ˆ s c − c ˆ s − ˆ s ∗ ˆ s c c c , (19)and A = c c c − c c ˆ s ˆ s ∗ − c ˆ s ˆ s ∗ c − ˆ s ˆ s ∗ c c c c c − c c ˆ s c ˆ s ∗ + c ˆ s ˆ s ∗ ˆ s ˆ s ∗ − c ˆ s c ˆ s ∗ c + ˆ s ˆ s ∗ c ˆ s ˆ s ∗ +ˆ s ˆ s ∗ ˆ s ˆ s ∗ c − ˆ s c ˆ s ∗ c c − c c ˆ s ˆ s ∗ − c ˆ s ˆ s ∗ c − ˆ s ˆ s ∗ c c c c c ,R = ˆ s ∗ c c ˆ s ∗ c ˆ s ∗ − ˆ s ∗ c ˆ s ˆ s ∗ − ˆ s ∗ ˆ s ˆ s ∗ c + c ˆ s ∗ c c − ˆ s ∗ ˆ s ˆ s ∗ + c ˆ s ∗ c c ˆ s ∗ − ˆ s ∗ c ˆ s c ˆ s ∗ + ˆ s ∗ ˆ s ˆ s ∗ ˆ s ˆ s ∗ − ˆ s ∗ ˆ s c ˆ s ∗ c − c ˆ s ∗ c ˆ s ˆ s ∗ − c ˆ s ∗ ˆ s ˆ s ∗ c + c c ˆ s ∗ c c − ˆ s ∗ ˆ s c ˆ s ∗ − c ˆ s ∗ ˆ s ˆ s ∗ + c c ˆ s ∗ c c c ˆ s ∗ , (20)where c ij ≡ cos θ ij and ˆ s ij ≡ e iδ ij sin θ ij with θ ij and δ ij (for 1 ≤ i < j ≤
6) being rotationangles and phase angles, respectively. One can see that V is just the standard parametriza-tion of the unitary neutrino mixing matrix (up to some proper phase rearrangements) [18],and thus non-vanishing A signifies the non-unitarity of V . One can also see that A and R involve the same parameters: nine rotation angles and nine phase angles . If all of them Note that none of the phases of R (or A ) can be rotated away by redefining the phases of threecharged-lepton fields, because such a phase redefinition will also affect the phases of A (or R ), asone can easily see from Eq. (8). R = and A = . In view of the fact that theunitarity violation of V must be very small effects (at most at the percent level as con-strained by current experimental data on neutrino oscillations, rare lepton-flavor-violatingor lepton-number-violating processes and precision electroweak tests [19]), one may treat A as a perturbation to V . The smallness of θ ij (for i = 1 , , j = 4 , ,
6) allows us tomake the following excellent approximations: A = − ( s + s + s ) 0 0ˆ s ˆ s ∗ + ˆ s ˆ s ∗ + ˆ s ˆ s ∗
26 12 ( s + s + s ) 0ˆ s ˆ s ∗ + ˆ s ˆ s ∗ + ˆ s ˆ s ∗ ˆ s ˆ s ∗ + ˆ s ˆ s ∗ + ˆ s ˆ s ∗
36 12 ( s + s + s ) + O ( s ij ) ,R = + ˆ s ∗ ˆ s ∗ ˆ s ∗ ˆ s ∗ ˆ s ∗ ˆ s ∗ ˆ s ∗ ˆ s ∗ ˆ s ∗ + O ( s ij ) (21)with s ij ≡ sin θ ij being real. Note that the approximation made in Eq. (15) is equivalentto A ≈ , leading to unitary V and U . One may therefore take V ≈ V when applying theapproximate seesaw relation in Eq. (2) or (16) to the phenomenology of neutrino mixingand leptogenesis. In this case, Eq. (9) is simplified to V c M ν V T ≈ − R c M N R T . (22)The total number of free parameters in c M ν , c M N , V (or V ) and R is thirty (six masses, twelvemixing angles and twelve CP-violating phases). But either Eq. (9) or Eq. (22) can givetwelve real constraint conditions. Hence we are left with eighteen independent parametersin the type-I seesaw mechanism.Given the approximate expression of R in Eq. (21), it is straightforward to obtainIm (cid:16) R c M N R T (cid:17) ij = − M s i s j sin (cid:16) δ i + δ j (cid:17) − M s i s j sin (cid:16) δ i + δ j (cid:17) − M s i s j sin (cid:16) δ i + δ j (cid:17) , (23)where 1 ≤ i < j ≤
3. In comparison, Eqs. (4) and (18) tell us that the CP-violatingasymmetries ε i (for i = 1 , ,
3) in unflavored leptogenesis are associated withIm (cid:16) R † R (cid:17) = X i =1 s i s i sin ( δ i − δ i ) , Im (cid:16) R † R (cid:17) = X i =1 s i s i sin ( δ i − δ i ) , Im (cid:16) R † R (cid:17) = X i =1 s i s i sin ( δ i − δ i ) . (24)We see that there are in general nine independent phase combinations in Eq. (23), whilethere are only six independent phase combinations in Eq. (24). It is possible to acquireIm( R c M N R T ) = by fine-tuning the free parameters in Eq. (23), such that Im( V c M ν V T ) ≈ V is real) as one can see from Eq. (22). In thisspecial case, there is no low-energy CP violation but viable unflavored leptogenesis is likelyto take place. To achieve a direct connection between the CP-violating phases of V andthe CP-violating asymmetries ε i , one should switch off as many phases of R as possible.Such a treatment can be realized in some specific type-I seesaw models [20], in which thetexture of Y ν (or M D ) might get constrained from a certain flavor symmetry in the basisof M R = c M N . But our general conclusion is that there is only indirect connection betweenunflavored leptogenesis and low-energy observables. The same conclusion as obtained above is true for flavored leptogenesis. When themass of the lightest heavy Majorana neutrino is lower than about 10 GeV, flavor-dependenteffects matter in leptogenesis [21] and have to be carefully handled [22]. In this case, theCP-violating asymmetries ε iα between N i → l α + H c and N i → l cα + H decays (for i = 1 , , α = e, µ, τ ) depend on the phases of M D (or Y ν ) in the following way [23]: ε iα = 18 πv X j = i F ( x ij ) Im h ( M † D M D ) ij ( M ∗ D ) αi ( M D ) αj i | ( M D ) αi | + 11 − x ij · Im h ( M † D M D ) ji ( M ∗ D ) αi ( M D ) αj i | ( M D ) αi | , (25)where the loop function F ( x ij ) with x ij ≡ M j /M i has been given below Eq. (4). Takingaccount of M D ≈ R c M N , we findIm h ( M † D M D ) ij ( M ∗ D ) αi ( M D ) αj i ≈ M i M j Im h ( R † R ) ij R ∗ αi R αj i , Im h ( M † D M D ) ji ( M ∗ D ) αi ( M D ) αj i ≈ M i M j Im h ( R † R ) ∗ ij R ∗ αi R αj i . (26)It has been shown in Eq. (24) that the quantities ( R † R ) ij (for i = j ) rely on six independentphase combinations of R . On the other hand, it is easy to check that the quantities R ∗ αi R αj (for α = e, µ, τ and i = j ) depend on the same phase combinations. Hence non-vanishing ε iα in Eq. (25) and ε i in Eq. (4) originate from the same source of CP violation, nomatter whether there are flavor effects or not. This point keeps unchanged even if resonantleptogenesis [22] is taken into account.If the CI parametrization in Eq. (3) is applied to the description of flavored leptogenesis,then V will show up in the expression of ε iα . The reason is simply that the elements of V cannot cancel out in ( M ∗ D ) αi ( M D ) αj , although they can cancel out in ( M † D M D ) ij . Thisobservation has been used by a number of authors to support the argument that viableflavored leptogenesis may result from V even in the case of O being a real orthogonalmatrix (see, e.g., Refs. [12–15]). Such an argument is certainly not wrong, but it is notprofound either [24]. In view of Eq. (17), we find that O can be real only when nontrivialCP-violating phases in V and R delicately combine to make V † R purely imaginary. Thisextremely special case means nothing but a very special correlation between V and R . Whileone may argue that flavored leptogenesis is linked to the neutrino mixing matrix V in thiscontrived case, one should keep in mind that both ε iα and the CP-violating phases of V actually originate from R and their direct connection can only be established when some7or most) of the phase parameters of R are switched off. In general, however, “there is nocorrelation between successful leptogenesis and the low-energy CP phase” [24] . The CI parametrization, in which the neutrino mixing matrix V and an orthogonalmatrix O are unjustifiedly assumed to be independent of each other, has often been appliedto the phenomenology of neutrino mixing and leptogenesis in the type-I seesaw mechanism.In the present work, we have clarified the physical meaning of O by establishing a relationshipbetween O and the observable quantities in a generic type-I seesaw model without any specialassumptions. We find that O depends not only on V but also on R , the matrix responsiblefor the charged-current interactions of heavy Majorana neutrinos. The CP-violating phasesof R govern the strength of CP violation at low energies and that in leptogenesis. Wehave examined the dependence of unflavored or flavored leptogenesis on R and analyzed thecorrelation between R and V . Our general conclusion is that both unflavored leptogenesisand flavored leptogenesis have no direct connection with low-energy CP violation.Let us finally give some remarks on R , which makes more sense than O in the analysis ofleptogenesis. If the type-I seesaw mechanism could be realized at the TeV scale, it might bepossible to measure or constrain the mixing angles of R at the Large Hadron Collider andprobe the CP-violating phases of R at a neutrino factory [25]. Because non-vanishing R isa clean signature of the unitarity violation of V , it can actually lead to rich phenomenologyof lepton-flavor-violating and lepton-number-violating processes. In particular, R bridgesa gap between high-energy neutrino physics (e.g., heavy neutrino decays and leptogenesis)and low-energy neutrino physics (e.g., neutrino mixing and neutrino oscillations).The author would like to thank S. Zhou for many useful discussions. This work wassupported in part by the National Natural Science Foundation of China under grant No.10425522 and No. 10875131. This conclusion was drawn in Ref. [24] from a very detailed analysis of the sensitivity of leptoge-nesis to the neutrino mixing matrix V by using the CI parametrization and allowing the elementsof O to take arbitrary values in the parameter space. Here we arrive at the same conclusion byclarifying the physical meaning of O in an analytic way. EFERENCES [1] SNO Collaboration, Q.R. Ahmad et al. , Phys. Rev. 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