Comment on Reparametrization Invariance of Quark-Lepton Complementarity
aa r X i v : . [ h e p - ph ] M a r Comment on Reparametrization Invarianceof Quark-Lepton Complementarity
Guan-Nan Li, Hsiu-Hsien Lin, and Xiao-Gang He
1, 2, 3, ∗ INPAC, Department of Physics, Shanghai Jiao Tong University, Shanghai, China CTS, CASTS, and Department of Physics, National Taiwan University, Taipei, Taiwan Department of Physics, National Tsing Hua University, and National Center for Theoretical Sciences, Hsinchu 300, Taiwan
We study the complementarity between quark and lepton mixing angles (QLC), the sum of anangle in quark mixing and the corresponding angle in lepton mixing is π/
4. Experimentally inthe standard PDG parametrization, two such relations exist approximately. These QLC rela-tions are accidental which only manifest themselves in the PDG parametrization. We proposereparametrization invariant expressions for the complementarity relations in terms of the magni-tude of the elements in the quark and lepton mixing matrices. In the exact QLC limit, it is foundthat | V us /V ud | + | V e /V e | + | V us /V ud || V e /V e | = 1 and | V cb /V tb | + | V µ /V τ | + | V cb /V tb || V µ /V τ | = 1.Expressions with deviations from exact complementarity are obtained. Implications of these rela-tions are also discussed. PACS numbers: 12.15.Ff, 14.60.-z, 14.60.Pq, 14.65.-q, 14.60.Lm
Mixing between different generations of fermions in weak interaction is one of the most interesting issues in particlephysics. The mixing is described by an unitary matrix in the charged current interaction of W-boson in the masseigen-state of fermions. Quark mixing is described by the Cabibbo [1]-Kobayashi-Maskawa [2](CKM) matrix V CKM ,and lepton mixing is described by the Pontecorvo [3]-Maki-Nakawaga-Sakata [4] (PMNS) matrix V PMNS with L = − g √ U L γ µ V CKM D L W + µ − g √ E L γ µ V PMNS N L W − µ + H.C. , (1)where U L = ( u L , c L , t L , ... ) T , D L = ( d L , s L , b L , ... ) T , E L = ( e L , µ L , τ L , ... ) T , and N L = ( ν , ν , ν , ... ) T are the left-handed fermion generations. For n-generations, V = V CKM or V PMNS is an n × n unitary matrix.A commonly used form of mixing matrix for three generations of fermions is given by [5, 6], V P DG = c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ s c s s − c c s e iδ − c s − s c s e iδ c c , (2)where s ij = sin θ ij and c ij = cos θ ij are the mixing angles and δ is the CP violating phase. This is the so calledstandard PDG parametrization. If neutrinos are of Majorana type, for the PMNS matrix one should include anadditional diagonal matrix with two Majorana phases diag( e iα / , e iα / ,
1) multiplied to the matrix from right inthe above. The two CP violating Majorana phases do not affect neutrino oscillations. To distinguish different CPviolating phases, the phase δ is sometimes called Dirac CP violating phase. In our later discussions, we will indicatethe mixing angles with superscriptions Q and L for quark and lepton sectors respectively when specification is needed.There are a lot of experimental data on the mixing patterns in both the quark and lepton sectors. For quark mixing,the ranges of the magnitudes of the CKM matrix elements have been very well determined with [6] . ± . . ± . . +0 . − . . ± . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . (3)From the above, we obtain the ranges for mixing angles θ Qij , θ Q = 13 . ◦ ± . ◦ , θ Q = 2 . ◦ ± . ◦ , θ Q = 0 . ◦ ± . ◦ . (4)The CP violating phase has also been determined with δ Q = 68 . ◦ [6]. ∗ Electronic address: [email protected]
Considerable experimental data on lepton mixing have also been accumulated including the recent data fromDAYA-BAY collaboration[8]. The global, 1 σ (3 σ ), fit from neutrino oscillation data pre-DAYA-Bay data gives [7], . +0 . − . ( +0 . − . ) 0 . +0 . − . ( +0 . − . ) 0 . +0 . − . ( +0 . − . )0 . +0 . − . ( +0 . − . ) 0 . +0 . − . ( +0 . − . ) 0 . +0 . − . ( +0 . − . )0 . +0 . − . ( +0 . − . ) 0 . +0 . − . ( +0 . − . ) 0 . +0 . − . ( +0 . − . ) . (5)and the mixing angles are given by [7] θ L = 33 . ◦ ± . ◦ ( ± . ◦ ) , θ L = 40 . ◦ ± . ◦ ( ± . ◦ ) ,θ L = 8 . ◦ ± . ◦ ( ± . ◦ ) . (6)The recent measured non-zero θ at a 5.2 σ level by the DAYA-BAY collaboration is[8] sin (2 θ ) = 0 . ± . stat. ) ± . syst ). Translating into the angle θ , it is θ = 8 . ◦ ± . ◦ . This values agrees with global fitvalue very well, but with a smaller error bar. At present there is no experimental data on the CP violating Diracphase δ L and Majorana phases α i .The mixing angles for quark and lepton sectors, a priori, are unrelated. If there is a way to connect the seeminglyindependent mixing angles in these two sectors, it would gain more insights about fermion mixing. Indeed there is avery nice way to make the connection via the so called quark-lepton complementarity (QLC) [9–12]. Here the QLCrelations refer to θ Q + θ L = π , θ Q + θ L = π . (7)The third angles are approximately zero, θ Q ∼ θ L ∼ θ Q + θ L = 46 . ◦ ± . ◦ , θ Q + θ L = 42 . ◦ ± . ◦ . (8)At present the above relations are still at the phenomenological level. It is far from having a complete theoreticalunderstanding although there are attempt to build theoretical models [12]. Even at the phenomenological level, thereare some questions to address about these relations. One of them is that there are different ways to parameterize themixing matrices. The relation between the angles may only hold in a particular parametrization [13]. Therefore theserelations seem to be parametrization dependent. Let us illustrate this point by working out the values of the anglesin the original KM parametrization [2], V KM = c − s c − s s s c c c c − s s e iδ c c s + s c e iδ s s c s c + c s e iδ c s s − c c e iδ . (9)Using the observed values for the mixing matrices, one would obtain θ Q = 13 . ◦ ± . ◦ , θ Q = 2 . ◦ ± . ◦ , θ Q = 0 . ◦ ± . ◦ θ L = 34 . ◦ ± . ◦ , θ L = 28 . ◦ ± . ◦ , θ L = 14 . ◦ ± . ◦ . (10)We see that only θ Q + θ L is close to π/
4, and the other two angles sums do not have the complementarity relations.It has been shown that there are nine independent ways to parameterize the mixing matrices [14]. We have checked,in details, whether similar complementarity relations hold in some of the other parameterizations. We find that onlythe PDG parametrization has the complementarity property for two different angles. To avoid the complementarityrelations be parametrization dependent, it would be more meaningful to find the relations using quantities which arereparametrization invariant. We will work the scenario that there are two complementarity relations discussed above.To this end we find the magnitudes of the elements in the mixing matrices convenient quantities. They can be usedto represent the complementarity relations. In the following we derive such relations.We will take the usual complementarity relations in eq.(7) as the starting point. By taking sine and cosine on bothsides, we obtain sin( θ Q + θ L ) = sin θ Q cos θ L + cos θ Q sin θ L = 1 √ , cos( θ Q + θ L ) = cos θ Q cos θ L − sin θ Q sin θ L = 1 √ θ Q + θ L ) = sin θ Q cos θ L + cos θ Q sin θ L = 1 √ , cos( θ Q + θ L ) = cos θ Q cos θ L − sin θ Q sin θ L = 1 √ . (11)Using the relations between the angles and elements in the mixing matrices, and taking the ratios of the first twoand the last two equations above, we havetan θ Q + tan θ L − tan θ Q tan θ L = | V us /V ud | + | V e /V e | − | V us /V ud || V e /V e | = 1 , tan θ Q + tan θ L − tan θ Q tan θ L = | V cb /V tb | + | V µ /V τ | − | V cb /V tb || V µ /V τ | = 1 . (12)The complementarity relations can now be written as | V us || V ud | + | V e || V e | + | V us || V ud | | V e || V e | = 1 , | V cb || V tb | + | V µ || V τ | + | V cb || V tb | | V µ || V τ | = 1 . (13)The above relations are the new reparametrization invariant complementarity relations. These relations may tellmore about the mixing matrix elements. One can solve these relations to express ratios of elements of lepton mixingmatrix in terms of the ratios of elements of quark mixing matrix to obtain | V e || V e | = 1 − | V us | / | V ud | | V us | / | V ud | , | V µ || V τ | = 1 − | V cb | / | V tb | | V cb | / | V tb | . (14)Similarly one can express ratios of quark mixing matrix elements in terms of the lepton mixing matrix elements.Experimentally, the quark mixing matrix elements are more precisely known, therefore one can take them as inputto predict the lepton mixing matrix element. We have | V e || V e | = 0 . ± . , | V µ || V τ | = 0 . ± . . (15)Notice that | V τ | > | V µ | . This is consistent with data from neutrino oscillation at 2 σ and 1 σ level for the first andthe second ratios in the above equations, respectively.The tribimaximal mixing is a good approximation for the lepton mixing matrix [15]. Data from global fit andthe recent DAYA-BAY measurement show that the tribimaximal mixing pattern, which predicts θ = 0, has to bemodified. There are models prefer that the elements in the second column to be all equal to 1 / √ V e = 1 / √ | V e | + | V e | >
1. This indicates that the QLC isnot consistent with this type of models. There are also speculations that other columns or row in the tribimaximalmixing is kept unaltered but other elements are modified [16]. If one keeps the first column of the tribimaximal mixingmatrix unaltered, one would have V e = 2 / √
6. Coming with eq.(15), one would obtain V e = 0 .
27. This predicts toolarge a V e outside the 1 σ region allowed by the pre-DAYA-BAY global data fit and more from present DAYA-BAYdata. Also if one keeps the second row of tribimaximal mixing unaltered, one would then predict | V µ | + | V τ | > V e = 0 . V e and V ub do not show up directly. To have some information about the 13entries of the mixing matrices, let us consider the unitarity of the first row and the third column. Eq.(14) suggestthat one can write | V e | = a √ | V ud | + | V us | ) , | V e | = a √ | V ud | − | V us | ) , | V τ | = b √ | V tb | + | V cb | ) , | V µ | = b √ | V tb | − | V cb | ) . (16)One then obtains X i | V ei | = a ( | V ud | + | V us | ) + | V e | = b ( | V cb | + | V tb | ) + | V e | = 1 . (17) a and b are related by b = a ( | V ud | + | V us | ) / ( | V cb | + | V tb | ) = (1 . ± . a .The size for V e depends on what a in the form: | V e | = p − a ( | V ud | + | V us | ). Using current 1 σ allowed valuefor | V e | , a is determined to be smaller than 0.99.There may be modifications to the complementarity relations. The modifications can be written as θ Q + θ L = π/ α .In this case the reparametrization invariant relations will be modified to | V us || V ud | + | V e || V e | + | V us || V ud | | V e || V e | = 1 + (cid:18) − | V us || V ud | | V e || V e | (cid:19) tan α − tan α , | V cb || V tb | + | V µ || V τ | + | V cb || V tb | | V µ || V τ | = 1 + (cid:18) − | V cb || V tb | | V µ || V τ | (cid:19) tan α − tan α . (18) - - Α H degree L R --- R = ¡ V Μ ¥ V Τ ¤ as the function of Α — R = V e2 ¤ V e1 ¤ as the function of Α FIG. 1: Ratios of | V e | / | V e | and | V µ | / | V τ | as functions of deviations α and α , respectively. Expressing lepton mixing matrix elements in terms of the quark mixing elements and the deviations, we have | V e || V e | = 1 + tan α − | V us | / | V ud | (1 − tan α )1 − tan α + | V us | / | V ud | (1 + tan α ) , | V µ || V τ | = 1 + tan α − | V cb | / | V tb | (1 − tan α )1 − tan α + | V cb | / | V tb | (1 + tan α ) . (19)In Fig. 1 we show how | V e | / | V e | and | V µ | / | V τ | depend on α and α . We see that a small deviation away fromthe complementarity relation can cause sizeable difference in the predicted neutrino mixing matrix elements. Withmore precisely measured mixing matrix elements in both the quark and lepton sectors, the QLC and deviations canbe studied more which may give some hints on theoretical models for quark and lepton mixing matrices.This work was supported in part by NSC and NCTS of ROC, NNSF and SJTU 985 grants of PRC. [1] N. Cabibbo, Phys. Rev. Lett. , 531 (1963).[2] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. , 652 (1973).[3] B. Pontecorvo, Sov. Phys. JETP , 429 (1957).[4] Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. , 870 (1962).[5] L.L. Chau and W.Y. Keung, Phys. Rev. Lett. , 1802 (1984).[6] K. Nakamura et al. [Particle Data Group Collaboration], J. Phys. GG , 075021 (2010).[7] G. Fogli et al., Phys. Review. D84 , 053007(2011).[8] F.P.An et al. [DAYA-BAY Collaboration], arXiv:1203.1669[hep-exp].[9] A.Y. Smirnov, Talk given at 2nd International Workshop on Neutrino Oscillations in Venice (NO-VE 2003), Venice, Italy,3-5 Dec 2003. Published in *Venice 2003, Neutrino oscillations* 1-2, arXiv:hep-ph/0402264.[10] H. Minakata and A.Y. Smirnov, Phys. Rev.
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