arXiv:0904.2301arXiv:0904.2301

Unitarity Tests of Mixing MatricesUnitarity Tests of Mixing Matrices

The quark sectorThe quark sector

More general W- interaction with quarksMore general W- interaction with quarks

Example: Example:

Left-Right symmetric model with more than Left-Right symmetric model with more than

3 generations can induce right-handed 3 generations can induce right-handed

current to Vcurrent to VCKM CKM and make the 3x3 first 3 and make the 3x3 first 3

generation Vgeneration VCKM CKM non-unitary.non-unitary.

Use unitary gauge if not known the full particle contentsUse unitary gauge if not known the full particle contents to to

calculate loops.calculate loops.He, Tandean & ValenciaHe, Tandean & Valencia

Mixing in Quarks and LeptonsMixing in Quarks and Leptons

Xiao-Gang HeXiao-Gang HeDepartment of Physics and Center for Theoretical SciencesDepartment of Physics and Center for Theoretical Sciences

NTU, TaipeiNTU, Taipei

1.1.Mixing in Quarks and NeutrinosMixing in Quarks and Neutrinos

2.2.Quark -Lepton ComplementarityQuark -Lepton Complementarity

3.3.Unitarity Tests of Mixing MatricesUnitarity Tests of Mixing Matrices

4.4.ConclusionsConclusions

1. Mixing in Quarks and Neutrinos1. Mixing in Quarks and Neutrinos

Quark mixingQuark mixingA convenient parameterization: The Wolfenstein parameterizationA convenient parameterization: The Wolfenstein parameterization

Neutrino MixingNeutrino Mixing

Three light neutrinos, Z decay, N = 2.983\pm 0.009, 3 neutrino mixingThree light neutrinos, Z decay, N = 2.983\pm 0.009, 3 neutrino mixing

Summary of mixing anglesSummary of mixing angles

Good approximation for neutrino mixing:Good approximation for neutrino mixing:

The tri-bimaximal matrixThe tri-bimaximal matrix

Harrison, Perkins & Scott; Zhi-Zhong Xing; Harrison, Perkins & Scott; Zhi-Zhong Xing; Xiao-Gang He & A. ZeeXiao-Gang He & A. Zee

Good approximation for quark mixing:Good approximation for quark mixing:

The unit matrixThe unit matrix

Very different mixing patterns for quarks and Very different mixing patterns for quarks and neutrinos!neutrinos!But: But:

Some interesting featuresSome interesting features

Natural Zero-th order mixing matricesNatural Zero-th order mixing matrices

The natural 0-th order mixing matrix for quarkThe natural 0-th order mixing matrix for quark

The natural 0-th order mixing matrix for neutrino The natural 0-th order mixing matrix for neutrino – tri-bimaximal mixing– tri-bimaximal mixing

Babu and He, He, Keum & VolkasBabu and He, He, Keum & Volkas

The natural 0-th order mixing matrix for neutrino The natural 0-th order mixing matrix for neutrino – tri-bimaximal mixing- Independent of lepton masses– tri-bimaximal mixing- Independent of lepton masses

Good approximation for neutrino mixing:Good approximation for neutrino mixing:

The tri-bimaximal matrixThe tri-bimaximal matrix

Harrison, Perkins & Scott; Zhi-Zhong Xing; Xiao-Gang He & A. ZeeHarrison, Perkins & Scott; Zhi-Zhong Xing; Xiao-Gang He & A. Zee

Good approximation for quark mixing:Good approximation for quark mixing:

The unit matrixThe unit matrix

But:But:

Hint some deeper reason? Q-L ComplementarityHint some deeper reason? Q-L ComplementarityHe, Li & MaHe, Li & Ma

2. Quark –Lepton Complementarity2. Quark –Lepton Complementarity

QLC - Start with unit matrix for quark mixingQLC - Start with unit matrix for quark mixing

If one takes unit matrix for the 0th quark mixingIf one takes unit matrix for the 0th quark mixing

N. Li & B.-Q. MaN. Li & B.-Q. Ma

The corresponding QLC predicted mixing for The corresponding QLC predicted mixing for

lepton is the bi-maximal mixing of the formlepton is the bi-maximal mixing of the form

The corresction is of order Wolfenstein parameters.The corresction is of order Wolfenstein parameters.

A better 0th order expansion for quarks?A better 0th order expansion for quarks?

A new proposal: Tri-minimal parameterizationA new proposal: Tri-minimal parameterizationS.-W.Li & Q.-Q. MaS.-W.Li & Q.-Q. Ma

Much faster convergence than Much faster convergence than Wolfenstein parameterization!Wolfenstein parameterization!

He Li & MaHe Li & Ma

Exact Q-L complementarityExact Q-L complementarity

With deviations With deviations

A theoretical understanding of Q-L A theoretical understanding of Q-L compplementarity is still lacking! compplementarity is still lacking!

3. Unitarity Tests of Mixing Matrices3. Unitarity Tests of Mixing Matrices

The quark sectorThe quark sector

More general W- interaction with quarksMore general W- interaction with quarks

Example: Example:

Left-Right symmetric model with more than Left-Right symmetric model with more than

3 generations can induce right-handed 3 generations can induce right-handed

current to Vcurrent to VCKM CKM and make the 3x3 first 3 and make the 3x3 first 3

generation Vgeneration VCKM CKM non-unitary.non-unitary.

Use unitary gauge if not known the full particle contents.Use unitary gauge if not known the full particle contents.

He, Tandean & Valencia, Xiao et al. ,,,He, Tandean & Valencia, Xiao et al. ,,,

There are rooms for violation of unitarity. Further tests are neededThere are rooms for violation of unitarity. Further tests are needed

Neutrino MixingNeutrino Mixing

Three light neutrinos, Z decay, N = 2.983\pm 0.009, 3 neutrino mixingThree light neutrinos, Z decay, N = 2.983\pm 0.009, 3 neutrino mixing

The lepton sectorThe lepton sector

Summary from Valle.Summary from Valle.

Still has large room for non-unitarity at 10 percent level , something Still has large room for non-unitarity at 10 percent level , something

new may be there to make it happen. Example: Seesaw models.new may be there to make it happen. Example: Seesaw models.

Large light and heavy neutrino mixing in Seesaw modelsLarge light and heavy neutrino mixing in Seesaw models

Can Seesaw Models cause large non-unitary Can Seesaw Models cause large non-unitary deviation in Udeviation in UPMNSPMNS??

Naively, No! … But …Naively, No! … But …

Constraints on elements in UConstraints on elements in U\nu N\nu N

Kerstin & Smirnov; Xing et al.; He, Oh, Tandean &WenKerstin & Smirnov; Xing et al.; He, Oh, Tandean &Wen

Possible to have large elements in UPossible to have large elements in U \nu N\nu N and and

therefore observable non-unitarity in lepton mixing!therefore observable non-unitarity in lepton mixing!

andeanandean

Interesting application of large UInteresting application of large U \nu N\nu N at the LHC at the LHCHe, Oh, Tandean & Wen; Li & HeHe, Oh, Tandean & Wen; Li & He

Model realization for large mixing between light and heavy neutrinosModel realization for large mixing between light and heavy neutrinos

X.-G. He and E. Ma, arXiv:0907.2737[hep-ph]X.-G. He and E. Ma, arXiv:0907.2737[hep-ph]

Seesaw 3+3 or 3+2?Seesaw 3+3 or 3+2?

The UThe UPMNS PMNS is not unitary in any case.is not unitary in any case.

5. Conclusions5. ConclusionsThe CKM and PMNS mixing matrices for quark and lepton sectors describe related The CKM and PMNS mixing matrices for quark and lepton sectors describe related phenomena well.phenomena well.

The quark mixing is approximated by a unit matrix and the lepton mixing is by the tri-The quark mixing is approximated by a unit matrix and the lepton mixing is by the tri-bimaximal matrix. The tri-bimaximal mixing can be understood from theoretical point bimaximal matrix. The tri-bimaximal mixing can be understood from theoretical point of view.of view.

There are interesting relations between quark and lepton mixing, the quark-lepton There are interesting relations between quark and lepton mixing, the quark-lepton complementarity. Theoretical understanding these relations are poor.complementarity. Theoretical understanding these relations are poor.

There are rooms of violating the unitarity of the mixing matrices both in quark and There are rooms of violating the unitarity of the mixing matrices both in quark and lepton sector. lepton sector.

Seesaw models can give large mixing between light and heavy neutrinos, and Seesaw models can give large mixing between light and heavy neutrinos, and therefore large violation of unitarity in lepton mixing. Theoretical models can be therefore large violation of unitarity in lepton mixing. Theoretical models can be constructed. There are interesting LHC physics may results. constructed. There are interesting LHC physics may results.

The FL symmetry can link 3+3 seesaw model to a 3+2 seesaw model.The FL symmetry can link 3+3 seesaw model to a 3+2 seesaw model.