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E. Witten – Opening Talk at Neutrino 00 [hep-ph/0006332] . Zhi-zhong Xing ( 邢志忠 ) IHEP, Beijing . Theoretical Aspects of Neutrino Mass. I’ll cover some aspects of interesting attempts. OCPA Workshop on Underground Science, 21-23 July 08, HK. when detectable? . under water or ice.

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theoretical aspects of neutrino mass

E. Witten–Opening Talk at Neutrino 00 [hep-ph/0006332]

Zhi-zhong Xing (邢志忠)

IHEP, Beijing

Theoretical Aspects of Neutrino Mass

I’ll cover some aspects of interesting attempts.

OCPA Workshop on Underground Science, 21-23July 08, HK


when detectable?



or ice



Neutrino Astronomy


What / where / how can we do?

中科院未来五十年战略发展: 重大交叉学科之“宇宙起源, 超弦理论, 中微子, 暗物质与暗能量”路线图: 建立国家地下实验室?

Underground Business

Physics Potential of a large multi-purpose underground infrastructure with a sufficiently big detector (100 – 1000 ktons?):

◆ Solar interior: various solar neutrinos and precision test of the SSM;

◆ Atmospheric neutrinos: high-statistics study;

◆ Earth interior: geo-neutrinos;

◆ Supernova neutrinos: information on the dynamics of SN explosion;

◆ Dark matter search: with / without the help of detecting neutrinos;

◆ Reactor and accelerator neutrinos: long-baseline oscillations;

◆ Proton decay: better sensitivity;

◆ ……


Part (B):heavy Majorana neutrinos: ≤TeV or ≥EeV?


My Focus

Part (A): free parameters of known (light) neutrinos:







CP phases


Part (A)

(Smirnov 07)


A pureDiracmass term added into the SM is theoretically not favored:

---- it worsens the problem of the large lepton & quark mass hierarchy;

---- it violates ‘t Hooft’s naturalness criterion, as a Majorana mass term of right-handed ’s is not forbidden by the SM gauge symmetry;

---- it imposes the ad hoc assumption of L conservation on the theory.

And thus most theorists believe that massive ’s are Majorana particles and their salient feature is lepton number violation.


Neutrinos aremasslessin the SM, a result of the SM’s simple structure:

---- it has no right-handed ’s (weak-interaction experiments have not required them, so theorists did not introduce them into the SM);

Dirac mass term is not allowed.

---- it conserves the SU(2)_L gauge symmetry, and it only contains the Higgs doublet (the SM accidently possesses (B-L) symmetry);

Majorana mass term is forbidden.


Latest global 3-oscillation analysis (Fogli et al., arXiv:0805.2517):


Masses: Known

(1) 3-masses m_1, m_2 and m_3 must have non-degenerate values;

(2) One of them must be about or larger than 0.05 eV;

(3) The upper bound of m_i is expected to be about 1 eV, or even less.

Absolute mass scale:

: single  decay;

0: neutrinoless  decay;

CMB: cosmological constraints.



HM claim



95% C.L.



Combined Constraints

Constraints from-oscillations++0 claim + CMB (Fogli et al., 08)


HM claim: 0.16 – 0.52 eV



95% C.L.


B:m_1 and m_3, which is bigger? ---- normal or inverted hierarchy?

Comment: Long-baseline neutrino oscillations + matter effect can tell.

C: Can one mass be vanishing (or vanishingly small)? ---- it is possible.

Comment: Flavor symmetry may work to assure m_1 = 0 or m_3 = 0.



Masses: Unknown

A: If the HM (Heidelberg-Moscow) claim is not true, then the absolute mass scale of 3 neutrinos remains an open question.

Comment: I personally like a near mass degeneracy of 3 neutrinos, in particular if neutrinos are Majorana particles.

Dirac (charged) fermions: strong mass hierarchy ---- limit 0 : 0 : 1; Majorana (neutral) neutrinos: near mass degeneracy ---- limit 1 : 1 : 1.


Translational FL symmetry:

Mass matrix:

one neutrino is massless!

Comment 1:it is easy to extend the FL symmetry to the Majorana case;

Comment 2:one may obtain an effective -mass matrix with m_1 = 0 / m_3 = 0 by combining the FL symmetry with the seesaw mechanism.

Xing, Zhang, Zhou (PLB 06)

Luo, Xing, (PLB 07)

Jarlskog (PRD 08) Liao et al (PRD08)

The FL symmetry forces one massless heavy Majorana neutrino to decouple: M_R rank 2.

m_1 = 0 or m_3 = 0

Example: Friedberg-Lee symmetry on the effective -mass operator at low energy scale (hep-ph/0606071):


Neutrino Flavor Violation

Lepton Flavor Violation

Phenomenological assumption: neutrino flavor mixing can be described by a 3 × 3 unitary matrix containing 3 mixing angles and 3 CPV phases.

Warning: the dynamics of flavor mixing relies on the theory of -mass generation. Hence we don’t know if this assumption is correct or not.


Flavor mixingsignifies a mismatch between the weak-interaction states and the mass (or free propagation) states of leptons.

Neutrino Oscillations


Latest global 3-oscillation analysis (Fogli et al., arXiv:0805.2517):


theta_23: seems < /4 (at < 1  C.L.);

theta_12: is quite close to ~ 35.3°;

theta_13: hints of > 0 (at ~ 1  C.L.).

95% C.L. (2)

(Fogli et al., arXiv:0806.2649)

Angles: Known

(a) Angle theta_23 is large and close to /4, suggestive of something?

(b) Angle theta_12 is large and seems to lie between /6and 35.3°.

(c) Angle theta_13 is not large and its upper bound is about 10°.


Hints of theta_13 > 0

Fogli et al., arXiv:0806.2649


B: Why theta_12 and theta_23 are large and close to 2 special values?

Comment: Very strong hints at a certain (underlying) flavor symmetry.

mysterious 35.3°

T.D. Lee,


Angles: Unknown

A: What is the value of theta_13? Is it small? And how small is small?

Comment: Reactor & accelerator -oscillation experiments can answer, but can they answer before a global fit yields definite “prediction”?


Example:0, 1, 2, 3.

(Harrison, Perkins, Scott 02; Xing 02)

Guiding Principle

Tri-bimaximal -mixing

Flavor Symmetry

S3 , S4 , A4 , Z2 , …... U(1)F , SU(2)F , .….. Friedberg-Lee, …...

A realization of the tri-bimaximal -mixing in the Friedberg-Lee model:

-symmetry breaking

Tri-bimaximal Mixing

One may play games with a few small integers & their square roots (an economical group language)



-mass matrix:

Tetra-maximal Mixing?

Noise to Tri-bimax:0, 1, 2 and i (Xing, arXiv:0805.0416, PRD in press):


Jarlskog parameter is a rephasing-invariant measure of CP / T violation in neutrino oscillations:

J is maximal when theta_12 = theta_23 = /4, theta_13 = 35.3°and delta = /2.

It will be impossible to see CP Violation if theta_13 is too small.


(a) Phase  is relevant to the strength of CP violation in -oscillations;

(b) Phases  and  are associated with the LN-violating 0-process;

(c) They are entangled with one another in the RGE running.

Three phases are entirely unconstrained by current experimental data.


Neutrino masses, if they are of Majorana nature, must have a different origin compared to the masses of charged leptons and quarks.

The observation of 0 must imply the Majorana nature of ’s, but it may not uniquely & directly point to -masses and mixing parameters.

Given the SM interactions, a massive Dirac neutrino can get tiny (one-loop) magnetic dipole moment:

A massive Majorana neutrino cannot have magnetic and electric dipole moments, because its anti-particle is just itself.

More on Majorana


Both Dirac and Majorana’s can have transition dipole moments, of a size comparable with _ .

The electro-magnetic dipoles of massive ’s can produce a variety of rare processes:

---- neutrino decays;

---- scattering with electrons;

---- interaction with an external magnetic field (sun, supernovae, red-giant stars, …);

---- contributions to neutrino masses.

Current experimental bound on dipole moments (Kayser, Neutrino 08):

New Physics?

More on Dipole Moment


A Theoretician’s Personal Roadmap of Model Building (King 08)

where are the signposts?


numerous -experiments

Model Building?


Part (B)

(P5 Report 08)


The Money Frontier



A natural theoretical way to understand why 3 -masses are very small.

Type-one Seesaw(Minkowski 77, Yanagida 79, Glashow 79, Gell-Mann, Ramond, Slanski 79, Mohapatra, Senjanovic 79).

Triplet Seesaw (Magg, Wetterich 80, Schechter, Valle 80, Lazarides, Shafi, Wetterich 80, Mohapatra, Senjanovic 80, Gelmini, Roncadelli 80).



Type-II Seesaw (a few right-handed Majorana neutrinos and one Higgs triplet are both added into the SM).

Why Seesaws?


Is theseesaw scalevery close to a fundamental physics scale?

How heavy are the heavy Majorana neutrinos or the Higgs triplet?



to unify strong, weak & electromagnetic forces?

Conventional (Type-one)SeesawPicture: close to the GUT scale

TeV SeesawIdea: driven by testability at LHC


to solve the unnatural gauge hierarchy problem?




Why TeV Seesaws?


0.01 eV

100 GeV

Unnatural case: large cancellation in the leading seesaw term.

0.01 eV

100 GeV

TeV-scale (right-handed) Majorana neutrinos: small masses of light Majorana neutrinos come from sub-leading perturbations.

Type-I Seesaw

Natural case: no large cancellation in the leading seesaw term.


Given diagonal M_R with 3 eigenvalues M_1, M_2 and M_3, the leading (i.e., type-I seesaw) term of the light neutrino mass matrix vanishes, if and only if M_D has rank 1,

and if

(Buchmueller, Greub 91; Ingelman, Rathsman 93; Heusch, Minkowski 94; ……; Kersten, Smirnov 07).

Tiny -masses can be generated from tiny corrections to this complete “structural cancellation”, by deforming M_D or M_R .

Simple example:

Structural Cancellation


Incomplete cancellation between two leading terms of the light neutrino mass matrix in type-II seesaw scenarios. The residue of this incomplete cancellation generates the neutrino masses:

tiny mass








(Chao, Luo, Z.Z.X., Zhou, PRD08)

Discrete flavor symmetries may be used to arrange the textures of two mass terms, but fine-tuning seems unavoidable in the (Big – Big) case.

Collider signatures: both heavy Majorana neutrinos and doubly-charged scalars are possible to be produced at the LHC (e.g., Azuleos et al06; del Aguila et al07; Han et al 07; ….). But decoupling between collider physics & the mechanism of neutrino mass generation is very possible.

Type-II Seesaw


L = 2like-sign dilepton events


  • Lesson 1:two necessary conditions to test a seesaw model with heavy right-handed Majorana neutrinos at the LHC:
  • Masses of heavy Majorana neutrinos must be of O (1) TeV or below;
  • (B) Light-heavy neutrino mixing (i.e., M_D/M_R) must be large enough.

Lesson 2:LHC-collider signatures of heavy Majorana ’s are essentially decoupled from masses and mixing parameters of light Majorana ’s.

Lesson 3:non-unitarity of the light neutrino flavor mixing matrix might lead to observable effects in neutrino oscillations and rare processes.

Lesson 4: nontrivial limits on heavy Majorana neutrinos can be derived at the LHC, if the SM backgrounds are small for a specific final state.


Lepton number violation: like-sign dilepton events at hadron colliders, such as Tevatron (~2 TeV) and LHC (~14 TeV).

collider analogue to 0 decay

dominant channel

N can be produced on resonance

Collider Signature


Chao, Si, Xing, Zhou arXiv:0804.1265



A single heavyN

(minimal Type-II)

Just for Illustration

Han, Zhang (hep-ph/0604064, PRL): cross sections are generally smaller for larger masses of heavy Majorana neutrinos.

Del Aguila et al (hep-ph/0606198): signal & background cross sections (in fb) as a function of the heavy Majorana neutrino mass (in GeV).


The scheme of Minimal Unitarity Violation (Antusch et al 07):

---- Only 3 light neutrino species are considered;

---- Sources of non-unitarity are allowed only in those terms of the SM Lagrangian which involve neutrinos.

Constraint on the3×3-mixing matrixV---- data on -oscillations, Wand Z decays, rare LFV modes and lepton universality tests, …... (Antusch et al 07):

Non-unitarity of V ?

Example A: light sterile neutrinos --- no good TH / EX motivation today.

Example B: heavy Majorana neutrinos --- popular seesaw mechanisms.

Example C: whole tower of KK states --- models with extra dimensions.




V = AV_0

Standard Parametrization of V and R(Xing, PLB08):

The deviation of A from 1 measures the strength of unitarity violation of V .

Correlated CC-interactions:

A and R contain 9 new mixing angles and 9 new CP-violating phases.

Charged Current Interactions


All 9 rotation angles are expected to be small, at most of O(0.1), 9 CPV phases may be large to generate new CP-violating effects.

A and R


If the unitarity violation of V is close to the percent level, then elements of Rcan reach order of 0.1, leading to appreciable collider signatures for TeV-scale Majorana neutrinos.

New CP-violating effects, induced by the non-unitarity of V, may show up in (short-baseline) neutrino oscillations.

Such a parametrization turns out to be very useful in –phenomenology.


Example:V_0takes the tri-bimaximal mixing pattern which has

Non-unitaryV takes the simple form

CP violation (9 Jarlskog invariants):



UV-induced CPV


Oscillation probability in vacuum (e.g.,Antusch et al06, Z.Z.X. 08):

Short- or medium-baseline experiments in the neglect of matter effects (Fernandez-Martinez et al07). In particular (Z.Z.X. 08),


UV-induced CPV at 1% level?

Neutrino Oscillations




Sensitivity ≤ 1% ?

Numerical Illustration

Example: an experiment with E  a few GeV&L ~ a few 100 km.


(Goswami, Ota 08; Luo 08)

Genuine CPV

Matter effect

The same matter-effect term appears in _ _ oscillations (Luo 08).

Matter Effects

Illustration: one heavy Majorana neutrinoand constant matter density.


Resonant enhancement with 4 heavy Majorana neutrinos. (Bray, Lee, Pilaftsis 07)

CPV at Colliders?

CP violation: interference between tree and one-loop amplitudes of N.


CP violation

L-number asymmetry

B-number asymmetry


Canonical idea(Fukugita, Yanagida 86):

●Lepton number violation at the tree level of Majorana neutrinodecays;

●Direct CP violation at the one-loop level of Majorana neutrino decays;

● At least 2 Majorana neutrinos are required.

Developments and variations(Davidson, Nardi, Nir, Phys. Rept. 08):

●Recent developments: spectator processes; finite temperature effects; flavor effects; N_2 leptogenesis; resonant (TeV) leptogenesis; ……

●Some variations: soft (SUSY) leptogenesis; type-II leptogenesis; Dirac leptogenesis; electromagnetic leptogenesis; ……


Naturalness of the SM implies that there should exist a kind of new physics at the TeV scale. We wonder whether it is also responsible for the neutrino mass generation ---- TeV seesaws.

It seems that theorists are struggling for a balance between THnaturalness and EXtestability as the guiding principle. Let’s hope so in the era of LHC + precision -experiments!

Concluding Remarks

Established new physics: at least 2 of 3 known ’s must be massive. We wonder whether their tiny masses imply new DoF such as heavy Majorana neutrinos (more exciting new physics).

It seems that theorists are facing a new problem in looking for the true theory of -mass generation, flavor mixing and CP violation ---- uniqueness (or more credit?).


Let’s Do Something Somewhere Underground!