NEUTRINOS - ICTP – SAIFR · 3 - Mass-Induced Neutrino Flavor Oscillations Neutrino Flavor change can arise out of several di erent mechanisms. The simplest one is to appreciate

Andre de Gouvea Northwestern

NEUTRINOS

Andre de Gouvea

Northwestern University

School and Workshop on Dark Matter and Neutrino Detection

ICTP-SAIFR

July 23 - August 3, 2018

July 23–26, 2018 Neutrinos


Tentative Outline for The Lectures

1. Brief History of the Neutrino;

2. Neutrino Puzzles – The Discovery of Neutrino Masses;

3. Neutrino Oscillations;

4. What We Know We Don’t Know;

5. Neutrino Masses As Physics Beyond the Standard Model;

6. Some Ideas for Tiny Neutrino Masses, and Some Consequences.

[note: Questions/Suggestions/Complaints are ALWAYS welcome]



Some Neutrino references (WARNING: Biased Sample)

• “Are There Really Neutrinos? – An Evidential History,” Allan Franklin, Perseus

Books, 2001. Good discussion of neutrino history.

• A. de Gouvea, “TASI lectures on neutrino physics,” hep-ph/0411274;

• R. N. Mohapatra et al., “Theory of neutrinos: A White paper,” Rept. Prog. Phys.

70, 1757 (2007) [hep-ph/0510213];

• R. N. Mohapatra, A. Yu. Smirnov,“Neutrino Mass and New Physics,” Ann. Rev.

Nucl. Part. Sci. 56, 569 (2006) [hep-ph/0603118];

• M. C. Gonzalez-Garcia, M. Maltoni, “Phenomenology with Massive Neutrinos,”

Phys. Rept. 460, 1 (2008) [arXiv:0704.1800 [hep-ph]];

• A. Strumia, F. Vissani, “Neutrino masses and mixings,” hep-ph/0606054 (2010);

• “The Physics of Neutrinos,”V. Barger, D. Marfatia, K. Whisnant, Princeton

University Press (2012);

• “J. Hewett et al., “Fundamental Physics at the Intensity Frontier,” arXiv:1205.267;

• A. de Gouvea et al., “Working Group Report: Neutrinos,” arXiv:1310:4340;

• A. de Gouvea, “Neutrino Mass Models,” Ann. Rev. Nucl. Part. Sci. 66, 197 (2016).



1 - Brief History of the Neutrino

1. 1896: Henri Becquerel discovers natural radioactivity while studying

phosphorescent properties of uranium salts.

• α rays: easy to absorb, hard to bend, positive charge, mono-energetic;

• β rays: harder to absorb, easy to bend, negative charge, spectrum?;

• γ rays: no charge, very hard to absorb.

2. 1897: J.J. Thompson discovers the electron.

3. 1914: Chadwick presents definitive evidence for a continuous β-ray

spectrum. Origin unkown. Different options include several different energy

loss mechanisms.

It took 15+ years to decide that the “real” β-ray spectrum was really

continuous. Reason for continuous spectrum was a total mystery:

• QM: Spectra are discrete;

• Energy-momentum conservation: N → N ′ + e− — electron energy and

momentum well-defined.



Nuclear Physics before 1930: nucleus = npp+ nee−.

Example: 4He = 4p+ 2e−, works well. However: 14N = 14p+ 7e− is expected to

be a fermion. However, it was experimentally known that 14N was a boson!

There was also a problem with the magnetic moment of nuclei: µN , µp � µe

(µ = eh/4mc). How can the nuclear magnetic moment be so much smaller than

the electron one if the nucleus contains electrons?

SOLUTION: Bound, nuclear electrons are very weird!

This can also be used to solve the continuous β-ray spectrum: energy need not

be conserved in nuclear processes! (N. Bohr)

“... This would mean that the idea of energy and its conservation fails in dealing

with processes involving the emission and capture of nuclear electrons. This

does not sound improbable if we remember all that has been said about peculiar

properties of electrons in the nucleus.” (G. Gamow, Nuclear Physics Textbook,

1931).



enter the neutrino. . .

1. 1930: Postulated by Pauli to (a) resolve the problem of continuous β-ray

spectra, and (b) reconcile nuclear model with spin-statistics theorem. ⇒

2. 1932: Chadwick discovers the neutron.

neutron 6= Pauli’s neutron = neutrino (Fermi);

3. 1934: Fermi theory of Weak Interactions – current-current interaction

H ∼ GF (pΓn) (eΓνe) , where Γ = {1, γ5, γµ, γµγ5, σµν}

Way to “see” neutrinos: νe + p→ e+ + n. Prediction for the cross-section -

too small to ever be observed...

4. 1935: (Yukawa postulates the existence of mesons (pions) as mediators of

the nuclear (strong) force: mπ ∼ 100 MeV.)

5. 1936/37: (“Meson” discovered in cosmic rays. Another long, tortuous story.

Turns out to be the muon...)

6. 1947: (Marshak, Bethe postulate the 2 meson hypothesis (π → µ). Pion

observed in cosmic rays.)





observing the unobservable:

1. 1956: “Discovery” of the neutrino (Reines and Cowan) in the Savannah

River Nuclear Reactor site. ⇒νe + p→ e+ + n. Measure positron (e+e− → γs) and neutron

(nN → N∗ → N + γs) in delayed coincidence in order to get rid of

backgrounds.

2. 1958: Neutrino Helicity Measured (Goldhaber et al.). Neutrinos are purely

left-handed. Interact only weakly (Parity violated maximally).

e− +152 Eu(J = 0)→152 Sm∗(J = 1) + ν →152 Sm(J = 1) + ν + γ

3. 1962: The second neutrino: νµ 6= νe (Lederman, Steinberger, Schwarts at

BNL). First neutrino beam.

p+ Z → π+X → µ+νµ ⇒νµ + Z → µ− + Y (“always”)

νµ + Z → e− + Y (“never”)

4. 2001: ντ directly observed (DONUT experiment at FNAL). Same strategy:

ντ + Z → τ− + Y . (τ -leptons discovered in the 1970’s). ⇒









What we Knew of Neutrinos: End of the 20th Century

• come in three flavors (see figure);

• interact only via weak interactions (W±, Z0);

• have ZERO mass – helicity good

quantum number;

• νL field describes 2 degrees of freedom:

– left-handed state ν,– right-handed state ν (CPT conjugate);

• neutrinos carry lepton number:– L(ν) = +1,

– L(ν) = −1.



2– Neutrino Puzzles – 1960’s to 2000’s

Long baseline neutrino experiments have revealed that neutrinos change

flavor after propagating a finite distance, violating the definitions in the

previous slide. The rate of change depends on the neutrino energy Eν and

the baseline L.

• νµ → ντ and νµ → ντ — atmospheric experiments [“indisputable”];

• νe → νµ,τ — solar experiments [“indisputable”];

• νe → νother — reactor neutrinos [“indisputable”];

• νµ → νother — from accelerator experiments [“indisputable”].









The SNO Experiment: conclusive evidence for flavor change

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

)-1 s-2 cm6

(10eφ

)-1

s-2

cm

6 (

10τµφ SNONCφ

SSMφ

SNOCCφSNO

ESφSNO Measures:

[CC] νe +2H → p+ p+ e−

[ES] ν + e− → ν + e−

[NC] ν +2H → p+ n+ ν

different reactions

sensitive to different

neutrino flavors.





UP 6= DOWN – neutrinos can tell time! → neutrinos have mass.



3 - Mass-Induced Neutrino Flavor Oscillations

Neutrino Flavor change can arise out of several different mechanisms. The

simplest one is to appreciate that, once neutrinos have mass, leptons

can mix. This turns out to be the correct mechanism (certainly the

dominant one), and only explanation that successfully explains all

long-baseline data consistently.

Neutrinos with a well defined mass:

ν1, ν2, ν3, . . . with masses m1,m2,m3, . . .

How do these states (neutrino mass eigenstates) relate to the neutrino

flavor eigenstates (νe, νµ, ντ )?

να = Uαiνi α = e, µ, τ, i = 1, 2, 3

U is a unitary mixing matrix. I’ll talk more about it later.



The Propagation of Massive Neutrinos

Neutrino mass eigenstates are eigenstates of the free-particle Hamiltonian:

|νi〉 = e−iEit|νi〉, E2i − |~pi|2 = m2

i

The neutrino flavor eigenstates are linear combinations of νi’s, say:

|νe〉 = cos θ|ν1〉+ sin θ|ν2〉.

|νµ〉 = − sin θ|ν1〉+ cos θ|ν2〉.

If this is the case, a state produced as a νe evolves in vacuum into

|ν(t, ~x)〉 = cos θe−ip1x|ν1〉+ sin θe−ip2x|ν2〉.

It is trivial to compute Peµ(L) ≡ |〈νµ|ν(t, z = L)〉|2. It is just like a two-level

system from basic undergraduate quantum mechanics! In the ultrarelativistic

limit (always a good bet), t ' L, Ei − pz,i ' (m2i )/2Ei, and

Peµ(L) = sin2 2θ sin2(

∆m2L4Eν

)



L(a.u.)

P eµ =

1-P

ee

sin22θ

Losc

π LLosc≡ ∆m2L

4E = 1.267(L

km

) (∆m2

eV2

) (GeVE

)

amplitude sin2 2θ{oscillation parameters:



CHOOZ experiment

Pee = 1− sin2 2θ sin2(

∆m2L4E

)

result: 1− Pee < 0.05

low ∆m2: 1− Pee ∝ sin2 2θ(∆m2)2

high ∆m2: 1− Pee ∝ 12

sin2 2θ



There is a long (and oftentimes confused and confusing) history behind

this derivation and several others. A comprehensive discussion can be

found, for example, in

E.K. Akhmedov, A. Yu. Smirnov, 0905.1903 [hep-ph]

In a nutshell, neutrino oscillations as described above occur whenever

• Neutrino Production and Detection are Coherent → cannot “tell” ν1

from ν2 from ν3 but “see” νe or νµ or ντ .

• Decoherence effects due to wave-packet separation are negligible →baseline not too long that different “velocity” components of the

neutrino wave-packet have time to physically separate.

• The energy released in production and detection is large compared to

the neutrino mass → so we can assign all of the effect to the neutrino

propagation, independent from the production process. Also assures

ultra-relativistic approximation good.



Pµµ ∼ 1

↓

Pµµ∼1− 12

sin2 2θ

↖

Pµµ = 1− sin2 2θ sin2(

∆m2L4E

)Works great for sin2 2θ ∼ 1 and ∆m2 ∼ 10−3 eV2



[Gonzalez-Garcia, PASI 2006]



Matter Effects

The neutrino propagation equation, in the ultra-relativistic approximation, can

be re-expressed in the form of a Shrodinger-like equation. In the mass basis:

id

dL|νi〉 =

m2i

2E|νi〉,

up to a term proportional to the identity. In the weak/flavor basis

id

dL|νβ〉 = Uβi

m2i

2EU†iα|να〉.

In the 2× 2 case,

id

dL

|νe〉

|νµ〉

=∆m2

2E

sin2 θ cos θ sin θ

cos θ sin θ cos2 θ

|νe〉

|νµ〉

,

(again, up to additional terms proportional to the 2× 2 identity matrix).



Fermi Lagrangian, after a Fiertz rearrangement of the charged-current terms:

L ⊃ νeLi∂µγµνeL − 2√

2GF (νeLγµνeL) (eLγµeL) + . . .

Equation of motion for one electron neutrino state in the presence of a

non-relativistic electron background, in the rest frame of the electrons:

〈eLγµeL〉 = δµ0Ne2

where Ne ≡ e†e is the average electron number density ( at rest, hence δµ0

term). Factor of 1/2 from the “left-handed” half.

Dirac equation for a one neutrino state inside a cold electron “gas” is (ignore

neutrino mass)

(i∂µγµ −√

2GFNeγ0)|νe〉 = 0.

In the ultrarelativistic limit, (plus√

2GFNe � E), dispersion relation is

E ' |~p| ±√

2GFNe, + for ν, − for ν



id

dL

|νe〉

|νµ〉

=

∆m2

2E



+

A 0

0 0

|νe〉

|νµ〉

,

A = ±√

2GFNe (+ for neutrinos, − for antineutrinos).

Note: Similar effect from neutral current interactions common to all (active)

neutrino species → proportional to the identity.

In general, this is hard to solve, as A is a function of L: two-level non-relativistc

quantum mechanical system in the presence of time dependent potential.

In some cases, however, the solution is rather simple.



Constant A: good approximation for neutrinos propagating through matter

inside the Earth [exception: neutrinos that see Earth’s internal structure (the

crust, the mantle, the outer core, the inner core)]

id

dL

|νe〉

|νµ〉

=

A ∆/2 sin 2θ

∆/2 sin 2θ ∆ cos 2θ

|νe〉

|νµ〉

, ∆ ≡ ∆m2/2E.

Peµ = sin2 2θM sin2

(∆ML

2

),

where

∆M =

√(A−∆ cos 2θ)2 + ∆2 sin2 2θ,

∆M sin 2θM = ∆ sin 2θ,

∆M cos 2θM = A−∆ cos 2θ.

The presence of matter affects neutrino and antineutrino oscillation differently.

Nothing wrong with this: CPT-theorem relates the propagation of neutrinos in

an electron background to the propagation of antineutrinos in a positron

background.



Enlarged parameter space in the presence of matter effects.

For example, can tell whether cos 2θ is positive or negative.

L(a.u.)

P eµ =

1-P

ee

sign(A)=sign(cos2θ)

A=0 (vacuum)

sign(A)=-sign(cos2θ)



The MSW Effect

Curiously enough, the oldest neutrino puzzle is the one that is most subtle

to explain. This is because solar neutrinos traverse a strongly varying

matter density on their way from the center of the Sun to the surface of

the Earth.

For the Hamiltonian∆



+A

1 0

0 0

,

it is easy to compute the eigenvalues as a function of A:

(remember, ∆ = ∆m2/2E)



A(a.u.)

λ(a.u.)

heavy

light

|νe〉 = |νH〉



A decreases “slowly” as a function of L ⇒ system evolves adiabatically.

|νe〉 = |ν2M 〉 at the core → |ν2〉 in vacuum,

PEarthee = |〈νe|ν2〉|2 = sin2 θ.

Note that Pee ' sin2 θ applies in a wide range of energies and baselines, as long

as the approximations mentioned above apply —ideal to explain the energy

independent suppression of the 8B solar neutrino flux!

Furthermore, large average suppressions of the neutrino flux are allowed if

sin2 θ � 1. Compare with P vacee = 1− 1/2 sin2 2θ > 1/2.

One can expand on the result above by loosening some of the assumptions. |νe〉state is produced in the Sun’s core as an incoherent mixture of |ν1M 〉 and |ν2M 〉.Introduce adiabaticity parameter Pc, which measures the probability that a

|νiM 〉 matter Hamiltonian state will not exit the Sun as a |νi〉 mass-eigenstate.



|νe〉 → |ν1M 〉, with probability cos2 θM ,

→ |ν2M 〉, with probability sin2 θM ,

where θM is the matter angle at the neutrino production point.

|ν1M 〉 → |ν1〉, with probability (1− Pc),

→ |ν2〉, with probability Pc,

|ν2M 〉 → |ν1〉 with probability Pc,

→ |ν2〉 with probability (1− Pc).

P1e = cos2 θ and P2e = sin2 θ so

PSunee = cos2 θM

[(1− Pc) cos2 θ + Pc sin2 θ

]+ sin2 θM

[Pc cos2 θ + (1− Pc) sin2 θ

].

For Ne = Ne0e−L/r0 , Pc, (crossing probability), is exactly calculable

Pc =e−γ sin2 θ − e−γ

1− e−γ, γ = 2πr0∆. (1)

Adiabatic condition: γ � 1, when Pc → 0.



Vacuum - Mattertransition

cos4θ13(1- sin22θ12) 1 2

|

cos4θ13sin2θ12

β=23/2GFcos2θ13neEν

∆m21 2

P

E0.0

0.2

0.4

0.6

0.8

1.0

We need:

• Pee ∼ 0.3 (8B neutrinos)

• Pee ∼ 0.6 (7Be, pp neutrinos)

⇒ sin2 θ ∼ 0.3

⇒ ∆m2 ∼ 10−(5 to 4) eV2

for a long time, there were many

other options!

(LMA, LOW, SMA, VAC)





Borexino, 1110.3230



“Final” SNO results, 1109.0763



Solar oscillations confirmed by Reactor experiment: KamLAND[arXiv:1303.4667]


∆m2L4E

)

phase= 1.27(

∆m2

5×10−5 eV2

)(5 MeVE

)(L

100 km

)

oscillatory behavior!



[Gonzalez-Garcia, PASI 2006]



Atmospheric Oscillations in the Electron Sector: Daya Bay, RENO, Double Chooz


∆m2L4E

)

phase= 0.64(

∆m2

2.5×10−3 eV2

)(5 MeVE

)(L

1 km

)



Summarizing:

Both the solar and atmospheric puzzles can be properly explained in

terms of two-flavor neutrino oscilations:

• solar: νe ↔ νa (linear combination of νµ and ντ ): ∆m2 ∼ 10−4 eV2,

sin2 θ ∼ 0.3.

• atmospheric: νµ ↔ ντ : ∆m2 ∼ 10−3 eV2, sin2 θ ∼ 0.5 (“maximal

mixing”).

• short-baseline reactors: νe ↔ νa (linear combination of νµ and ντ ):

∆m2 ∼ 10−3 eV2, sin2 θ ∼ 0.02.



Putting it all together – 3 flavor mixing:

νe

νµ

ντ

=

Ue1 Ue2 Ue3

Uµ1 Uµ2 Uµ3

Uτ1 Ueτ2 Uτ3

ν1

ν2

ν3

Definition of neutrino mass eigenstates (who are ν1, ν2, ν3?):

• m21 < m2

2 ∆m213 < 0 – Inverted Mass Hierarchy

• m22 −m2

1 � |m23 −m2

1,2| ∆m213 > 0 – Normal Mass Hierarchy

tan2 θ12 ≡ |Ue2|2

|Ue1|2 ; tan2 θ23 ≡ |Uµ3|2|Uτ3|2 ; Ue3 ≡ sin θ13e

−iδ

[For a detailed discussion see AdG, Jenkins, PRD78, 053003 (2008)]



NuFIT 3.2 (2018)

Normal Ordering (best fit) Inverted Ordering (∆χ2 = 4.14) Any Ordering

bfp ±1σ 3σ range bfp ±1σ 3σ range 3σ range

sin2 θ12 0.307+0.013−0.012 0.272→ 0.346 0.307+0.013

−0.012 0.272→ 0.346 0.272→ 0.346

θ12/◦ 33.62+0.78

−0.76 31.42→ 36.05 33.62+0.78−0.76 31.43→ 36.06 31.42→ 36.05

sin2 θ23 0.538+0.033−0.069 0.418→ 0.613 0.554+0.023

−0.033 0.435→ 0.616 0.418→ 0.613

θ23/◦ 47.2+1.9

−3.9 40.3→ 51.5 48.1+1.4−1.9 41.3→ 51.7 40.3→ 51.5

sin2 θ13 0.02206+0.00075−0.00075 0.01981→ 0.02436 0.02227+0.00074

−0.00074 0.02006→ 0.02452 0.01981→ 0.02436

θ13/◦ 8.54+0.15

−0.15 8.09→ 8.98 8.58+0.14−0.14 8.14→ 9.01 8.09→ 8.98

δCP/◦ 234+43

−31 144→ 374 278+26−29 192→ 354 144→ 374

∆m221

10−5 eV2 7.40+0.21−0.20 6.80→ 8.02 7.40+0.21

−0.20 6.80→ 8.02 6.80→ 8.02

∆m23`

10−3 eV2 +2.494+0.033−0.031 +2.399→ +2.593 −2.465+0.032

−0.031 −2.562→ −2.369

[+2.399→ +2.593−2.536→ −2.395

]

Three Flavor Mixing Hypothesis Fits All∗ Data Really Well.

∗Modulo a handful of 2σ to 3σ anomalies.

[Esteban et al, JHEP 01 (2017) 087, http://www.nu-fit.org]



H

0

60

120

180

240

300

360

d CP

NO

H H

0.016 0.02 0.024 0.028

sin2q

13

0

60

120

180

240

300

360

d CP

IO

0.4 0.5 0.6 0.7

sin2q

23

2.4 2.5 2.6 2.7

|Dm2

atm| [10

-3 eV

2]

NuFIT 3.2 (2018)

[Esteban et al, JHEP 01 (2017) 087, http://www.nu-fit.org]



4 - Understanding Neutrino Oscillations: What is Left to Do?

(∆m2)sol

(∆m2)sol

(∆m2)atm

(∆m2)atm

νe

νµ

ντ

(m1)2

(m2)2

(m3)2

(m1)2

(m2)2

(m3)2

normal hierarchy inverted hierarchy

• What is the νe component of ν3?(θ13 6= 0!)

• Is CP-invariance violated in neutrinooscillations? (δ 6= 0, π?) [‘yes’ hint]

• Is ν3 mostly νµ or ντ? [θ23 6= π/4 hint]

• What is the neutrino mass hierarchy?(∆m2

13 > 0?) [NH weak hint]

⇒ All of the above can “only” be

addressed with new neutrino

oscillation experiments

Ultimate Goal: Not Measure Parameters but Test the Formalism (Over-Constrain Parameter Space)



(∆m2)sol

(∆m2)sol

(∆m2)atm

(∆m2)atm

νe

νµ

ντ

(m1)2

(m2)2

(m3)2

(m1)2

(m2)2

(m3)2


The Neutrino

Mass Hierarchy

which is the right picture?



Why Don’t We Know the Neutrino Mass Hierarchy?

Most of the information we have regarding θ23 and ∆m213 comes from

atmospheric neutrino experiments (SuperK). Roughly speaking, they

measure

Pµµ = 1− sin2 2θ23 sin2

(∆m2

13L

4E

)+ subleading.

It is easy to see from the expression above that the leading term is simply

not sensitive to the sign of ∆m213.

On the other hand, because |Ue3|2 ∼ 0.02 and∆m2

12

∆m213∼ 0.03 are both small,

we are yet to observe the subleading effects.



Determining the Mass Hierarchy via Oscillations – the large Ue3 route

Again, necessary to probe νµ → νe oscillations (or vice-versa) governed by

∆m213. This is the oscillation channel that (almost) all next-generation,

accelerator-based experiments are concentrating on, including the ongoing

experiments T2K and NOνA.

In vaccum

Pµe = sin2 θ23 sin2 2θ13 sin2

(∆m2

13L

4E

)+ “subleading”,

so that, again, this is insensitive to the sign of ∆m213 at leading order. However,

in this case, matter effects may come to the rescue.

As I discussed already, neutrino oscillations get modified when these propagate

in the presence of matter. Matter effects are sensitive to the neutrino mass

ordering (in a way that I will describe shortly) and different for neutrinos and

antineutrinos.



If ∆12 ≡ ∆m212

2E terms are ignored, the νµ → νe oscillation probability is

described, in constant matter density, by

Pµe ' Peµ ' sin2 θ23 sin2 2θeff13 sin2

(∆eff

13L2

),

sin2 2θeff13 =

∆213 sin2 2θ13

(∆eff13 )2 ,

∆eff13 =

√(∆13 cos 2θ13 −A)2 + ∆2

13 sin2 2θ13,

∆13 =∆m2

13

2E ,

A ≡ ±√

2GFNe is the matter potential. It is positive for neutrinos and

negative for antineutrinos.

Pµe depends on the relative sign between ∆13 and A. It is different for the

two different mass hierarchies, and different for neutrinos and

antineutrinos.



L(a.u.)

P eµ =

1-P

ee

sign(A)=sign(cos2θ)

A=0 (vacuum)

sign(A)=-sign(cos2θ)

replace sign(cos 2θ) → sign(∆m213)

Requirements:

• sin2 2θ13 large enough – otherwise there is nothing to see!

• |∆13| ∼ |A| – matter potential must be significant but not overwhelming.

• ∆eff13L large enough – matter effects are absent near the origin.



The “Holy Graill” of Neutrino Oscillations – CP Violation

In the old Standard Model, there is only onea source of CP-invariance

violation:

⇒ The complex phase in VCKM , the quark mixing matrix.

Indeed, as far as we have been able to test, all CP-invariance violating

phenomena agree with the CKM paradigm:

• εK ;

• ε′K ;

• sin 2β;

• etc.

Recent experimental developments, however, provide strong reason to

believe that this is not the case: neutrinos have mass, and leptons mix!

amodulo the QCD θ-parameter, which will be “willed away” henceforth.



Golden Opportunity to Understand Matter versus Antimatter?

The SM with massive Majorana neutrinos accommodates five irreducible

CP-invariance violating phases.

• One is the phase in the CKM phase. We have measured it, it is large,

and we don’t understand its value. At all.

• One is θQCD term (θGG). We don’t know its value but it is only

constrained to be very small. We don’t know why (there are some

good ideas, however).

• Three are in the neutrino sector. One can be measured via neutrino

oscillations. 50% increase on the amount of information.

We don’t know much about CP-invariance violation. Is it really fair to

presume that CP-invariance is generically violated in the neutrino sector

solely based on the fact that it is violated in the quark sector? Why?

Cautionary tale: “Mixing angles are small”



CP-invariance Violation in Neutrino Oscillations

The most promising approach to studying CP-violation in the leptonic

sector seems to be to compare P (νµ → νe) versus P (νµ → νe).

The amplitude for νµ → νe transitions can be written as

Aµe = U∗e2Uµ2

(ei∆12 − 1

)+ U∗e3Uµ3

(ei∆13 − 1

)

where ∆1i =∆m2

1iL2E , i = 2, 3.

The amplitude for the CP-conjugate process can be written as

Aµe = Ue2U∗µ2

(ei∆12 − 1

)+ Ue3U

∗µ3

(ei∆13 − 1

).

[remember: according to unitarty, Ue1U∗µ1 = −Ue2U∗µ2 − Ue3U∗µ3]



In general, |A|2 6= |A|2 (CP-invariance violated) as long as:

• Nontrivial “Weak” Phases: arg(U∗eiUµi) → δ 6= 0, π;

• Nontrivial “Strong” Phases: ∆12, ∆13 → L 6= 0;

• Because of Unitarity, we need all |Uαi| 6= 0 → three generations.

All of these can be satisfied, with a little luck: given that two of the three

mixing angles are known to be large, we need |Ue3| 6= 0. (X)

The goal of next-generation neutrino experiments is to determine the

magnitude of |Ue3|. We need to know this in order to understand how to

study CP-invariance violation in neutrino oscillations!



In the real world, life is much more complicated. The lack of knowledge

concerning the mass hierarchy, θ13, θ23 leads to several degeneracies.

Note that, in order to see CP-invariance violation, we need the

“subleading” terms!

In order to ultimately measure a new source of CP-invariance violation,

we will need to combine different measurements:

– oscillation of muon neutrinos and antineutrinos,

– oscillations at accelerator and reactor experiments,

– experiments with different baselines,

– etc.



4– What We Know We Don’t Know (ii): How Light is the Lightest Neutrino?

(∆m2)sol

(∆m2)sol

(∆m2)atm

(∆m2)atm

νe

νµ

ντ

(m1)2

(m2)2

(m3)2

(m1)2

(m2)2

(m3)2


m2 = 0 ——————

——————↑↓

m2lightest = ?

So far, we’ve only been able to measure

neutrino mass-squared differences.

The lightest neutrino mass is only poorly

constrained: m2lightest < 1 eV2

qualitatively different scenarios allowed:

• m2lightest ≡ 0;

• m2lightest � ∆m2

12,13;

• m2lightest � ∆m2

12,13.

Need information outside of neutrino oscillations.



The most direct probe of the lightest neutrino mass –

precision measurements of β-decay

Observation of the effect of non-zero neutrino masses kinematically.

When a neutrino is produced, some of the energy exchanged in the process

should be spent by the non-zero neutrino mass.

Typical effects are very, very small – we’ve never seen them! The most sensitive

observable is the electron energy spectrum from tritium decay.

3H→3He + e− + ν

Why tritium? Small Q value, reasonable abundances. Required sensitivity

proportional to m2/Q2.

In practice, this decay is sensitive to an effective “electron neutrino mass”:

m2νe ≡

∑i

|Uei|2m2i



Experiments measure the shape of the end-point of the spectrum, not the

value of the end point. This is done by counting events as a function of

a low-energy cut-off. note: LOTS of Statistics Needed!

E0 = 18.57 keV

t1/2 = 12.32 years

e

e



NEXT GENERATION: The Karlsruhe Tritium Neutrino (KATRIN) Experiment:

(not your grandmother’s table top experiment!)

sensitivity m2νe> (0.2 eV)2



Big Bang Neutrinos are Warm Dark Matter

• Constrained by the Large Scale

Structure of the Universe.

Constraints depend on

• Data set analysed;

• “Bias” on other parameters;

• . . .

Bounds can be evaded with

non-standard cosmology. Will we

learn about neutrinos from

cosmology or about cosmology

from neutrinos?[Z. Hou et al. arXiv:1212.6267]




• Constrained by the Large Scale

Structure of the Universe.

Constraints depend on

• Data set analysed;

• “Bias” on other parameters;

• . . .

Bounds can be evaded with

non-standard cosmology. Will we

learn about neutrinos from

cosmology or about cosmology

from neutrinos?[Z. Hou et al. arXiv:1212.6267]






4– What We Know We Don’t Know (iii) – Are Neutrinos Majorana Fermions?

νL

you

νR? ν

L?

you

__

A massive charged fermion (s=1/2) isdescribed by 4 degrees of freedom:

(e−L ← CPT→ e+R)

l Lorentz

(e−R ← CPT→ e+L)

A massive neutral fermion (s=1/2) isdescribed by 4 or 2 degrees of freedom:

(νL ← CPT→ νR)

l Lorentz “DIRAC”

(νR ← CPT→ νL)

(νL ← CPT→ νR)

“MAJORANA” l Lorentz

(νR ← CPT→ νL)How many degrees of freedom are requiredto describe massive neutrinos?



Why Don’t We Know the Answer (Yet)?

If neutrino masses were indeed zero, this is a nonquestion: there is no

distinction between a massless Dirac and Majorana fermion.

Processes that are proportional to the Majorana nature of the neutrino

vanish in the limit mν → 0. Since neutrinos masses are very small, the

probability for these to happen is very, very small: A ∝ mν/E.

The “smoking gun” signature is the observation of LEPTON NUMBER

violation. This is easy to understand: Majorana neutrinos are their own

antiparticles and, therefore, cannot carry any quantum numbers —

including lepton number.

The deepest probes are searches for Neutrinoless Double-Beta Decay.



Weak Interactions are Purely Left-Handed (Chirality):

For example, in the scattering process e− +X → νe +X, the electron

neutrino is, in a reference frame where m� E,

|νe〉 ∼ |L〉+(mE

)|R〉.

If the neutrino is a Majorana fermion, |R〉 behaves mostly like a “νe,”

(and |L〉 mostly like a “νe,”) such that the following process could happen:

e− +X → νe +X, followed by νe +X → e+ +X, P '(mE

)2

Lepton number can be violated by 2 units with small probability. Typical

numbers: P ' (0.1 eV/100 MeV)2 = 10−18. VERY Challenging!



Search for the Violation of Lepton Number (or B − L)

10−4 10−3 10−2 10−1 1lightest neutrino mass in eV

10−4

10−3

10−2

10−1

1

|mee

| in

eV

90% CL (1 dof)

∆m232 > 0

disfavoured by 0ν2β

disfavouredby

cosmology

∆m232 < 0

Helicity Suppressed Amplitude ∝ meeE

Observable: mee ≡∑i U

2eimi

⇐ no longer lamp-post physics!

Best Bet: search for

Neutrinoless Double-Beta

Decay: Z → (Z + 2)e−e−×

←(next)

←(next-next)



How many new CP-violating parameters in the neutrino sector?

If the neutrinos are Majorana fermions, there are more physical

observables in the leptonic mixing matrix.

Remember the parameter counting in the quark sector:

9 (3× 3 unitary matrix)

−5 (relative phase rotation among six quark fields)

4 (3 mixing angles and 1 CP-odd phase).



If the neutrinos are Majorana fermions, the parameter counting is quite

different: there are no right-handed neutrino fields to “absorb” CP-odd

phases:

9 (3× 3 unitary matrix)

−3 (three right-handed charged lepton fields)

6 (3 mixing angles and 3 CP-odd phases).

There is CP-invariance violating parameters even in the 2 family case:

4− 2 = 2, one mixing angle, one CP-odd phase.



L ⊃ eLUWµγµνL − eL(Me)eR − νcL(Mν)νL +H.c.

Write U = E−iξ/2U ′Eiα/2, where Eiβ/2 ≡ diag(eiβ1/2, eiβ2/2, eiβ3/2),

β = α, ξ

L ⊃ eLU ′WµγµνL − eLEiξ/2(Me)eR − νcL(Mν)E−iανL +H.c.

ξ phases can be “absorbed” by eR,

α phases cannot go away!

on the other hand

Dirac Case:

L ⊃ eLUWµγµνL − eL(Me)eR − νR(Mν)νL +H.c.

L ⊃ eLU ′WµγµνL − eLEiξ/2(Me)eR − νR(Mν)E−iα/2νL +H.c.

ξ phases can be “absorbed” by eR, α phases can be “absorbed” by νR,



VMNS =

Ue1 Ue2 Ue3

Uµ1 Uµ2 Uµ3

Uτ1 Ueτ2 Uτ3

′

eiα1/2 0 0

0 eiα2/2 0

0 0 eiα3/2

.

It is easy to see that the Majorana phases never show up in neutrino

oscillations (A ∝ UαiU∗βi).Furthermore, they only manifest themselves in phenomena that vanish in

the limit mi → 0 – after all they are only physical if we “know” that

lepton number is broken.

A(αi) ∝ mi/E → tiny!



NEUTRINOS

HAVE MASS

10-5

10-4

10-3

10-2

10-1

1

10

10 2

10 3

10 4

10 5

10 6

10 7

10 8

10 9

10 10

10 11

10 12

0 1 2 3 4generation

mas

s (e

V)

t

bτ

µ

c

s

du

e

ν3

ν2

ν1

TeV

GeV

MeV

keV

eV

meV

albeit very tiny ones...

SO WHAT?



5 - Only∗ “Palpable” Evidence of Physics

Beyond the Standard Model

The SM we all learned in school predicts that neutrinos are strictly

massless. Hence, massive neutrinos imply that the the SM is incomplete

and needs to be replaced/modified.

Furthermore, the SM has to be replaced by something qualitatively

different.

——————∗ There is only a handful of questions our model for fundamental physics cannot

explain (these are personal. Feel free to complain).

• What is the physics behind electroweak symmetry breaking? (Higgs X).

• What is the dark matter? (not in SM).

• Why is there more matter than antimatter? (Not in SM).

• Why does the Universe appear to be accelerating? Why does it appear that the

Universe underwent rapid acceleration in the past? (not in SM).



Standard Model in One Slide, No Equations

The SM is a quantum field theory with the following defining

characteristics:

• Gauge Group (SU(3)c × SU(2)L × U(1)Y);

• Particle Content (fermions: Q, u, d, L, e, scalars: H).

Once this is specified, the SM is unambiguously determined:

• Most General Renormalizable Lagrangian;

• Measure All Free Parameters, and You Are Done! (after several

decades of hard experimental work. . . )

If you follow these rules, neutrinos have no mass. Something has to give.



What is the New Standard Model? [νSM]

The short answer is – WE DON’T KNOW. Not enough available info!

m

Equivalently, there are several completely different ways of addressing

neutrino masses. The key issue is to understand what else the νSM

candidates can do. [are they falsifiable?, are they “simple”?, do they

address other outstanding problems in physics?, etc]

We need more experimental input, and it looks like it may be coming in

the near/intermediate future!



Neutrino Masses, EWSB, and a New Mass Scale of Nature

The LHC has revealed that the minimum SM prescription for electroweak

symmetry breaking — the one Higgs double model — is at least approximately

correct. What does that have to do with neutrinos?

The tiny neutrino masses point to three different possibilities.

1. Neutrinos talk to the Higgs boson very, very weakly (Dirac neutrinos);

2. Neutrinos talk to a different Higgs boson – there is a new source of

electroweak symmetry breaking! (Majorana neutrinos);

3. Neutrino masses are small because there is another source of mass out

there — a new energy scale indirectly responsible for the tiny neutrino

masses, a la the seesaw mechanism (Majorana neutrinos).

Searches for 0νββ help tell (1) from (2) and (3), the LHC, charged-lepton flavor

violation, et al may provide more information.



6 - νSM – One Path

SM as an effective field theory – non-renormalizable operators

LνSM ⊃ −yij LiHLjH

2Λ+O

(1

Λ2

)+H.c.

There is only one dimension five operator [Weinberg, 1979]. If Λ� 1 TeV, it

leads to only one observable consequence...

after EWSB LνSM ⊃ mij2νiνj ; mij = yij

v2

Λ.

• Neutrino masses are small: Λ� v → mν � mf (f = e, µ, u, d, etc)

• Neutrinos are Majorana fermions – Lepton number is violated!

• νSM effective theory – not valid for energies above at most Λ.

• What is Λ? First naive guess is that Λ is the Planck scale – does not work.

Data require Λ ∼ 1014 GeV (related to GUT scale?) [note ymax ≡ 1]

What else is this “good for”? Depends on the ultraviolet completion!



Note that this VERY similar to the “discovery” weak interactions.

Imagine the following scenario:

U(1)E&M + e(q = −1), µ(q = −1), νe(q = 0), νµ(q = 0).

The most general renormalizable Lagrangian explains all QED phenomena

once all couplings are known (α,mf ).

New physics: the muon decays! µ− → e−νeνµ. This can be interpreted as

evidence of effective four fermion theory (nonrenormalizable operators):

−4GF√2

∑

γ

gγ (eΓγν) (νΓγµ) , Γγ = 1, γ5, γµ, . . .

Prediction: will discover new physics at an energy scale below√1/GF ' 250 GeV. We know how this turned out ⇒ W±, Z0 discovered

slightly below 100 GeV!



Full disclosure:

All higher dimensional operators are completely negligible, except those

that mediate proton decay, like:

λBM2

QQQL

The fact that the proton does not decay forces M/λB to be much larger

than the energy scale required to explain neutrino masses.

Why is that? We don’t know. . .



Example: the (Type I) Seesaw Mechanism

A simplea, renormalizable Lagrangian that allows for neutrino masses is

Lν = Lold − λαiLαHN i −3∑

i=1

Mi

2N iN i +H.c.,

where Ni (i = 1, 2, 3, for concreteness) are SM gauge singlet fermions. Lνis the most general, renormalizable Lagrangian consistent with the SM

gauge group and particle content, plus the addition of the Ni fields.

After electroweak symmetry breaking, Lν describes, besides all other SM

degrees of freedom, six Majorana fermions: six neutrinos.

aOnly requires the introduction of three fermionic degrees of freedom, no new inter-

actions or symmetries.



What We Really Know About M and λ:

• M = 0: the six neutrinos “fuse” into three Dirac states. Neutrino mass

matrix given by µαi ≡ λαiv.

The symmetry of Lν is enhanced: U(1)B−L is an exact global symmetry of

the Lagrangian if all Mi vanish. Small Mi values are ’tHooft natural.

• M � µ: the six neutrinos split up into three mostly active, light ones, and

three, mostly sterile, heavy ones. The light neutrino mass matrix is given

by mαβ =∑i µαiM

−1i µβi [m ∝ 1/Λ ⇒ Λ = M/µ2].

This the seesaw mechanism. Neutrinos are Majorana fermions. Lepton

number is not a good symmetry of Lν , even though L-violating effects are

hard to come by.

• M ∼ µ: six states have similar masses. Active–sterile mixing is very large.

This scenario is (generically) ruled out by active neutrino data

(atmospheric, solar, KamLAND, K2K, etc).

• M � µ: neutrinos are quasi-Dirac fermions. Active–sterile mixing is

maximal, but new oscillation lengths are very long.



Accommodating Small Neutrino Masses

If µ = λv �M , below the mass scale M ,

L5 =LHLH

Λ.

Neutrino masses are small if Λ� 〈H〉. Data require Λ ∼ 1014 GeV.

In the case of the seesaw,

Λ ∼ M

λ2,

so neutrino masses are small if either

• they are generated by physics at a very high energy scale M � v

(high-energy seesaw); or

• they arise out of a very weak coupling between the SM and a new, hidden

sector (low-energy seesaw); or

• cancellations among different contributions render neutrino masses

accidentally small (“fine-tuning”).



10-14

10-12

10-10

10-8

10-6

10-4

10-2

1

10-12

10-10

10-8

10-6

10-4

10-2

1 102

104

106

108

1010

1012

MN (eV)

sin2 !

as

Experimentally Excluded10-1

10-2

10-5

m"=.....eV

Constraining the Seesaw Lagrangian

(νss

live

here)

(νss

live

here)

[Cosmology!?. . . ]

[AdG, Huang, Jenkins, arXiv:0906.1611]



High-Energy Seesaw: Brief Comments

• This is everyone’s favorite scenario.

• Upper bound for M (e.g. Maltoni, Niczyporuk, Willenbrock, hep-ph/0006358):

M < 7.6× 1015 GeV ×(

0.1 eV

mν

).

• Hierarchy problem hint (e.g., Casas et al, hep-ph/0410298; Farina et al, ; 1303.7244; AdG et

al, 1402.2658): M < 107 GeV.

• Leptogenesis! “Vanilla” Leptogenesis requires, very roughly, smallest

M > 109 GeV.

• Stability of the Higgs potential (e.g., Elias-Miro et al, 1112.3022): M < 1013 GeV.

• Physics “too” heavy! No observable consequence other than leptogenesis.

Will we ever convince ourselves that this is correct? (Buckley et al, hep-ph/0606088)



“Higher Order” Neutrino Masses from ∆L = 2 Physics

Imagine that there is new physics that breaks lepton number by 2 units at

some energy scale Λ, but that it does not, in general, lead to neutrino

masses at the tree level.

We know that neutrinos will get a mass at some order in perturbation

theory – which order is model dependent!

For example:

• SUSY with trilinear R-parity violation – neutrino masses at one-loop;

• Zee models – neutrino masses at one-loop;

• Babu and Ma – neutrino masses at two loops;

• Chen et al, 0706.1964 – neutrino masses at two loops;

• Angel et al, 1308.0463 – neutrino masses at two loops;

• etc.



9

TABLE I: Dimension-five through dimension-eleven LNV operators analyzed in this survey. The first two columns display theoperator name and field structure, respectively. Column three presents the induced neutrino mass expressions, followed bythe inferred scale of new physics, Λν . Column five lists favorable modes of experimental exploration. Column six describes anoperator’s current status according to the key U (Unconstrained), C (Constrained) and D (Disfavored). See text for details.

O Operator mαβ Λν (TeV) Best Probed Disfavored

4a LiLjQiucHkεjk

yu

16π2

v2

Λ 4 × 109 ββ0ν U

4b LiLjQkucHkεijyug2

(16π2)2v2

Λ 6 × 106 ββ0ν U

5 LiLjQkdcH lHmHiεjlεkmyd

(16π2)2v2

Λ 6 × 105 ββ0ν U

6 LiLjQkucH lHkHiεjlyu

(16π2)2v2

Λ 2 × 107 ββ0ν U

7 LiQj ecQkHkH lHmεilεjm y%β

g2

(16π2)2v2

Λ

“1

16π2 + v2

Λ2

”4 × 102 mix C

8 LiecucdcHjεij y%β

ydyu

(16π2)2v2

Λ 6 × 103 mix C

9 LiLjLkecLlecεijεkly2

"(16π2)2

v2

Λ 3 × 103 ββ0ν U

10 LiLjLkecQldcεijεkly"yd

(16π2)2v2

Λ 6 × 103 ββ0ν U

11a LiLjQkdcQldcεijεkly2

dg2

(16π2)3v2

Λ 30 ββ0ν U

11b LiLjQkdcQldcεikεjly2

d(16π2)2

v2

Λ 2 × 104 ββ0ν U

12a LiLjQiucQjuc y2

u(16π2)2

v2

Λ 2 × 107 ββ0ν U

12b LiLjQkucQlucεijε

kl y2ug2

(16π2)3v2

Λ 4 × 104 ββ0ν U

13 LiLjQiucLlecεjl

y"yu

(16π2)2v2

Λ 2 × 105 ββ0ν U

14a LiLjQkucQkdcεijydyug2

(16π2)3v2

Λ 1 × 103 ββ0ν U

14b LiLjQiucQldcεjl

ydyu

(16π2)2v2

Λ 6 × 105 ββ0ν U

15 LiLjLkdcLiucεjkydyug2

(16π2)3v2

Λ 1 × 103 ββ0ν U

16 LiLjecdcecucεijydyug4

(16π2)4v2

Λ 2 ββ0ν, LHC U

17 LiLjdcdcdcucεijydyug4

(16π2)4v2

Λ 2 ββ0ν, LHC U

18 LiLjdcucucucεijydyug4

(16π2)4v2

Λ 2 ββ0ν, LHC U

19 LiQjdcdcecucεij y%β

y2dyu

(16π2)3v2

Λ 1 ββ0ν, HElnv, LHC, mix C

20 LidcQiucecuc y%β

ydy2u

(16π2)3v2

Λ 40 ββ0ν, mix C

21a LiLjLkecQlucHmHnεijεkmεlny"yu

(16π2)2v2

Λ

“1

16π2 + v2

Λ2

”2 × 103 ββ0ν U

21b LiLjLkecQlucHmHnεilεjmεkny"yu

(16π2)2v2

Λ

“1

16π2 + v2

Λ2

”2 × 103 ββ0ν U

22 LiLjLkecLkecH lHmεilεjmg2

(16π2)3v2

Λ 4 × 104 ββ0ν U

23 LiLjLkecQkdcH lHmεilεjmy"yd

(16π2)2v2

Λ

“1

16π2 + v2

Λ2

”40 ββ0ν U

24a LiLjQkdcQldcHmHiεjkεlmy2

d(16π2)3

v2

Λ 1 × 102 ββ0ν U

24b LiLjQkdcQldcHmHiεjmεkly2

d(16π2)3

v2

Λ 1 × 102 ββ0ν U

25 LiLjQkdcQlucHmHnεimεjnεklydyu

(16π2)2v2

Λ

“1

16π2 + v2

Λ2

”4 × 103 ββ0ν U

26a LiLjQkdcLiecH lHmεjlεkmy"yd

(16π2)3v2

Λ 40 ββ0ν U

26b LiLjQkdcLkecH lHmεilεjmy"yd

(16π2)2v2

Λ

“1

16π2 + v2

Λ2

”40 ββ0ν U

27a LiLjQkdcQidcH lHmεjlεkm

g2

(16π2)3v2

Λ 4 × 104 ββ0ν U

27b LiLjQkdcQkdcH lHmεilεjmg2

(16π2)3v2

Λ 4 × 104 ββ0ν U

28a LiLjQkdcQjucH lHiεkl

ydyu

(16π2)3v2

Λ 4 × 103 ββ0ν U

28b LiLjQkdcQkucH lHiεjlydyu

(16π2)3v2

Λ 4 × 103 ββ0ν U

28c LiLjQkdcQlucH lHiεjk

ydyu

(16π2)3v2

Λ 4 × 103 ββ0ν U

29a LiLjQkucQkucH lHmεilεjmy2

u(16π2)2

v2

Λ

“1

16π2 + v2

Λ2

”2 × 105 ββ0ν U

29b LiLjQkucQlucH lHmεikεjm

g2

(16π2)3v2

Λ 4 × 104 ββ0ν U

30a LiLjLiecQkucHkH lεjly"yu

(16π2)3v2

Λ 2 × 103 ββ0ν U

30b LiLjLmecQnucHkH lεikεjlεmn y"yu

(16π2)2v2

Λ

“1

16π2 + v2

Λ2

”2 × 103 ββ0ν U

31a LiLjQidcQkucHkH lεjl

ydyu

(16π2)2v2

Λ

“1

16π2 + v2

Λ2

”4 × 103 ββ0ν U

Effective

Operator

Approach

AdG, Jenkins,

0708.1344 [hep-ph]

(there are 129

of them if you

discount different

Lorentz structures!)

classified by Babu

and Leung in

NPB619,667(2001)



να νβ

LNVOperator

(a)(b)

νβνα

yv yv

(c)

νβνα

yv yv

yH−

eyβ

(d) (e)

γ, g

W, Z

να νβ

vv

W, Z

γ, g

νβνα

yyH−

e e

v v

H+

yβyαh0h0h0

h0



−1 0 1 2 3 4 5 6 7 8 9 10 11 120

5

10

15

20

25

30

35

40

45

Log( Λ/TeV)

Num

ber

Of O

pera

tors

Dim 5Dim 7Dim 9Dim 11



−1 0 1 2 3 4 5 6 7 8 9 10 11 120

5

10

15

20

25

30

35

40

45

Log( Λ/TeV)

Num

ber

Of O

pera

tors

Dim 5Dim 7Dim 9Dim 11

“Directly Accessible”

Out of “direct” reach if not weakly-coupled (?)

|||||||

Colliders

g − 2 CLFVEDM ⇓

(seesaw)



Dirac Neutrinos – Enhanced Symmetry!(Symmetries?)

Back to

Lν = Lold − λαiLαHN i −3∑

i=1

Mi

2N iN i +H.c.,

where Ni (i = 1, 2, 3, for concreteness) are SM gauge singlet fermions.



Dirac Neutrinos – Enhanced Symmetry!(Symmetries?)

If all Mi ≡ 0, the neutrinos are Dirac fermions.

Lν = Lold − λαiLαHN i +H.c.,

where Ni (i = 1, 2, 3, for concreteness) are SM gauge singlet fermions. In

this case, the νSM global symmetry structure is enhanced. For example,

U(1)B−L is an exactly conserved, global symmetry. This is new!

Downside: The neutrino Yukawa couplings λ are tiny, less than 10−12.

What is wrong with that? We don’t like tiny numbers, but Nature seems

to not care very much about what we like. . .

More to the point, the failure here is that it turns out that the neutrino

masses are not, trivially, qualitatively different. This seems to be a

“missed opportunity.”



There are lots of ideas that lead to very small Dirac neutrino masses.

Maybe right-handed neutrinos exist, but neutrino Yukawa couplings are

forbidden – hence neutrino masses are tiny.

One possibility is that the N fields are charged under some new symmetry

(gauged or global) that is spontaneously broken.

λαiLαHN i → καi

Λ(LαH)(N iΦ),

where Φ (spontaneously) breaks the new symmetry at some energy scale

vΦ. Hence, λ = κvΦ/Λ. How do we test this?

E.g., AdG and D. Hernandez, arXiv:1507.00916

Gauged chiral new symmetry for the right-handed neutrinos, no Majorana

masses allowed, plus a heavy messenger sector. Predictions: new stable massive

states (mass around vΦ) which look like (i) dark matter, (ii) (Dirac) sterile

neutrinos are required. Furthermore, there is a new heavy Z′-like gauge boson.

⇒ Natural Conections to Dark Matter, Sterile Neutrinos, Dark Photons!



VMNS ∼

0.8 0.5 0.2

0.4 0.6 0.70.4 0.6 0.7

VCKM ∼

1 0.2 0.001

0.2 1 0.01

0.001 0.01 1

1

Understanding Fermion Mixing

The other puzzling phenomenon uncovered by the neutrino data is the

fact that Neutrino Mixing is Strange. What does this mean?

It means that lepton mixing is very different from quark mixing:

[|(VMNS)e3| < 0.2]

WHY?

They certainly look VERY different, but which one would you label

as “strange”?



“Left-Over” Predictions: δ, mass-hierarchy, cos 2θ23

[Albright and Chen, hep-ph/0608137]

| || || || || || || || || |Daya Bay

(3 σ)

↔

↔

↔



10 anarchical mixing matrices, plus the “real” one|Ue1|2 |Ue2|2 |Ue3|2

· · · · · · |Uµ3|2

· · · · · · |Uτ3|2

=

0.69 0.29 0.02

· · · · · · 0.40

· · · · · · 0.58

,

0.36 0.35 0.29

· · · · · · 0.68

· · · · · · 0.03

,

0.83 0.11 0.06

· · · · · · 0.87

· · · · · · 0.07

,

0.71 0.13 0.16

· · · · · · 0.20

· · · · · · 0.64

,

0.24 0.47 0.29

· · · · · · 0.58

· · · · · · 0.13

,

0.16 0.35 0.49

· · · · · · 0.13

· · · · · · 0.38

,

0.63 0.24 0.13

· · · · · · 0.73

· · · · · · 0.14

,

0.12 0.35 0.53

· · · · · · 0.12

· · · · · · 0.35

,

0.22 0.55 0.23

· · · · · · 0.12

· · · · · · 0.65

,

0.21 0.37 0.42

· · · · · · 0.08

· · · · · · 0.50

,

0.54 0.44 0.02

· · · · · · 0.54

· · · · · · 0.44

.



00.0050.010.0150.020.0250.030.0350.040.0450.05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1sin2e23

sin2e 1

3Anarchy vs. Order — more precision required!

Order: sin2 θ13 = C cos2 2θ23, C ∈ [0.8, 1.2] [AdG, Murayama, 1204.1249]



Piecing the Neutrino Mass Puzzle

Understanding the origin of neutrino masses and exploring the new physics in the

lepton sector will require unique theoretical and experimental efforts, including . . .

• understanding the fate of lepton-number. Neutrinoless double beta decay!

• a comprehensive long baseline neutrino program, towards precision oscillation

physics.

• other probes of neutrino properties, including neutrino scattering.

• precision studies of charged-lepton properties (g − 2, edm), and searches for rare

processes (µ→ e-conversion the best bet at the moment).

• collider experiments. The LHC and beyond may end up revealing the new physics

behind small neutrino masses.

• cosmic surveys. Neutrino properties affect, in a significant way, the history of the

universe. Will we learn about neutrinos from cosmology, or about cosmology from

neutrinos?

• searches for baryon-number violating processes.



Concluding Remarks

The venerable Standard Model sprung a leak in the end of the last

century: neutrinos are not massless! [and we are still trying to patch it. . . ]

1. We still know very little about the new physics uncovered by neutrino

oscillations. In particular, the new physics (broadly defined) can live almost

anywhere between sub-eV scales and the GUT scale.

2. Neutrino masses are very small – we don’t know why, but we think it

means something important.

3. Neutrino mixing is “weird” – we don’t know why, but we think it

means something important.

4. What is going on with the short-baseline anomalies?

5. There is plenty of room for surprises, as neutrinos are very deep probes

of all sorts of physical phenomena. Neutrino oscillations are “quantum

interference devices,” potentially sensitive to whatever else might be out

there (keep in mind, neutrino masses might be physics at Λ ' 1014 GeV).


Documents

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