Andre de Gouvea Northwestern
NEUTRINOS(Lectures 1–3)
Andre de Gouvea
Northwestern University
Summer School on Particle Physics
ICTP-Trieste
June 10 – 21, 2019
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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]
June 12–14, 2019 Neutrinos
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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]];
• “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).
• Neutrino Unbound (C. Giunti) – http://www.nu.to.infn.it
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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.
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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).
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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.)
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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, Schwartz 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). ⇒
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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.
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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”].
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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.
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UP 6= DOWN – neutrinos can tell time! → neutrinos have mass.
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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.
June 12–14, 2019 Neutrinos
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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
)
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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:
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CHOOZ experiment
Pee = 1− sin2 2θ sin2(
∆m2L4E
)
[by-now-old result: 1− Pee < 0.05]
low ∆m2: 1− Pee ∝ sin2 2θ(∆m2)2
high ∆m2: 1− Pee ∝ 12
sin2 2θ
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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.
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Pµµ ∼ 1
↓
Pµµ∼1− 12
sin2 2θ
↖
Pµµ = 1− sin2 2θ sin2(
∆m2L4E
)Works great for sin2 2θ ∼ 1 and ∆m2 ∼ 10−3 eV2
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[Gonzalez-Garcia, PASI 2006]
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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).
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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
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id
dL
|νe〉
|νµ〉
=
∆m2
2E
sin2 θ cos θ sin θ
cos θ sin θ cos2 θ
+
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.
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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.
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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)
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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∆
sin2 θ cos θ sin θ
cos θ sin θ cos2 θ
+A
1 0
0 0
,
it is easy to compute the eigenvalues as a function of A:
(remember, ∆ = ∆m2/2E)
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A(a.u.)
λ(a.u.)
heavy
light
|νe〉 = |νH〉
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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.
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|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.
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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)
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Borexino, 1110.3230
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“Final” SNO results, 1109.0763
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Solar oscillations confirmed by Reactor experiment: KamLAND[arXiv:1303.4667]
Pee = 1− sin2 2θ sin2(
∆m2L4E
)
phase= 1.27(
∆m2
5×10−5 eV2
)(5 MeVE
)(L
100 km
)
oscillatory behavior!
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[Gonzalez-Garcia, PASI 2006]
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Atmospheric Oscillations in the Electron Sector: Daya Bay, RENO, Double Chooz
Pee = 1− sin2 2θ sin2(
∆m2L4E
)
phase= 0.64(
∆m2
2.5×10−3 eV2
)(5 MeVE
)(L
1 km
)
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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.
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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)]
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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]
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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]
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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 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)
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(∆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
The Neutrino
Mass Hierarchy
which is the right picture?
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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.
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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.
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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.
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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.
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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.
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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”
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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]
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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!
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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.
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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
normal hierarchy inverted hierarchy
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.
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
NEXT GENERATION: The Karlsruhe Tritium Neutrino (KATRIN) Experiment:
(not your grandmother’s table top experiment!)
sensitivity m2νe> (0.2 eV)2
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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]
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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]
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
Big Bang Neutrinos are Warm Dark Matter
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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?
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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.
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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!
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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)
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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).
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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.
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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,
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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!
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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?
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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).
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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.
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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!
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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.
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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!
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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!
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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. . .
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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.
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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.
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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”).
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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]
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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)
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
“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.
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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)
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
−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
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
−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)
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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.
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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.”
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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!
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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”?
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
“Left-Over” Predictions: δ, mass-hierarchy, cos 2θ23
[Albright and Chen, hep-ph/0608137]
| || || || || || || || || |Daya Bay
(3 σ)
↔
↔
↔
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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
.
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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]
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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.
June 12–14, 2019 Neutrinos
Andre de Gouvea Northwestern
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).
June 12–14, 2019 Neutrinos