Neutrino Physics (I)

Mariam Tórtola

IFIC, Universitat de València/CSIC

International Summer Workshop on High Energy Physics 2018 TAE 2018 - Benasque (Spain), Sep 02 - Sep 15

Neutrino Physics (I)

Outline

๏ Historical introduction to neutrino physics

๏ Neutrinos in the Standard Model

๏ Neutrino masses beyond the Standard Model

๏ Neutrino oscillations in vacuum and matter

๏ Three-flavour neutrino oscillations

๏ Beyond three-neutrino flavours: sterile neutrinos

๏ The absolute scale of neutrino mass

๏ Future prospects in neutrino oscillations

๏ Neutrino physics beyond the Standard Model

Outline

๏ Historical introduction to neutrino physics

๏ Neutrinos in the Standard Model

๏ Neutrino masses beyond the Standard Model

๏ Neutrino oscillations in vacuum and matter

๏ Three-flavour neutrino oscillations

๏ Beyond three-neutrino flavours: sterile neutrinos

๏ The absolute scale of neutrino mass

๏ Future prospects in neutrino oscillations

๏ Neutrino physics beyond the Standard Model

What is a neutrino?

๏ spin 1/2 particle

๏ massless particle (almost)

๏ neutral

๏ 3 flavors (mixing)

Anything else?

๏ our body emits 400 neutrinos/s ( ⁴⁰ K decay)

๏ the Universe contains ∼ 330 neutrinos/cm ³ Every second we are traversed by:

๏ 400x10 ¹² neutrinos from the Sun

๏ 50x10 ⁹ neutrinos from natural radioactivity

๏ 10x10 ⁹ neutrinos from nuclear power plants

Moreover:

Why neutrinos are so important?

๏ they can probe environments that other techniques cannot: SN explosions, core of the Sun,...

๏ their role is crucial for the evolution of the universe (Big Bang Nucleosynthesis, structure formation)

๏ they could help explaining the matter-antimatter asymmetry of the Universe (leptogenesis mechanism)

๏ they could be a component of the dark matter of the universe.

๏ they provide the first evidence for physics beyond the SM!!!

Why neutrinos are so important?

๏ they can probe environments that other techniques cannot: SN explosions, core of the Sun,...

๏ their role is crucial for the evolution of the universe (Big Bang Nucleosynthesis, structure formation)

๏ they could help explaining the matter-antimatter asymmetry of the Universe (leptogenesis mechanism)

๏ they could be a component of the dark matter of the universe.

๏ they provide the first evidence for physics beyond the SM!!!

However: there are still many open questions in

neutrino physics

Historical introduction

to neutrino physics

‣ 1930: Pauli introduced the neutrino to explain continuous electron spectrum in nuclear beta decay.

The proposal of the neutrino

c"

‣ 1933: Fermi postulated the first theory of nuclear beta decay , the theory of weak interactions

interactions

➡ new name for particle: neutrino

n ! p + e + ¯ ^e

+?

G F

Pauli, 1930

But, where was the neutrino?

‣ 1934: Bethe and Peierls calculated the cross section σ for the processes:

+ n ! p + e

¯ + p ! n + e ⁺

According to Fermi theory, they obtained (for antineutrino + proton):

(to be compared with σ ^γp ∼ 10 ^-25 cm ² ) !!!

σ ∼ 5x10 ^-44 cm ² for a 2 MeV neutrino

Difficult but not impossible!

factor 2 missing!

Discovery of the neutrino

‣ 1956: First observation of reactor ν ^e by Reines and Cowan.

Telegram to Pauli on 12/06/1956

1995 Nobel Prize in Physics to

Reines

2 tanks with 200 liters H 2 O

+

40 kg CdCl 2

3 scintillator

layers with PMTs

More than one neutrino flavour?

‣ 1962: Discovery of ν ^μ by Lederman, Schwartz and Steinberger

1988 Nobel Prize in Physics

‣ 1959: Pontecorvo suggested the existence of a different neutrino, associated to muon decay and proposed an experiment to check it.

acc + n ! p + (e or µ ?)

⇥ ⁺ ! µ ⁺ + _µ

µ + n ! p + µ

not e ^-

‣ 2000: Discovery of ν ^τ by the DONUT Collaboration.

More than two neutrino flavours?

‣ 1989: LEP measurements of the invisible decay width of Z boson

‣ 1978: Discovery of τ at SLAC → imbalance of energy in τ decay suggests existence of a third neutrino.

inv ⌘ ^{Z had} 3 _lep

N = _inv / _SM (Z ! ⁱ ¯ _i )

→ N ^ν = 2.984 ± 0.008

800 GeV p → D ^s meson (≡cs) → ν ^τ - beam → τ detected

‣ 1962: Maki, Nakagawa and Sakata proposed flavor neutrino oscillations.

Neutrino oscillations

‣ 1957: Pontecorvo suggests oscillations between neutrinos &

antineutrinos (only ν ^e ).

2 ν mixing

B. Pontecorvo, J. Exp. Theor. Phys. 33 (1957)549.

B. Pontecorvo, J. Exp. Theor. Phys. 34 (1958) 247.

Z. Maki, M. Nakagawa, S. Sakata, Prog. Theor. Phys. 28 (1962) 870.

‣ 1969: Gribov & Pontecorvo calculated the neutrino oscillation probability (in vacuum) for the first time

true

neutrinos weak

neutrinos

V. Gribov, B. Pontecorvo, Phys. Lett. B28 (1969) 493.

First indication of ν oscillations

‣ 1968: First observation of solar neutrinos by R. Davis in Homestake.

2002 Nobel Prize in Physics

∼30% ∼ 50% ∼ 40%

→ confirmed by the following experiments

Explanation?

→ theory (SM, SSM) was wrong

→ experiments were wrong (all of them?)

→ something was happening to neutrinos

➡ 1/3 of the Standard Solar Model prediction !!

e + ³⁷ Cl ! ³⁷ Ar + e

‣ 1998: Discovery of atmospheric neutrino oscillations in Super-Kamiokande.

The atmospheric ν anomaly

‣ 1985: First indications of a deficit in the observed number of atmospheric ν ^μ at the IMB experiment.

‣ 1994: Kamiokande finds the ν ^μ deficit depends on the distance travelled by the neutrino.

oscillation channel ν ^{μ →} ν ^τ

➡ first evidence for non-zero neutrino masses .

Other important dates

‣ 2011-2012: neutrino oscillations observed in solar, atmospheric, reactor and accelerator neutrino experiments.

2002 Nobel Prize in Physics

‣ 1987: Supernova neutrino detection from supernova 1987A in Kamiokande & IMB.

‣ 2001: Sudbury Neutrino Observatory (SNO) confirms a change of flavor in solar ν ^e flux.

‣ 2002: KamLAND experiment confirms solar neutrino oscillations using neutrinos from nuclear reactors

M. Koshiba

Neutrinos in the Standard

Model

Neutrinos in the Standard Model

• neutrinos come in 3 flavours ,

corresponding to the charged lepton associated

• they belong to SU(2) lepton doublets

✓

e

e

◆

L

,

✓

µ

µ

◆

L

✓

⇥

◆

L

• In the SM, there are no SU(2) neutrino singlets (alike e R , μ ^R , τ ^R )

• neutrinos are left handed and antineutrinos right handed

Helicity and Chirality (handedness)

‣ Helicity is the projection of spin along the momentum direction

→ Lorentz-invariant only for massless particles

‣ Chirality is an asymmetry property:

a chiral object is not identical to its mirror image, cannot be superimposed on it.

→ Lorentz-invariant although not directly measurable

P _L,R = 1 ⌥ ⁵

2 , ⇥ _L,R = P _L,R ⇥

→ conserved in time

→ not conserved: mass terms mix LH and RH chiral states

H = ˆ ~ · ~p

|~p|

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Massless particles: Helicity = Chirality

Massive particles: Chiral states contain contributions from both helicity states Ultra-relativistic

particles: LH (RH) chiral projection dominated by a - (+) helicity state

Neutrino interactions in the SM

Charged Current (CC):

Neutral Current (NC): ^{Z 0 ! ↵ + ¯ ↵}

W ! l ↵ + ¯ _↵ W ⁺ ! l ↵ ⁺ + _↵

= e, µ, ⇥

‣ interactions conserve total Lepton Number L

‣ family lepton numbers L e , L μ , L τ are also conserved (1998: nu oscill !!) L ^CC int = g

⇥ 2

X ⇥ ¯ _{L ⇥} l _L W ^⇥ + h.c.

!

in the SM, only LH neutrinos and RH

antineutrinos

participate in weak interactions

L ^NC int = g

4 cos ⇥ _W

X ⇤ ¯ _⇥ (1 ₅ )⇤ Z ^⇥ + h.c.

!

‣ neutrinos interact only through the weak force

Neutrino mass in the Standard Model

‣ In the SM, fermion masses appears in the lagrangian in the term:

decomposing into its chiral states:

→ mass couples L and R chiral states of a particle: flips chirality

→ OK for most of particles but SM neutrino has only a L-chiral state

→ But in the SM there are no R-chiral states for neutrinos, N ^R

→ Dirac mass term

m ¯

⇥ = ⌘ ^L + N _R

L ^D = m _D ¯ = m _D ( _L + N _R )( _L + N _R ) = m _D ( _L N _R + N _{R L} )

‣ Therefore, neutrinos are massless in the SM

Neutrino masses: Majorana neutrinos

‣ Other option: try to make a mass term from ν ^L alone

→ a R-chiral field from a L-chiral field by charge conjugation:

R ⌘ L ^C = ˆ C _L ^T

→ the total neutrino field is: = _L + _R = _L + _L ^C

C = ( _L + _L ^C ) ^C = _L ^C + _L =

→ taking the charge conjugate

neutrino = antineutrino

‣ Majorana mass term:

= ⌫ = ⌫ _L + ⌫ _L ^C

C = i ˆ ^{2 0}

L ^M = 1

2 m( _{L L} ^c + _{L L} ^c )

However: this mass term not invariant under weak isospin

Majorana, ∼1930

2 degrees of freedom

Dirac mass term Majorana mass term L ^M = 1

2 m( _{L L} ^c + _{L L} ^c ) under U(1) transformation:

! e ⁱ , ! e ⁱ

invariant not invariant

→ conserves all charges (Q, L, B) → breaks all charges in 2 units

1) charged particles must be Dirac OR only neutral particles can be Majorana

→ neutrino, with Q(ν) =0, can be Majorana

2) if neutrinos are Majorana, total lepton number is not conserved L ^D = m _D ( _L N _R + N _{R L} )

3) if neutrinos are Dirac, L conservation has to be imposed by hand However: none of the terms can be constructed in SM

→ no N ^R in SM → ⌫ _L ^c ⌫ _L forbidden by weak isospin

Neutrinos are massless within the SM

But from oscillations we know neutrinos do have mass!!

m ν ∼ 0 - 1 eV

Neutrino masses beyond the

Standard Model

Dirac mass term Majorana mass term L ^M = 1

2 m( _{L L} ^c + _{L L} ^c ) under U(1) transformation:

! e ⁱ , ! e ⁱ

invariant not invariant

→ conserves all charges (Q, L, B) → breaks all charges in 2 units

1) charged particles must be Dirac OR only neutral particles can be Majorana

→ neutrino, with Q(ν) =0, can be Majorana

2) if neutrinos are Majorana, total lepton number is not conserved L ^D = m _D ( _L N _R + N _{R L} )

3) if neutrinos are Dirac, L conservation has to be imposed by hand However: none of the terms can be constructed in SM

→ no N ^R in SM → ⌫ _L ^c ⌫ _L forbidden by weak isospin Neutrinos are massless within the SM

add Higgs triplet

add N R

Dirac mass term

-> decomposing into its chiral states:

→ “sterile” neutrino

‣ 4 components Dirac neutrino:

⇥ = ⌘ ^L + N _R

L ^D = m _D ¯ = m _D ( _L + N _R )( _L + N _R ) = m _D ( _L N _R + N _{R L} )

L , _L , N _R , N _R

Minimal extension SM: add N R

L ^Yukawa = Y _⌫ (¯ ⌫ _e e) ¯ _L

✓ ₀ ◆

N _R + h.c.

-> after SSB: ^{< >= p 1}

2

✓ v 0

◆

‣ ^{From 𝝂 oscill: m ⌫ q m 2} 31 = 0.05 eV

much smaller than other Yukawas: Y _e ' 10 ⁵

4 degrees of freedom

m _D = Y _⌫ v

p 2 ! Y ^⌫ ' 10 ¹³

‣ Add a right handed neutrino singlet under SU(2)xU(1):

⌫ = ⌫ _L + ⌫ _L ^C N = N _R + N _R ^C

‣ Most general mass term:

L = L ^D + L ^M = 1

2 ^L N _R ^c

✓ 0 m _D m _D M _R

◆ ✓ _c

L

N _R

◆

+ h.c.

not mass eigenstates 1

2 N

✓ M ₁ 0 0 M ₂

◆ ✓

N

→ diagonalization: ◆

for ^{M 1 '}

m ² _D M _R M ₂ ' M ^R

→ seesaw mechanism SU(2) forbidden

M _R m _D :

(m _D ' vY ^⌫ )

Seesaw mechanism for neutrino mass

Seesaw mass models

⇒ ν masses are generated through mixing with heavy particles

Low energy seesaw models

Inverse seesaw model Mohapatra and Valle, PRD 34 (1986) 1642

Extended lepton content:

SU(2) singlets

(⌫, ⌫ ^c , S) L=(+1,-1,+1)

m _⌫ = M _D (M ^T ) ¹ µ M ¹ M _D ^T

M _⌫ =

0

@ 0 M _D 0

M _D ^T 0 M 0 M ^T µ

1 A

- 𝛍 breaks L and generates neutrino mass (massless for 𝛍=0) - m ^𝛎 can be very light even if M is far below GUT scale:

with 𝛍～ keV and M～10 ³ GeV → m ^𝛎 ～eV

Radiative models of neutrino masses

extension of scalar sector of the SM

neutrino masses can be generated through loops

⇒ loop suppression accounts for the smallness of m ν

Zee model

+ singlet scalar h ⁺

Zee-Babu model

+ extra Higgs doublet H + singlet scalar k ⁺⁺

+ singlet scalar h ⁺

Zee, PLB 93 (1980) 389 Zee, NPB 264 (1986) 99; Babu, PLB 203 (1988) 132

The flavour problem

‣ Why do fermion masses show these hierarchical relations ?

‣ Why quark and lepton mixings are so different?

‣ seesaw models explain the smallness of neutrino masses However, they can not explain:

θ ¹² ≃ 13°

θ ¹³ ≃ 0.2°

θ ²³ ≃ 2.4°

θ ¹² ≃ 34°

θ ¹³ ≃ 9°

θ ²³ ≃ 49°

m _e ⌧ m ^µ ⌧ m ^⌧

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m _u , m _d ⌧ m ^c , m _s ⌧ m ^t , m _b

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⇒ One can add new symmetries of leptons to the Standard Model

Neutrino oscillations

Neutrino mixing

‣ leptonic weak charged current:

‣ Mixing is described by the Maki-Nakagawa-Sakata (MNS) matrix:

‣ NxN unitary matrix: NxN mixing parameters

L = X

k

U _{i kL}

→ N(N-1)/2 mixing angles + N(N+1)/2 phases

‣ Lagrangian invariant under global phase transformations of Dirac fields:

! e ⁱ , ⇥ _k ! e ^{i⇥ k} ⇥ _k

j _⇤ ^{CC †} ! 2 X

,k

L e ^{i(⇥ e ⌅ 1 )} e ^{i(⇥ ⇥ e )} ⇥ _⇤ U _k e ^{i(⌅ k ⌅ 1 )} ⇤ _kL j _⇥ ^{CC †} = 2 X

=e,µ,⇤

L ⇥ _⇥ ⇤ _L = 2 X

=e,µ,⇤

X 3 k=1

L ⇥ _⇥ U _k ⇤ _kL

N-1 N-1

1

→ 2N-1 phases can be eliminated: (N-1)(N-2)/2 physical phases

U = U _l ^† U

‣ For Majorana neutrinos , the lagrangian is NOT invariant under global phase transformations of the Majorana fields:

→ N(N-1)/2 physical phases:

k ! e ^{i k k kL T C † kL e 2i k kL T C † kL}

→ only N phases can be eliminated by rephasing charged lepton fields:

j _⇤ ^{CC †} ! 2 X

,k

L e ^{i⇥ ↵} ⇥ _⇤ U _k ⇤ _kL

N

(N-1) Majorana phases relevant for 0 νββ effect in ν oscil.

→

Neutrino mixing

(N-1)(N-2)/2 Dirac phases

→

Neutrino mixing

‣ 2-neutrino mixing depends on 1 angle only (+1 Majorana phase)

‣ 3-neutrino mixing is described by 3 angles and 1 Dirac (+2 Majorana) CP violating phases.

atmospheric + LBL

measurements solar + KamLAND

measurements reactor disapp + LBL

appearance searches

Neutrino oscillations

i d

dt | ⇥ = H| ⇥

‣ Neutrino evolution equation:

in the neutrino mass eigenstates basis ν ^{j :}

L = X

k

U _{i kL}

‣ flavour states are admixtures of mass eigenstates:

for relativistic neutrinos: t = L equal momentum approx:

| ^j ⇥ e ^{iE j t} | ^j ⇥

E _j ' p + m ² _j

2p ' p + m ² _j 2E

H =

0

@ E ₁ 0 0 0 E ₂ 0 0 0 E ₃

1 A

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|⌫ ^j i ! e ^ipL e ⁱ

m2 j L

2E |⌫ ^j i ! e ⁱ

m2 j L

2E |⌫ ^j i

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| = X

j

j |U ^j

Production Propagation Detection

coherent superposition of massive states

projection over flavour eigenstates

different propagation phases change ν ^j

composition

| = X

j

U ^⇤ _j | ^j

i d

dt | ⇥ = H| ⇥

Neutrino oscillations picture

⌫ _j : e ⁱ

m2 j L 2E

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Neutrino oscillation amplitude:

A ^⇤ ↵ !⇤ = _⇥ (t) | (0)⇥ = X

j

⇥ | ^j (t) ⇥ ^j (t) | ^j (0) ⇥ ^j (0) | ⇥

Neutrino oscillation probability:

P _⇥ = _⇥ 4 X

i>j

Re(U ^⇤ _i U _j U _⇥i U _⇥j ^⇤ ) sin ² m ² _ij L 4E

! +

+2 X

i>j

Im(U ^⇤ _i U _j U _⇥i U _⇥j ^⇤ ) sin m ² _ij L 2E

!

= X

j

U _j e ⁱ

m2 j L

2E U _↵j ^⇤

P _{⌫ ↵ !⌫} = X

j

U _j e ⁱ

m2 j L

2E U _↵j ^⇤

2

production detection

propagation

Neutrino oscillation probability

m ² _kj = m ² _k m ² _j

‣ Conservation of probability:

X P ( _↵ ! ) = 1

‣ Neutrino oscillations violate flavour lepton number conservation (expected from mixing) but conserve total lepton number

‣ Neutrino oscillations do not depend on the absolute neutrino mass scale and Majorana phases.

‣ Neutrino oscillations are sensitive only to mass squared differences:

General properties of neutrino oscillations

‣ For antineutrinos: U →U*

‣ Complex phases in the mixing matrix induce CP violation:

P ( _↵ ! ) 6= P ( ^↵ ! )

Two possible mass orderings:

- ^{Δm 231} : atmospheric + long-baseline

- ^{Δm 221} : solar + KamLAND (we know it is positive)


Oscillation experiments favor NO with ∼ 3 𝞂 (2018)

2-neutrino oscillations

‣ 2-neutrino mixing matrix:

‣ 2-neutrino oscillation probability ( α≠β):

P ( ! ^⇥ ) = U ₁ U _⇥1 ^⇤ + U ₂ U _⇥2 ^⇤ e ^{i m2 2E 21 L}

2

✓ cos ✓ sin ✓ sin ✓ cos ✓

◆

= m ² ₂₁ L

4E = 1.27 m ² ₂₁ [eV ² ]L[km]

E[GeV ]

‣The oscillation phase:

→ short distances, << 1: oscillations do not develop, P αβ = 0

→ long distance, ∼ 1: oscillations are observable

→ very long distances, >> 1: oscillations are averaged out:

P _↵ ' 1

2 sin ² (2 )

= sin ² (2 ) sin ²

✓ m ² L 4E

◆

2-neutrino oscillation probability

first oscillation maximum:

averaged oscillations oscillation

amplitude

L _osc = 4 E m ²

oscillation length:

P _↵ = sin ² (2 ) sin ²

✓ m ² L 4E

◆

Appearance vs disappearance experiments

‣ a ppearance experiments:

↵ 6=

→ appearance of a neutrino of a new flavour β in a beam of ν ^α

‣ disappearance experiments:

→ measurement of the survival probability of a neutrino of given flavour P _↵ = sin ² (2 ) sin ²

✓ m ² L 4E

◆

P _↵↵ = 1 sin ² (2 ) sin ²

✓ m ² L 4E

◆

Matter effects on neutrino oscillations

‣ When neutrinos pass trough matter, the interactions with the particles in the medium induce an effective potential for the neutrinos.

[ → the coherent forward scattering amplitude leads to an index of refraction for neutrinos. L. Wolfenstein, 1978 ]

→ modifies the mixing between flavor states and propagation states as

well as the eigenvalues of the Hamiltonian, leading to a different oscillation

probability with respect to vacuum oscillations.

Effective matter potential

‣ Effective four-fermion interaction Hamiltonian (CC+NC)

H _int ^{⇥ ↵} = G _F

p 2 ⇥ _µ (1 ₅ )⇥ X

j

f ^µ (g _V ^,f g _A ^,f ₅ )f

in ordinary matter: f=e ^- ,p,n

non-relativistic unpolarised

neutral

for a medium: hf ^µ f i = 1

2 N _f ⇥ _µ,0 hf ^{5 µ} f i = 0

N _e = N _p

To obtain the matter-induced potential we integrate over f-variables:

J _matt ^µ

J _matt ^µ = 1

2 [N _e (g _V ^,e + g _V ^,p ) + N _n g _V ^,n ]

Effective matter potential

J _matt ^µ = 1

2 [N _e (g _V ^,e + g _V ^,p ) + N _n g _V ^,n ]

J _matt ^µ = (N _e 1

2 N _n , 1

2 N _n , 1

2 N _n )

V _matt = p

2G _F diag(N _e 1

2 N _n , 1

2 N _n , 1

2 N _n )

‣ only ν ^e are sensitive to CC (no μ,τ in ordinary matter)

‣ NC has the same effect for all flavours → it has no effect on evolution (however it can be important in presence of sterile neutrinos)

‣ for antineutrinos the potential has opposite sign

2-neutrino oscillations in matter

‣ Effective hamiltonian in matter H _f ^matt = H _f ^vac + V _e⇥ =

m ²

4E cos 2 + V _{CC 4E} ^{m 2} sin 2

m ²

4E sin 2 _4E ^{m 2} cos 2

!

Diagonalizing the Hamiltonian, we identify the mixing angle and mass splitting in matter:

H _f ^matt = M ² 4E

✓ cos 2 _M sin 2 _M sin 2 _M cos 2 _M

◆

V _CC = p

2G _F N _e

In general: N e =N e (x), so θ ^M and ΔM ² will be function of x as well

→ however, in some cases analytical solutions can be obtained

‣ Hamiltonian in vacuum in the flavour basis:

H _f ^vac = U H _m U ^† = m ² 4E

✓ cos 2✓ sin 2✓

sin 2✓ cos 2✓

◆

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→ we can use vacuum expression for oscillation probability, replacing “vacuum” parameters by “matter” parameters:

2- ν oscillations in constant matter

‣ If N e is constant (good approximation for oscillations in the Earth crust):

P _⇥ = sin ² (2 _M ) sin ²

✓ M ² L 4E

◆

M ² = m ² q

sin ² 2 + (cos 2 A) ²

A = 2EV m ²

There is a resonance effect for A = cos2 θ → MSW effect

→ θ ^M and ΔM ² are constant as well

Wolfenstein, 1978

Mikheyev & Smirnov, 1986

sin ² 2 _M = sin ² 2

sin ² 2 + (cos 2 A) ²

2- ν oscillations in constant matter

‣ A << cos2 θ, small matter effect → vacuum oscillations: θ ^M = θ

‣ A >> cos2 θ, matter effects dominate → oscillations suppressed: θ ^M ≈ π/2

‣ A = cos2 θ, resonance takes place → maximal mixing θ ^M ≈ π/4

A = 2EV m ²

mixing angle in matter:

→ resonance condition is satisfied for neutrinos for Δm ² > 0

for antineutrinos for Δm ² < 0 sin ² 2 _M = sin ² 2

sin ² 2 + (cos 2 A) ²

Tutorials: matter effects in solar

neutrinos

Matter effects in atmospheric ν’ ^s

➙ Matter effects on the atmospheric neutrino flux are sensitive to the mass ordering.

• atmospheric neutrinos interact with the Earth mantle and core

tan 2✓ _m =

m ²

4E sin 2✓

m ²

4E cos 2✓ ⌥ p

2G _F N _e

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✓ MSW resonance in ν ^μ →ν ^{e channel}

✓ no matter effects in ν ^μ →ν ^{τ channel}

(-) neutrinos (+)antineutrinos

Matter effects in atmospheric ν’ ^s

At E∼ 3-8 GeV: MSW resonance for neutrinos and NO mass spectrum.

If IO ⟹ resonance for antineutrinos

NO IO

de Salas et al, arXiv:1806.11051