Graduiertentag Kepler Center for Astro and Particle Physics · Lepton mixing Flavour neutrinos A...

GraduiertentagKepler Center for Astro and Particle PhysicsNeutrino Theory - I: Neutrino Oscillation Phenomenology

Thomas Schwetz-Mangold

Max-Planck-Institut für Kernphysik, Heidelberg

13 May 2011

T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 1 / 52

Neutrinos oscillate...

... and have mass ⇒ physics beyond the Standard Model

I Lecture I: Neutrino Oscillation phenomenologyI Lecture II: Neutrinos and physics beyond the Standard Model


Neutrinos oscillate...

... and have mass ⇒ physics beyond the Standard Model

I Lecture I: Neutrino Oscillation phenomenologyI Lecture II: Neutrinos and physics beyond the Standard Model


Outline

Lepton mixing

Neutrino oscillationsOscillations in vacuumOscillations in matterVarying matter density and MSW

Global data and 3-flavour oscillations

Outlookθ13CPV, mass hierarchy


Lepton mixing

Outline

Lepton mixing





Lepton mixing

The Standard Model

Flavours: 1 2 3

u c tQuarks: d s b

νe νµ ντLeptons: e µ τ

Fermions in the Standard Model come in three generations (“Flavours”)

Neutrinos are the “partners” of the charged leptonsmore precisely: they form a doublet under the SU(2) gauge symmetry


Lepton mixing

Flavour neutrinos

A neutrino of flavour α is defined by the charged current interaction withthe corresponding charged lepton:

LCC = − g√2

W ρ∑

α=e,µ,τ

ναLγρ`αL + h.c.

for example

π+ → µ+νµ

the muon neutrino νµ comes together with the charged muon µ+


Lepton mixing

Flavour neutrinos

A neutrino of flavour α is defined by the charged current interaction withthe corresponding charged lepton:

LCC = − g√2

W ρ∑

α=e,µ,τ

ναLγρ`αL + h.c.

for example

π+ → µ+νµ

the muon neutrino νµ comes together with the charged muon µ+


Lepton mixing

Flavour neutrinos

W

lα

W

lα

νανα

detectorneutrino source

"short" distance


Lepton mixing

Let’s give mass to the neutrinos

Majorana mass term:

LM = −12

∑α,β=e,µ,τ

νTαLC−1MαβνβL + h.c.

M: symmetric mass matrix

In the basis where the CC interaction is diagonal the mass matrix is ingeneral not a diagonal matrix

any complex symmetric matrix M can be diagonalised by a unitary matrix

UTν MUν = m , m : diagonal, mi ≥ 0


Lepton mixing

Lepton mixing

LCC = − g√2

W ρ∑

α=e,µ,τ

3∑i=1

νiLU∗αiγρ`αL + h.c.

LM = −12

3∑i=1

νTiL C−1νiLmν

i −∑

α=e,µ,τ

¯αR`αLm`

α + h.c.

The unitary lepton mixing matrix:

(Uαi ) ≡ UPMNS = V Dirac DMaj

DMaj = diag(e iαi /2)


Neutrino oscillations

Outline

Lepton mixing







W

lα

W

lβ

νβνα

detectorneutrino source

"long" distance

neutrino oscillations

|να〉 = U∗αi |νi 〉 e−i(Ei t−pi x) |νβ〉 = U∗

βi |νi 〉


Neutrino oscillations Oscillations in vacuum

Neutrino oscillations in vacuum

oscillation amplitude:

Aνα→νβ= 〈νβ| propagation |να〉

=∑i ,j

Uβj〈νj | e−i(Ei t−pi x) |νi 〉U∗αi =

∑i

UβiU∗αie

−i(Ei t−pi x)

oscillation probability:Pνα→νβ

=∣∣Aνα→νβ

∣∣2

To derive the oscillation probability rigorously one needs either awave-packet treatment or field theory Akhmedov, Kopp, JHEP 1004:008 (2010) [1001.4815]






=∑i ,j


∑i

UβiU∗αie

−i(Ei t−pi x)


=∣∣Aνα→νβ

∣∣2







=∑i ,j


∑i

UβiU∗αie

−i(Ei t−pi x)


=∣∣Aνα→νβ

∣∣2




A hand-waving derivation for two flavoursoscillation phase:

φ = (E2 − E1)t − (p2 − p1)x with E 2i = p2

i + m2i

define: ∆E = E2 − E1, ∆E 2 = E 22 − E 2

1 , E = (E1 + E2)/2

then: ∆E 2 = 2E∆E (similar for p and m)

φ = ∆Et − ∆p2

2px = ∆Et − ∆E 2 −∆m2

2px

= ∆Et − 2E2p

∆Ex +∆m2

2px

use “average velocity” of the neutrino v = p/E and x ≈ vt:

φ ≈ ∆m2

2px ≈ ∆m2

2Ex




φ = (E2 − E1)t − (p2 − p1)x with E 2i = p2

i + m2i

define: ∆E = E2 − E1, ∆E 2 = E 22 − E 2

1 , E = (E1 + E2)/2


φ = ∆Et − ∆p2

2px = ∆Et − ∆E 2 −∆m2

2px

= ∆Et − 2E2p

∆Ex +∆m2

2px


φ ≈ ∆m2

2px ≈ ∆m2

2Ex




φ = (E2 − E1)t − (p2 − p1)x with E 2i = p2

i + m2i

define: ∆E = E2 − E1, ∆E 2 = E 22 − E 2

1 , E = (E1 + E2)/2


φ = ∆Et − ∆p2

2px = ∆Et − ∆E 2 −∆m2

2px

= ∆Et − 2E2p

∆Ex +∆m2

2px


φ ≈ ∆m2

2px ≈ ∆m2

2Ex




φ = (E2 − E1)t − (p2 − p1)x with E 2i = p2

i + m2i

define: ∆E = E2 − E1, ∆E 2 = E 22 − E 2

1 , E = (E1 + E2)/2


φ = ∆Et − ∆p2

2px = ∆Et − ∆E 2 −∆m2

2px

= ∆Et − 2E2p

∆Ex +∆m2

2px


φ ≈ ∆m2

2px ≈ ∆m2

2Ex



The oscillation probability in vacuum

Aνα→νβ=

∑i

UβiU∗αie

−i(Ei t−pi x)

Pνα→νβ(L) =

∑jk

UαjU∗βjU

∗αkUβk exp

[−i

∆m2kjL

2 Eν

]

∆m2kj ≡ m2

k −m2j : oscillations are sensitive only to mass-squared

differences (not to absolute mass!)

to observe oscillations one needsI non-trivial mixing Uαi

I non-zero mass-squared differences ∆m2kj

I a suitable value for L/Eν



The oscillation phase

φ =∆m2L4Eν

= 1.27∆m2[eV2] L[km]

Eν [GeV]

I “short” distance: φ � 1: no oscillations can develop and Pνα→νβ= δαβ

because of∑

j UαjU∗βj = δαβ .

I “long” distance: φ & π/2: oscillations are observable

I “very long” distance: φ � 2π: oscillations are averaged out(indep. of L and Eν):

Pνα→νβ=∑

j

|Uαj |2|Uβj |2



2-neutrino oscillations

Two-flavour limit:

U =

(cos θ sin θ− sin θ cos θ

), P = sin2 2θ sin2 ∆m2L

4Eν

0.1 1 10 100L / Eν (arb. units)

0

0.2

0.4

0.6

0.8

1

Pαβ

4π / ∆m2

sin2 2θ

"short"distance

"long"distance

"very long"distance



Appearance vs. disappearance

I appearance experiments:

Pνα→νβ, α 6= β

“appearance” of a neutrino of a new flavour β 6= α in a beam of να

I disappearance experiments:

Pνα→να

measurement of the “survival” probability of a neutrino of givenflavour



General properties of vacuum oscillations

Pνα→νβ=∑jk

UαjU∗βjU

∗αkUβk exp

[−i

∆m2kjL

2 Eν

]

I Unitarity:∑

β Pνα→νβ= 1

I For anti-neutrinos replace Uαi → U∗αi

I Pνα→νβ= Pνβ→να (CPT invariance)

I Phases in U induce CP violation: Pνα→νβ6= Pνα→νβ

I there is no CP violation in disappearance experiments:

Pνα→να = Pνα→να =∑k,j

|Uαk |2|Uαj |2e−i∆m2kj L/2E



Eff. Schrödinger equation

The evolution of the flavour state can be described by an effectiveSchrödinger equation:

iddt

(aeaµaτ

)= Hvac

(aeaµaτ

)

where

Hνvac = Udiag

(0,

∆m221

2Eν,∆m2

312Eν

)U†

H νvac = U∗diag

(0,

∆m221

2Eν,∆m2

312Eν

)UT


Neutrino oscillations Oscillations in matter

The matter effect

When neutrinos pass through matter the interactions with the particles inthe background induce an effective potential for the neutrinos

The coherent forward scattering amplitude leads to an index of refractionfor neutrinos

L. Wolfenstein, Phys. Rev. D 17, 2369 (1978); ibid. D 20, 2634 (1979)



Effective Hamiltonian in matter

Hνmat = Udiag

(0,

∆m221

2Eν,∆m2

31

2Eν

)U† + diag(

√2GF Ne , 0, 0)

H νmat = U∗diag

(0,

∆m221

2Eν,∆m2

31

2Eν

)UT︸︷︷︸

Hvac

− diag(√

2GF Ne , 0, 0)︸︷︷︸Vmat

Ne(x): electron density along the neutrino path

for non-constant matter the Hamiltonian depends on time:

iddt

a = Hmat(t)a



Effective matter potential - 1

Effective 4-point interaction Hamiltonian in the SM

Hναint =

GF√2

ναγµ(1− γ5)να

∑f

f γµ(gα,fV − gα,f

A γ5)f︸︷︷︸Jµ

mat

ordinary matter: e−, p, n

non-relativistic: 〈f γµf 〉 = 12Nf δµ0

unpolarised: 〈f γ5γµf 〉 = 0

neutral: Ne = Np




Jµmat =

12δµ0

∑f =e,p,n

Nf gα,fV

=12δµ0[Ne(gα,e

V + gα,pV)

+ Nngα,nV]

gV e− p nνe 2 sin2 ΘW + 1

2 −2 sin2 ΘW + 12 −1

2νµ,τ 2 sin2 ΘW − 1

2 −2 sin2 ΘW + 12 −1

2

⇒ Vmat ∝(

Ne −12

Nn,−12

Nn,−12

Nn

)




Vmat =√

2GF diag (Ne − Nn/2,−Nn/2,−Nn/2)

ν l

l ν

W

ν ν

f f

Z0

CC NC

I only νe fell CC (there are no µ, τ in normal matter)

I NC is the same for all flavours ⇒ potential proportional to identiy has noeffect on the evolution

I NC has no effect for 3-flavour active neutrinos, but is important in thepresence of sterile neutrinos



Neutrino oscillations in constant matter

diagonalize the Hamiltonian in matter:

Hνmat = Udiag

(0,

∆m221

2Eν,∆m2

31

2Eν

)U† + diag(

√2GF Ne , 0, 0)

= Umdiag (λ1, λ2, λ3) U†m

Same expression for oscillation probability, but replace “vacuum”parameters by “matter” parameters



2-neutrino oscillations in constant matter

Two-flavour case:

Pmat = sin2 2θmat sin2 ∆m2matL

4E

with

sin2 2θmat =sin2 2θ

sin2 2θ + (cos 2θ − A)2

∆m2mat = ∆m2

√sin2 2θ + (cos 2θ − A)2

A ≡ 2EV∆m2



2-neutrino oscillations in constant matter

sin2 2θmat =sin2 2θ

sin2 2θ + (cos 2θ − A)2A ≡ 2EV

∆m2

resonance for cos 2θ = A: “MSW resonance” Mikheev, Smirnov, Sov. J. Nucl. Phys. 42, 913 (1985)

0.01 0.1 1 10 100A

0

0.2

0.4

0.6

0.8

1

sin2 2θ

mat

small matter effectvacuum osc.

strong matter effectosc. are suppressed

reso

nanc

e

sin22θ

vac = 0.3


Neutrino oscillations Varying matter density and MSW

Varying matter density: example solar neutrinos

The electron density in the sun:



The LMA-MSW mechanism

evolution is adiabatic if(

1θm

dθm

dx

)−1

� Losc

using ∆m2 = 8× 10−5 eV2 the oscillation length is

Losc =4πE∆m2 ' 30 km

(E

MeV

)for large mixing angles (sin2 θ12 ' 0.3):(

1θm

dθm

dx

)−1

∼(

1V

dVdx

)−1

∼ size of sun � 30 km

⇒ adiabatic evolution




the electron neutrino is born at the center of the sun as

|νe〉 = cos θm|ν1〉+ sin θm|ν2〉

then |ν1〉 and |ν2〉 evolve adiabatically to the Earth

Pee = Pprode1 Pdet

1e + Pprode2 Pdet

2e

Pprode3 ≈ sin2 θ13 ≈ 0, interference term averages out

Pprode1 = cos2 θm , Pdet

1e = cos2 θ

Pprode2 = sin2 θm , Pdet

2e = sin2 θ

⇒ Pee = cos2 θm cos2 θ + sin2 θm sin2 θ




in the center of the sun we have

A ≡ 2EV∆m2 ' 0.2

(Eν

MeV

)(8× 10−5 eV2

∆m2

)

resonance occurs forA = cos 2θ = 0.4

⇒ Eres ' 2 MeV




Pee = cos2 θm cos2 θ

+ sin2 θm sin2 θ

0.1 1 10Eν [MeV]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 - 1/2 sin22θ

sin2θ

Ere

s

strongmatter

vacuum

Pee

sin

2 θ m

Pee =

{c4 + s4 = 1− 1

2 sin2 2θ vacuum (θm = θ)sin2 θ strong matter (θm = π/2)



LMA-MSW and the solar neutrino spectrum

0.2

0.3

0.4

0.5

0.6

Pee

1 - 1/2 sin22θ

sin2θ



Evidence for LMA-MSW

Measurements of the solar neutrino rate at SNO and Borexino

0.1 1 10Eν [MeV]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Pee

1 - 1/2 sin22θ

sin2θ

Ere

s

strongmatter

vacuum

SNO CC/NC

Borexino



Outline

Lepton mixing






Evidences for neutrino oscillationsI solar neutrinos Homestake, SAGE+GNO, Super-K, SNO, Borexino

νe → νµ,τ LMA-MSW with ∆m2 ∼ 7× 10−5eV2

I Kamland reactor neutrino experiment (180 km)νe disappearance with Eν/L ∼ 7× 10−5eV2

I atmospheric neutrinos Super-Kamiokandelong-baseline accelerator experiments K2K (250 km), MINOS (735 km)νµ → ντ with Eν/L ∼ 2× 10−3eV2

• fits beautifully in 3-flavour framework with∆m2

21 ≈ 7× 10−5eV2 and ∆m231 ≈ 2× 10−3eV2

• no oscillations of νe seen so far with Eν/L ∼ 2× 10−3eV2

• controversial hints for oscillations with Eν/L ∼ eV2

(LSND, MiniBooNE, SBL reactor exps) "sterile neutrinos"?








21 ≈ 7× 10−5eV2 and ∆m231 ≈ 2× 10−3eV2











21 ≈ 7× 10−5eV2 and ∆m231 ≈ 2× 10−3eV2











21 ≈ 7× 10−5eV2 and ∆m231 ≈ 2× 10−3eV2






3-flavour oscillation parameters

νeνµ

ντ

=

Ue1 Ue2 Ue3Uµ1 Uµ2 Uµ3Uτ1 Uτ2 Uτ3

ν1ν2ν3

∆m231 ∆m2

21

U =

1 0 00 c23 s230 −s23 c23

c13 0 e−iδs130 1 0

−e iδs13 0 c13

c12 s12 0−s12 c12 0

0 0 1

atmospheric+LBL Chooz solar+KamLAND

3-flavour effects are suppressed: ∆m221 � ∆m2

31 and θ13 � 1 (Ue3 = s13e−iδ)

⇒ dominant oscillations are well described by effective two-flavour oscillations⇒ CP-violation is suppressed by θ13



3-flavour oscillation parameters

νeνµ

ντ

=

Ue1 Ue2 Ue3Uµ1 Uµ2 Uµ3Uτ1 Uτ2 Uτ3

ν1ν2ν3

∆m2

31 ∆m221

U =

1 0 00 c23 s230 −s23 c23

c13 0 e−iδs130 1 0

−e iδs13 0 c13

c12 s12 0−s12 c12 0

0 0 1

atmospheric+LBL Chooz solar+KamLAND

3-flavour effects are suppressed: ∆m221 � ∆m2

31 and θ13 � 1 (Ue3 = s13e−iδ)

⇒ dominant oscillations are well described by effective two-flavour oscillations⇒ CP-violation is suppressed by θ13



Three flavour oscillation parameters summary

TS, Tortola, Valle, 11




★

★

0 0.25 0.5 0.75 1

{sin2θ12, sin

2θ23}

10-4

10-3

{∆m

2 21, ∆

m2 31

} [e

V2 ]

Solar+KamLAND

Atmospheric+LBL




two possibilities for the neutrino mass spectrum

INVERTEDNORMAL

[mas

s]2

3ν

ν2

ν1

ν2ν1

ν3

νe

µν

ντ

∆m231 > 0 ∆m2

31 < 0



The 1-2 mass ordering

We know that the mass state containing most of νe is the lighter of thetwo “solar mass” states

∆m221 ≡ m2

2 −m21 > 0 and θ12 < 45o

thanks to the observation of the matter effect in the sun:

resonance condition:

∆m221 cos 2θ12 = 2EνV ⇒ ∆m2

21 cos 2θ12 > 0



The 1-3 mass ordering

We do not know the sign of ∆m231

(normal or inverted mass ordering)

No matter effect has been observed for oscillations with ∆m231, only

“vacuum” νµ → νµ(ντ ) oscillations:

Pµµ ≈ 1− sin2 2θ23 sin2 ∆m231L

4E

Has to look for matter effect in νe ↔ νµ oscillations due to ∆m231, θ13

⇒ future long-baseline experiments



Pνe→νµ

10-2

10-1

100

101

E [GeV]

101

102

103

104

L [

km]

< 0.1%

0.1% − 1%

1% − 5%

5% − 10%

> 10%

Contours of constant Peµ

1st osc m

ax with

∆m2

31

1st osc m

ax with

∆m2

21

LV = π

sin22θ

13 = 0.05 , ∆m

2

31 = 0.002 eV

2 , α = 0.026 , δ = 0



Pνe→νµ

10-2 10-1 100 101

E [GeV]

101

102

103

104

L [k

m]

Reactor-θ13

KamLANDT2K

WBB FNL-HS

MINOS

NOνA

CNGS

NuFact

βB100SPL

βB350

< 0.1%0.1% − 1%1% − 5%5% − 10%> 10%

Contours of constant Peµ

1st osc m

ax with

∆m2

31

1st osc m

ax with

∆m2

21

LV = π

sin22θ13 = 0.05 , ∆m231 = 0.002 eV2 , α = 0.026 , δ = 0


Outlook

Outline

Lepton mixing





Outlook θ13

Measuring θ13

two complementary approaches towards θ13:

I νe → νe disappearance reactor experiments:Double Chooz, Daya Bay, RENO

“clean” measurement of θ13:

Pee ≈ 1− sin2 2θ13 sin2 ∆m231L

4Eν+O

(∆m2

21∆m2

31

)2

⇒ bring stat. and syst. errors below % level

I long-baseline accelerator experiments looking forνµ → νe appearance: T2K, NOνA

Pµe is a complicated function of all 3-flavour parametersθ13 is correlated with other parameters (CP-phase δ, sign of ∆m2

31)


Outlook θ13

Measuring θ13


I νe → νe disappearance reactor experiments:Double Chooz, Daya Bay, RENO“clean” measurement of θ13:


4Eν+O

(∆m2

21∆m2

31

)2


I long-baseline accelerator experiments looking forνµ → νe appearance: T2K, NOνA

Pµe is a complicated function of all 3-flavour parametersθ13 is correlated with other parameters (CP-phase δ, sign of ∆m2

31)


Outlook θ13

Measuring θ13


I νe → νe disappearance reactor experiments:Double Chooz, Daya Bay, RENO“clean” measurement of θ13:


4Eν+O

(∆m2

21∆m2

31

)2


I long-baseline accelerator experiments looking forνµ → νe appearance: T2K, NOνAPµe is a complicated function of all 3-flavour parametersθ13 is correlated with other parameters (CP-phase δ, sign of ∆m2

31)


Outlook θ13

The LBL appearance oscillation probability

Pµe ' sin2 2θ13 sin2 θ23sin2(1− A)∆

(1− A)2

+ sin 2θ13 α sin 2θ23sin(1− A)∆

1− Asin A∆

Acos(∆ + δCP)

+ α2 cos2 θ23sin2 A∆

A2

with ∆ ≡ ∆m231L

4Eν, α ≡ ∆m2

21

∆m231

sin 2θ12 , A ≡ 2EνV∆m2

31

anti-ν: δCP → −δCP, A → −A, Peµ: δCP → −δCPother hierarchy: ∆ → −∆, A → −A, α → −α


Outlook θ13

Reactor vs Beam

assume sin2 2θ13 = 0.1, δ = π/2


Outlook θ13

The race for θ13

2010 2012 2014 2016 2018year

10-2

10-1

sin22

θ13

Discovery potential at 3 σ for NH

T2K

NOνADayaBay

DoubleChoozRENO

1

3

4

5

2

1 3 4 52

Mezzetto, TS 10

about 1 order of magnitude improvement within ∼5 yearsT. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 48 / 52

Outlook CPV, mass hierarchy

The ultimate goals

I measure the value of δCP ⇒ establish CP violation

measure Pνα→νβvs Pνα→νβ

cross section and fluxes are different for ν and ν,matter effect is CP violating

I determine the neutrino mass hierarchy, i.e. sgn(∆m231)

observe matter effect



The ultimate goals

I measure the value of δCP ⇒ establish CP violationmeasure Pνα→νβ

vs Pνα→νβ






The ultimate goals

I measure the value of δCP ⇒ establish CP violationmeasure Pνα→νβ

vs Pνα→νβ






Determination of the mass hierarchy

the vacuum oscillation probability is invariant under

∆m231 → −∆m2

31 δCP → π − δCP

→ the key to resolve the hierarchy degeneracy is the matter effectresonance condition for νµ → νe oscillations:

± 2EV∆m2

31= cos 2θ13 ≈ 1

can be fulfilled forneutrinos if ∆m2

31 > 0 (normal hierarchy)anti-neutrinos if ∆m2

31 < 0 (inverted hierarchy)



The size of the matter effect

A ≡∣∣∣∣ 2EV∆m2

31

∣∣∣∣ ' 0.09(

EGeV

)(|∆m2

31|2.5× 10−3 eV2

)−1

for experiments at the 1st osc. max, |∆m231|L/2E ' π, and

A ' 0.02(

L100 km

)

need L & 1000 km and Eν & 3 GeV in order to reach the regime of strongmatter effect A & 0.2.

terms linear in A do not break the degeneracy →have to be sensitive to higher order terms in A TS, hep-ph/0703279



Subsequent generation of LBL experiments

I superbeam upgardesI beta beamsI neutrino factory

under intense study e.g.

EUROν http://www.euronu.org

NF-IDS http://www.ids-nf.org


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