THE TALE OF NEUTRINO OSCILLATIONSSanjeev Kumar Verma

WHAT ARE NEUTRINOS?

Neutrino νe νμ ντ

Charged lepton

e μ τ

• The neutrinos are elementary particles very similar to the charged leptons except that they do not carry electromagnetic charge.• They do not have electromagnetic interactions and interact with matter only through the week interaction.• They come in three flavors. Every charged lepton has an associated neutrino as its neutral companion.• The three flavors are differentiated through their charged current weak interactions.

SOURCES OF NEUTRINOS

NEUTRINO OSCILLATIONS : A CHANGE IN FLAVOR?

Imagine that Alice bought an ice-cream of vanilla flavor. As she travels home, the ice-cream changes to strawberry flavor on the way. A little further, it changes back to vanilla …

Experiments found that similar things happen with neutrinos: an muon neutrino on its way transforms into a tau neutrino and after traveling some more it transforms back to muon neutrino. This is what neutrino oscillation is.

HOW CAN A PARTICLE CHANGE ITS IDENTITY?

Neutrino oscillations involve all the three flavors. For simplicity, we will often consider only two flavors.

If we view the electron and muon neutrinos as two different particles, such a transformation may seam strange.

If we view them as quantum mechanical states, then neutrino oscillations are a simple consequence of principle of superposition in quantum mechanics.

Example: two level system in quantum mechanics

TWO LEVEL SYSTEM IN QM

Consider a beam of electrons filter to have sz = +1/2.

Let’s measure its spin along x-axis. The measurement will give +1/2 in 50% of

the cases and -1/2 in another 50% of the cases.

Lesson: the state |sz = +1/2> is an admixture of the states |sx = +1/2> and |sx = -1/2>:

€

sz = +1

2=

1

2sx = +

1

2−

1

2sx = −

1

2

€

A SIMILAR SITUATIONS FOR THE NEUTRINOS COMING FROM THE SUN

The thermonuclear reactions in the Sun, apart from producing light, also produce neutrinos.

The high energy part of the solar neutrino spectrum is an admixture of νe, νμ and ντ in almost equal proportions:

Consequently, a detector on Earth, designed to see only νe will see only one third of the expected flux!

€

€

ν2 =1

3ν e +

1

3ν μ +

1

3ντ

FOR THIS OBSERVATION, DAVIS WON THE NOBEL PRIZE IN 2002

Raymond Davis Jr and John N. Bahcall in late 1960s started developing a detector to detect the neutrinos coming from the Sun in a deep underground mine at Homestake (US), by trapping them in a huge tank of the cleansing liquid with an entirely different aim: to confirm the source of the Sun’s energy.

The flux measured by Davis was about one third of thetheoretical prediction of Bahcall using Standard Solar Model!

THE HOMESTAKE EXPERIMENT

NEUTRINOS IN THE MASS BASIS

Suppose we have two neutrino states ν1 andν2 with masses m1 and m2 .

The energies will be

The time-evolution equation is

€

E1 ≅ pc +m1

2c 3

2p,E2 ≅ pc +

m22c 3

2p

E2 − E1 ≈m2

2 −m12

2p /c 3 ≡Δm12

2

2p /c 3

€

ih∂

∂tν = Hν

NEUTRINOS IN THE MASS BASIS

The solutions are simple:

The basis is called mass basis as the states ν1 andν2 propagate with definite masses m1 and m2 . They are called mass-eigenstates.

We are not equating them with νe and νμ as they needn’t propagate with definite masses.

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ih∂

∂t

ν 1

ν 2

⎛

⎝ ⎜

⎞

⎠ ⎟=E1 0

0 E2

⎛

⎝ ⎜

⎞

⎠ ⎟ν 1

ν 2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

ν1(t) =ν1(0)e−iE1t / h ,ν 2(t) =ν 2(0)e−iE2t / h .

NEUTRINOS IN THE FLAVOR BASIS

The states νe and νμ are not the states that take part in the week interactions.

They needn’t be the eigenstates of Hamiltonian in general, though nature might have chosen so. Thus, they do not propagate with definite masses like mass eigenstates.

The time evolution in the flavor basis will be of the form:

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ih∂

∂t

ν eν μ

⎛

⎝ ⎜

⎞

⎠ ⎟=Hee Heμ

Heμ Hμμ

⎛

⎝ ⎜

⎞

⎠ ⎟ν eν μ

⎛

⎝ ⎜

⎞

⎠ ⎟.

NEUTRINOS IN THE FLAVOR BASIS

The off-diagonal term Heμ is called the mixing term.

If this mixing term were absent: The states νe and νμ would be identical with the

mass eigenstates ν1 andν2. The neutrinos would not oscillate. Experiments tell us that nature doesn’t take this

route.

Because of the mixing term, νe can pick up a νμ component as it propagates. It take one oscillation length (Lo) for νe to transform into νμ and νμ to transform back into νe .

LINKING THE TWO BASIS: THE MIXING MATRIX

The flavor eigenstates and mass eigenstates are related as

Here, the matrix U diagonalizes H in the flavor basis and is called mixing matrix.

So, νe is a quantum mechanical admixture of ν1 andν2 with amplitudes cosθ and sinθ.

Example: Maximal mixing (θ= π/4)

€

νeν μ

⎛

⎝ ⎜

⎞

⎠ ⎟=U

ν1

ν 2

⎛

⎝ ⎜

⎞

⎠ ⎟=

cosθ sinθ

−sinθ cosθ

⎛

⎝ ⎜

⎞

⎠ ⎟ν1

ν 2

⎛

⎝ ⎜

⎞

⎠ ⎟.

€

νe =1

2ν1 +

1

2ν 2

THE MIXING MATRIX IN CASE OF THREE FLAVORS

TIME EVOLUTION OF ELECTRON NEUTRINO As

we have

So, if we start with an electron neutrino at t=0, it will not remain an electron neutrino at time t as the amplitudes of the eigenstates ν1 andν2 are oscillating with time.

The oscillating amplitudes will, at time t = Lo/2c, convert the neutrino to νμ and then back to νe at time t = Lo/c, periodically.

It’s like two musical instruments vibrating at two almost equal frequencies ω1 and ω2 and producing beats with frequency ω2-ω1 = (E2 – E1)/ħ = (c3 Δm2

12 )/(2ħp).

This simple analogy gives us Lo = 2π/(ω2-ω1 ) = (4πħp)/(c3Δm212 ).

€

νe (t) = cosθ ν1(t) + sinθ ν 2(t) ,

€

νe (t) = e−iE1t / h cosθ ν1(0) + e−iE2t / h sinθ ν 2(0) .

PROBABILITIES FOR SURVIVAL AND TRANSITION

The probability that an electron neutrino will survive as electron neutrino after traveling distance L is (disappearance experiments)

The probability that an electron neutrino will transform to a muon neutrino after traveling distance L is (appearance experiments)

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Pee = ve (t = L /c) ν e (t = 0) =1− sin2 2θ sin2 πL

Lo

⎛

⎝ ⎜

⎞

⎠ ⎟.

€

Pee = vμ (t = L /c) ν e (t = 0) = sin2 2θ sin2 πL

Lo

⎛

⎝ ⎜

⎞

⎠ ⎟.

PROBABILITIES FOR SURVIVAL AND TRANSITION

Note that the electron neutrino will be completely converted to muon neutrino only in the case of maximal mixing: Invoking the analogy with beat formation, the

sound disappears completely if the two amplitudes are equal.

The period of oscillations (oscillation length) is directly proportional to E and inversely proportional to Δm2 12 .

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Lo = 2.47E[MeV ]

Δmij2[eV 2]

m = 2.47E[GeV ]

Δmij2[eV 2]

km

TWO CRITERIA FOR OSCILLATIONS To observe oscillations, we must have1. Non-zeroΔm2

ij (controls the period of oscillations) and

2. Non-zero θij (controls the amplitude of oscillations). The source-detector distance to observe oscillations

should be at least of the order of oscillation length for an appreciable transition:

1. IfΔm2L/E <<1, we are so close to the source that oscillations will not be observed.

2. IfΔm2L/E >>1, we are so far away from the source that oscillations are averaged out. Then, Pee= 1- ½sin2 2θ. We will be able to measure only the mixing angle in this case. There will be no sensitivity to Δm2.

€

Lo = 2.47E[MeV ]

Δmij2[eV 2]

m = 2.47E[GeV ]

Δmij2[eV 2]

km

ΔM2 SENSITIVITY OF AN EXPERIMENT

When people started building experiments to detect neutrino oscillations, they had no idea what values of Δm2 nature had chosen.

They chose many combinations of L and E. If an experiment observed oscillations, it would mean that Δm2 ≈ E (MeV)/L(m) eV2.

Consider experiments like LSND and KARMEN with E ≈ 1 MeV and L ≈ 10m that tried to observe transition of muon antineutrino to electron antineutrino. They would be sensitive to Δm2 ≈ 0.1 eV2. They didn’t observed any oscillations and it implied that Δm2 < 0.1 eV2.

ATMOSPHERIC NEUTRINOSNeutrinos produced by cosmic rays as they interact with atmosphere

Atmospheric neutrinos have a flavor ratio of νe: νμ = 2 : 1. Any departure from it would signal oscillations.

DETECTION OF ATMOSPHERIC NEUTRINOS AT SUPER-KAMIOKANDE EXPERIMENT

L ≈ 104 km, E ≈ 10 GeV, Δm2 ≈ 10-3 eV2.

Disappearance experiment in the channel νμντ.

Measured Δm223 ≈ 2 ×

10-3 eV2 and θ23 ≈ 45o.

SUPER-KAMIOKANDE EXPERIMENT

A view of SK detector A neutrino event

REACTOR NEUTRINOS These experiments utilize

the neutrino flux produced in the nuclear reactors.

1. Short baseline (10m)2. Long baseline (1km):

Δm2 ≈ 10-3 eV2 (CHOOZ,DAYA BAY)

3. Very long baseline (100 km): Δm2 ≈ 10-5 eV2 (KamLAND)

KamLAND has successfully observed oscillations and measured Δm2

12.

DAYA BAY EXPERIMENT

Detects electron antineutrinos via inverse beta decay.

Number of events are counted by coincidence. measurements of positrons and energy released from capture of neutrons by gadolinium.

Uses near and far detectors.

KAMLAND EXPERIMENT

The detectorObserved flux/ Expected flux

KAMLAND RESULTS SUPERIMPOSED ON THE SOLAR NEUTRINO RESULTS

Solar neutrinos sensitivity:

L ≈ 1011 km E ≈ 10 MeV Δm2 ≈ 10-12 eV2. Yet, Δm2

12 ≈ 8×10-5 eV2. So,Δm2L/E >>1, and

oscillations are averaged out.

So, how do we have sensitivity to Δm2 ?

Solar neutrino facts

SOLAR NEUTRINOS

Thermonuclear reactions in the core of the Sun, apart from producing light, also produce neutrinos.

Neutrino image of the Sun produced by Super-Kamikande

NEUTRINO PROPAGATION IN MATTER

When neutrinos propagate in matter, νe and νμ interact differently with matter.

This modifies the time evolution equation. The modified Hamiltonian in the flavor basis is

€

H =1

4E

−Δm2 cos2θ + 2 2GFNeE Δm2 sin2θ

Δm2 sin2θ Δm2 cos2θ − 2 2GFNeE

⎛

⎝ ⎜

⎞

⎠ ⎟

MIKHEYEV - SMIRNOV- WOLFENSTEIN (MSW) EFFECT The mixing angle gets modified to

Since, θm depends on Δm2, we can determine Δm2! The mass eigenvalues and probabilities get

modified as well. For small energies, θm ≈ θ.

There is a value of energy for which θm = π/4, no matter however small θ is. This is called MSW resonance.

For high energies, θm ≈ π, no matter what θ is. This implies, νe = -ν2 (matter dominance).

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tan2θm =Δm2 sin2θ

Δm2 cos2θ − 2 2GFNeE

SOLAR NEUTRINO SPECTRUM

Neutrinos produced in different nuclear reactions

Survival probability and results from various solar neutrino experiments. One can see (i) the averaged out vacuum oscillations, (ii) MSW resonance and (iii) the matter dominated case

SAGE SAGE is a solar neutrino

experiment based on the reaction 71Ga + n 71Ge + e-.

The 71Ge atoms are chemically extracted from a 50-metric ton target of Ga metal and concentrated in a sample of germane gas mixed with xenon.

The atoms are then individually counted by observing their decay back to 71Ga in a small proportional counter.

SOLAR NEUTRINO DATA FROM SUPERKAMIKANDE EXPERIMENT

SUNDBURY NEUTRINO OBSERVATORY (SNO)

MEASUREMENT OF ELECTRONIC AND NON-ELECTRONIC NEUTRON FLUXES AT SNO

CC flux:

ES flux

NC flux€

φCC = φee

€

φEC = φee + 0.1553φeμ

€

φNC = φee +φeμ

CURRENT UNDERSTANDING OF NEUTRINO PARAMETERS

CURRENT UNDERSTANDING OF NEUTRINO PARAMETERS

INDIA-BASED NEUTRINO (INO) SITE

ICAL DETECTOR AT INO A neutrino event gives

rise to a hadron shower and a muon track.

Hadron energy is measured by calibrating it with number of hits.

Muon energy is measured by track length and curvature in magnetic field.

Muon direction can also be reconstructed from the track.

INO SENSITIVITIES

INO SENSITIVITIES

THE NOVA EXPERIMENT

THE NOVA EXPERIMENT

Fermi National Accelerator Laboratory generates a beam of neutrinos to send to a 14,000-ton detector in Ash River, Minnesota.

The particles complete the 500-mile interstate trip in less than three milliseconds. Because neutrinos rarely interact with other matter, they travel straight through the Earth without a tunnel.

If very few electron neutrinos pass through the near detector and a larger percentage of electron neutrinos pass through the far detector, it will mean that the muon neutrinos from the NuMI beam have become electron neutrinos during the trip.

NUMI [NEUTRINOS AT THE MAIN INJECTOR] NEUTRINO BEAM NuMI beam is generated by firing protons from

Fermilab’s Main Injector into a graphite target resembling a long roll of quarters.

Many different types of fundamental particles come out of the collision between the protons and the target, including pions.

The pions are steered through in the direction we want the neutrinos to travel by magnetic fields.

The pions eventually decay into muons and muon neutrinos, which continue on the same path the pions were traveling.

The neutrino beam is aimed downward at a 3.3 ° angle. Although the beam starts out 150 feet below ground at Fermilab, it will pass as much as six miles below the surface as it travels toward Ash River.

THE NOVA DETECTORS NOvA experiment uses two detectors: a 330 metric-ton

near detector at Fermilab and a much larger 14 metric-kiloton far detector in Minnesota just south of the U.S.-Canada border.

The detectors are made up of 344,000 cells of extruded, highly reflective plastic PVC filled with liquid scintillator.

Each cell in the far detector measures 3.9 cm wide, 6.0 cm deep and 15.5 meters long.

When a neutrino strikes an atom in the liquid scintillator, it releases a burst of charged particles.

As these particles come to rest in the detector, their energy is collected using wavelength-shifting fibers connected to photo-detectors.

Using the pattern of light seen by the photo-detectors, scientists can determine what kind of neutrino caused the interaction and what its energy was.

MASS HIERARCHY SENSITIVITY OF NOVA

ICECUBE

ICECUBE

IceCube is a particle detector at the South Pole that records the interactions of a nearly massless subatomic particle called the neutrino.

IceCube is the world’s largest neutrino detector, encompassing a cubic kilometer of ice.

IceCube searches for neutrinos from the most violent astrophysical sources: events like exploding stars, gamma-ray bursts, and cataclysmic phenomena involving black holes and neutron stars.

In addition, exploring the background of neutrinos produced in the atmosphere, IceCube studies the neutrinos themselves; their energies far exceed those produced by accelerator beams.

ICECUBE

PINGU

PINGU extends the energy range of IceCube down to energies that allow it to use low-energy atmospheric neutrinos to determine the neutrino mass hierarchy. IceCube is optimized for much higher energies, and it does not have enough sensitivity at lower energies to make this determination. Similarly, PINGU will not have enough sensitivity for the high-energy astrophysical neutrinos recently discovered by IceCube.

PINGU: PRECISION ICECUBE NEXT GENERATION UPGRADE