india-based neutrino observatory – facts and fiction · india-based neutrino observatory –...

India-based Neutrino Observatory –Facts and Fiction

D. Indumathi

Institute of Mathematical Sciences, Chennai

(indu@imsc.res.in)

IIT (M), Chennai, Oct 3, 2018 – p. 1

Outline of talk

Our Sun and Other Stars

Neutrinos; Neutrino Oscillations

INO: Status and Prospects

How does the Sun shine?

Nuclear FusionBy 1938, von Weizäcker and then Bethe completed the detailed

calculation of the evolution of the Sun: he showed

p+ p+ p+ p → 4He+2 e++2νe+26.7 MeV .

Question 1: How can two protons (which have the same electric

charge) come close enough together for them to fuse?

Quantum mechanics allows this; the Gamow factor calculates this

probability of overlap.

Question 2: How is so much energy emitted?

10−10 m (1 Å) ↔ 10

−15 m (1 fm)

1 : 10−5

Generally, E ∼ 1/distance or E(nuclear) ∼ 105 E(atomic).

Note: Baryon number conservation, charge conservation, lepton

number conservation and Energy-momentum conservation.IIT (M), Chennai, Oct 3, 2018 – p. 4

What is the world

made up of?

What is a quantum field theory?A QFT involves relativity, quantum mechanics and a symmetry

principle.

A QFT describes the properties and interaction of particles. All

particles interact by exchanging other particles which are the carriers

of the interaction.

Eg: Quantum Electrodynamics QED (Feynman): interaction of

charged particles and radiation (photons); most precise theory known

today. Eg: two electrons repel, electron and proton attract.

Interaction Mediator Matter Physical Consequence

EM Photon e, p Atoms formed

Weak W, Z e, µ, quarks Radioactivity

Strong gluons quarks Nucleus formed

Gravity graviton All matter UniverseIIT (M), Chennai, Oct 3, 2018 – p. 6

Leptons and the Standard ModelParticle electro-magnetic strong weak

p+ ✔ ✔ ✔

n0 ✔ ✔ ✔

e− ✔ ✘ ✔

νe ✘ ✘ ✔

Leptons come in three flavours or types or generations:

µ and τ heavier versions of e. Rea-

son for their existence (and no. of

generations) a mystery.

All neutrinos are assumed massless within the Standard Model.

The νe neutrino is precisely that which is predicted to occur in the

fusion processes in the Sun.

The pp Chain: p, ..., 8B

Does the Sun really

shine in neutrinos?

The proof of the pudding . . .

. . . is in looking for, and finding the neutrinos!

Neutrinos are notoriously hard to detect because of weak int.

First attempts were made as early as 1960’s and eventually

successful.IIT (M), Chennai, Oct 3, 2018 – p. 10

Solar Neutrinos and Physics NobelsThe Nobel Prize in Physics 2002 was awarded to Raymond Davis Jr.

and Masatoshi Koshiba “for pioneering contributions to astrophysics,

in particular for the detection of cosmic neutrinos".

The Nobel Prize in Physics 2015 was awarded to Takaaki Kajita,

Arthur B. McDonald (+1) “for the discovery of neutrino oscillations,

which shows that neutrinos have mass”

Actually, to summarise: that neutrinos show quantum mechanical

mixing and have non-zero mass. Both are required to enablen

neutrino oscillations to occur.

What led to these discoveries and this excitement?

Early solar neutrino experimentsDavis and collaborators, first results in 1968. Awarded the Nobel

prize in 2002 for observing solar neutrinos, decades after they were

predicted!

600 tons of perchloroethylene (drycleaning fluid!) containing Chlorine.

νe+37Cl→37Ar+e− .

Event rate about 1 in 3 days.

RCC = Number of events observedNumber of events expected

≃ 13 .

Here CC means charged current interactions:

An important criticism of Davis’ experiment:

no guarantee that these neutrinos were indeed from the Sun.IIT (M), Chennai, Oct 3, 2018 – p. 12

Are they solar neutrinos?Koshiba, Totsuka and collaborators, 1986, Kamioka in Japan,

followed by Super-Kamioka, used a tank of water to detect neutrinos.

The detection is by elastic scattering of neutrinos on water:

νX + e → νX + e .

X = νe : νµ :: 6 : 1

R = 0.45

Raymond Davis Jr. and Masatoshi Koshiba won the Nobel prize in

Physics 2002 for their work.IIT (M), Chennai, Oct 3, 2018 – p. 13

Neutrino oscillationsNeutrinos come in more than one flavour or type. Consider, for

simplicity, two-flavours, νe and νµ.

If neutrinos are massive (different masses), and, further, show the

quantum mechanical phenomenom called flavour mixing, then

neutrinos can oscillate between flavours.

Real νe(t) = cos θ �+ sin θ �

How do you prove that the puzzle does not have a different origin?

The final denouementAn obvious test of the oscillation hypothesis is to look for the other

flavours of neutrinos from the Sun.

The SNO detector, Sudbury, Canada, 1000 tons of heavy water D2O,

announced their first results in 2002, and then in 2003.

RCC(νe) = Number of events observedNumber of events expected

≃ 13 (Cl and Ga).

RES(νX) ≃ 12 (Super-K).

RNC(All ν) ≃ 1 .

Here NC stands for the neutral current process:

n,p n,p

Hence the Standard Solar Model is vindicated

in the neutral current sector.IIT (M), Chennai, Oct 3, 2018 – p. 15

Other observations of neutrinosSo far, discussed Solar neutrinos : US, Russia/USSR, Japan, Canada

Atmospheric neutrinos : US, Japan, Mediterranean Sea, South Pole

Reactor neutrinos : France, Korea, China

Beam neutrinos : US, Japan, Switzerland, Italy

Geo neutrinos : US, Japan

There are also neutrinos which are a part of the cosmic background.

Relics of the Big Bang, estimated about 300 neutrinos per cc in the

Universe! Very difficult to detect, but indirect detection.

Tremendous interest in neutrino physics due to its implications in

particle physics, astrophysics, cosmology, including the origin and

evolution of the Universe.

What is the state of our current knowledge?IIT (M), Chennai, Oct 3, 2018 – p. 16

A Schematic of Neutrino PropertiesNeutrino masses are not well-known. Oscillation studies only determine

mass-squared differences: ∆m2ij = m2

i −m2j and mixing angles θij .

solar~8×10−5eV2

atmospheric~2×10−3eV2

νµντ

solar~8×10−5eV2

∆m221 ∼ 0.8× 10−4 eV2 ;

|∆m232| ∼ 2.0× 10−3 eV2

imi < 0.7–2 eV.

θ12 ∼ 34◦

θ23 ∼ 45◦: octant?

θ13 ∼ 8.5◦;

⇒ CP violation possible

δCP =?:

diff. between interaction

of ν, ν with matter

INO being built in this context IIT (M), Chennai, Oct 3, 2018 – p. 17

India-based NeutrinoObservatory Project

At Pottipuram, Theni

The INO ProjectA mega Science Project funded by the Dept. of Science and

Technology and Dept. of Atomic Energy, Govt. of India

Immediate goal: Creation of an underground laboratory for research

in neutrino physics at Pottipuram

Will develop into a full fledged underground laboratory over the years

for other studies in physics, biology and geology

Main detector proposed is magnetised Iron CALorimeter (ICAL) to

study primarily atmospheric neutrinos

Will incorporate a centre for particle physics and detector technology

and its varied applications at Madurai

The INO graduate training program is now 11 years old

The INO Collaborating Institutions• American College

• AMU • BARC

• BHU • Calcutta U

• Calicut U • CMEMS

• Delhi U • HRI

• IGCAR • IITB

• IITG • IITI

• IITM • IMSc

• Jammu U • Kashmir U

• Lucknow U • Mysore U

• Panjab U • RMVU

• Sambalpur U • SINP

• TIFR • VECC

Project Director: Prof. Vivek Datar; Host Institute: TIFR

Jointly funded by DAE and DST: not many people are aware that DAE is oneof the main funding agencies for theoretical physics and mathematics inIndia. IIT (M), Chennai, Oct 3, 2018 – p. 20

The ICAL detector50 kton iron, magnetised to ∼ 1.5 T with 150 layers of 5.6 cmplates in three modules

Each module = 16× 16× 14.4 m3

��

16m16m

Specifications of the ICAL detector

No. of modules 3Module dimension 16 m × 16 m × 14.4 mDetector dimension 48 m × 16 m × 14.4 mNo. of layers 150Iron plate thickness 5.6 cmGap for RPC trays 4.0 cmMagnetic field 1.5 Tesla

RPC unit dimension 2 m × 2 mReadout strip width 3 cmNo. of RPC units/Road/Layer 8No. of Roads/Layer/Module 8No. of RPC units/Layer 192Total no. of RPC units ∼ 30,000No. of electronic readout channels 3.9 × 106

Needs large industry interface.

Example of RPC/Electronics

Detector StructureMechanical design and assembly project report prepared by Tata ConsultingEngineers (TCE), Mumbai

ICAL Prototype status

Cosmic Muons in 5 layers

Cosmic Muons in 8 layers

ICAL and Atmospheric NeutrinosStudy of atmospheric neutrinos with magnetised ironcalorimeter detector, ICAL

Hence cosmic rays produce νµ (νµ) and νe (νe) inabout 2:1 ratio. Any deviation from this raio, especiallyas a function of direction (path length travelled in theEarth) is a signature of oscillations.

The CC interactions of interest are:νµ +N → µ− +X ,

νµ +N → µ+ +X , where X is any hadronic debris.

Crucially, the magnetic field differentiates betweenνµ-induced and νµ-induced events.

Hence we can use the data to precisely determine theoscillation parameters, including the mass hierarchy.

Physics reach of INOPrecision Measurements

Note ICAL yet to be built!

Result marginalised over magnitude of ∆m2, as well as θ23 and θ13.

Physics reach of INOMass Ordering

Note improvement with addition of hadrons

Result marginalised over magnitude of ∆m2, as well as θ23 and θ13.

Global status: Mass hierarchy• Matter effect / mass hierarchy is thecentrepiece of ICAL physics.

• It has a major role to play in under-standing models of neutrino mass andmixing. It also impacts the determi-nation of whether neutrinos are Majo-rana or Dirac type of fermions.

What is the role of other experiments indetermining this quantity?

Apart from INO, MINOS, T2K, NOνA, PINGU/Icecube, JUNO, DUNE, Hyper-K,LBNE all will/are probing mass hierarchy. Each is an amazing experiment.

Most have to disentangle effects of CP phase from the hierarchy measurement;can accomplish this only for a fraction of possible δ from −π to π.

Additional Synergies

Other PhysicsInclusion of electron neutrinos. Hard to do since need to separate

electron shower from hadron shower.

Inclusion of tau neutrinos. Very interesting possibility. Work in

progress.

Inclusion of sterile neutrinos. Hard to do since need to measure all

neutrino flavours.

Instead can think of models in which neutrinos decay (via Majorons)

into steriles—indirect.

Can have non-standard neutrino interactions. Probe by looking for

deviations from usual oscillations.

Look for signatures of CP violation, Lorentz invariance violation, etc.

Can use the detectors to probe non-oscillation, non-neutrino physics:

cosmic muons, dark matter, etc.

Bottom line: lots of exciting possibilities!

Outreach: A never-ending saga

Facts versus Fiction

Why go underground?• Huge cosmic ray background on thesurface, consisting of mostly protonsand heavier nuclei. Origins not known.

• These nuclei interact with moleculesin Earth’s atmosphere (CO2, N2, O2,etc) to produce pions which decay intomuons π → µνµ.

• Muons are minimum ionising parti-cles and extremely penetrating, unlikeelectrons

• Earth acts as a shield and absorbs alarge fraction of these muons: expect3,000 muons per hour versus 3 neu-trino events of interest per day

Fiction: INO will be built underground to store nuclear wasteIIT (M), Chennai, Oct 3, 2018 – p. 40

INO Site Selection FactsThe depth requirement is determined

by the physics

The primary criteria are safety (and

hence good rock quality) and

minimal environmental impact

Charnockite rock in India found

mostly below 13◦ N latitude

Low rainfall/humidity for best working

of detectors implies that the possible

sites are in Tamil NaduA search in Tamil Nadu satisfying these requirements yielded the

Pottipuram region in Bodi West Hills as the one having minimal

ecological impact.

Fiction: “INO site selection was not transparent".IIT (M), Chennai, Oct 3, 2018 – p. 41

Schematic of INO Underground Lab

Fiction: INO tunnel will be vertical and will deplete the groundwater. IIT (M), Chennai, Oct 3, 2018 – p. 42

Environmental Impact andManagement

Environmental IssuesEnvironmental Impact Assessment done by SACON, (under MOEF)

Portal located in poromboke land outside the RF. All surface activitiesoutside the forest. The tunnel and laboratory caverns remain undergroundbelow the RF.

In all, 517 species of plants, and 232 species of vertebrates and 59 speciesof butterflies recorded. While a few species are in the endangered orvulnerable category, none of the species is limited to the study area and arewidely distributed. Very few of these are found in the portal area.

Summary of EIA Report: Most of the construction work of the proposedproject will be carried out deep underground... However, noting that wildlifeis rarely reported in the portal area, the impacts of the activities on them willbe effectively negligible (SACON Report-p4).

INO to adopt best practices with active environmental monitoring.

Environmental Management: HighlightsThis is expected to be significant only during construction; negligible

impact during operation phase

All major facilities deep underground: minimum impact.

Noise levels : controlled blasting and use of new vehicles.

Tunnel muck disposal; About 10 % to be reused and rest stored in

temporary muck yard; disposed of as and when generated, to the

extent possible.

Local species planting and greening.

No contract labour settlement in forest area—temporary colony in

Poromboke land; LPG to be supplied.

Proper waste disposal and sanitation.

Proposed lab will be in rain-shadow region with scant wildlife,

vegetation; also 2 km from nearest village; cannot impact people,

flora, fauna IIT (M), Chennai, Oct 3, 2018 – p. 45

Construction phase: noise and vibrations

200 400 600 800 1000 1200

Rock Cover (meters)

INO site is in seismic zone 2, the lowest in India.

Controlled blasting in 2/3 pulls per day, each lasting a few seconds

Vibration felt a few hundred meters from site is less than 1mm

Contrast conventional blasting in quarries; lots of quarries in

neighbourhood.

Fiction: INO construction will damage nearby villages/fields that are

more than 2 km away. IIT (M), Chennai, Oct 3, 2018 – p. 46

Underground Tunnels and safetyTunneling is a routine activity that does not normally affect nearby structures(metro rail in cities). About 104 km of tunnels/caverns have been constructed byTNEB in TN Western Ghats

PUSHEP Powerhouse at Singara in Nilgiris: Pykara dam in the Nilgiris is about10-11 kms from PUSHEP (Pykara ultimate stage hydro-electric powerproject) underground hydro-electric project which was commissioned in2004.

Nearly 13 kms of tunnels were constructed under the northern edge of Nilgirimountains for the PUSHEP project below Glenmorgan Forebay bringingwaters from the Pykara dam. No danger to Pykara dam was envisaged andhas been reported.

Singara heritage power house (built 1932) is about 200m from the mainaccess tunnel portal to the underground project. No damage was caused tothis heritage dam during the construction of the tunnels and theunderground powerhouse which took more than ten years to complete.

Underground Tunnels, cont’dIdukki dam and power station: The Idukki Project harnesses a major portion ofthe power potential of Periyar, the largest river in Kerala, by the creation of areservoir of 2,000 M.cum (2 Billion Tonnes) capacity.

Idukki commissioned in 1973.

A hydel power project was planned later

The water is diverted using a 2 km tunnel to the two underground pressureshafts 956 m long to an underground power house (141.1 m × 19.8 m ×34.6 m) 669 m below (Muvattupuzha Valley) and along 1 km tail race tunnel.

Tunnels, etc., built and hydel project was completed in 1976 and was stillunder construction while the dam was commissioned.

Railway tunnels in South India: More than 30 km of shallow tunnels inMaharashtra, Goa and Karnataka; mainly along Konkan railway.

Fiction: “Dams can be impacted ... [by] explosion induced seismicity”

Muck Storage Yard

Muck from blasting about 2.30 lakh cu.m; mostly in chunks of rock

useful as construction materialAbout 10% utilised for project, rest will be stored on temporary site

within INO land and disposed offRetaining walls providedWind screen to be provided above the retaining walls to prevent

possible aerial muck dispersalTo be covered with vegetation to prevent run-off after construction.Fiction: INO muck will cover fields and crops and destroy them.

INO disaster impact assessmentFire: Very low - The detectors consists of passive components made

up of steel plates and sandwiched RPC detectors. Inflammable

materials are not used except in small quantities.

Floods: No reservoir of large capacity is required for the project. Only

underground sump.

Release of toxic gases: Gas mixture used in the RPC detector is non

toxic, non-inflammable and not ozone depleting. It is in a closed loop

system. There is no use /release of toxic gases in the systems and

processes employed.

Radiation: No radio active materials are used or produced in the

system/processes.

Fiction: “Dams can be impacted [by] Radioactive contamination of water”

Future Physics at INOMention of neutrino factories due to “magic baseline”: very precise

measurements of mixing angle θ13 can be made even if very small, if

the distance travelled is abut 7500 km. No more interest since θ13

large and already determined in 2012.

However, these factory neutrinos are the same type as natural

atmospheric ones with somewhat higher energy. Note that neutrinos

are neutral (and are hence often confused with neutrons, with which

they have no similarity); they cannot be made into a mono-energetic

Fiction: “High energy neutrinos manufactured in Neutrino Factory at

Chicago in USA will be beamed towards INO ... The neutrinos made in

Neutrino Factories are different from the natural neutrinos ... [in] Energy

Level, Intensity level and Mono-energetic"

Fiction: “A collimated high energy neutrino beam ... is a defense-less

weapon” IIT (M), Chennai, Oct 3, 2018 – p. 51

Public Outreach

INO Cell for public awareness formed at American College, Madurai in January

2010. Many outreach meetings held till now.

A major public meeting held on 8 July 2010 by Collector with 1200 local villagers

from Pottipuram panchayat (and about 40 press members).

“With the full support of local community, the INO will come up on the

declivity of western ghats, roughly three km away from Pottipuram

Panchayat" Collector Shri P. Muthuveerran, IAS (Hindu 9/7/10).Fiction: “INO is a secret project." IIT (M), Chennai, Oct 3, 2018 – p. 52

In Conclusion, our VisionINO is projected to be a world class underground science laboratorystraddling many fields. Nearly 100 scientists from 25 research institutes andUniversities all over India. Testifies to the collaborative spirit of the scientificcommunity.Main aim to study naturally occurring particles—neutrinos. INO willgalvanise physics research around the country. Expertise gained here willcontribute to other physics projects around the world.World-wide interest due to implications: pure physics, possible technologyspin-offs, and human resource developmentScience is truly international: in future possibly scientists from other parts ofthe world may collaborate with us. For this to happen, we must prove ourcompetence by building this experiment in a competitive and timely manner!

Fiction: “INO’s is a part of the US Fermilab project. Its mandate is to provide information onthe quality of neutrinos detected at INO to the US lab, more or less like a hospital undertakingdrug trials. The project proposal was written by scientists of Fermilab and submitted to theIndian Planning Commission for funding in Feb 2006. US is not likely to share the weapondeveloped with India. Details of this collaboration with US are not available in any of thedocument or official websites in India."

A few last wordsINO has been conceived on a scale that no other basic sciences

project in India has attempted. It is a testimony to the enthusiasm and

collaborative spirit shown by the scientific community in India and is

unique in this sense.

We also believe that INO will galvanise science across the country by

offering opportunities for science students to work in a cutting-edge

research atmosphere.

We reiterate that INO is committed to use best practises during the

construction phase and will adhere to a sound environmental policy

of sustainable development.

“INO scientists believe that the study of Nature’s innermost workings need not be at

loggerheads with Nature itself. Models of S & T development that are sensitive to

environmental conservation thus assume importance. The proposed India-based

Neutrino Observatory (INO) offers immense opportunities and a challenge for

realising such a model.” IIT (M), Chennai, Oct 3, 2018 – p. 54

T H A N K Y O U

Additional Slides

Production of Carbon—Beyond8B

Hoyle state is a resonant excitation of carbon-12 predicted by Hoyle.

Key point is energy of Be and He almost exactly equal to this excited

state, so transition probability increases.

Otherwise τBe = 6.7× 10−17 s: decays before it can interact!

Hoyle state essential for the nucleosynthesis of carbon in

helium-burning red giant stars, and for existence of life!IIT (M), Chennai, Oct 3, 2018 – p. 57

Beyond CarbonCarbon interacts with hydrogen in three steps to form 13N and 14N .

The nitrogen fuses with hydrogen in three steps to form oxygen.

The CNO are catalysts in the fusion reaction of 4H → He.

Eventually, C(12,13), N(13, 14, 15) and O(15,16,17) are produced.

In massive stars, F and Ne can also be produced.

How Stars DieOur Sun is just another star. For billions of years, it will be in

hydrodynamic equilibrium, where the internal gas pressure acting

outwards balances the attractive force of gravity.

If the star has a mass less than ∼ 8M⊙ it simply cools into a white

dwarf.

For masses in the range ∼ 8–20M⊙, the star collapses and

supernovae explosions occur. Briefly, their brightness exceeds that of

an entire galaxy. Remnant is a neutron star.

Stars much heavier than these collapse into black holes.

Large Stars and

Supernovae

Large stars and supernovaeOnce the hydrogen in the core is depleted by being transformed into helium,the hydrogen fusion stops.

Gravity takes over, compressing the core, leading to a rise in temperature,until ∼ 108 K, when helium fusion starts.

This chain of events continues; each stage of fuel burning is shorter than theprevious one, with continually increasing core temperatures and densities.

The star takes a typical “onion-skin” pattern, with an outer (relatively cool)shell of hydrogen, enveloping, in turn, layers of helium, carbon, neon,oxygen, silicon and iron.

With the formation of the stable iron nucleus, the process of energyproduction through thermonuclear fusion comes to an abrupt end.

Properties of an18M⊙ starEvent Properties

Primordial gas 77% hydrogen; 23% helium

Hydrogen burning Core temperature = 4 × 107 K; core density = 7 gm/cc;lasts about 10–15 million years; radius = 23× 106 km

Helium burning Core temperature = 1.6 × 108 K; core density = 1500gm/cc; lasts about 1 million years

Carbon burning Lasts about 100,000 years

Neon burning Lasts less than a year

Oxygen burning Lasts about 100,000 years

C, O, Neon Together have a radius = 3.6 × 104 km at the end ofsilicon burning

Si/iron core Silicon burns in a day to produce an inert core of iron850 km in size, surrounded by silicon to a radius of4,000 km. Central temperatures greater than 109 K; av.densities ∼ 107 gm/cc

The last momentsAt the final stages, the core (Fe/Si) has T ∼ 109 K.

More importantly, the core attains a mass of around 1.4M⊙ ; this is

the famous Chandrasekhar limit.

The core starts to contract for the last time. Temperature increase

imparts a kinetic energy to the particles of around 0.5 MeV.

This enables the electron capture process:

e− + p → n+ νe .

This is the first source of (electron)-neutrino production in the star.

Results in neutronisation of the core.

The consequent neutrino burst lasts only a few milliseconds with the

neutrinos having an average energy of 10 MeV.

Stellar Collapse

How much energy?The total energy emitted in the explosion is simply the gravitational

binding energy of the core:EB ∼ GNM2⊙

With a radius of about R = 10 km, this is about 3× 1053 ergs. About

99% is radiated away by neutrinos, which cool the core.

p+ e− → n+ νe ,

n+ e+ → p+ νe ,

n+ n → n+ n+ ν + ν ,

e+ + e− → ν + ν .

Approximately equal amounts of energy are emitted in each type of

neutrino (and its antiparticle).

Neutrinos produced during this cooling phase of the core are the

second source of neutrinos from supernovae explosions.IIT (M), Chennai, Oct 3, 2018 – p. 65

Detection of supernova neutrinosThe only detection on Feb 24, 1987 (SN1987A), 50 Kpc away.

About 20 neutrinos, lasting about 10 seconds, were detected by

Kamiokande (Japan) and IMB (US).

From time-of-flight measurements, a direct mass bound: mν < 16 eV,

was placed.

From number and type, gross features of supernova neutrino spectra

were verified.

No details on neutrino oscillation could be verified.

Eagerly awaiting the next supernova explosion!

Supernovae and Life

OutlookNeutrinos probe some of the most important frontiers of physics,

astrophysics and cosmology.

Neutrinos are the cleanest probes of weak interactions, which are

least studied and understood today.

Neutrinos are the key to understanding many features about the

origin and evolution of our Universe (though this was not really

discussed in this talk).

Neutrinos are therefore a new window into our Universe. About 50

experiments world-wide are devoted to study neutrinos from different

sources.

With thanks, and apologies for the lack of any experimental details.

Neutrino Oscillations

|να〉 =∑

Uαi|νi〉 .(1)

Here U is the 3× 3 unitary matrix which may be parametrised as (ignoringMajorana phases):

c12c13 s12c13 s13e−iδCP

−c23s12 − s23s13c12eiδCP c23c12 − s23s13s12e

iδCP s23c13

s23s12 − c23s13c12eiδCP −s23c12 − c23s13s12e

iδCP c23c13

δCP is the CP violating (Dirac) phase and Mν is diagonalised in thecharged-lepton mass basis by U :

U †MνU = diag(m1,m2,m3).(3)

Matter EffectsFirst consider matter of constant density ρ (in gms/cc). Then we can replace thevacuum values by the corresponding matter-modified effective ones obtained bydiagonalising the matter dependent matrix (Hamiltonian):

0 ∆m221 0

0 0 ∆m231

U † +

A = 2√2GFneE = 7.63× 10−5 eV2 ρ(gm/cc) E(GeV) eV2.(5)

Mixing angles in matterFurther simplification arises because ∆m2

21 ≪ ∆m231 and we can treat the

propagation in matter as a one mass-scale problem involving only∆m2

32 ≈ ∆m231. The matter dependent mixing angle θ12,m may be approximately

written as

sin 2θ12,m ≈ sin 2θ12√

(cos 2θ12 − (A/∆m221) cos

2 θ13)2 + sin2 2θ12

The effect of matter on the angle θ13 is non-trivial and is given by

sin 2θ13,m =sin 2θ13

(cos 2θ13 − (A/∆m231))

2 + (sin 2θ13)2.(7)

sin 2θ23,m ≈ sin 2θ23 .(8)

india-based neutrino observatory – facts and fiction · india-based neutrino observatory –...

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