trabajo final de grado física(uv): angular distribution and energy spectrum of boosted off-axis...

Angular distribution and enegy spectrum of boostedoff-axis neutrinos

Christian Roca CatalaSupervised by: Jose Bernabeu Alberola

Universidad de Valencia (UV)

[email protected]

July 22, 2013

What is the scope of this presentation?

“I have done a terrible thing, I have postulated a particle that cannot bedetected”Wolfgang Ernst Pauli, 1930

Fortunately he was WRONG and neutrinos can be detected and thus,their oscillations!Christian Roca Catala (UV) Trabajo Fin de Grado July 22, 2013 2 / 42


Question: What are we going to study?

Answer: The project presented today it’s about measuring the neutrinooscillations. The performance of new generation experiments it’s capitalin order to disentangle the open problems in neutrino physics nowadays.

Question: Why we want to measure neutrino oscillations?

Answer: To solve the open problems in the neutrino sector:

Neutrino mass hierarchy: sign(∆m223) → neutrino mass picture

would be completed

CP violation in leptonic sector: asymmetry matter-antimatter inearly universe (leptogenesis)

Christian Roca Catala (UV) Trabajo Fin de Grado July 22, 2013 3 / 42


Question: How we may perform the measurements?

Answer: Using the off-axis method → Improving the energy resolutionof the neutrino beams detected it’s crucial for measuring the neutrino

oscillations parameters.

Question: Who is using this performances?

Answer: Long base-lines accelerator experiments of neutrino appearancelike T2K and NOνA

At the end → new results released 3 days ago by T2K!!



Accelerator Experiments

NOνA (Fermilab)

T2K (Japan)

MINERνA (Fermilab)

Reactor Experiments

Double Chooz (France)

RENO (South Korea)

Daya Bay (China)


Contents

1 What is the scope of this presentation?

2 Neutrino Oscillations in a nutshellWhat are Neutrino Oscillations?Reactor ExperimentsAccelerator experiments

CP Violation: parameter δNeutrino mass hierarchy

A key technique: off-axis neutrinos

3 Neutrino kinematicsPion rest frame - Center of MassLab Frame - Boosted Pion

Boost of the pionAngular distribution

Relation between Eν and Eπ

Pion energy distributionSummary

4 Conclusions

Neutrino Oscillations in a nutshell What are Neutrino Oscillations?

Neutrino Oscillations is aphenomenon

BEYOND THE STANDARDMODEL

predicted by Bruno Pontecorvo in1957

Basically it consists inFLAVOUR MIXING

of the different neutrino familiesOnly make sense if mν 6= 0!!

not predicted by SM



Flavour mixing... not like this!



... but more like this |νe〉|νµ〉|ντ 〉

=

Ue1 Ue2 Ue3

Uµ1 Uµ2 Uµ3

Uτ1 Uτ2 Uτ3

· |ν1〉|ν2〉|ν3〉

Neutrinos from a weak decay →well-defined flavour

Flavour eigenstates (e, µ, τ) 6= mass eigenstates (1,2,3) well-defined kinematics

Flavour basis and mass basis correlated by mixing matrix U (PMNS Matrix):

|να(x , t)〉 =∑

i

Uαi |νi (x , t)〉

Mass eigenstates →their evolution is given by pi ,Ei (Schrodinger img):

|νi (x , t)〉 = eipi xe−iEi t |νi 〉 = eiφi x |νi 〉 t ∼ x

NOTE!

t ∼ x since neutrinos are ultrarelativistic

Ei 6= pi for oscillations to happen, that is, mν 6= 0



Question: thus, what are the neutrino oscillations?

Answer: an effect whereby neutrinos created with a well-defined lepton flavour (e, µ, τ)can later be measured to have a different flavour:

|ψα(x , t)〉 =∑

iβ′ U†β′ i Uαi eiφi x |νβ′〉

The oscillation is determined, thus, by the matrix Elements Uαj . This elements dependon what is called Oscillation Parameters or Oscillation Angles: θ12,θ23 and θ13

First family

Ue1 = c12c13

Uµ1 = −s12c23 − c12s23s13eiδ

Uτ1 = s12s23 − c12c23s13eiδ

Second family

Ue2 = s12c13

Uµ2 = c12c23 − s12s23s13eiδ

Uτ2 = −c12s23 − s12c23s13eiδ

Third family

Ue3 = s13e−iδ

Uµ3 = s23c13

Uτ3 = c23c13

NOTE!

The parameter δ appears in thePMNS matrix as the CP violationparameter



Question: Which is the probability that a neutrino oscillate aftertravelling an interval L?

Answer: This depends on the medium the neutrinos travel through. The electrondensity Ne influence the cross section of charged current weak interactions of νe . Let’s

take the concrete example (νµ → νe) → it allows to measure θ13 and δ:

Oscillations through vacuum

P(νµ → νe ) = sin2 2θ13 sin2 θ23 sin2

(∆m2

13L

4E13

)+ subleading eff.

Dependence on:

|∆m2|: absolute value of squared mass difference

Oscillation parameters θ13, θ23

Subleading effects: very important for the analysis of CP violation (seen later)

The dependences give us CRUCIAL information about what can we expect fromoscillations through vacuum/matter.

NOTE!

The oscillations will always depend on the mass differences between mass familiesrelated to the oscillation ∆m2

13 = m21 −m2

3 → we will refer to it as mass difference.



Question: Which is the probability that a neutrino oscillate aftertravelling an interval L?

Answer: This depends on the medium the neutrinos travel through. The electrondensity Ne influence the cross section of charged current weak interactions of νe .

Oscillations through matter

P(νµ → νe ) = sin2 2θ13 sin2 θ23 sin2(

∆eff13 L/2

)+ subleading eff.

Dependence on:

∆eff13 =

√(∆13 cos 2θ13 − A)2 + ∆2

13 sin2 2θ13

The sign of ∆m2: sign(∆m2)=signA

Subleading effects: very important for the analysis of CP violation (seen later)

NOTE!

The oscillations will always depend on the mass differences between mass familiesrelated to the oscillation ∆m2

13 = m21 −m2

3 → we will refer to it as mass difference.



Summary: which implications do all this have?

Neutrino oscillation is a phenomenon beyond the SM → mν 6= 0

Flavour mixing happens due to different mass-flavour eigenstates

Oscillation probabilities depend highly on the media the neutrinos are travellingthrough

∆m2 can only be measured in experiments where neutrinos travel through matter

CPV can only be measured looking at the subleading effects (appearanceexperiments, seen later)


Contents









4 Conclusions

Neutrino Oscillations in a nutshell Reactor Experiments

Reactor Experiments

Neutrino energies ∼ MeV

Modest base-line ∼ km

Solar/atmospheric neutrinooscillation parameters via...

...Antineutrino disappearanceexperiments

Oscillations through vacuum


plain Neutrino Oscillations in a nutshell Reactor Experiments

Question: What can reactor experiments measure?

Answer: Reactor experiments searching for νe disappearancemake neutrinos to oscillate into vacuum, thus the precision in

measuring θ13 is very high. A non-zero value for θ13 is aprerequisite to... →

Question: What can not reactor experiments measure

Answer: ← ... to measure the open problems in acceleratorexperiments:

Measure the mass hierarchy of neutrinos: need oscillationsin matter.

Probe CP violation in the leptonic sector leading to thepossibility that neutrino mixing violates matter/anti-mattersymmetry: need appearance experiments.


plainContents









4 Conclusions

Neutrino Oscillations in a nutshell Accelerator experiments

Accelerator Experiments

Neutrino energies ∼GeV

Long base-line ∼hundreds km

Neutrino appearanceexperiments

Oscillations throughmatter


plain Neutrino Oscillations in a nutshell Accelerator experiments

CP Violation comes from subleading effects in P(νµ → νe):

Pδ(να → νβ) ∝ Jr sin δ

Remember PMNS Matrix U → the factor eδalways comes along with sin θ13.

Thus, θ13 must be measured with sensibility!! (Reactor experiments)

Question: What is CP Violation?

Answer: There are not the same physicsfor particles and antiparticles → Particle

and antiparticle symmetry is broken!!U† describes antineutrino oscillations,

eδ → e−δ:

Pδ(να → νβ) ∝ Jr (− sin δ)

Thus P(να → νβ) 6= P(να → νβ)

NOTE!

Jr = 0 IF α = β, thus disappearance experiments cannot measure CPV!!In other words, survival experiments can’t reconstruct neutrino interference generatingthose subleading effects terms.



Question: What is masshierarchy?

Answer: Mass hierarchy is theunknown order of the several

neutrino mass families.There are two possibles hierarchies,

depending on the sign of ∆m213:

normal hierarchy (m21 < m2

3)

inverted hierarchy (m23 < m2

1)

→ Oscillations in matter: dependon the parameter A ∝ ∆m2, thus

are sensitive to sign(∆m2)!!

NOTE!

Mass difference ∆m212 has been completely measured by solar neutrino. Since the

oscillations inside the sun are considered to occur through matter, the sign have beendetermined.


plainContents









4 Conclusions

plain Neutrino Oscillations in a nutshell A key technique: off-axis neutrinos

Detecting neutrinos is not impossible, butthe truth is they are very elusive!

Damn you, Pauli!

→ That’s why several techniques havebeen developed in order to attain higherenergy resolution.

“I told you!”Wolfgang Ernst Pauli

The scope of the second part of my work is to show how the off-axis neutrinos methodworks, in the same way it’s used in actual experiments like T2K and NOνA.



Question: What is off-axis neutrino technique?

Answer: We say a neutrino detector is placed off-axis when it subtends a determinednon-zero angle respect the travel line of the neutrino beam. Indeed, the neutrino beambehaves as a wave package, and it spreads out around this line. For a given neutrino’ssource energy, there is an angle off-axis where the neutrino flux have a well-defined

energy.



You may not believe me, but T2K and NOνA do! Let’s analyse theneutrino kinematics and discover the goodness of the off-axis

technique!

NOνA placed the fardetector at an off-axis angle

θ = 14mrad.

T2K placed the far detectorat an off-axis angle

θ = 44mrad.Christian Roca Catala (UV) Trabajo Fin de Grado July 22, 2013 21 / 42

plainContents









4 Conclusions

plain Neutrino kinematics Pion rest frame - Center of Mass

Question: What are we going to study?

Answer: The process to analyse is the decay of a pion into muon and neutrino:

π− → µ− + νµ

π+ → µ+ + νµ

We’ll attack this problem from two points of view:

massless neutrino approximation → mν = 0

massive neutrino first order correction →∼ m2ν in energy and momentum

The pion at the same tame come from the collision between a beam of acceleratedprotons towards a fixed target:

p + X → π− + X + Y

NOTE!

The masses are mπ = 139.57MeV and mµ = 105.66MeV


plain Neutrino kinematics Pion rest frame - Center of Mass

Massless neutrinos(arXiv:1005.0574)

In the first approximation we takemν = 0 and thus E = P

Pν = Eν = E =m2

π −m2µ

2mπ

Massive neutrinos

Taking the first order corrections∼ m2

nu, the result gives

Ecm = E +m2

i

2mπ

Pcm = E − m2i

2Eε


plainContents









4 Conclusions

plain Neutrino kinematics Lab Frame - Boosted Pion

Question: How do we do the change of coordinates?

Answer: Change from CoM Frame (pion at rest) → to Lab Frame (pion at flight)through a Lorentz boost γ for a pion travelling at β in the z-axis:

Λ =

γ 0 0 γβ0 1 0 00 0 1 0γβ 0 0 γ

Energy and momentum in Lab Frame

Applying the boost to the 4-momentum in CoM Pσlab = Λσ

δPδcm:

Elab = γ(Ecm + βPcm cos θcm)Plab sin θlab = Pcm sin θcm

Plab cos θlab = γ(Pcm cos θcm + βEcm)

We have used spherical coordinates with cylindrical symmetry → Independent ofazimutal degree of freedom ϕ

NOTE!

It does not matter to take β nor −β, that is, in direction z or −z : the results areindeed equivalent.



Question: How can we obtain the angular distribution of the neutrino?

Answer: From the CoM angular distribution and the Jacobian of the transformation(Ecm, cos θcm → Elab, cos θlab:

1

Γ

d2Γ

dϕlabdcos θlab=

1

4πJ(cos θcm, ϕcm; cos θlab, ϕlab)

The jacobian of the transformation is easy to obtain as:

J(cos θcm, ϕcm; cos θlab, ϕlab) =

∣∣∣∣∣∣∣∣∂ cos θcm

∂ cos θlab

∂ cos θcm

∂ϕlab

∂ϕcm

∂ cos θlab

∂ϕcm

∂ϕlab

∣∣∣∣∣∣∣∣ =∂ cos θcm

∂ cos θlab

We need to calculate de derivatives of the cosines!

NOTE!

The pion decays isotropically in rest, therefore the angulardistribution in CoM:

1

Γ

dΓ

dΩcm=

1

4π



Thus the angular distribution can be obtained from the derivatives of the cosines:

First Approximation: mi = 0,Ecm = Pcm (arXiv:1005.0574)

cos θlab =

cos θcm + β

1 + β cos θcm

cos θcm =cos θlab − β

1− β cos θlab

∂cos θcm

∂cos θlab= γ2

(1

1− β cos θlab

)2

Angular distribution:

1

Γ

dΓ

dΩlab=

1

4πγ2

(1

1− β cos θlab

)2

First order corrections ∼ m2i , Ecm 6= Pcm

cos θlab =

Pcm cos θcm + βEcm

Ecm + βPcm cos θcm

cos θcm =Ecm

Pcm

cos θlab − β1− β cos θlab

∂ cos θcm

∂ cos θlab=

Ecm

Pcmγ2

(1

1− β cos θlab

)2

Angular distribution:

1

Γ

dΓ

dΩlab=

1

4π

Ecm

Pcmγ2

(1

1− β cos θlab

)2



Neutrino Flux in terms of θlab for several Eπ

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

2000

4000

6000

8000

10000

12000

θlab

(rad)

νµ F

lux (

Arb

itra

ry U

nits)


Either for the approximation mν = 0 or for the first order corrections ∼ m2ν →

same results obtained!

For a fixed Eπ, neutrino flux peaks at θlab = 0


plainContents









4 Conclusions

plain Neutrino kinematics Relation between Eν and Eπ

Question: Why is important to look at the relationship between Eνand Eπ?

Answer: This relationship will lead us a hint about the energy distribution Eν of theneutrinos that we will treat in further sections.

Remember Elab obtained from the Lorentz boost → put in terms of θlab

Elab = γ(Ecm + βPcm cos θcm) =Ecm

γ

(1

1− β cos θlab

)

Maximum neutrino energy

The maximum energy is given by∂Eν

∂Eπ= 0 and this lead us the conditions:

β = cos θlab

γ = 1/ sin θlab

Thus, the maximum energy is Emax = Ecmγ



Neutrino Energy in Lab Frame in terms of Pion Energy

0 5000 10000 150000

1000

2000

3000

4000

5000

6000

7000

Eπ (MeV)

Eν (

Me

V)

θ = 0 rad

θ = 0.008 rad

θ = 0.02 rad

θ = 0.06 rad

θ = 0.044 rad

θ = 0.03 rad


For a fixed θlab, Eν has a maximum for β = cos θlab

There is a “stationarity” of Eν around Emax tending asymptotically to a constant

valueChristian Roca Catala (UV) Trabajo Fin de Grado July 22, 2013 29 / 42


Neutrino Energy in Lab Frame in terms of Pion Energy

0 5000 10000 150000

1000

2000

3000

4000

5000

6000

7000

Eπ (MeV)

Eν (

Me

V)

θ = 0 rad

θ = 0.008 rad

θ = 0.02 rad

θ = 0.06 rad

θ = 0.044 rad

θ = 0.03 rad


Neutrinos emitted will be bunched in a energy region ∼ Emax .

Neutrino flux needs to be peaked near Emax .


plainContents









4 Conclusions

plain Neutrino kinematics Pion energy distribution

Question: What does it happen if pion has not a definite energy buta given energy spectrum?

Answer: What we get from the collision proton → fixed target is a non-linear energyspectrum for pions (arXiv:1005.3692). This spectrum have to be implemented in the

analysis of the neutrino distribution: (Eπ, cos θcm)→ (Elab, cos θlab):

1

Γ

d2Γ

dEdΩlab=∝ (Ep − Eπ)5 · J(Eπ, cos θcm; Elab, cos θlab)

This time the jacobian is:

J(Eπ, cos θcm; Elab, cos θlab) =

∣∣∣∣∣∣∣∣∂Eπ

∂Elab

∂ cos θcm

∂Elab

∂Eπ

∂cosθlab

∂cosθcm

∂ cos θlab

∣∣∣∣∣∣∣∣NOTE!

The energy spectrum for the pions F(Eπ,Ep) we have taken comes from the results ofNA61/SHINE Collaboration (arXiv:1005.3692):

1

Γ

d2Γ

dEπdΩcm∝ (Ep − Eπ)5



Neutrino Flux in terms of Eπ for Off-axis Angles θlab > 0.01

2000 2500 3000 3500 4000 4500 50000

0.5

1

1.5

2

2.5x 10

4

Eπ (MeV)

νµ F

lux (

Arb

itra

ry U

nits)

θ = 0.06 rad θ = 0.05 rad

θ = 0.044 rad

θ = 0.04 rad

θ = 0.03 rad

1

Γ

d2Γ

dEdΩlab∝ (Ep − Eπ)5 mπβ

Pcm(cos θlab − β)



Neutrino Flux in terms of Eπ for Off-axis Angles θlab > 0.01

2000 2500 3000 3500 4000 4500 50000

0.5

1

1.5

2

2.5x 10

4

Eπ (MeV)

νµ F

lux (

Arb

itra

ry U

nits)

θ = 0.06 rad θ = 0.05 rad

θ = 0.044 rad

θ = 0.04 rad

θ = 0.03 rad


For any Off-axis angle only a small region of Eπ contributes.

Singularity at cos θlab = β appears: maximum energy Emax condition!Christian Roca Catala (UV) Trabajo Fin de Grado July 22, 2013 33 / 42


Neutrino Flux in terms of Elab for θlab = 0.044

660 665 670 675 680 685 690 695 7000

100

200

300

400

500

600

700

800

900

1000

Eν (MeV)

νµ f

lux (

arb

itra

ry u

nits)


Off-axis angles peaks the neutrino flux in a narrow neutrino energy region.

Neutrinos with higher energy than Emax are essentially absent.



Neutrino Flux in terms of Elab for near on-axis angles θlab < 0.01

0 2000 4000 6000 8000 10000 12000 140000

5

10

15

20

25

30

35

40

Eν (MeV)

νµ flu

x (

arb

itra

ry u

nits)

θ = 0

θ = 0.001

θ = 0.002

θ = 0.003

θ = 0.004


On-axis angles give higher integrated flux but less energy resolution.

On-axis angles also peaks the flux but for energies higher than parent pion energy→ cos θlab ∼ 1.


plainContents









4 Conclusions

plain Neutrino kinematics Summary

Summary

Massless neutrinos mν = 0 and first order correction ∼ m2ν approximations are

equivalent

Neutrinos attain a maximum energy Emax , independent of the pion energy, forβ = cos θlab.

Neutrinos bunch in a small energy region Eν ≤ Emax

For every energetic region for the parent pion there is an off-axis angle whichpeak the neutrino flux for β = cos θlab

The relative narrowness of the off-axis beam increase the energy resolution ofthe neutrino beam.


plainContents









4 Conclusions

plain Conclusions

New results from SuperK far detector of T2K - 19th July,2013 (3 days ago!)

νe appearance confirmation at the 7.5σ level of significance.“Observation of this new type of neutrino oscillation leads the way to new studies ofcharge-parity (CP) violation which provides a distinction between physical processes

involving matter and antimatter.”T2K announcement


plain Conclusions

We DO know neutrinos exist.

We DO know about their flavour oscillations.

We DO know matter and antimatter annihilate.


plain Conclusions

Next generation accelerator experiments would discovereventually the divergence between neutrino-antineutrino oscillations

→ CP symmetry breaking.

This asymmetry would generate the so-called Leptogenesis,happening responsible of the residual existing after

matter-antimatter annihilation just after the Big Bang... butcalling it residual seems a bit pejorative, isn’t it?


plain Conclusions

Why not to call it...Universe?


plain Conclusions

THANKS FOR WATCHING!

“This is not even wrong!” Wolfgang Ernst Pauli, again...


plain Conclusions

Kirk T. McDonald (6 November 2001) “An Off-Axis Neutrino Beam” Princetonhttp://www.hep.princeton.edu/~mcdonald/examples/offaxisbeam.pdf

Jean-Michel Levy (6 May 2010). “Kinematics of an off axis neutrino beam”http://arxiv.org/abs/1005.0574

Carlo Giunti (4 January 2008) “Neutrino Flavor States and the Quantum Theory ofNeutrino Oscillations” http://arxiv.org/pdf/0801.0653v1.pdf

Hiroshi Nunokawa, Stephen Parke, Jose W. F. Valle (2 October 2007) “CPViolation and Neutrino Oscillations” http://arxiv.org/pdf/0710.0554v2.pdf

Gina Rameika (20 May 2006) “Off-Axis Neutrinos” Fermilabhttp://www.phy.bnl.gov/~diwan/talks/talks/nusag-may-20/NuSAG_052006_

_offaxis.pdf

The T2K Collaboration (8 June 2011). “The T2K Experiment”http://arxiv.org/abs/1106.1238

The T2K Collaboration (3 April 2013). “Evidence of Electron Neutrino Appearancein a Muon Neutrino Beam”http://arxiv.org/abs/1304.0841

The T2K Collaboration (6 November 2005) “ND280 Conceptual Design Report(Internal Report)” www.nd280.org/documents/cdr.pdf/download


http://www.hep.princeton.edu/~mcdonald/examples/offaxisbeam.pdf

http://arxiv.org/abs/1005.0574

http://arxiv.org/pdf/0801.0653v1.pdf


http://www.phy.bnl.gov/~diwan/talks/talks/nusag-may-20/NuSAG_052006__offaxis.pdf

http://www.phy.bnl.gov/~diwan/talks/talks/nusag-may-20/NuSAG_052006__offaxis.pdf



www.nd280.org/documents/cdr.pdf/download

plain Conclusions

John N. Bahcall and Raymond Davis Jr. (1976) “Solar Neutrinos: A ScientificPuzzle”, Science, 191, 264

NOνA Collaboration (21 March 2005) “The NOνA Experiment”http:

//nova-docdb.fnal.gov/0005/000593/001/NOvA_P929_March21_2005.pdf

CHOOZ Collaboration (15 November 1999) “Initial Results from the CHOOZ LongBaseline Reactor Neutrino Oscillation Experiment”http://arxiv.org/pdf/hep-ex/9711002v1.pdf

SuperKamiokande Collaboration (14 May 2001) “Super-Kamiokande atmosphericneutrino results” http://arxiv.org/pdf/hep-ex/0105023v1.pdf

SNO Collaboration (2 August 2004) “Results from the Sudbury NeutrinoObservatory”http://www.slac.stanford.edu/econf/C040802/papers/WET001.PDF

CHOOZ Collaboration (13 June 2003) “Search for neutrino oscillations on a longbase-line at the CHOOZ nuclear power station”http://arxiv.org/pdf/hep-ex/0301017v1.pdf

Daya Bay Collaboration (2012) “Observation of electron-antineutrinodisappearance at Daya Bay” http://arxiv.org/pdf/1203.1669.pdf


http://nova-docdb.fnal.gov/0005/000593/001/NOvA_P929_March21_2005.pdf

http://nova-docdb.fnal.gov/0005/000593/001/NOvA_P929_March21_2005.pdf

http://arxiv.org/pdf/hep-ex/9711002v1.pdf


http://www.slac.stanford.edu/econf/C040802/papers/WET001.PDF


http://arxiv.org/pdf/1203.1669.pdf

plain Conclusions

RENO Collaboration (8 April 2012) “Observation of Reactor Electron AntineutrinoDisappearance in the RENO Experiment”http://arxiv.org/pdf/1204.0626v2.pdf

Andre de Gouvea, James Jenkins and Boris Kayser (23 March 2005) “NeutrinoMass Hierarchy, Vacuum Oscillations, and Vanishing |Ue3|” Fermilabhttp://arxiv.org/pdf/hep-ph/0503079v2.pdf

Hisakazu Minakata, Hiroshi Nunokawa, Stephen Parke (23 January 2013) “TheComplementarity of Eastern and Western Hemisphere Long-Baseline NeutrinoOscillation Experiments” http://arxiv.org/abs/hep-ph/0301210

K. Nakamura (2010). “Review of Particle Physics”

Double Chooz Collaboration (30 October 2006) “Double Chooz: A Search for theNeutrino Mixing Angle θ13” http://arxiv.org/pdf/hep-ex/0606025v4.pdf

Rabindra N. Mohapatra and Palash B. Pal (November 1990) “Massive neutrinos inphysics and astrophysics” World Scientific

NA61 Collaboration (2012) “Hadron production measurement from NA61/SHINE”University of Genevahttp://indico.cern.ch/getFile.py/access?contribId=0&resId=

0&materialId=3&confId=183449Christian Roca Catala (UV) Trabajo Fin de Grado July 22, 2013 42 / 42


http://arxiv.org/pdf/hep-ph/0503079v2.pdf

http://arxiv.org/abs/hep-ph/0301210


http://indico.cern.ch/getFile.py/access?contribId=0&resId=0&materialId=3&confId=183449

http://indico.cern.ch/getFile.py/access?contribId=0&resId=0&materialId=3&confId=183449

trabajo final de grado física(uv): angular distribution and energy spectrum of boosted off-axis...

Science

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dierent neutrino families

t mass eigenstates

mass basis

mass families