Trabajo Final de Grado Física(UV): Angular distribution and energy spectrum of boosted off-axis neutrinos

Angular distribution and enegy spectrum of boosted
off-axis neutrinos
Christian Roca Catalá
Supervised by: José Bernabeu Alberola
Universidad de Valencia (UV)
crisroc2@alumni.uv.es
July 22, 2013

What is the scope of this presentation?
“I have done a terrible thing, I have postulated a particle that cannot be
detected”
Wolfgang Ernst Pauli, 1930
Fortunately he was WRONG and neutrinos can be detected and thus,
Christian Roca Catal´a (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 neutrino
oscillations. The performance of new generation experiments it’s capital
in 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(∆m2
23) → neutrino mass picture
would be completed
CP violation in leptonic sector: asymmetry matter-antimatter in
early universe (leptogenesis)

Question: How we may perform the measurements?
Answer: Using the oﬀ-axis method → Improving the energy resolution
of 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 appearance
like 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 nutshell
What are Neutrino Oscillations?
Reactor Experiments
Accelerator experiments
CP Violation: parameter δ
Neutrino mass hierarchy
A key technique: oﬀ-axis neutrinos
3 Neutrino kinematics
Pion rest frame - Center of Mass
Lab Frame - Boosted Pion
Boost of the pion
Angular distribution
Relation between Eν and Eπ
Pion energy distribution
Summary
4 Conclusions

Neutrino Oscillations in a nutshell What are Neutrino Oscillations?
Neutrino Oscillations is a
phenomenon
BEYOND THE STANDARD
MODEL
predicted by Bruno Pontecorvo in
1957
Basically it consists in
FLAVOUR MIXING
of the diﬀerent neutrino families
Only make sense if mν = 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) = 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 (Schrödinger img):
|νi (x, t) = eipi x
e−iEi t
|νi = eiφi x
|νi t ∼ x
NOTE!
t ∼ x since neutrinos are ultrarelativistic
Ei = pi for oscillations to happen, that is, mν = 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 depend
on 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 the
PMNS matrix as the CP violation
parameter

Question: Which is the probability that a neutrino oscillate after
travelling an interval L?
Answer: This depends on the medium the neutrinos travel through. The electron
density 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 from
oscillations through vacuum/matter.
NOTE!
The oscillations will always depend on the mass differences between mass families
related to the oscillation ∆m2
13 = m2
1 − m2
3 → we will refer to it as mass difference.

Question: Which is the probability that a neutrino oscillate after
travelling an interval L?
Answer: This depends on the medium the neutrinos travel through. The electron
density Ne influence the cross section of charged current weak interactions of νe .
Oscillations through matter
P(νµ → νe ) = sin2
2θ13 sin2
θ23 sin2
∆eff
13 L/2 + subleading eff.
Dependence on:
∆eff
13 = (∆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 families
related to the oscillation ∆m2
13 = m2
1 − 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 = 0
Flavour mixing happens due to different mass-flavour eigenstates
Oscillation probabilities depend highly on the media the neutrinos are travelling
through
∆m2
can only be measured in experiments where neutrinos travel through matter
CPV can only be measured looking at the subleading effects (appearance
experiments, seen later)

Neutrino Oscillations in a nutshell Reactor Experiments
Reactor Experiments
Neutrino energies ∼ MeV
Modest base-line ∼ km
Solar/atmospheric neutrino
oscillation parameters via...
...Antineutrino disappearance
experiments
Oscillations through vacuum

plain Neutrino Oscillations in a nutshell Reactor Experiments
Question: What can reactor experiments measure?
Answer: Reactor experiments searching for ¯νe disappearance
make neutrinos to oscillate into vacuum, thus the precision in
measuring θ13 is very high. A non-zero value for θ13 is a
prerequisite to... →
Question: What can not reactor experiments measure
Answer: ← ... to measure the open problems in accelerator
experiments:
Measure the mass hierarchy of neutrinos: need oscillations
in matter.
Probe CP violation in the leptonic sector leading to the
possibility that neutrino mixing violates matter/anti-matter
symmetry: need appearance experiments.

plain
Contents
1 What is the scope of this presentation?
2 Neutrino Oscillations in a nutshell
What are Neutrino Oscillations?
Reactor Experiments
Accelerator experiments
CP Violation: parameter δ
Neutrino mass hierarchy
A key technique: oﬀ-axis neutrinos
3 Neutrino kinematics
Pion rest frame - Center of Mass
Lab Frame - Boosted Pion
Boost of the pion
Angular distribution
Relation between Eν and Eπ
Pion energy distribution
Summary
4 Conclusions

Neutrino Oscillations in a nutshell Accelerator experiments
Accelerator Experiments
Neutrino energies ∼
GeV
Long base-line ∼
hundreds km
Neutrino appearance
experiments
Oscillations through
matter

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 eδ
always 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 physics
for 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) = 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 generating
those subleading eﬀects terms.

Question: What is mass
hierarchy?
Answer: Mass hierarchy is the
unknown order of the several
neutrino mass families.
There are two possibles hierarchies,
depending on the sign of ∆m2
13:
normal hierarchy (m2
1 < m2
3)
inverted hierarchy (m2
3 < m2
1)
→ Oscillations in matter: depend
on the parameter A ∝ ∆m2
, thus
are sensitive to sign(∆m2
)!!
NOTE!
Mass diﬀerence ∆m2
12 has been completely measured by solar neutrino. Since the
oscillations inside the sun are considered to occur through matter, the sign have been
determined.

plain Neutrino Oscillations in a nutshell A key technique: oﬀ-axis neutrinos
Detecting neutrinos is not impossible, but
the truth is they are very elusive!
Damn you, Pauli!
→ That’s why several techniques have
been developed in order to attain higher
energy resolution.
“I told you!”
Wolfgang Ernst Pauli
The scope of the second part of my work is to show how the oﬀ-axis neutrinos method
works, 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 determined
non-zero angle respect the travel line of the neutrino beam. Indeed, the neutrino beam
behaves as a wave package, and it spreads out around this line. For a given neutrino’s
source 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 the
neutrino kinematics and discover the goodness of the off-axis
technique!
NOνA placed the far
detector at an off-axis angle
θ = 14mrad.
T2K placed the far detector
at an off-axis angle
θ = 44mrad.

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 →∼ m2
ν in energy and momentum
The pion at the same tame come from the collision between a beam of accelerated
protons 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 take
mν = 0 and thus E = P
Pν = Eν = E =
m2
π − m2
µ
2mπ
Massive neutrinos
Taking the ﬁrst order corrections
∼ m2
nu, the result gives
Ecm = E +
m2
i
2mπ
Pcm = E −
m2
i
2E
ε

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 γβ
0 1 0 0
0 0 1 0
γβ 0 0 γ




Energy and momentum in Lab Frame
Applying the boost to the 4-momentum in CoM Pσ
lab = Λσ
δ Pδ
cm:



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 of
azimutal degree of freedom ϕ
NOTE!
It does not matter to take β nor −β, that is, in direction z or −z: the results are
indeed 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
d2
Γ
dϕlabdcos θlab
=
1
4π
J(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 angular
distribution 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
4π
γ2 1
1 − β cos θlab
2
First order corrections ∼ m2
i , Ecm = Pcm



cos θlab =
Pcm cos θcm + βEcm
Ecm + βPcm cos θcm
cos θcm =
Ecm
Pcm
cos θlab − β
1 − β cos θlab
∂ cos θcm
∂ cos θlab
=
Ecm
Pcm
γ2 1
1 − β cos θlab
2
Angular distribution:
1
Γ
dΓ
dΩlab
=
1
4π
Ecm
Pcm
γ2 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 2
0
2000
4000
6000
8000
10000
12000
θ
lab
(rad)
ν
µ
Flux(ArbitraryUnits)
Either for the approximation mν = 0 or for the first order corrections ∼ m2
ν →
same results obtained!
For a fixed Eπ, neutrino flux peaks at θlab = 0

plain Neutrino kinematics Relation between Eν and Eπ
Question: Why is important to look at the relationship between Eν
and Eπ?
Answer: This relationship will lead us a hint about the energy distribution Eν of the
neutrinos 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 15000
0
1000
2000
3000
4000
5000
6000
7000
Eπ
(MeV)
E
ν
(MeV)
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
value

Neutrino Energy in Lab Frame in terms of Pion Energy
0 5000 10000 15000
0
1000
2000
3000
4000
5000
6000
7000
Eπ
(MeV)
E
ν
(MeV)
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 .

plain Neutrino kinematics Pion energy distribution
Question: What does it happen if pion has not a deﬁnite energy but
a given energy spectrum?
Answer: What we get from the collision proton → ﬁxed target is a non-linear energy
spectrum 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
d2
Γ
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 of
NA61/SHINE Collaboration (arXiv:1005.3692):
1
Γ
d2
Γ
dEπdΩcm
∝ (Ep − Eπ)5

Neutrino Flux in terms of Eπ for Oﬀ-axis Angles θlab > 0.01
2000 2500 3000 3500 4000 4500 5000
0
0.5
1
1.5
2
2.5
x 10
4
Eπ
(MeV)
ν
µ
θ = 0.06 rad θ = 0.05 rad
θ = 0.044 rad
θ = 0.04 rad
θ = 0.03 rad
1
Γ
d2
Γ
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 5000
0
0.5
1
1.5
2
2.5
x 10
4
E
π
(MeV)
ν
µ
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!

Neutrino Flux in terms of Elab for θlab = 0.044
660 665 670 675 680 685 690 695 700
0
100
200
300
400
500
600
700
800
900
1000
E
ν
(MeV)
ν
µ
flux(arbitraryunits)
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 14000
0
5
10
15
20
25
30
35
40
E
ν
(MeV)
ν
µ
flux(arbitraryunits)
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.

plain Neutrino kinematics Summary
Summary
Massless neutrinos mν = 0 and first order correction ∼ m2
ν approximations are
equivalent
Neutrinos attain a maximum energy Emax , independent of the pion energy, for
β = 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 which
peak the neutrino flux for β = cos θlab
The relative narrowness of the off-axis beam increase the energy resolution of
the neutrino beam.

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 of
charge-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 discover
eventually 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... but
calling 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” Princeton
http://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 of
Neutrino Oscillations” http://arxiv.org/pdf/0801.0653v1.pdf
Hiroshi Nunokawa, Stephen Parke, Jose W. F. Valle (2 October 2007) “CP
Violation and Neutrino Oscillations” http://arxiv.org/pdf/0710.0554v2.pdf
Gina Rameika (20 May 2006) “Off-Axis Neutrinos” Fermilab
http://www.phy.bnl.gov/~diwan/talks/talks/nusag-may-20/NuSAG_052006_
_offaxis.pdf
The T2K Collaboration (8 June 2011). “The T2K Experiment”
The T2K Collaboration (3 April 2013). “Evidence of Electron Neutrino Appearance
in a Muon Neutrino Beam”
The T2K Collaboration (6 November 2005) “ND280 Conceptual Design Report
(Internal Report)” www.nd280.org/documents/cdr.pdf/download

plain Conclusions
John N. Bahcall and Raymond Davis Jr. (1976) “Solar Neutrinos: A Scientiﬁc
Puzzle”, 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 Long
Baseline Reactor Neutrino Oscillation Experiment”
http://arxiv.org/pdf/hep-ex/9711002v1.pdf
SuperKamiokande Collaboration (14 May 2001) “Super-Kamiokande atmospheric
neutrino results” http://arxiv.org/pdf/hep-ex/0105023v1.pdf
SNO Collaboration (2 August 2004) “Results from the Sudbury Neutrino
Observatory”
http://www.slac.stanford.edu/econf/C040802/papers/WET001.PDF
CHOOZ Collaboration (13 June 2003) “Search for neutrino oscillations on a long
base-line at the CHOOZ nuclear power station”
http://arxiv.org/pdf/hep-ex/0301017v1.pdf
Daya Bay Collaboration (2012) “Observation of electron-antineutrino
disappearance at Daya Bay” http://arxiv.org/pdf/1203.1669.pdf

plain Conclusions
RENO Collaboration (8 April 2012) “Observation of Reactor Electron Antineutrino
Disappearance in the RENO Experiment”
http://arxiv.org/pdf/1204.0626v2.pdf
André de Gouvêa, James Jenkins and Boris Kayser (23 March 2005) “Neutrino
Mass Hierarchy, Vacuum Oscillations, and Vanishing |Ue3|” Fermilab
http://arxiv.org/pdf/hep-ph/0503079v2.pdf
Hisakazu Minakata, Hiroshi Nunokawa, Stephen Parke (23 January 2013) “The
Complementarity of Eastern and Western Hemisphere Long-Baseline Neutrino
Oscillation 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 the
Neutrino Mixing Angle θ13” http://arxiv.org/pdf/hep-ex/0606025v4.pdf
Rabindra N. Mohapatra and Palash B. Pal (November 1990) “Massive neutrinos in
physics and astrophysics” World Scientific
NA61 Collaboration (2012) “Hadron production measurement from NA61/SHINE”
University of Geneva
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 neutrinos

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