Mass hierarchy and CP violation with upgraded NOA and T2K...

Mass hierarchy and CP violation with upgradedNOνA and T2K in light of large θ13

Suprabh Prakash

Department of PhysicsIndian Institute of Technology Bombay

Mumbai, India

September 26, 2012

Suprabh Prakash (IITB) INFN, Padova visit - Sep 2012 1 / 39

This talk is based on the following works:

SP, S. K. Raut, S. U. Sankar, Getting the best out of T2K and NOνA ,Phys. Rev. D 86, 033012 (2012) arXiv:1201.6485

S. K. Agarwalla, SP, S. K. Raut, S. U. Sankar, Potential of optimizedNOνA for large θ13 and combined performance with a LArTPC and T2K,arXiv:1208.3644


Outline

Introduction

Recent reactor neutrino results

The superbeam experiments: Pµe channel

Hierarchy-δCP degeneracy

Reoptimized NOνA, for large θ13 and NOνA upgrades

Mass hierarchy sensitivity

Leptonic CP violation sensitivity

Results and Summary


The physics of neutrino oscillations

Neutrinos are massive and they mix

As they travel, they can oscillate from one flavor to another

The physics of flavor oscillations can solely be explained by quantummechanics Quant. Mech. description

Neutrino oscillations are sensitive to 6 parameters:

3 mixing angles: θ12, θ13 and θ23

A leptonic CP violating phase: δCP

2 mass squared differences between the 3 mass eigenvalues:∆21 = m2

2 − m21 and ∆31 = m2

3 − m21


Present status of neutrino oscillations

Best fit points, 1σ & 3σ ranges for various neutrino parameters (after Neutrino2012):

[Gonzalez-Garcia,Maltoni,Salvado,Schwetz, arXiv:hep-ph/1209.3023v2]Global sin2 2θ13 best-fit: 0.089Best final precision on sin2 2θ13: Daya Bay: ∼5%


The unknowns in neutrino oscillations physics

Mass hierarchy

We do not know whether the hierarchyis

Normal: m1 < m2 � m3 or

Inverted: m3 � m1 < m2 ?

Leptonic CP violation

Whether CP is violated in the leptonic sector?i.e. is P(α→ β) 6= P(α→ β) ?


Recent θ13 measurements by reactor neutrino experiments

Updated results from Reactor Neutrino Experiments (reported at Neutrino2012):

Daya Bay: sin2 2θ13 = 0.089± 0.011(stat)± 0.005(syst) , 7.7σ fornon-zero θ13

RENO: sin2 2θ13 = 0.113± 0.013(stat)± 0.019(syst) , 4.9σ fornon-zero θ13

Double CHOOZ: sin2 2θ13 = 0.109± 0.030(stat)± 0.025(syst) , 3.1σfor non-zero θ13


Implications of a large θ13

Nature has been very friendly to us with θ13

Present and future generation Superbeam experiments have a very goodscope of performing well

With present generation superbeam experiments, one can get significanthints about both hierarchy and δCP

And, if we are lucky with δCP also, the present generation superbeamexperiments will be able to establish hierarchy as well

However, establishing CP violation and measuring the phase δCP mightnot be possible even with such a large θ13


The superbeam experiments

An intense beam of neutrinos directed towards a detector

Most experiments plan to run in both ν as well as ν mode

The beams primarily consists of the µ-type ν/ν; contaminations are alsopresent which lead to backgrounds

The beam is directed towards a detector which in general is locatedfar-off and the beam travels entirely through the earth which leads tomatter effects

The distance that the beam travels i.e. the baseline, is important

The beam profile i.e. the Eν at which the flux peaks, is also important

Finally, in the detector, one looks for electron appearance i.e. νµ whichhave oscillated to νe


The Pµe oscillation channel

νµ → νe oscillation probability, expanded perturbatively in α = ∆21/∆31:

Pµe = sin2 2θ13 sin2 θ23sin2 ∆(1− A)

(1− A)2

+α cos θ13 sin 2θ12 sin 2θ13 sin 2θ23 cos(∆ + δCP)sin ∆A

A

sin ∆(1− A)

1− A

+α2 sin2 2θ12 cos2 θ13 cos2 θ23sin2 ∆A

A2

where

∆ = ∆31L/4E

A = A/∆31

Wolfenstein matter term: A(eV2) = 0.76× 10−4ρ(gm/cc)E(GeV).


Dependence of Pµe on the baseline

Pµe(as a band in δCP) vs. EnergyLeft panel: L=295 km & Right panel: L=810 km

0

0.03

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0.09

0.12

0.15

1 2 3 4 5

Pro

bab

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y(n

eutr

ino

s)

E(GeV)

sin22θ13 = 0.1

295 km

NHIH

δCP = +90o

δCP= -90o

0

0.03

0.06

0.09

0.12

0.15

1 2 3 4 5

Pro

bab

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y(n

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E(GeV)

sin22θ13 = 0.1

810 km

NHIH

δCP = +90o

δCP= -90o

The value of Pµe scales with LThe position of 1st Oscillation maxima also scales with L

Go Back


Various LBL superbeam experiments

Present generation LBL superbeam experiments

MINOS: 735 km

T2K: 295 km

NOνA: 810 km

Next generation LBL superbeam experiments

CERN-Frejus: 130 km

T2HK: 295 km

CERN-Canfranc: 630 km

CERN-Gran Sasso: 730 km

CERN-Homestake: 1300 km

CERN-Pyhasalmi: 2300 km


The Pµe oscillation channel: degeneracies

The Pµe channel

Has sensitivity to all the neutrino parameters

Pros: This feature, in principle, makes measurement of all neutrinoparameters possible using superbeam

Cons: This very feature handicaps it of various degeneracies

A measurement of Pµe will have

Intrinsic or (δCP, θ13)-degeneracy

Hierarchy or sign(∆31)-degeneracy

Octant or θ23-degeneracy

This leads to an eight-fold degeneracy in determination of θ13 and δCP.[Barger et. al., hep-ph/0112119]


The Pµe oscillation channel: degeneracies

But, the reactor neutrino experiments are independent of these degeneraciesas they are independent of δCP and matter-effects (hence hierarchy) andtherefore, they have provided a very clean measurement of θ13.

In this work, we have assumed a precise information of θ13. Hence thedegeneracies relevant for us are:

hierarchy-δCP degeneracy

Octant or θ23-degeneracy

We further assume the maximal value of sin2 2θ23 = 1 and, for the moment,ignore the octant degeneracies. So, the degeneracy relevant for this work is

hierarchy-δCP degeneracy


Hierarchy-δCP degeneracy

Pµe(as a band in δCP) vs. EnergyLeft panel: Neutrinos & Right panel: Anti-neutrinos

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0.03

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0.09

0.12

0.15

1 2 3 4 5

Pro

bab

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y(n

eutr

ino

s)

E(GeV)

sin22θ13 = 0.1

810 km

NHIH

δCP = +90o

δCP= -90o

0

0.03

0.06

0.09

0.12

0.15

1 2 3 4 5

Pro

bab

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y(a

nti

-neu

trin

os)

E(GeV)

sin22θ13 = 0.1

810 km

NHIH

δCP = +90o

δCP= -90o

Lower half plane (−180◦ ≤ δCP ≤ 0): favourable plane for NH

Upper half plane (0 ≤ δCP ≤ 180◦): favourable plane for IH


Resolving the hierarchy-δCP degeneracy

The hierarchy-δCP degeneracy can be resolved in following ways:1 Including atmospheric data2 The Magic baseline proposal [Barger et. al., hep-ph/0112119; Huber and

Winter, hep-ph/0301257]3 By combining data from more that one channels like: νe → νµ orνe → ντ [Donini et. al., hep-ph/0206034]

4 The Bimagic baseline proposal5 By combining data from more than one LBL superbeam experiments

[Barger et. al., hep-ph/0112119]

Option 1 is a free source, but for a good hierarchy sensitivity, huge statisticsare required.While options 2 and 3 are futuristic and challenging to implement, options 4and 5 are more feasible


The Bimagic baseline

Pµe(as a band in δCP) vs. Energy for 2540 kmLeft panel: Neutrinos & Right panel: Anti-neutrinos

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0.06

0.12

0.18

0.24

0.3

2 4 6 8 10

Pro

bab

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y(n

eutr

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s)

E(GeV)

sin22θ13 = 0.1

2540 km

NHIH

δCP = +90o

δCP= -90o

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0.06

0.12

0.18

0.24

0.3

2 4 6 8 10

Pro

bab

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y(a

nti

-neu

trin

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E(GeV)

sin22θ13 = 0.1

2540 km

NHIH

δCP = +90o

δCP= -90o

Clean separation between NH and IH around 1st Osc. maximaOne of the hierarchy loses its δCP dependence around 3 GeV and theother is amplified


The Bimagic baseline

References for 2540 km:[Raut et. al. arXiv:0908.3741, Phys.Lett.B696:227-231,2011][Dighe et. al. arXiv:1009.1093, Phys.Rev.Lett.105:261802,2010][Joglekar et. al. arXiv:1011.1146, Mod.Phys.Lett.A26:2051-2063,2011]

CERN-Pyhasalmi (L=2300 km), one strong contender for the LBLsuperbeam experiment in Europe, is very close to the bimagic baseline

Very good sensitivity to mass hierarchy even with very moderateexposures [Dusini et. al. arXiv:1209.5010]


NOνA and T2K

Characteristic NOνA T2KBaseline

810 km 295 kmLocation Fermilab-Ash River J-PARC-KamiokaBeam NuMI beam 0.8◦ off

- axisJHF beam 2.5◦ off -axis

Beam power 0.7 MW 0.75 MWBeam peaks at

2 GeV 0.6 GeVPµe 1st Osc. Maximum 1.5 GeV 0.55 GeVDetector TASD, 14 kton Water Cerenkov,

22.5 ktonRuntime

3 in ν+3 in ν 5 in ν

Simulation Details


Hierarchy sensitivity of NOνA

NOνA with various design statisticsLeft panel: NH is true & Right panel: IH is true

-180

-120

-60

0

60

120

180

0 0.05 0.1 0.15 0.2

δC

P(t

rue)

sin2 2θ13(true)

True:NH / Test:IH, 90% C.L.

NOνA1.5*NOνA

3*NOνA

-180

-120

-60

0

60

120

180

0 0.05 0.1 0.15 0.2

δC

P(t

rue)

sin2 2θ13(true)

True:IH / Test:NH, 90% C.L.

NOνA1.5*NOνA

3*NOνA

For the bestfit, 90% C.L. hint for the whole of favourable half planeAlmost no sensitivity in the unfavourable half plane, even with largerstatistics


Hierarchy sensitivity of NOνA + T2K

NOνA + T2K with nominal design statisticsLeft panel: NH is true & Right panel: IH is true

-180

-120

-60

0

60

120

180

0 0.05 0.1 0.15 0.2

δC

P(t

rue)

sin2 2θ13(true)

True:NH / Test:IH, 90% C.L.

NOνA+T2K

-180

-120

-60

0

60

120

180

0 0.05 0.1 0.15 0.2

δC

P(t

rue)

sin2 2θ13(true)

True:IH / Test:NH, 90% C.L.

NOνA+T2K

Adding the T2K data helps in the unfavourable half plane; butinsufficient for the bestfitNo difference in the favourable half plane


Resolving hierarchy-δCP degeneracy using two differentLBL experiments

Event sample observed in the detector has to be analyzed assuming a truemass hierarchy. If the assumption is right, we get a right estimate of δCP

(∆χ2min occuring at right δCP). If the assumption is wrong, we get a wrong

estimate of δCP (∆χ2min occuring at wrong δCP).

measurement has degenerate solutions: (hierright, δCPright) and

(hierwrong, δCPwrong)

For a given experiment, (sin δCPright − sin δCP

wrong) ∝ A ∝ E ∝ L forthat experiment [O. Mena and S. Parke, arXiv:hep-ph/0408070]

E810 kmpeak flux ' 3 · E295 km

peak flux

(sin δCPright − sin δCP

wrongNOνA) ' 3 · (sin δCP

right − sin δCPwrongT2K )

δCPwrong predicted by two LBL experiments with different matter effects

are different (also evident from the Pµe vs. E plots )


Resolving hierarchy-δCP degeneracy using two differentLBL experiments

[∆χ2min]NOνA at (hierright, δCP

right) and (hierwrong, δCPwrongNOνA)

[∆χ2min]T2K at (hierright, δCP

right) and (hierwrong, δCPwrongT2K )

[∆χ2]NOνA not minimal at (hierwrong, δCPwrongT2K )

[∆χ2]T2K not minimal at (hierwrong, δCPwrongNOνA)

[∆χ2min]NOνA+T2K not at (hierwrong, δCP

wrongNOνA) or (hierwrong, δCP

wrongT2K )

=⇒ wrong solutions excluded

Caveat: In order to have reasonable excluding sensitivity, ∆χ2min value at

(hierwrong, δCPwrong) should be much larger than ∆χ2

min value at (hierright,δCP

right). Therefore, the experiments must have enough statistics.


How does combining data from two LBL help?

χ2 vs. δCP(test) plots for NOνALeft panel: favourable half plane & Right panel: unfavourable half plane

0

10

20

30

40

50

60

70

-180 -120 -60 0 60 120 180

χ2

δCP(test)

1.5*NOνA True sin22θ13 : 0.1

True:NH / Test:IH

True δCP = -180o

True δCP = -135o

True δCP = -90o

True δCP = -45o

0

10

20

30

40

50

60

70

-180 -120 -60 0 60 120 180

χ2

δCP(test)

1.5*NOνA True sin22θ13 : 0.1

True:NH / Test:IH

True δCP = 0

True δCP = 45o

True δCP = 90o

True δCP = 135o

χ2min always occurs around −90◦

χ2min large for fav. half plane, very small for unfav. half plane



χ2 vs. δCP(test) plots for T2KLeft panel: favourable half plane & Right panel: unfavourable half plane

0

5

10

15

20

25

30

-180 -120 -60 0 60 120 180

χ2

δCP(test)

2*T2K True sin22θ13 : 0.1

True:NH / Test:IH

True δCP = -180o

True δCP = -135o

True δCP = -90o

True δCP = -45o

0

5

10

15

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30

-180 -120 -60 0 60 120 180

χ2

δCP(test)

2*T2K True sin22θ13 : 0.1

True:NH / Test:IH

True δCP = 0

True δCP = 45o

True δCP = 90o

True δCP = 135o

Right panel: large χ2 where χ2min of NOνA was almost 0

Right panel: χ2min occurs at test δCP different from that of NOνA



χ2 vs. δCP(test) plots for 1.5*NOνA +2*T2KLeft panel: 1.5*NOνA only & Right panel: 1.5*NOνA +2*T2K

0

5

10

15

20

25

30

35

40

-180 -120 -60 0 60 120 180

χ2

δCP(test)

True (sin22θ13, δCP) = (0.1, 90

o)

1.5*NOνA

χ2 = 2.71, 90% C.L.

True:NH / Test:NHTrue:NH / Test:IH

0

5

10

15

20

25

30

35

40

-180 -120 -60 0 60 120 180

χ2

δCP(test)

True (sin22θ13, δCP) = (0.1, 90

o)

1.5*NOνA+2*T2K

χ2 = 2.71, 90% C.L.

True:NH / Test:NHTrue:NH / Test:IH

Left panel: A large range of wrong δCP compatible with wrong hierarchyRight panel: Adding T2K data almost eliminates this wrong δCP range



χ2 vs. δCP(test) plots for 1.5*NOνA +2*T2KLeft panel: 1.5*NOνA only & Right panel: 1.5*NOνA +2*T2K

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χ2

δCP(test)

True (sin22θ13, δCP) = (0.1, -90

o)

1.5*NOνA

χ2 = 2.71, 90% C.L.

True:IH / Test:NHTrue:IH / Test:IH

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10

15

20

25

30

35

40

-180 -120 -60 0 60 120 180

χ2

δCP(test)

True (sin22θ13, δCP) = (0.1, -90

o)

1.5*NOνA+2*T2K

χ2 = 2.71, 90% C.L.

True:IH / Test:NHTrue:IH / Test:IH

Left panel: A large range of wrong δCP compatible with wrong hierarchyRight panel: Adding T2K data almost eliminates this wrong δCP range


Reoptimization and upgrades in NOνA, post-large θ13

We study the physics reach of the reoptimized NOνA (in conjunctionwith T2K) to determine the mass hierarchy and CP violation

There is considerable enthusiasm in the NOνA and LBNE collaborationsfor an additional small liquid argon detector at the Ash River site

We consider the possiblility of such a module


Reoptimized NOνA

Previous optimization: done under the assumption sin2 2θ13 = 0 withobjective to have least background

New optimization: for moderately large θ13 with relaxed cuts for eventselection criteria which allow more signal and background [talk by R.Patterson in NuFACT 12]

The main differences between the old criteria and new criteria are:

The signal efficiencies for new NOνA are higher than that of old one byroughly a factor of 2 for neutrino events

The background acceptance has also increased

The NC spectrum shift to the measured energies is implemented throughmigration matrices. [LBNE Collaboration, T. Akiri et al. arXiv:1110.6249]

We have taken care of the neutrino contamination in the anti-neutrinobeam


Liquid Argon TPC

A 10 kt liquid argon time projection chamber (LArTPC) constructedclose to NOνA

A signal efficiency of 80% for e± compared to 45% for TASD [LBNECollaboration, T. Akiri et al. arXiv:1110.6249]

The energy resolution and background rejection for the LArTPC andTASD are comparable

Such a detector will come on line much later than NOνA

In considering NOνA + LArTPC, we assume equal 6 years ν and ν runsfor NOνA and equal 3 years ν and ν runs for LArTPC


Mass hierarchy sensitivity with upgraded NOνA and T2K

Hierarchy discovery for various set-upsLeft panel: TASD vs. LArTPC for NH true & Right panel: Combined set-upfor NH true

Mass Hierarchy Discovery, NH true

δ

χ

(true)CP

∆

95% C.L.

90% C.L.

2

0

8

10

12

14

16

−180 −90 0 90 180

NOvA (3+3)

5 kt LAr (3+3)

10 kt LAr (3+3)

14 kt LAr (3+3)

4

2

6

Mass Hierarchy Discovery, NH true

χ

CP

∆2

δ

90% C.L.

95% C.L.

(true)

NOvA (6+6) + T2K (5) + 5 kt LAr (3+3)

20

25

30

−180 −90 0 90 180

10

5

0

NOvA (6+6) + T2K (5) + 10 kt LAr (3+3)

NOvA (3+3) + T2K (5+0)

15

Left panel: LArTPC of mass 10 kt can be as effective as 14 kton TASDRight panel: With the addition of a 10 kt LArTPC, close to 95% C.L.hierarchy discrimination becomes possible for all the values of δCP


Mass hierarchy sensitivity with upgraded NOνA and T2K

Hierarchy discovery for various set-upsLeft panel: TASD vs. LArTPC for IH true & Right panel: Combined set-upfor IH true

90% C.L.

δ (true)CP

∆ χ

Mass Hierarchy Discovery, IH true

95% C.L.

2

0

8

10

12

14

16

−180 −90 0 90 180

4

2

6

Mass Hierarchy Discovery, IH true

(true)CP

∆ χ2

90% C.L.

95% C.L.

δ

0

20

25

30

−180 −90 0 90 180

10

5

15

Left panel: LArTPC of mass 10 kt can be as effective as 14 kton TASDRight panel: With the addition of a 10 kt LArTPC, close to 95% C.L.hierarchy discrimination becomes possible for all the values of δCP


Sensitivity of upgraded NOνA and T2K for CP violation

Leptonic CP violation discovery for various set-upsLeft panel: TASD vs. LArTPC for NH true & Right panel: Combined set-upfor NH true

95% C.L.

(true)CP

∆2

δ

CP Violation Discovery, NH true

90% C.L.χ

0

4

5

6

−180 −90 0 90 180

NOvA (3+3)

5 kt LAr (3+3)

10 kt LAr (3+3)

14 kt LAr (3+3)

2

1

3

90% C.L.

χ∆2

δ (true)

CP Violation Discovery, NH true

95% C.L.

CP

0

8

10

12

14

16

−180 −90 0 90 180

4

2

NOvA (3+3) + T2K (5+0)

NOvA (6+6) + T2K (5+0) + 5 kt LAr (3+3)

NOvA (6+6) + T2K (5+0) + 10 kt LAr (3+3)

6

Left panel: LArTPC of mass 10 kt can be as effective as 14 kton TASDRight panel: Addition of a 10 kt LArTPC significantly improves range oftrue δCP for which CP conservation can be ruled out at 95% C.L.


Sensitivity of upgraded NOνA and T2K for CP violation

Leptonic CP violation discovery for various set-upsLeft panel: TASD vs. LArTPC for IH true & Right panel: Combined set-upfor IH true

90% C.L.

CP

∆ χ2

δ

CP Violation Discovery, IH true

95% C.L.

(true)

0

4

5

6

−180 −90 0 90 180

2

1

3

CP Violation Discovery, IH true

2

δ (true)CP

χ

95% C.L.

90% C.L.

∆

0

8

10

12

14

16

−180 −90 0 90 180

4

2

6

Left panel: LArTPC of mass 10 kt can be as effective as 14 kton TASDRight panel: Addition of a 10 kt LArTPC significantly improves range oftrue δCP for which CP conservation can be ruled out at 95% C.L.


Can hier-δCP degeneracy be resolved by first measuring δCP?

Can a short LBL experiment such as T2K(295 km) or CERN-Frejus(130km) measure the δCP phase independent of hierarchy with electronappearance channel?

A short baseline has small matter effects and therefore is relatively freeof hierarchy-δCP degeneracy. Therefore, it is expected that it can providea clean δCP measurement.

But it cannot!!



Allowed δCP plots for T2K[3+3]Left panel: Hierarchy known & Right panel: Hierarchy not known

-180

-120

-60

0

60

120

180

-180 -120 -60 0 60 120 180

δC

P (

test

)

δCP (true)

True:NH/Test:NH, 90% C.L.

-180

-120

-60

0

60

120

180

-180 -120 -60 0 60 120 180

δC

P (

test

)

δCP (true)

True:NH/Test:IH, 90% C.L.

Left panel: allowed range of test δCP is around true δCP

Right panel: large region of the wrong half-plane alsoSuprabh Prakash (IITB) INFN, Padova visit - Sep 2012 36 / 39


Allowed δCP plots for CERN-Frejus(130 km)Left panel: Hierarchy known & Right panel: Hierarchy not known

-180

-120

-60

0

60

120

180

-180 -120 -60 0 60 120 180

δC

P (

test

)

δCP (true)

True:NH/Test:NH, 90% C.L.

-180

-120

-60

0

60

120

180

-180 -120 -60 0 60 120 180

δC

P (

test

)

δCP (true)

True:NH/Test:IH, 90% C.L.

Left panel: allowed range of test δCP is around true δCP

Right panel: large region of the wrong half-plane alsoSuprabh Prakash (IITB) INFN, Padova visit - Sep 2012 37 / 39

δCP phase measurement with upgraded NOνA and T2K

δCP phase measurementLeft panel: NOνA +T2K & Right panel: NOνA +T2K+LArTPC

-180

-120

-60

0

60

120

180

-180 -120 -60 0 60 120 180

δC

P (

test

)

δCP (true)

True:NH/Test:NH 90% C.L.

-180

-120

-60

0

60

120

180

-180 -120 -60 0 60 120 180

δC

P (

test

)

δCP (true)

True:NH/Test:NH 90% C.L.

A precise δCP phase measurement will require a future facility


Summary

A 10 kt LArTPC is found to be equivalent to new NOνA in its ability toexclude the wrong hierarchy and CP violation discovery

For increased exposure for NOνA, the combination of data from NOνA,LArTPC and T2K (nominal) can determine hierarchy at almost 2σ levelfor all the values of δCP

NOνA by itself can discover CP violation only for a small fraction of thefavourable half-plane, and only at 90% C.L.

Addition of data from T2K causes a remarkable increase in the range ofδCP for which CP violation can be established

With the combination of additional data from the LArTPC detector withboosted NOνA and nominal T2K, CP violation can be discovered at 95%confidence level for around 60% of the entire δCP range


n flavor eigenstates |να〉; connected to n mass eigenstates |νi〉 through unitaryPMNS mixing matrix U.

|να〉 =∑

i

Uαi|νi〉

νe

νµντ

= U

ν1ν2ν3

U =

Ue1 Ue2 Ue3Uµ1 Uµ2 Uµ3Uτ1 Uτ2 Uτ3

U = U23 (θ23)× U13 (θ13, δ)× U12 (θ12)× UMaj

Schroedinger’s Eq. in matter : i ddt |να〉 = Hflav

matt|να〉

P (α→ β) =

δαβ − 4<∑

j>i U∗αiUβiUαjU∗

βj sin2 ∆jiL4E + 2=

∑j>i U∗

αiUβiUαjU∗βj sin ∆jiL

2E

∆ji = m2j − m2

i

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