Tomography by neutrino pair beam Hisashi Okui, Takehiko Asaka, - - PowerPoint PPT Presentation

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Tomography by neutrino pair beam Hisashi Okui, Takehiko Asaka, - - PowerPoint PPT Presentation

Sep 04, 2022 285 likes •613 views

24th International Summer Institute SI2018 on Phenomenology of Elementary Particle Physics and Cosmology 17 August, Friday Tomography by neutrino pair beam Hisashi Okui, Takehiko Asaka, Minoru Tanaka A , Motohiko Yoshimura B Niigata Univ, A

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SLIDE 1

Tomography by neutrino pair beam

Hisashi Okui, 
 Takehiko Asaka, Minoru TanakaA, Motohiko YoshimuraB

12-17 August

SI2018

24th International Summer Institute

  • n Phenomenology of Elementary Particle Physics and Cosmology

@Tianjin Niigata Univ, AOsaka Univ, BOkayama Univ arXiv:1805.10793[hep-ph] 17 August, Friday

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SLIDE 2

Introduction

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SLIDE 3

SI2018

3

Development of the neutrino physics

Our understanding of neutrino has been improved greatly since the end of the last century. Especially, the observation of flavor oscillations of neutrino has shown the presence of new physics beyond the standard model. This is inconsistent with the prediction of the Standard Model that predicts massless neutrinos. This is a clear signature of new physics beyond the Standard Model.

The Nobel Prize in Physics 2015

Takaaki Kajita (SK) Arthur B. McDonald (SNO)

For the discovery of neutrino oscillation, which shows the neutrinos have mass.

Takaaki Kajita It’s me !

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SLIDE 4

SI2018

4

Neutrino Oscillation Parameter

sij = sin θij, cij = cos θij P =   1 eiα eiβ  

PMNS matrix

UP MNS =   1 c23 s23 −s23 c23     c13 s13e−iδ 1 −s13e−iδ c13     c12 s12 −s12 c12 1   P

Reactor neutrino Accelerator neutrino Atmospheric neutrino Solar neutrino Majorana phase CP phase

m1 < m2 < m3

Normal Ordering

m3 < m2 < m1

Inverted Ordering

has been measured accurately.

θij ∆mij

From neutrino oscillation experiments ＊ Absolute value of neutrino mass, CP phase, Majorana phase, mass ordering have not yet determined.

So, we consider seriously the application of neutrino physics to various fields of basic science.

4

NuFIT 3.0 (2016) Normal Ordering (best fit) Inverted Ordering (∆χ2 = 0.83) Any Ordering bfp ±1σ 3σ range bfp ±1σ 3σ range 3σ range sin2 θ12 0.306+0.012

0.012

0.271 → 0.345 0.306+0.012

0.012

0.271 → 0.345 0.271 → 0.345 θ12/ 33.56+0.77

0.75

31.38 → 35.99 33.56+0.77

0.75

31.38 → 35.99 31.38 → 35.99 sin2 θ23 0.441+0.027

0.021

0.385 → 0.635 0.587+0.020

0.024

0.393 → 0.640 0.385 → 0.638 θ23/ 41.6+1.5

1.2

38.4 → 52.8 50.0+1.1

1.4

38.8 → 53.1 38.4 → 53.0 sin2 θ13 0.02166+0.00075

0.00075

0.01934 → 0.02392 0.02179+0.00076

0.00076

0.01953 → 0.02408 0.01934 → 0.02397 θ13/ 8.46+0.15

0.15

7.99 → 8.90 8.49+0.15

0.15

8.03 → 8.93 7.99 → 8.91 δCP/ 261+51

59

0 → 360 277+40

46

145 → 391 0 → 360 ∆m2

21

105 eV2 7.50+0.19

0.17

7.03 → 8.09 7.50+0.19

0.17

7.03 → 8.09 7.03 → 8.09 ∆m2

3`

103 eV2 +2.524+0.039

0.040

+2.407 → +2.643 −2.514+0.038

0.041

−2.635 → −2.399  +2.407 → +2.643 −2.629 → −2.405

  • Thanks to the remarkable efforts of various experiments
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SLIDE 5

SI2018

5

The idea of Neutrino Tomography

Imaging of the Earth’s interior structure using the neutrino. Neutrino can easily transmit the Earth due to the weakness of its interaction.

ν

Source

ν

Detector

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SLIDE 6

SI2018

6

Neutrino Tomography

3 different methods of Neutrino Tomography

  • 1. Neutrino Absorption Tomography
  • 2. Neutrino Oscillation Tomography

(3. Neutrino Diffraction Tomography)

  • Using the absorption of neutrino by matter.
  • Same mechanism to the X-ray computed tomography.
  • This method needs the high energy neutrinos (Eν > 10 TeV).
  • Using the matter effect of neutrino oscillation.
  • Measure the diffraction pattern of crystalline


matter in the deep interior of the Earth.

  • Not realistic yet.

In this talk, we discuss about this type !

There is no precise tomography method. There is no powerful source. There is no established reconstruction method.

  • T. Ohlsson andW.Winter, Europhys. Lett. 60 (2002) 34
  • E. K. Akhmedov, M. A. Tortola and J.W. F. Valle, JHEP 0506, 053 (2005)
  • W.Winter, Nucl. Phys. B 908 (2016) 250
  • A.N. Ioannisian and A. Y. Smirnov, Phys. Rev. D 96 (2017) no.8, 083009

And more …

  • L. V. Volkova and G. T. Zatsepin, Bull. Acad. Sci. USSR, Phys. Ser. 38 (1974) 151.

And more …

  • A.D. Fortes, I. G.Wood, and L. Oberauer, Astron. 

  • Geophys. 47(2006) 5.31–5.33.
  • R. Lauter, Astron. Nachr. 338 (2017) no.1, 111.
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SLIDE 7

SI2018

7

Neutrino Oscillation

Neutrino oscillation is phenomenon that the neutrino flavor will vary with distance. It is caused by the quantum mechanical superposition. Neutrino flavor eigenstates is written by superposition of the mass eigenstates.   νe νµ ντ   = UP MNS   ν1 ν2 ν3  

flavor eigenstate Mass eigenstate

P(νe ! νµ; E, t) = |hνµ|νe(t)i|2 = | sin θ cos θ(1 e−i(E2−E1)t)|2 = sin2(2θ) sin2(∆m2 4E t)

500 1000 1500 2000 2500 3000 0.2 0.4 0.6 0.8 1.0

∆m2 = 2.524 × 10−3 [eV ] θ = 41.6 180 π E = 1 [GeV ] x [km]

~1000km Ex) 2 flavor case Mass eigenstates evolve respectively in time. Then, because of the interference between the mass state, the flavor transition probability behaves oscillatory.

UP MNS

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Pontecorvo-Maki-Nakagawa-Sakata matrix

slide-8
SLIDE 8

SI2018

8

Neutrino Oscillation in Matter

W νe e e νe

Z νe, νµ, ντ νe, νµ, ντ e, p, n e, p, n

i d dx ~ A(x) = [HF

0 + V F ] ~

A(x) Evolution equation of transition amplitudes of neutrino flavors is written as follow. In matter, additional effective potential is added to the vacuum Hamiltonian.

Vacuum contribution Additional effective potential

  • Neutrino interacts with the electron, proton, neutron in matter, 


through the CC and NC interaction.

  • The contribution of the NC interaction is common to all flavors, and eliminated by the 


common phase shift.

  • Therefore, the main contribution to the potential is the CC interaction and 


effective potential depend on the electron number density.

In 2 flavor case

Aβα(x) = hνβ|να(x)i

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~ A(x) = (Aνe→νe(x), Aνe→νµ(x))T

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slide-9
SLIDE 9

SI2018

9

Neutrino Oscillation in Matter

Density Profile Oscillation Probability (subtracted the vacuum contribution)

Energy Spectrum of the Oscillation Probability change by the density profile.

50 100 150 200 250 300 2 4 6 8 10 x@kmD Ρ@gêcm3D

0.002 0.005 0.010 0.020 0.050 0.100

  • 0.05

0.00 0.05 0.10 En@GeVD PHn eÆn eL-PH0LHn eÆn eL

For simplicity, we consider the 2 flavor neutrino oscillation

i d dx ✓Aνe→νe Aνe→νµ ◆ =  U ✓0

∆m2 2E

◆ U † + ✓VCC(x) ◆ ✓Aνe→νe Aνe→νµ ◆

Pνα→νβ(Eν, x) = |Aνα→νβ(Eν, x)|2

Effective potential is written as The electron number density is translated into the matter density. Probability is calculated as follow

VCC(x) = √ 2GF ne(x) ne(x) ' ρ(x) 2mp

slide-10
SLIDE 10

SI2018

10

Neutrino Oscillation Tomography

50 100 150 200 250 300 2 4 6 8 10 x@kmD Ρ@gêcm3D 50 100 150 200 250 300 2 4 6 8 10 @kmD D

0.002 0.005 0.010 0.020 0.050 0.100

  • 0.05

0.00 0.05 0.10 En@GeVD PHn eÆn eL-PH0LHn eÆn eL

Normal calculation Tomography Density Profile OscillationProbability (subtracted the vacuum contribution) There include the information

  • f the density profile

It is required the precise measurement of the energy spectrum. So, powerful neutrino source is required.

slide-11
SLIDE 11

Method

slide-12
SLIDE 12

SI2018

12

Neutrino Pair Beam

The pair beam, which has been proposed recently, can produced a large amount of neutrino pairs from the circulating partially stripped ions.

[Yoshimura, Sasao, Phys. Rev. D 92, 073015 (2015)]

Characteristics of the Neutrino Pair Beam

It generates the all flavor neutrino pairs

(νe, νe), (νµ, νµ), (ντ, ντ)

Very high intensity flux of neutrino beam High beam directivity

  • Fig. 7. Neutrino energy spectrum rate at the forward direction of solid angle

area π/γ 2. Assumed parameters are ρϵeg = 1014, N = 108 and ϵeg = 50 keV,

γ = 4000 in solid black, 5000 in dashed red, 6000 in dash-dotted blue. (For in-

terpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

[Asaka,Tanaka,Yoshimura, Phys.Lett. B760 (2016) 359-364 ]

Neutrino Tomography requires the precise measurement of the energy spectrum for the precise reconstruction of the density profile. This high event rate (high flux) is essential.

slide-13
SLIDE 13

SI2018

13

Toy model

ρ(x) = ¯ ρ +(ρl − ¯ ρ)exp " − ° x − L

2

¢2 D2

l

# ,

We consider the symmetric exponential type of the density profile.

L : length of the baseline Dl : width of the lump L = 300 km

50 100 150 200 250 300 2 4 6 8 10

x@kmD Ρ@gêcm3D

We assume the huge liquid Argon as the neutrino detector. Fiducial volume 105 m3 We consider the low energy oscillation.

¯ νe → ¯ νe

Eν : 2 ∼ 100 [MeV]

Source Point Detection Point Lump

L = 300 [km]

slide-14
SLIDE 14

SI2018

14

Statistical Test

6 8 10 12 14 16 18 25 50 75 100 125 ρ* [g/cm3] D* [km] (A) (B) (C)

  • - : 1 σ region

ー : 3 σ region Δχ2 contour

= ∆χ2 =

Nb

X

i=1

£ N(Ei)|D§,ρ§ ° N(Ei)|Dl,ρl §2 σ2(Ei) , N = flux × oscillation probability × detection rate

We perform the χ2 analysis.

Nb = 100 : the number of energy bin

(A) D§ = 50+5.9

°5.9 km

and ρ§ = 8.0+0.62

°0.48 g cm°3 ,

(B) D§ = 50+2.5

°2.4 km

and ρ§ = 16+0.58

°0.53 g cm°3 ,

(C) D§ = 100+8.2

°7.1 km

and ρ§ = 8.0+0.22

°0.21 g cm°3 ,

The pair beam can probe the lump at the 1 σ level as

We estimate how precisely the width( ) and density( ) of the lump can be reconstructed under this set up. D∗ ρ∗

We assume the 3 density profile.

It is seen that the neutrino pair beam can provide the measurement of the density profile.

ρ(x) = ¯ ρ +(ρl − ¯ ρ)exp " − ° x − L

2

¢2 D2

l

# ,

¯ ρ = 2.7[g/cm3]

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We assume 1 year as experimental running time.

slide-15
SLIDE 15

SI2018

15

Neutrino Oscillation Tomography

50 100 150 200 250 300 2 4 6 8 10 x@kmD Ρ@gêcm3D 50 100 150 200 250 300 2 4 6 8 10 @kmD D

0.002 0.005 0.010 0.020 0.050 0.100

  • 0.05

0.00 0.05 0.10 En@GeVD PHn eÆn eL-PH0LHn eÆn eL

?

How reconstruct the density profile from the energy spectrum of the neutrino oscillation?

slide-16
SLIDE 16

SI2018

16

Density Profile Reconstruction Method

χ2 = X

i=1,NE

£ N obs(Ei)° N th(Ei) §2 σ2(Ei) , p

  • 1. We discretize the neutrino baseline into the NL segments.

L NL

  • 2. We consider the matter densities for these segments as free parameters .

L ρ1 ρ2 ρ3 ρNL ρNL−1 ρNL−2

We assume that the each density is constant within each segment.

  • 3. We also divide the energy range into the NE parts, and define the χ2 function
  • 4. We determine those density by minimizing the χ2 function by comparing the 


experimental data for a given original profile ρ(x) with the theoretical 
 prediction from unknown parameters .

Nobs(Ei) Nth(Ei)

ρj

where σ(Ei) = p N obs(Ei).

Source Point Detection Point Lump

L = 300 [km]

ρj

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j = 1, · · · , NL

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slide-17
SLIDE 17

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17

Perturbation Formula

We introduce the perturbation formula of the neutrino oscillation probability which is used for the theoretical prediction from unknown parameters .

Nth(Ei)

ρj Nth(Ei) = flux × P¯

νe→¯ νe(E, L) × detection rate

Neutrino Oscillation Probability

i d dx ~ A(x) = [HF

0 + V F ] ~

A(x)

Neutrino Oscillation Probability is calculated from the this evolution equation. Then we assume the relation HF

0 > V F

And calculate the oscillation probability by perturbation. Ex) perturbation formula at 1st order is written as P (1)(Ei) ∝ X ρ(xj)  sin ⇢∆m2 2EI L

  • − sin

⇢∆m2 2EI xj

  • − sin

⇢∆m2 2Eν (L − xj)

  • We find the 2nd order perturbation is important for the successful reconstruction.

Pαβ = |A(0)

βα + A(1) βα + A(2) βα + · · · |2

= |A(0)

βα|2 + A(0)∗ βα A(1) βα + A(0) βαA(1)∗ βα + |A(1) βα|2 + A(0)∗ βα A(2) βα + A(0) βαA(2)∗ βα + · · ·

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0th 1st 2nd

slide-18
SLIDE 18

SI2018

18

Assumption

ρj ≥ 0 And reconstruct the density profile.

N = flux × oscillation probability × detection rate

i d dx ✓Aνe→νe Aνe→νµ ◆ =  U ✓0

∆m2 2E

◆ U † + ✓VCC(x) ◆ ✓Aνe→νe Aνe→νµ ◆

ne(x) ' ρ(x) 2mp VCC(x) = √ 2GF ne(x)

Nobs(Ei)

We assume as event rate by the calculation from evolution equation with original matter density profile. ★ We assume about the fitting parameter (matter density in the each segment) ★

χ2 = X

i=1,NE

£ N obs(Ei)° N th(Ei) §2 σ2(Ei) , p

Minimize ★ N th(Ei, ρj)

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slide-19
SLIDE 19

Results of reconstruction

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SLIDE 20

SI2018

20

Result of the flat density

50 100 150 200 250 300 1 2 3 4 5 x@kmD Ρ@gêcm3D 50 100 150 200 250 300 1 2 3 4 5 x@kmD Ρ@gêcm3D

  • : Original density profile

・: reconstructed density profile Result with using the 1st order formula Result with using the 2nd order formula ¯ ρ = 2.7 [g/cm3]

Reconstruction of 60 points 100 Energy bin The 2nd order matter effect is important for the reconstruction

slide-21
SLIDE 21

SI2018

21

Result with using the 2nd order formula

50 100 150 200 250 300 2 4 6 8 10 x@kmD Ρ@gêcm3D 50 100 150 200 250 300 1 2 3 4 5 x@kmD Ρ@gêcm3D

  • : Original density profile

・: reconstructed density profile

  • : Original density profile

ρ(x) = ¯ ρ +(ρl − ¯ ρ)exp " − ° x − L

2

¢2 D2

l

# ,

Original density profile

¯ ρ = 2.7 [g/cm3] ρl = 8.0 [g/cm3] ρl = 1.0 [g/cm3] The lump of iron The lump of water

Reconstruction of 60 points 100 Energy bin We could reconstruct the density profile of lump.

slide-22
SLIDE 22

SI2018

22

Result of the exotic density profile

50 100 150 200 250 300 1 2 3 4 5 x@kmD Ρ@gêcm3D

  • : Original density profile

・: reconstructed density profile using the 2nd order formula 4 lump

Reconstruction of 60 points 100 Energy bin We could reconstruct the exotic density profile.

slide-23
SLIDE 23

Summary

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SLIDE 24

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24

Summary

We have investigated the oscillation tomography by the neutrino pair beam. The neutrino pair beam is powerful source to the probe of the Earth’s interior. The reconstruction method with the 2nd order perturbation formula is powerful tool. Toward to the realization of the neutrino tomography the realistic 3 flavor oscillation. the method with the reconstruction of the asymmetric density profile. This talk the more realistic set up.
 (- uncertainty of the real experiment)
 (- realistic target of the neutrino tomography)
 ex.) Earth’s core and mantle, mineral, oil, etc… It has been demonstrated that the profile can be reconstructed well by including the 2nd order correction. We believe that these two ingredients give considerable progress toward the realization of the neutrino tomography.

slide-25
SLIDE 25

Back Up

slide-26
SLIDE 26

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26

If we consider the 2 flavor oscillation, probability degenerate by the Unitarity.

P(νe → νe) + P(νe → νµ) = 1 P(νe → νe) + P(νµ → νe) = 1 ∴ P(νµ → νe) = P(νe → νµ)

So, Oscillation probability with asymmetric density profile coincide with the another one.

50 100 150 200 250 300 x @kmD 3.0 3.5 4.0 4.5 5.0 Ρ @gêcm3 D 50 100 150 200 250 300 x @kmD 3.0 3.5 4.0 4.5 5.0 Ρ @gêcm3 D

Density profile with left side lump Density profile with right side lump

Same Oscillation Probability

Degeneracy of the 2 flavor Neutrino Oscillation

slide-27
SLIDE 27

SI2018

Condition of perturbation

27

2 4 6 8 10 12 14 0.0 0.2 0.4 0.6 0.8 1.0

ρ [g/cm3] Eν [GeV ]

∆m2

SOL = 7.5 × 10−5[eV ]

HF

0 > V F

HF

0 = V F

HF

0 < V F

Perturbative

i d dx ~ A(x) = [HF

0 + V F ] ~

A(x) HF

0 > V F

Evolution equation Condition of perturbation

Neutrino Energy Density

slide-28
SLIDE 28

SI2018

Perturbation formulae

28

P (1)(νe → νe) = GF 2 √ 2mp sin2 2θ cos 2θ Z L dx ρ(x)[sin{∆m2 2E L} − sin{∆m2 2E x} − sin{∆m2 2E (L − x)}]

1st order 2nd order

P (2)(νe → νe; t) = P (2a)(νe → νe; t) + P (2b)(νe → νe; t) P (2a)(νe → νe; t) = [cos8 θ + sin8 θ + 2 cos4 θ sin4 θ cos (Φt)]G1(t)2 + cos4 θ sin4 θ[G2(t)2 + G3(t)2] + 2(cos4 θ + sin4 θ) cos2 θ sin2 θG1(t)G2(t) P (2b)(νe → νe; t) = −2 Z t dt1 Z t1 dt2VCC(t1)VCC(t2) × {+ cos8 θ + sin8 θ + cos2 θ sin2 θ(cos4 θ + sin4 θ)[cos (Φt) + cos (Φt2) + cos (Φ(t2 − t1)) + cos (Φ(t1 − t))] + 2 cos4 θ sin4 θ[cos (Φ(t2 − t)) + cos (Φt1) + cos (Φ(t2 − t1 + t))]} G1(t) = Z t dt1VCC(t1) G2(t) = Z t dt1VCC(t1)[cos (Φt1) + cos (Φ(t − t1))] G3(t) = Z t dt1VCC(t1)[sin (Φt1) + sin (Φ(t − t1))]

slide-29
SLIDE 29

SI2018

Reconstruction with 1st order perturbation

29

The result when the number of devisors is increased. NE = 300 NL = 300

50 100 150 200 250 300 2 4 6 8

ρ [g/cm3] x [km]

  • : Original density profile

・: reconstructed density profile

slide-30
SLIDE 30

SI2018

Reconstruction with 2nd order perturbation

30

We see the dependence of density.

50 100 150 200 250 300 x@kmD 1.5 2.0 2.5 3.0 Ρ@gêcm3 D

50 100 150 200 250 300 x@kmD 1.5 2.5 3.0 Ρ@gêcm3 D

50 100 150 200 250 300 x@kmD 1.5 2.0 3.0 Ρ@gêcm3D 50 100 150 200 250 300 x@kmD 1.0 1.5 2.0 2.5 3.0 Ρ@gêcm3D 50 100 150 200 250 300 x@kmD 1.0 1.5 2.0 3.0 3.5 4.0 4.5 Ρ@gêcm3D 50 100 150 200 250 300 x@kmD 2 4 6 8 Ρ@gêcm3 D

We can’t reconstruct if the density becomes too small in this method.

  • : reconstructed density profile

・: Original density profile

slide-31
SLIDE 31

SI2018

31

50 100 150 200 250 300 1 2 3 4 5 6 7 8 50 100 150 200 250 300 1 2 3 4 5 6 7 8 50 100 150 200 250 300 2 4 6 8 50 100 150 200 250 300 2 4 6 8 50 100 150 200 250 300 2 4 6 8

Reconstruction with 2nd order perturbation

We see the dependence of width of lump.

We can’t reconstruct if the density becomes too narrow in this method.

  • : reconstructed density profile

・: Original density profile