Neutrino Physics - Lecture 3 Steve Elliott LANL Staff Member UNM Adjunct Professor 505-665-0068, [email protected]

Neutrino Physics - Lecture 3

Steve Elliott

LANL Staff Member

UNM Adjunct Professor

505-665-0068, [email protected]

Spring 2007 Steve Elliott, UNM Seminar Series

2

Lecture 3 Outline

• Matter enhanced oscillations• 3 flavor oscillations• Neutrinos from the Sun

The neutrinosPast experimentsWhat we know and what we want to learn


3

What about MSW?

The Sun is mostly electrons (not muons).

e can forward scatter from electrons via the charged or neutral current.

can only forward scatter via the neutral current.

The e picks up an effective mass term, which acts on the weak eigenstates.


4

The MSW H term.

€

HMSW =2GFNe 0

0 0

⎛

⎝ ⎜

⎞

⎠ ⎟

€

i∂∂t

ν α = UHU−1ν α + HMSWν α

This extra term results in an oscillation probability that can have a resonance. Thus even a small mixing angle, , can have a large oscillation probability.


5

Similar algebra as before

€

∂α∂t

= iA+ x B

B A+ D

⎛

⎝ ⎜

⎞

⎠ ⎟ν α

x = 2GFNe


6

Constant Density Solutions

€

e ν e (t)2

= sin2 2θm sin2 πRLm

⎛

⎝ ⎜

⎞

⎠ ⎟

Note similar form to vacuumOscillations.

€

sin2 2θm =sin2 2θ

sin2 2θ + (LL0

− cos2θ )2

Lm =L

sin2 2θ + (LL0

− cos2θ )2

L =4πp

δm2 ; L0 =2π

2GFNe

Note that sin22m can be 1 even when sin22 is small. That is when:L/L0 = cos2


7

Variable Density

• Integrate over the changing density (such as in a star or supernova or Earth).


8

Three Formulism

€

α = Uαii

∑ ν i

U = U(θ 23 )U(δ)U(θ13 )U(θ12 )

U(θ 23 ) =

1 0 0

0 c23 s23

0 −s23 c23

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ U(θ13 ) =

c13 0 s13

0 1 0

−s13 0 c13

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

U(θ12 ) =

c12 s12 0

−s12 c12 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ U(δ) =

1 0 0

0 1 0

0 0 e iδ

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟


9

Transition Probability

€

α =Uα 1 ν 1 + Uα 2 ν 2 + Uα 3 ν 3

at time t :

ν α (t) = Uα 1e−iE1t ν 1 + Uα 2e−iE2t ν 2 + Uα 3e−iE3t ν 3

ν α (t) = Uαm∑ e−iEmt ν m

Pαβ = ν β ν α2


10

Transition Probability

€

Pαβ = Uβm* Uαme−iEmt

m

∑2

= ( Uβm* Uαme−iEmt

m

∑ )( UβkUαk* e+iEk t

k

∑ )

= δαβ − 4 UαkUβkUαmUβm sinm>k

∑ 2 πRLkm

⎛

⎝ ⎜

⎞

⎠ ⎟

Lkm =πEν

1.27δmkm2

Real U’s


11

Complex U’s

If U is complex, then we have the possibility

€

Pαβ ≠ Pα β

The measurment of

ΔPαβ = Pαβ − Pα β measures cp violation

Pαβ − Pβα measures t violation


12

Oscillation Experiments

Appearance: look for when none are expected

Disappearance: look for decrease in flux of α

€

α →

€

α → α


13

SolarReactorAtmospheric

Maki, Nakagawa, Sakata, Pontecorvo


14

PDB parameterization

€

U =

Ue1 Ue2 Ue3

Uμ1 Uμ2 Uμ 3

Uτ 1 Uτ 2 Uτ 3

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

c12c13 s12c13 s13

−s12c23 − s23c12s13 c23c12 − s23s12s13 s23c13

s23s12 − s13c23c12 −s23c12 − s12s13c23 c23c13

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟


15

Phases in the mixing matrix

For nxn unitary matrix (U): 2n2 parameters in a complex matrix -n2 unitarity constraints -(2n-1) unphysical phases: that can be absorbed into the fields, and =(n-1)2 parameters (1/2)(n-1)n of these are rotation angles

€

αUiα ν α

Note however, that for Majorana fields, the phases of and are related. Hence there are only n unphysical phases.


16

CP violation

€

U =

c12c13 s12c13 s13

−s12c23 − s23c12s13 c23c12 − s23s12s13eiδ s23c13eiδ

s23s12 − s13c23c12 −s23c12 − s12s13c23eiδ c23c13eiδ

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

€

ΔP ≡ P ν μ → ν e( ) − P ν μ → ν e( )

4JCPν sin D12 + sin D23 + sin D31( )

€

Dij = δmij2 R

2E


17

The Jarlskog Invariant

€

JCPν = s12s13s23c12c23c13

2 sinδ

Note the product of the sin of all the angles. If any angle is 0, CP violation is not observable.

Note that I have seen different values of the leading constant. (taken to be 1 here)


18

CP violation - Vacuum

• 1st term associated with atmos m2

• 2nd term associated with solar m2

• The interference term changes signs for anti-neutrinos, because U is replaced by U*


19

CP violation

he

p-p

h/0

30

62

21


20

There are only 2 independent m2 for 3

€

m122 +δm23

2 +δm312

= (m12 − m2

2 )+ (m22 − m3

2 )+ (m32 − m1

2 )

= 0

This will be important when we discuss LSND.


21

Solar Neutrinos and Oscillations

• Solar neutrinos– Few MeV, L~1011 m– Electron neutrinos– Most are disappearance expts. (Except

SNO NC and SK’s slight NC sensitivity)


22

Solar Model Assumptions

• Hydrostatic Equilibrium– radiative & particle pressures balance

gravity• Energy Transport

– photon diffusion in deep interior• Energy Generation

– nuclear fusion• Isotopic/Elemental Abundances

– changes occur only by nuclear reactions– initially agrees with current surface

observations


23

The pp fusion reactions

p + e- + p 2H + e

2H + p 3He +

3He + p 4He + e+ +e

3He + 4He 7Be +

7Be + e- 7Li + +e

7Li + p α + α

3He + 3He 4He + 2p

99.75% 0.25%

85% ~15%

0.02%15.07%

~10-5%

7Be + p 8B +

8B 8Be* + e+ + e

p + p 2H + e+ + e


24

The energy spectrum


25

MSW at high energies, vacuum at low.

€

=2 2GFne Eν

δm2

Ratio of oscillation length in matter to vacuum


26

Matter and Vacuum

matter

vacuum

About 1 MeVFor the Sun

If < cos212 ~0.4then the oscillation probability is given by the vacuum averaged result.

he

p-p

h/0

30

51

59


27

The Cl Experiment

Ray Davis and his tank of cleaning fluid


28

Cl Operations

• 615 tons of C2Cl4 , 814-keV threshold• Homestake Gold Mine• Bubble He gas through to extract Ar• Ar trapped in cold trap.• PC filled with Ar gas (7% methane)• 37Cl is 24% abundant.

€

37Cl ν e ,e( )37 Ar


29

The Homestake Mine


30

The Cl Apparatus

Documents

Neutrino Physics - Lecture 3 Steve Elliott LANL Staff Member UNM Adjunct Professor 505-665-0068, [email protected]