Degeneracy and strategies of LBL

Osamu Yasuda

Tokyo Metropolitan University

NuFACT04 workshop 　July 28, 2004 at Osaka

Univ.

Contents

1. Introduction

2. θ13

3. δ

4. Summary

hep-ph/0405005 hep-ph/0405222

Based on

● Discussions on probabilities only w/o statistical and systematic errors

● Discussions in sect. 2 &3 assume JPARC at @ Osc. Max.νν

scenarios 10 years from now

Notation in this talk

Notations in this talk:

Oscillation Maximum (OM)

4EL|Δm|

Δ231

2π

4EL|Δm|

Δ231

)ννP(P eμ

eμ ννPP

21m

22m

23m

23m

21m

22m

0Δm231

0Δm231

normal hierarchy

inverted

hierarchy

1. Introduction

● intrinsic (δ, θ13)degeneracy

● Δm231⇔-Δm2

31 　 degeneracy

● θ23⇔π/2-θ23 degeneracy

Even if we know and in a long baseline accelerator experiments with approximately monoenergetic neutrino beam, precise determination of θ13, sign(Δm2

31) and δ is difficult because of the 8-fold parameter degeneracy.

eμ ννP eμ ννP

straight lines

hyperbolas

(or ellipses)

(P=const& =const’ on OM )P

(P=const& =const’ off OM )

(P=const, δ=const)

P( =const, δ=const) P

Plots in (sin22θ13, 1/s223) plane

The way curves intersectis easy to see

● θ23⇔π/2-θ23 degeneracy

● intrinsic (δ, θ13)　 degeneracy

● Δm231⇔-Δm2

31 　 degeneracy

0351

(b)0(a) |Δm|

Δm|Δm|

Δm231

221

231

221

0(b)cos2θ0(a)cos2θ 2323

0(b)AL/20(a)AL/2

1/2000kmeFNG2 A

JPARC 　　 is almost enough, since (a) there is no intrinsic (δ,θ13) degeneracy, and (b) sign(Δm2

31)degeneracy is small.

12θsin(1) 232

νν 12θsin(2) 23

2

Ambiguity due toθ23⇔π/2-θ23 degeneracy is significant.

2. Determination of θ13

Assumption: and will be measured at JPARC (@OM, 4MW, HK) .

eμ νν e

νμ

ν

Question: Will that be enough to determine |Ue3|?

Yes! No!

(A)reactor measurement of θ13

(B) LBL measurement of (or)

(C) measurement of

To resolve θ23 ambiguity, possible ways are:

eμ νν μe νν

τe νν

The reference values used here are:

sin22θ23=0.96, sin22θ13=0.05, δ=π/4, Δm231>0

eνeν

(A) reactor measurement of θ13

One can resolve

θ23 ambiguity at

90%CL.

To compete with

accelerator experiments, improvements in the sensitivity is necessary.

eνeν

One possible way to improve sensitivity of reactor measurements (theorist’s personal speculation) O.Y. LENE3@Niigata, March 20, 2004If one puts N near detectors and N far detectors with the same σu, then theoretically sensitivity becomes:

σu : the uncorrelated systematic error ( ～ 0.005 by optimistic estimate)

uonlysys

limit132

nL,fL2.8σ2θsinmin

uonlysys

limit132

nL,fLσ

12.82θsinmin

N

@90%CL

@90%CL

χ2 ⇒ Nχ2

Consider 3rd measurement of (or )

in addition to JPARC .

(B) LBL measurement of (or )

νν

δ+w δ-c

δ-w

δ+c

The value of δ for each point can be deduced (up to δ⇔π-δ) from

AL/2Δ/AL/2sing

,AL/2Δ1/AL/2Δsinff,,sin2θc|/ΔmΔm|y

,sin2θsx

,2fgxy

ygxfPsinδ

1223231

221

1323

2222

correct assumption wrong assumption on mass hierarchy

μe νν

μe νν

eμ νν

eμ νν

↓(exaggerated figure)

we can get a unique line (a hyperbola or an ellipse) in (sin22θ13,

1/s223) plane for

or .

Then from the equation for the probability of (or ) in the 3rd experiment

2

23[cw]13

2

2

23[cw]13

2

s,δ,2sinPP

s,δ,2sinPP

true

true

πθ

or

θ

true

2

23truetrue13

2 )(s,δ,)2(sinPP

where

true θ

eμ νν μe νν

[cw]δ

[cw]δ- π

JPARC

● δ⇔π-δ ambiguity

Δδcos2f

PC2

δdYdX

Δδcos2f

PC2

δ-πdY

dX

Δcoscosδ2f

PC4

δ-πdYdX

δdYdX

Difference in the gradients is large for Δ=0 or π

Assuming for simplicity P>>C

AL/2Δ/AL/2sing

2θsing/ΔmΔmC 122222

31221

Δcoscosδ

● sign(Δm231) ambiguity

(a) L: AL～ L/1900km

→ L>2000km is good to identify sgn(Δm2

31 )

(b) E: → low energy is advantageous

Enhancement of matter effect for π/2<Δ<π

i1322θsin

n1322θsin

tanΔ1

Δ1

2AL1n

i

132

132

2θsin

2θsin

X-intercepts

In my personal opinion nova should run with lower E!

● θ23 ambiguity

Resolution of θ23 ambiguity

(a) has to be small

(b) has to be large

δ is unknown at first, so it is

impossible to design to optimize

this resolution.

It may happen that this ambiguity

can be resolved as a byproduct.

Δ)cos(δxyfg

2

AL/2Δ-1

AL/2-Δsinf

Δ)cos(δ

The situation doesn’t change much for

if Δ π/2.≦e

νμ

ν

JPARC

Off-axis NuMI νfactory

νν

ν

(C) measurement of

Curves intersect with the JPARC line almost orthogonally.

τe νν

● θ23 ambiguity may be

resolved.

● δ⇔π-δ ambiguity may be resolved.

● sign(Δm231) ambiguity

may be resolved .

This channel may be interesting to be combined with JPARC in the future.

3. δ

Assumption: at JPARC (@OM, 4MW, HK)

and will be measured.eμ νν

Question:

Will that be enough to determine arg(Ue3)?

Answer: In general no.

Resolution of sign(Δm231)

ambiguity is important.

eν

μν

If δ0 =0, then at JPARC

δ0

δ1δ0 : by correct assumption

=true value

δ1 : by wrong assumption

on sign(Δm231)

Difference between δ0 & δ1 turns out to be large.

131 sin2θ2.2sinδ

= -0.5 (if sin22θ13=0.05)

Identification of sign(Δm231) is important.

(1) Ambiguity due to sign(Δm231)

3σ sensitivity to δ

Kobayashi

@Nufact02

Assuming Δm231>0

↑modified from

Minakata-Sugiyama (PLB580,216)

Assuming Δm231<0

↑modified from

Minakata-Sugiyama

(PLB580,216)

(2) Ambiguity of δ due to θ23 :

The Ambiguity due to θ23 is not serious.

132

132

23

232

2θsin

15001

2θsin

1t

cot2θ

2001

sinδ

)(ν1.02θsin0.9 atm232

outside of red or blue lines

JPARC@OM

is almost enough

In addition to JPARC

@OM,

is necessary to resolve θ23 ambiguity

inside of red or blue lines

In addition to JPARC

@OM,

LBL w/ L>～ 1000km

is necessary to resolve sign(Δm2

23 )

ambiguity

In addition to JPARC

@OM,

(A)

is necessary to resolve θ23 ambiguity

(B) LBL w/ L>～ 1000km

is necessary to resolve sign(Δm2

23 ) ambiguity

12θsin 232 12θsin 23

2

τe νν eνeν &

or

ν&ν

ν&νν&ν

eμ νν eν

μν

)2θsin (sinδ, 132

)2θsin (sinδ, 132

eνeν or τe νν

4.Summary

13θδ

Phase II Phase I

● for determination of θ13

to resolve θ23 ambiguity

if .

● for determination of δ

to resolve sign(Δm231)

ambiguity.

It is important

12θsin 232

If nova runs with lower E, then it will become complementary to JPARC, and only in this case it will play an important role.