Measurement of spectral function in the decay τ − → π π ν − 0 τ

Measurement of spectral function in the decay τ ⁻ → π π ν ^{− 0} _τ

Yukiko Hirano

(Nara Women’s Uni. , High Energy Physics Lab.)

For the Belle collaboration

1. Motivation

~ Muon Anomalous Magnetic Moment ~

2. Event selection

3. mass spectrum (unfolding)

4. Evaluation of

5. result

2 g

−

ππ

a

Outline

Motivation Motivation

~ ~ Muon Muon Anomalous Magnetic Moment ( ) ~ Anomalous Magnetic Moment ( ) ~ ^{g µ − 2}

Muon Anomalous Magnetic Moment ： 2

a g 2

µ = µ −

¾ The prediction of Standard Model

γ

µ µ

Z ,

SM QED EW had had LBL

a _µ = a _µ + a _µ + a _µ + a _µ ^γ

γ

µ µ

hadron

γ

µ µ

a _µ had

hadron

γ

µ largest error from

Hadron vacuum polarization

and and τ → ππ ν ⁰ _τ decay decay

a _µ had

the contribution of hadrom Vacuum polarization ^{（ ）}

obtained from Experimental Data.

γ

µ µ

ν

τ

W

π

π

ρ e

e

π

π

a

difficult to obtain from first principle ！

hadron e e

→

Cross section of

τ Semi-Leptonic decay

Hadron system

Hadron system

decay is

useful to determine the term ,

0 τ

τ → ππ ν

a _µ had

73% from system 2 π

and decay and decay

a _µ ^ππ τ → ππ ν ⁰ _τ

Hadron Vacuum polarization term from 2 system ( ) a

0 2

2

4

(0) ( )

em ( )

M K s v

a ds

s s

π

ππ ππ µ

α π

= ∫ ∞ _{s = M}

Spectral function

0 0

0 0

2 1

2 2 2

( ) 1 1 2

6

1

ud EW e

M s s dN

v s

M

B

N

V S M B d s

ππ ππ

ππ τ

τ τ

ππ

π

    

=   −  +  

  

 

ππ 0 mass square spectrum

π

K(s) is known function.

is measured in this experiment.

N

ds

π ⁰

1 dN

ππ

π

Present status ;

Present status ; Muon Muon Anomalous Magnetic Moment ( ) Anomalous Magnetic Moment ( ) ^g

− ²

¾ Exp. ・・・ measured by BNL (g-2) experiment. ^（ 2002.9 ）

¾ Theoretical prediction ・・・ new data ^(CMD-2) and data ^(ALEPH)

（ 2003.1 ）

exp 10

(11659203 8) 10 a

= ± ×

+ −

e e ^{+ −} τ

( )

exp 10

11659193.6 5.9( ) 3.5( ) (9.4 10.5) 10

0.4 ( ) 10

SM

SM

had LBL QED EW

a

a a

µ

µ µ ⁻

= ± ± ± +

− = ×

×

± ⇒

• τ base

0.9 σ

agree within

( )

exp 1

10 0

11659169.3 7.0( ) 3.5( ) 0.4( ) 10

(33.7 11.2) 1 0

SM

SM

had LBL QED EW

a

a a

µ

µ µ

−

−

= ± ± ± +

− ×

×

± ⇒

=

• e e base

3.0 σ

difference by

・Hadron Vacuum Polarization. term is different between and base predictions.

・ Cross check is important !

e e

τ

event selection event selection e e ^{+ −} → τ τ ^{+ −}

•Number of charged tracks ： 2 or 4

•All charge ( ) = 0

•event vertex position :

•Separate the event into 2 hemisphere by the event axis.

•Event axis direction :

•Back Ground rejection (next slide)

•Physics trigger

∆ Q

2.5 cm , 0.5 cm

z r

V < V <

35

< θ

< 145

VJGQVJGTJGOKURJGTG

QPGJGOKURJGTG

ǫ

ǫ

e e

→ τ τ

event selection criteria

(corresponding to production.) 4.0 10 ×

^{τ τ}

Event selection Event selection

Data : accumulated from 2000.10 to 2000.12 at Belle. 4.43 fb ^{− 1}

event selection (Back ground rejection) event selection (Back ground rejection)

e e ^{+ −} → τ τ ^{+ −}

( )

2

track

e e

p p p

MM =

+

− ∑ − ∑ p

‹

Missing mass and Missing angle cut ( plot)

Missing angle

Missing Mass(MM)

pe+

track

p

θmiss

track

p

p_γ p_γ

pe−

•Bhabha , and two photon rejection e e

→ µ µ γ

( )

•Hadron( ) rejection e e

→ qq

‹

Reject high-multiplicity event ( X

≡ ⁽ n

+ n

⁾

× ⁽ n

+ n

⁾

≤ ²⁵ )

V.S. mi s

MM θ

θ^∗miss

MC

(degree)

Missing Mass (Gev)

0 2 4 6 8 10

0 25 50 75 100 125 150 175 200

Missing Mass ( G eV)

Missing Angle

MC ( τ τ

)

Two photon BG

Bhabha BG

0 2 4 6 8 10

0 25 50 75 100 125 150 175 200 θ^∗miss

Data

(degree)

Missing Mass (Gev)

Data

Missing Mass ( G eV)

Missing Angle

τ τ

+ −

→

Event selection Event selection

0 τ

τ → ππ ν

π

selection criteria

0 τ

τ → π π ν

•one charged track in hemisphere.

•one in the hemisphere.

gamma condition : gamma-like shower shape :

• veto the additional gamma

(with high momentum (more than 200 MeV/c))

π 0

0.08

E

>

＊ We do analysis each hemispheres. τ → ππ ν

signal signal

π 0

(m_γγ - m_π⁰)/σ_γγ

number of entries / 0.19

DATA MC(signal) MC(non-τ B.G.)

Normalization:

# of entries

0 5000 10000 15000 20000 25000

-8 -6 -4 -2 0 2 4 6 8

( m m

) S

σ

≡ −

0

: (134.98

) m

π

:

: m

m

γγ

γγ γγ

γγ σ

invariant mass distribution resolution of

6 S

5

− < <

Signal region

9 7

7 9

S S

γγ γγ

− < < −

< <

right：

left：

Side-band region

320,000 τ → ππ ν

_τ events.

Side-band region are used for

estimation of BG in signal region.

( ) 11

4

left right

si total

sig an l sig de side

N = N −    N + N ×   

π

10 10² 10³ 10⁴

0.5 1 1.5 2 2.5 3

(M_π±π⁰)² (GeV)²

number of entries

DATA MC(signal) τ-feed across (Kπ⁰) τ-feed across (others) non-tau B.G.

Normalization:

# of entries

mass spectraum mass spectraum ππ 0

B.G. fraction

Total

( ) ( ) Feed across B.G.

Two photon non- B.G.( )

B.G. fraction source

Clear peak of .

is dominant .

(770) ρ

τ → ρν

peak.

is also included . (1450)

ρ

τ → ρν ^′

ee → qq 2.3 ^± 0.06 % 0.14 ^± 0.01 %

6.1 ^± 0.10 % 1.74 ^± 0.09 % 5.43 ^± 0.08 % K π

2

h ≥ π ν

τ

Unfolding Unfolding

Acceptance and bin-by-bin migration effects are corrected via Singular-Value-Decomposition method.

0 0.5 1 1.5 2 2.5 3

0 0.5 1 1.5 2 2.5 3

(M_ππ⁰)² (observe) (GeV)²

(M ππ0)2 (generate) (GeV)2

0 0

2 2

V.S.

gen obs

M

M

0 0.05 0.1 0.15 0.2 0.25 0.3

0.5 1 1.5 2 2.5 3

(M_ππ⁰)²

acceptance

(GeV)²

acceptance

Acceptance include both the

tau-pair and pipi0 selection. Mass square resolution : 0.03 GeV

Unfolded mass spectrum Unfolded mass spectrum

1 10 10

10

10

10

0 0.5 1 1.5 2 2.5

(M_ππ⁰)² (GeV)²

mass spectrum (unfolded) Belle

number of entries /0.05(GeV)

Data

G & S Fit

(ρ₍₇₇₀₎ + ρ₍₁₄₅₀₎)

Red line :

Breit Wigner fitting function ( and are included. ) ρ ρ ^′

ρ

ρ ^′

Breit

Breit Wigner Wigner fitting form fitting form

( )

2

2 2

2 3

'

( )

( )

1 1 2

1 ( ) 12

( ) 1

1

i i

s s

M M

F s

F s BW e

dN v s d

B s

W v

e

s A _{τ τ}

π π

π φ

φ ρ ρ

β

β

β

   

=  −   +  ⋅

   

=

= + ⋅

+

0

0

2

2

;

( ) ; M s

s dN ds

v

M

ππ

=

mass distribution

Spectral function

, , , , , ,

A M _ρ Γ _ρ M _{ρ ′} Γ _{ρ ′} β φ

free parameter

2

(

) ( ) ( )

GS

d

BW s f s i s

M M

M s

ρ ρ ρ

ρ ρ

ρ

+ ⋅ ⋅

= − + − ⋅

Γ

Γ

Gounaris and Sakurai (G&S) Model

GS model is known that it can fit wilder

mass region that the commonly used BW.

fit result and compare with previous Experiments fit result and compare with previous Experiments

Mρρ

Γ

βM

ρρ′′

Γ2

..

dof

χ

0.5 1.1

775.3 150.5

±

± 7 26

1365 356

±

±

0.108

− ^{± 0.007}

26.8/24

(fixed)

180.0

≡

0.9 1.6

776.4 150.5

±

± 16

(fixed)

1400 310

±

≡

0.077

− ^{± 0.008}

54/65

(fixed)

180.0

≡

ALEPH CLEO

Belle

Fit Parameter

M _ρ Γ ρ

(MeV) (MeV)

M _ρ

ρ

′

Γ ′

(MeV) (MeV)

0.4 0.7

773.9 152.4

±

± 21 40

1398 450

±

±

35.6/42

0.085 ^{± 0.010}

181.0 ^± 6.2

β

φ (degree)

2 d o f. .

χ

• parameters : good agreement with previous Exp.

• parameters : Belle results are most precise.

ρ

ρ ^′

Evaluation of Evaluation of ^{a ππ µ}

0 0

0 0

2 1

2 2 2

1 1 2

6

( ) 1

ud EW e

s s B dN

M

M M B

V S s

s N

v d

ππ ππ

π

τ τ

π τ π

π

π

    

=   −  +  

  

 

0 2

2

4

(0) ( )

em ( )

M K s v

ds s

a s

π

ππ ππ µ

α π

= ∫ ∞

(tau mass)

(Electro-Weak correction factor) (element of CKM matrix)

(Br. of )

e

E ud

e e

W

M S V

B

τ

→

Systematic error (

Systematic error ( Ⅰ Ⅰ ) )

External systematics ~Normalization factors~

total

value source

1.0199 0.0006 0.9734 0.0008

±

±

( )

( )

%

%

17.84 0.06 25.41 0.11

±

±

a ^ππ _µ

∆

(10⁻10)

S

V

B

B

0.32 0.42

±

±

1.82 2.30

±

±

± 2.98

Largest error from B

Systematic error (

Systematic error ( Ⅱ Ⅱ ) )

Internal systematics 1.Non- BG.

・ estimated by B.G. MC.

・ control data sample are used for the calibration.

2. Feed-across

・ of measured Br.

3. Energy scale

・ uncertainty estimated from mass peak.

4. Selection

・ estimated from the uncertainty of side-band.

5. minimum energy

1 σ

π

π

γ τ

efficiency

B.G. fraction B.G. estimation

total

Use side-band

non- BG

80MeV – 200MeV

Minimum energy Energy scale

Feed across BG

2 photon hadron

non- BG

comment

source

K

π

2

h ≥ π ν

π

γ τ

a ^ππ _µ

∆

(10⁻10)

2.3

% 0.14

%

1.74

% 5.43

%

0.05 0.4

±

± 0.3 1.2

±

±

± 0.1

1.8 0.8

±

±

± 2.36

π0

/ 0.2% ( 0)

E E π

∆ = ±

result result

Result of is … a

( ^{541.3 2.0 ( stat .) 2.36 ( sys .) 2.98 ( sys ext . .)} ) ^{10 10}

a _µ ^ππ = ± ± ^± × ⁻

2 2 2

Integrated mass sqr. region : 4 m

to (1.8 )

preli mina ry

cf. ALEPH

( 533.86 3.57( ) 2.36( ) ) 10

a

= ± stat ± sys ×

( base) τ

2 2 2

Integrated mass sqr. region : 4 m

to (1.8 )

( )

2 2

7.4 1.6 4.7

ALEPH

Belle ALEP Be e

H

a

a

σ σ

− = =

− excluding common error

Consistent within error

Backup slide

Analysis Analysis

Flow of this analysis Flow of this analysis

Data from Belle

Unfolding (reject detector contribution)

Spectral function measurement

Calculation of the term hadron vacuum polarization event selection

0 τ

τ → ππ ν

event selection e e

→ τ τ

mass spectrum of Data

ππ

decide mass spectrum ππ

2.Event selection 2.Event selection

Exp. Data

Monte Carlo simulation

Belle detector

Data taking system Raw Data

Simulator of event production

Detector simulation

Data reconstruction event selection

τ τ

event selection

0 τ

τ ππ ν →

event

0 τ

τ → ππ ν

event τ τ

τ τ

Used Data

data accumulated from 2000.10 to 2000.12 at Belle

data :

4.44 fb ⁻ 1

4.0 10 ×

event selection event selection e e ^{+ −} → τ τ ^{+ −}

• There are few charged tracks in the event.

e

τ

e

τ

e

τ

e

τ

decay to 1 charged track ： about 85%

decay to 3 charged tracks ： about 15%

τ τ

72% 13%

• There are missing of momentum and mass because of neutrino ( ) . ν

decay

τ

ντ

τ

W⁻

π

π0

ρ

missing

event selection (Back ground rejection) event selection (Back ground rejection)

e e ^{+ −} → τ τ ^{+ −}

•Bhabha , and two photon rejection e e

→ µ µ γ

( )

‹

clean Bhabha and mumu event rejection : ∑ P ≤ 9.0

, ∑ E ≤ 9.0

( )

2

track

e e

p p p

MM =

+

− ∑ − ∑ p

0 2 4 6 8 10

0 25 50 75 100 125 150 175 200 θ^∗miss

Data

(degree)

Missing Mass (Gev)

θ^∗miss

MC

(degree)

Missing Mass (Gev)

0 2 4 6 8 10

0 25 50 75 100 125 150 175 200

‹

Missing mass and Missing angle cut

Missing angle

Missing Mass(MM) pe+

track

p

θ

track

p

pγ

p_γ

pe−

•Hadron( ) rejection e e

→ qq

( ) ( ) 25

part track one track other

X ≡ n + n _γ × n + n _γ ≤

‹

Low-multiplicity event :

Then ,we obtained about 1,300,000 event of . e e

→ τ τ

Missing mass VS. Missing angle Missing mass VS. Missing angle

0 2 4 6 8 10

0 50 100 150 200 0 2 4 6 8 10

0 50 100 150 200

0 2 4 6 8 10

0 50 100 150 200 0 2 4 6 8 10

0 50 100 150 200

θ^∗miss (degree) (1)Data

Missing Mass (Gev)

θ^∗miss (degree) (2)MC(τsample)

Missing Mass (Gev)

θ^∗miss (degree) (3)MC(Bhabha,mumu)

Missing Mass (Gev)

θ^∗miss (degree) (4)MC(2photon)

Missing Mass (Gev)

Data MC(tau)

MC(Bhabha) MC(2photon)

Time dependence

10 cm

BELLE

0.246 0.248 0.25 0.252 0.254 0.256 0.258 0.26 0.262 0.264

8 9 10 11 12 13 14 15 16 17 18 19 20

∆R/R_ave= ± 0.5 %

average=0.2568

2000 2001 2002

Exp. Number R = N

/ N

data at Belle detector

τ

Time dependence of event ππ

is stable within 0.5%.

N

N

ππ ττ

Time dependence

0.246 0.248 0.25 0.252 0.254 0.256 0.258 0.26 0.262 0.264

8 9 10 11 12 13 14 15 16 17 18 19 20

∆R/R_ave= ± 0.5 %

average=0.2568

2000 2001 2002

Exp. Number R = N

/ N

Time dependence of event ππ

is stable within 0.5%.

N

N

ππ ττ

Momentum of and

Momentum of and ^{π 0} π ^±

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

P_h^±(CM) / E_Beam (GeV)

number of entries / 0.01

DATA MC(signal) non-tau B.G.

Normalization:

number of entries

π ^±

0 2000 4000 6000 8000 10000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P_π⁰ (CM) / E_Beam (GeV)

number of entries / 0.01

DATA MC(signal) non-tau B.G.

Normalization:

number of entries

π 0

Good agreement between Data and MC .

Fitting result of Breit Wigner model Fitting result of Breit Wigner model

1 10 10² 10³ 10⁴ 10⁵

0 0.5 1 1.5 2 2.5

(M_ππ⁰)² (GeV)²

mass spectrum (unfolded) Belle

number of entries /0.05(GeV)2

Data

G&S Fit(ρ₍₇₇₀₎)

G&S Fit(ρ₍₇₇₀₎ + ρ₍₁₄₅₀₎)

1 10 10² 10³ 10⁴ 10⁵

0 0.5 1 1.5 2 2.5

(M_ππ⁰)² (GeV)²

mass spectrum (unfolded) Belle

number of entries /0.05(GeV)2

Data

K&S Fit⁽ρ₍₇₇₀₎)

K&S Fit(ρ₍₇₇₀₎ + ρ₍₁₄₅₀₎)

G&S model

K&S model

Fitting result Fitting result

G&S G&S

K&S K&S

0.39 0.68 18.9 41.5 0.020

9.05 38.8/ 42

773.07 150.76 1421.7 542.28 0.14 188.4

0.93

±

±

±

±

±

±

=

'

( is real)

ρ ρ β + ρ ρ β φ +

( , used) ρ ρ β +

( is real) ρ ρ β φ +

( , used)

'

'

2 do f . .

M

M

ρ ρ

ρ ρ

β φ

χ

Γ Γ

0.36 0.66 6.6

29.6 0.005

40.9/ 43

773.25 150.58 1397.8 514.77

0.120

0.93

±

±

±

±

±

=

−

0.35 0.69 6.3 28.5

0.004

35.6/ 43

773.94 152.37 1395.0

445.9 0.084

0.83

±

±

±

±

±

=

−

0.37 0.71

20.9 39.9 0.010 6.17 35.6/ 42

773.9 152.4 1398.2

450.4 0.085 180.0

0.85

±

±

±

±

±

±

=

Unfolding of invariant mass Unfolding of invariant mass

The observed distribution includes contribution of detector acceptance ,and smeared .

We can obtain true distribution by using Unfolding.

About Unfolding

Detector

unfolding

initial distribution obserbed distribution

Unfolding is carried out by

Singular Value Decomposition (SVD) method.

method a la ALEPH , A.Hocker, V.Karvelishvili,N.I.M. 372(1996)469

Spectral function Spectral function

10

10

10

1

0 0.5 1 1.5 2 2.5

(M_ππ⁰)² (GeV)²

Belle

v

Belle CLEO

Systematic detail 1 Systematic detail 1

(1) BG estimation (two photon )

Data MC (two photon)

Two photon B.G. estimated by Data using control sample.

(2) BG estimation (hadron)

Hadron BG contribution also estimated by Data using control sample.

( ) ( ) 25 for selection

part track one track other

X ≡ n + n

× n + n

≤ τ

for hadron selecti

( ) ( ) 2 5 o n

part track one track other

X ≡ n + n

× n + n

>

Systematic detail2 Systematic detail2

(3) Energy scale

2% uncertainty of mass spectrum is assumed. π

(4) Gamma energy threshold

(5) side-band subtraction π

Uncertainty of as gamma-threshold function. B h ( π

)

24.6 24.8 25 25.2 25.4 25.6 25.8 26 26.2 26.4

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

∆B/B= 1.0 %

gamma theshold energy B(hπ0 )

(%)

(GeV)

Use control sample of side-band . π

Hadron Vacuum polarization and

Hadron Vacuum polarization and e e ^{+ −} Data Data

e e

→ hadron

The term of hadron vacuum polarization

e

γ

e

q

q

γ ^e

+

e

q

q γ

q

γ q γ

Data and

Data and Data Data

e e ^{+ −} τ

ν

τ

W

π

π

u

ρ d

τ semi-Leptonic dacayc decay

e e

→ hadron

e

γ

e

q

q

Iso-spin 、 Cnserve of Vectro Current

We can treat data as same condition as data. τ e e