CVC test in e+e−→KK cross section and data on τ−→K−K0ντ decay

Konstantin BeloborodovBudker Institute of Nuclear Physics

Novosibirsk, Russia

International Workshop on e+e- collisions from Phi to Psi

Laboratori Nazionali di Frascati, Italy, 7 - 10 April 2008

—

Outline

• Introduction• Experimental data• VDM, relations, parameters• Fit results• CVC test• Conclusions

What is a goal of this work?1. Do simultaneous fit of e+e–→K+K– and e+e–→KSKL cross sections in wide

energy region (2E0=1.03÷2.2 GeV)2. Test and improve cross section description, based on Vector Dominance

Model.3. Measure parameters of vector mesons4. As a result of the fit, extract isovector and isoscalar parts of the amplitude:

5. Compare the isovector part of K-meson form factor and spectral function, obtained from τ decay

Introduction

2

01

22

**** IIKKKKKKKKAAAAAA

2

01

22

00*00*00*00* IIKKKKKKKKAAAAAA

Experimental data

Channel Experiment Energy, GeV Points Luminosity, pb–1 Reference

e+e–→K+K–

SND (scan1) 1.010÷1.060 14 3.754 Phys.Rev.D 63, 072002 (2001)

SND (scan2) 1.010÷1.060 14 3.988 Phys.Rev.D 63, 072002 (2001)

SND 1.040÷1.380 34 6.7 Phys. Rev. D 76, 072012 (2007)

DM2* 1.350÷2.085 21 1.582 Z.Phys.C39:13,1988

e+e–→KSKL

SND (scan1) 1.004÷1.060 15 3964 Phys.Rev.D 63, 072002 (2001)

SND (scan2) 1.004÷1.060 15 4169 Phys.Rev.D 63, 072002 (2001)

SND 1.040÷1.380 22 8.857 Zh.Eksp.Teor.Fiz.103:831-839,2006

DM1 1.441÷2.140 10 1.429 Phys.Lett.B99:261,1981

* DM2 data was corrected by a factor of 1/0.7~1.4:1. Correction parameter for DM2 cross section data was free and was obtained about 1.42±0.162. Similar situation in the π+π–π0 channel: In the paper hep-ex/0201040 v2

Theoretical framework: Vector Dominance Model

e+

e-γ* V

K+

K-

gγV gVKK

e+

e-γ* V

KS

KL

gγV gVKK

)()(

)(3

8)(

2

32/5

2

sZesD

ggsq

ss Vi

V V

KVKVKKKK

)()( 2 ssisMsD VVV

n2S+1lJ I=1 I=0

13S1 ρ(770) φ(1020) ω(783)

13D1 ρ(1700) – ω(1650)

23S1 ρ(1450) φ(1680) ω(1420)

n2S+1lJ ρ(+/0) ω φ

13S1 +/– + +

13D1 +/– + +

23S1 –/+ – –

][ KVKV ggSign

f

VfVfV RsCssVf

);()()( 2

Parameters

Decays 2π 3π ωπ ωππ K+K- KSKL KK* πγ ηγ e+e–

ρ(770) + + + + +

ω(782) + + + + + +

φ(1020) + + + + + +

ρ(1450) + + + + +

ω(1650) + + + + +

φ(1680) + + + +

ρ(1700) + + + + +

ω(1650) + + + + +

+,+ not used in total width+ fixed from PDG+ free parameters+ gρωπ=gωρπ=16.8 GeV–1

V: MV and ΓV – fixed from PDGφ(1020): Mφ and Γφ – free parametersV‘, V“: MV and ΓV – free parameters within errors from PDG

Relations

2:1:9:: eeeeee 1:9: eeee

2:1:9)07.012.2(:)03.00.1(:)12.07.11(:: eeeeee

Leptonic widths

009.0975.0)()(

3

3

00

0

00

K

K

KK

KK

q

q

KKBrKKBr

g

g

3

3002

0

)(975.0)(K

K

q

qKKBrKKBr

),,,,(,00 VggKVKKVK

3

300

0

)()(K

K

q

qKKVBrKKVBr

SU(2)

00KVKKVK

– phases for charge and neutral channels are the same

KKKKKK ggg 2

1 )(21

)()( sss KKKKKK

KKKKKK ggg 2

1

KKKKKK ggg 2

1

)(21

)()( sss KKKKKK

)(21

)()( sss KKKKKK

SU(3)

4:1:9)(:)(:)( sss KKeeKKeeKKee

1:9)(:)( ss KKeeKKee

LW+SU(3)

Problems in the approximation of cross sections

• Parameters of vector meson excitations are known very approximately• Energy dependence of total widths is not well known• SU(3) relations are not precise (violation ~20%)• Quark model ratios between leptonic widths are not exact• Mixing between vector mesons exists• Additional amplitude exists due to rescattering in the final states• Experimental data above 1.4 GeV have low precision• Technical problem: there are many solutions (local minima), up to 2N-1 in case of N resonances.

Data approximations: variants

Variant SU(2) LW+SU(3) θV R=RVf add-ons1 + + {0,π} ≠0

2 + + free ≠0

3 + + free =0

4 + – free ≠0

5 + – free =0

6 + – free ≠0 without ρ", ω"

7 + + free ≠0 without ω, ω', ω"

8 + – free ≠0 MV±3σMv, ΓV±3σΓv

9 + – free ≠0 ΓV=const(s) for excitations

1. Find any possible solutions2. In case of constant widths of the resonances find all 2N-1 solutions 3. Use these 2N-1 solutions as starting positions for minimization procedure4. Choose a solution with smallest χ2

e-print (2007) 0710.5627

Ambiguity of resonances parameters determination: solution of the problem

Data approximation: results

Variant θρφ,, º θρ'φ,, º θω'φ,, º θφ'φ,, º θρ"φ,, º θω"φ,, º χ2/ndf

1 69±3 0 0 0 0 0 210/137

2 -54±6 -18±6 0±14 -6±27 105±11 -5±52 135/132

3 -33±4 -6±8 -50±16 26±16 101±9 36±27 143/133

4 -54±1 -18±1 -40±3 -6±3 106±2 -4±7 135/132

5 -32±2 -7±1 -52±4 26±2 100±2 36±4 143/133

6 72±2 -21±3 24±2 51±3 — — 153/136

7 -60±1 29±1 — -16±4 105±1 — 202/134

8 -47±1 9±1 -41±2 152±1 57±1 157±2 116/132

9 -54±1 -30±2 -49±3 -20±3 99±2 3±6 145/132

Variant R, GeV–1ΓVeeΓVKK/ ΓVeeΓVKK(SU(3))

ρ' ω‘ ρ"σφ‘KK, nb σω“KK, nb Γφ'eeΓφ'KK/Γφ'

2,·10–6 Γω"eeΓω"KK/Γω"2,·10–7

1 1.5±0.2 1 1 1 0.6±0.1 →0 0.11±0.02 0

2 2.0±0.3 1 1 1 2.0±0.4 0.2±0.1 0.37±0.08 0.4±0.2

3 0 1 1 1 2.0±0.6 0.4±0.2 0.37±0.11 0.7±0.4

4 2.0±0.1 0.99±0.03 1.07±0.09 0.97±0.07 2.0±0.2 0.20±0.04 0.37±0.03 0.4±0.1

5 0 1.03±0.04 1.07±0.09 0.93±0.07 2.0±0.2 0.4±0.1 0.37±0.03 0.7±0.1

6 0.8±0.2 0.49±0.02 1.23±0.09 1 1.3±0.1 0 0.25±0.02 0

7 1.8±0.1 1 1 1 0.8±0.1 0 0.15±0.02 0

8 0.8±0.1 1.04±0.02 1.08±0.05 0.92±0.04 6.1±0.3 1.6±0.2 1.15±0.05 2.9±0.3

9 1.9±0.1 1.02±0.03 1.06±0.09 0.95±0.07 2.1±0.2 0.18±0.04 0.39±0.03 0.4±0.1

Data approximation: results

Variant 1

Variant 2Variant 3Variant 4Variant 5Variant 6

Variant 7Variant 8Variant 9

Variant 1

Variant 2Variant 3Variant 4Variant 5Variant 6

Variant 7Variant 8Variant 9

Data• SND: e+e–→K+K–

▪ DM2: e+e–→K+K–

◦ SND: e+e–→KSKL

▫ DM1: e+e–→KSKL

Systematic №2–№5 №8–№9

+ №6,№7

Data• SND: e+e–→K+K–

▪ DM2: e+e–→K+K–

◦ SND: e+e–→KSKL

▫ DM1: e+e–→KSKL

Systematic №2–№5 №8–№9

+ №6,№7

Data approximation: systematic error

Data approximation: cross section

• SND: e+e–→K+K–

▪ DM2: e+e–→K+K–

◦ SND: e+e–→KSKL

▫ DM1: e+e–→KSKL

ρ(770) ω(782) φ(1020) ρ'(1450) ω'(1420) φ'(1680) ρ''(1700) ω''(1650)

• SND: e+e–→K+K–

▪ DM2: e+e–→K+K–

◦ SND: e+e–→KSKL

▫ DM1: e+e–→KSKL

ρ(770) ω(782) φ(1020) ρ'(1450) ω'(1420) φ'(1680) ρ''(1700) ω''(1650)

Data approximation: summary

1. In the charged channel there is a dip in the cross section in the energy range 1.05-1.15 GeV, which can be explained by an addition to the amplitude due to rescattering in the final state

2. Amplitudes of radial excitations of vector mesons have opposite sign with respect to the amplitudes of ρ, ω and φ vector mesons, as expected

3. Presence of ω mesons in the description is necessary4. Presence of orbital excitations of vector mesons in the description is necessary5. Experimental data above 1.7 GeV are not well described: it is evident from the

data that there is an interference structure 6. Phases θV, deviate from 0, showing that more sophisticated model taking into

account mixing between vector mesons and rescattering effects in final state should be used

Data• CLEO: τ−→K−K0ντ

Fit variants: №2–№5 №8–№9

№6№7

Data• CLEO: τ−→K−K0ντ

Fit variants: №2–№5 №8–№9

№6№7

CVC: comparison of e+e−→ρ,ρ',ρ"→KK and τ−→K−K0ντ

120 I

KKKFF

2222

3

2

22

2

2

2

2

2

2

2

0

)(4

12

12

12

)()(

10 QFQ

Q

m

mQ

mQ

m

V

Qd

KKdBrBr KK

ud K

Hep-ph/0409080 v2

Conclusions

• Simultaneous fit of e+e–→K+K– and e+e–→KSKL cross sections was performed in the framework of vector dominance model in a wide energy range (2E0=1.01÷2.2 GeV)• The following results were obtained:

• dip in the e+e–→K+K– experimental cross section in the energy range from 1.05 to 1.15 GeV • phases of radial excitations of light vector mesons are close to prediction• Γφ'eeΓφ'KK/Γφ‘

2=(0.37±0.08)·10–6 • Γω"eeΓω"KK/Γω“

2=(0.4÷0.7)·10–7

• Range parameter R=2.0±0.2 GeV–1

• Isovector and isoscalar parts of the K-meson form factor were obtained• Comparison of isovector contribution and experimental data on spectral function, obtained from τ−→K−K0ντ decay, was done• More sophisticated VMD model, including mixing between vector mesons and amplitude corrections due to rescattering in final state, should be used