1. CVC test in e+e−→KK cross section and data on τ−→K−K0ντ decay
International Workshop on e+e- collisions from Phi to Psi
CVC test in e+e−→KK cross section and data on τ−→K−K0ντ decay —
Konstantin Beloborodov
Budker Institute of Nuclear Physics
Novosibirsk, Russia
Laboratori Nazionali di Frascati, Italy, April 2008

2. Outline
Introduction
Experimental data
VDM, relations, parameters
Fit results
CVC test
Conclusions

3. Introduction What is a goal of this work?
Do simultaneous fit of e+e–→K+K– and e+e–→KSKL cross sections in wide energy region (2E0=1.03÷2.2 GeV)
Test and improve cross section description, based on Vector Dominance Model.
Measure parameters of vector mesons
As a result of the fit, extract isovector and isoscalar parts of the amplitude:
Compare the isovector part of K-meson form factor and spectral function, obtained from τ decay

4. Experimental data Phys.Rev.D 63, 072002 (2001)
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, (2001)
SND (scan2)
3.988
SND
1.040÷1.380 34
6.7
Phys. Rev. D 76, (2007)
DM2*
1.350÷2.085 21
1.582
Z.Phys.C39:13,1988
e+e–→KSKL
1.004÷1.060 15
3964
4169 22
8.857
Zh.Eksp.Teor.Fiz.103: ,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:
Correction parameter for DM2 cross section data was free and was obtained about 1.42±0.16
Similar situation in the π+π–π0 channel: In the paper hep-ex/ v2

5. Theoretical framework: Vector Dominance Modelγ* V K+ K-
gγV
gVKK e+ e- γ* V KS KL
gγV
gVKK
n2S+1lJ
ρ(+/0) ω φ
13S1
+/– +
13D1
23S1
–/+ –
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)

6. Parameters + + fixed from PDG + free parameters + gρωπ=gωρπ=16.8 GeV–1
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)
+,+ not used in total width
+ fixed from PDG
+ free parameters
+ gρωπ=gωρπ=16.8 GeV–1
V: MV and ΓV – fixed from PDG
φ(1020): Mφ and Γφ – free parameters
V‘, V“: MV and ΓV – free parameters within errors from PDG

7. – phases for charge and neutral channels are the same
Relations
Leptonic widths
SU(2)
– phases for charge and neutral channels are the same
SU(3)
LW+SU(3)

8. 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.

9. Data approximations: variants
SU(2)
LW+SU(3) θV
R=RVf
add-ons 1 +
{0,π} ≠0 2
free 3 =0 4 – 5 6
without ρ", ω" 7
without ω, ω', ω" 8
MV±3σMv, ΓV±3σΓv 9
ΓV=const(s) for excitations
Find any possible solutions
In case of constant widths of the resonances find all 2N-1 solutions
Use these 2N-1 solutions as starting positions for minimization procedure
Choose a solution with smallest χ2
e-print (2007)
Ambiguity of resonances parameters determination: solution of the problem

10. 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
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 5
-32±2
-7±1
-52±4
26±2
100±2
36±4 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
-30±2
-49±3
-20±3
99±2
3±6
145/132
Variant
R, GeV–1
ΓVeeΓVKK/ 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
0.6±0.1 →0
0.11±0.02 2
2.0±0.3
2.0±0.4
0.2±0.1
0.37±0.08
0.4±0.2 3
2.0±0.6
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
1.03±0.04
0.93±0.07
0.7±0.1 6
0.8±0.2
0.49±0.02
1.23±0.09
1.3±0.1
0.25±0.02 7
1.8±0.1
0.8±0.1
0.15±0.02 8
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

11. Data approximation: results
Variant 1
Variant 1
Variant 2
Variant 2
Variant 3
Variant 3
Variant 4
Variant 4
Variant 5
Variant 5
Variant 6
Variant 6
Variant 7
Variant 7
Variant 8
Variant 8
Variant 9
Variant 9

12. Data approximation: systematic error
• 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

13. 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)

14. Data approximation: summary
In the charged channel there is a dip in the cross section in the energy range GeV, which can be explained by an addition to the amplitude due to rescattering in the final state
Amplitudes of radial excitations of vector mesons have opposite sign with respect to the amplitudes of ρ, ω and φ vector mesons, as expected
Presence of ω mesons in the description is necessary
Presence of orbital excitations of vector mesons in the description is necessary
Experimental data above 1.7 GeV are not well described: it is evident from the data that there is an interference structure
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

15. CVC: comparison of e+e−→ρ,ρ',ρ"→KK and τ−→K−K0ντ
Data
• CLEO: τ−→K−K0ντ
Fit variants:
№2–№5
№8–№9 №6 №7
Hep-ph/ v2

16. 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