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, 7 - 10 April 2008

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

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

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, 072002 (2001) SND (scan2) 3.988 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 1.004÷1.060 15 3964 4169 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: 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/0201040 v2

Theoretical framework: Vector Dominance Model γ* 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)

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

– 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)

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 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) 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±3 210/137 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

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

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

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)

Data approximation: summary 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 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

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/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