9
ciency ε = 16.02% in the K+K−π0 mode and the event selection efficiency. We estimate the systemat-
K+K−π0
ε = 15.93% in the K0K±π∓ mode, respective- ic uncertainty associated with the resonance parameters
KS0K±π∓ S
ly. In the MC simulation, the mass and width of the of η (2S) by varying its mass and width by one stan-
c
η (2S) are quoted from the world averaged values [11]. dard deviation relative to the world average values [11].
c
Therefore, we determine the products of the BFs to Additionally, we consider the influence of intermediate
be (ψ(3686) γη (2S)) (η (2S) K0K+π− + resonances by generating new mixed MC samples with
c.c.)B =(3.23 0→ .20) c 10−6a× ndB (c ψ(3686→ ) γS η (2S)) additionaldecaychannels,namelyη (2S) K∗(1430)K¯
(η (2S) ± K+K−× π0)=(1.61B 0.10) 10→−6. Tc herat× io and η (2S) K∗(1430)K¯, using fc ractio→ ns in0 spired by
B c → ± × c → 2
of the BFs agrees well with the isospin symmetry expec- the LHCb study [14]. The difference in efficiency be-
tationof2:1betweenK0K±π∓andK+K−π0. Wedeter- tween the new mixed MC samples and the nominal MC
S
mine the mass and width of the η (2S) to be M = samples is taken as the corresponding systematic uncer-
c ηc(2S)
(3637.8 0.8) MeV/c2 and Γ = (10.5 1.7) MeV. tainty, which is estimated to be 2.9%.
±
ηc(2S)
±
The uncertainties are statistical only. The uncertainties related to the simultaneous fit in-
Considering isospin conservation, the product BF for clude those due to the signal resolution function, the ef-
ψ(3686) γη (2S) with η (2S) KK¯π can be ficiency curve, the line-shapes of the backgrounds, and
c c
obtained → by doubling the sum of t→ he K0K±π∓ and the damping function. We estimate the systematic un-
S
K+K−π0 BFs to obtain (ψ(3686) γη (2S)) certainty due to the resolution function by changing the
c
(η (2S) KK¯π)=(0.97 B 0.06) 10−5→ ,wheretheun× - parametersofthemassandthemassresolutioninthesin-
c
B → ± ×
certainty takes into account the correlation between the gleGaussianfunctionandtheconvolveddoubleGaussian
twomeasuredBFsfromthesimultaneousfit. HereKK¯π function by 1σ. The largest difference is 0.6%, and we
indicatesthesumofsevenchannelsK+K−π0, K K π0, take it as the uncertainty due to the resolution function.
S S
K K π0, K K+π−, K K−π+, K K+π−, K K−π+. The uncertainty from the efficiency curve is estimated
L L S S L L
If only consider the isospin symmetry, the ratios of the by comparing the difference in results with and without
BFs between them should be 2:1:1:2:2:2:2. the efficiency curve. We find a difference of 0.1% and
take it as the systematic uncertainty due to the efficien-
cy curve. We estimate the systematic uncertainties due
VI. SYSTEMATIC UNCERTAINTIES totheline-shapesofthebackgroundsbyusingtheshapes
fromalternative ones. The line shape of one background
component is changed each time, and the largest differ-
There are multiple sources of systematic uncertainties
ence is used for the uncertainty. We change the relative
in determining the resonant parameters of the η (2S)
c
ratio f by 1σ to evaluate the systematic uncertain-
and the product of branching fractions (ψ(3686) FSR
γη (2S)) (η (2S) KK¯π). B → ty due to FSR. We change the Novosibirsk function to a
c c
×B → Gaussianfunctiontoestimatetheuncertaintyduetothe
Systematic uncertainties associated with event selec-
tion, such as tracking, PID, photon reconstruction, K0 shapeoftheπ0KK¯π backgrounds,andchangetheshape
S fromthe inclusive MC sample to the Argus function [34]
reconstruction, and kinematic fit are all estimated us-
to estimate the uncertainty due to the shape of oth-
ing control samples. To account for each source’s un-
er backgrounds. We assume isospin symmetry between
certainty, we vary the efficiency accordingly and calcu-
K+K−π0 andK0K±π∓ channelsinthenominalfit,and
late the difference in the final BF. This difference rep- S
takethedifferencetotheresultwithoutisospinconserva-
resents the systematic uncertainty attributed to track-
tionastheuncertaintyassociatedwiththeisospinconser-
ing, PID, and photon reconstruction, respectively. From
the study with two control samples, J/ψ pp¯π+π− vationassumption. The systematicuncertaintyassociat-
and e+e− π+π−K+K−, the uncertaintie→ s of track- ed with the damping function is estimated in two ways.
→ One is varying the parameter m by 1σ in KEDR’s
ingandPIDare1%,respectively,foreachpionandkaon ηc(2S)
damping function [28]. The other one is modifying the
track[29,30]. Accordingtoastudywiththecontrolsam-
functionfromthe defaultKEDR’sformula[28]tothe al-
ple J/ψ ρπ, the uncertainty due to photon detection
→ ternative CLEO’s exp( E2/8β2) [35], where β = 0.033.
efficiency is 1%per photonandis additive [31]. The sys- − γ
tematic uncertainty from the K0 reconstruction is stud- The maximum differences from the nominal results are
iedusingtwocontrolsamples,J/S ψ K∗±(892)K∓ with taken as the relevant systematic uncertainties.
K∗±(892) K0π± and J/ψ φ→ K0K±π∓ [32]. We The uncertainty due to the number of ψ(3686) events
estimateth→ esystS ematicuncerta→ intyfroS mK0 reconstruc- in the data sample is determined with inclusive hadron-
S
tion to be 1.2% for the K0K±π∓ mode. To study the ic ψ(3686) decays. The 0.5% is determined to be the
S
uncertainty associatedwith the kinematic fit, we correct systematic uncertainty associated with the number of
the track helix parameters in the MC simulation [33]. ψ(3686) events [15].
The efficiency difference 2.8% and 5.1% before and af- Theuncertaintyduetothecontinuumprocessincludes
ter the correction is taken as the systematic uncertainty the number of events and the line shape. We vary
related to the kinematic fit for the K+K−π0 mode and the number of continuum events assuming it satisfies a
K0K±π∓ mode,respectively. Theresonanceparameters Possiondistribution, andtake the difference to the nom-
S
of η (2S) and the intermediate resonances also impact inal fit result as the corresponding uncertainty. We also
c