1 Introduction
Two of the most surprising aspects of high-energy deep inelastic scattering (DIS) observed at the HERA ep collider have been the sharp rise of the proton structure function, F2, with decreasing value of Bjorken x and the abundance of events with a large rapidity gap in the hadronic final state 1. The latter are identified as due to diffraction in the deep inelastic regime. A contribution to the diffractive cross section arises from the exclusive production of vector mesons (VM).
High-energy exclusive VM production in DIS has been postulated to proceed through two-gluon exchange 23, once the scale, usually taken as the virtuality Q2 of the exchanged photon, is large enough for perturbative Quantum Chromodynamics (pQCD) to be applicable. The gluons in the proton, which lie at the origin of the sharp increase of F2, are also expected to cause the VM cross section to increase with increasing photon proton centre-of-mass energy, W, with the rate of increase growing with Q2. Moreover, the effective size of the virtual photon decreases with increasing Q2, leading to a flatter distribution in t, the four-momentum-transfer squared at the proton vertex. All these features, with varying levels of significance, have been observed at HERA 45678910 in the exclusive production of ρ0, ω, φ, and J/ψ mesons.
This paper reports on an extensive study of the properties of exclusive ρ0-meson production,
γ*p → ρ0p,
based on a high statistics data sample collected with the ZEUS detector during the period 1996–2000, corresponding to an integrated luminosity of about 120 pb-1.
2 Theoretical background
Calculations of the VM production cross section in DIS require knowledge of the qq¯
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdghaXjqbdghaXzaaraaaaa@2DE3@ wave-function of the virtual photon, specified by QED and which depends on the polarisation of the virtual photon. For longitudinally polarised photons, γL∗
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciab=n7aNnaaDaaaleaacqWGmbataeaacqGHxiIkaaaaaa@2EE0@, qq¯
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdghaXjqbdghaXzaaraaaaa@2DE3@ pairs of small transverse size dominate 3. The opposite holds for transversely polarised photons, γT∗
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciab=n7aNnaaDaaaleaacqWGubavaeaacqGHxiIkaaaaaa@2EF0@, where qq¯
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdghaXjqbdghaXzaaraaaaa@2DE3@ configurations with large transverse size dominate. The favourable feature of exclusive VM production is that, at high Q2, the longitudinal component of the virtual photon is dominant. The interaction cross section in this case can be fully calculated in pQCD 11, with two-gluon exchange as the leading process in the high-energy regime. For heavy vector mesons, such as the J/ψ or the ϒ, perturbative calculations apply even at Q2 = 0, as the smallness of the qq¯
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdghaXjqbdghaXzaaraaaaa@2DE3@ dipole originating from the photon is guaranteed by the mass of the quarks.
Irrespective of particular calculations 12, in the region dominated by perturbative QCD the following features are predicted:
• the total γ*p → Vp cross section, σγ*p, exhibits a steep rise with W, which can be parameterised as σ ~ Wδ, with δ increasing with Q2;
• the Q2 dependence of the cross-section, which for a longitudinally polarised photon is expected to behave as Q-6, is moderated to become Q-4 by the rapid increase of the gluon density with Q2;
• the distribution of t becomes universal, with little or no dependence on W or Q2;
• breaking of the s-channel helicity conservation (SCHC) is expected.
In the region where perturbative calculations are applicable, exclusive vector-meson production could become a complementary source of information on the gluon content of the proton. At present, the following theoretical uncertainties have been identified:
• the calculation of σ(γ*p → Vp) involves the generalised parton distributions 1314, which are not well tested; in addition 15, it involves gluon densities outside the range constrained by global QCD analyses of parton densities;
• higher-order corrections have not been fully calculated 16; therefore the overall normalisation is uncertain and the scale at which the gluons are probed is not known;
• the rapid rise of σγ*p with W implies a non-zero real part of the scattering amplitude, which is not known;
• the wave-functions of the vector mesons are not fully known.
In spite of all these problems, precise measurements of differential cross sections separated into longitudinal and transverse components 17, should help to resolve the above theoretical uncertainties.
It is important in these studies to establish a region of phase space where hard interactions dominate over the non-perturbative soft component. If the relative transverse momentum of the qq¯
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdghaXjqbdghaXzaaraaaaa@2DE3@ pair is small, the colour dipole is large and perturbative calculations do not apply. In this case the interaction looks similar to hadron-hadron elastic scattering, described by soft Pomeron exchange as in Regge phenomenology 18.
The parameters of the soft Pomeron are known from measurements of total cross sections for hadron-hadron interactions and elastic proton-proton measurements. It is usually assumed that the Pomeron trajectory is linear in t:
α
ℙ
(
t
)
=
α
ℙ
(
0
)
+
α
′
ℙ
t
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFXoqydaWgaaWcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqGFzecuaeqaaOGaeiikaGIaemiDaqNaeiykaKIaeyypa0Jae8xSde2aaSbaaSqaaiab+LriqbqabaGccqGGOaakcqaIWaamcqGGPaqkcqGHRaWkcuWFXoqygaqbamaaBaaaleaacqGFzecuaeqaaOGaemiDaqNaeiOla4caaa@46A8@
The parameter αℙ(0) determines the energy behaviour of the total cross section,
σ
tot
~
(
W
2
)
α
ℙ
(
0
)
−
1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaWgaaWcbaGaeeiDaqNaee4Ba8MaeeiDaqhabeaakiabc6ha+jabcIcaOiabdEfaxnaaCaaaleqabaGaeGOmaidaaOGaeiykaKYaaWbaaSqabeaacqWFXoqydaWgaaadbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqGFzecuaeqaaSGaeiikaGIaeGimaaJaeiykaKIaeyOeI0IaeGymaedaaaaa@4759@
and α′ℙ
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciqb=f7aHzaafaWaaSbaaSqaamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbaceaGae4xgHafabeaaaaa@3783@ describes the increase of the slope b of the t distribution with increasing W. The value of α′ℙ
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciqb=f7aHzaafaWaaSbaaSqaamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbaceaGae4xgHafabeaaaaa@3783@ is inversely proportional to the square of the typical transverse momenta participating in the exchanged trajectory. A large value of α′ℙ
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciqb=f7aHzaafaWaaSbaaSqaamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbaceaGae4xgHafabeaaaaa@3783@ suggests the presence of low transverse momenta typical of soft interactions. The accepted values of αℙ(0) 19 and α′ℙ
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciqb=f7aHzaafaWaaSbaaSqaamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbaceaGae4xgHafabeaaaaa@3783@20 are
α
ℙ
(
0
)
=
1.096
±
0.003
α
′
ℙ
=
0.25
Gev
−
2
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqadeGabaaabaacciGae8xSde2aaSbaaSqaamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbaceaGae4xgHafabeaakiabcIcaOiabicdaWiabcMcaPiabg2da9iabigdaXiabc6caUiabicdaWiabiMda5iabiAda2iabgglaXkabicdaWiabc6caUiabicdaWiabicdaWiabiodaZaqaaiqb=f7aHzaafaWaaSbaaSqaaiab+LriqbqabaGccqGH9aqpcqaIWaamcqGGUaGlcqaIYaGmcqaI1aqncqqGGaaicqqGhbWrcqqGLbqzcqqG2bGDdaahaaWcbeqaaiabgkHiTiabikdaYaaakiabc6caUaaaaaa@5527@
The non-universality of αℙ(0) has been established in inclusive DIS, where the slope of the γ*p total cross section with W has a pronounced Q2 dependence 21. The value of α′ℙ
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciqb=f7aHzaafaWaaSbaaSqaamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbaceaGae4xgHafabeaaaaa@3783@ can be determined from exclusive VM production at HERA via the W dependence of the exponential b slope of the t distribution for fixed values of W, where b is expected to behave as
b
(
W
)
=
b
0
+
4
α
′
ℙ
ln
W
W
0
,
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGIbGycqGGOaakcqWGxbWvcqGGPaqkcqGH9aqpcqWGIbGydaWgaaWcbaGaeGimaadabeaakiabgUcaRiabisda0GGaciqb=f7aHzaafaWaaSbaaSqaamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbaceaGae4xgHafabeaakiGbcYgaSjabc6gaULqbaoaalaaabaGaem4vaCfabaGaem4vaC1aaSbaaeaacqaIWaamaeqaaaaacqGGSaalaaa@4832@
where b0 and W0 are free parameters. The value of α′ℙ
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciqb=f7aHzaafaWaaSbaaSqaamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbaceaGae4xgHafabeaaaaa@3783@ can also be derived from the W dependence of dσ/dt at fixed t,
d
σ
d
t
(
W
)
=
F
(
t
)
W
2
[
2
α
ℙ
(
t
)
−
2
]
,
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaiabdsgaKHGaciab=n8aZbqaaiabdsgaKjabdsha0baakiabcIcaOiabdEfaxjabcMcaPiabg2da9iabdAeagjabcIcaOiabdsha0jabcMcaPiabdEfaxnaaCaaaleqabaGaeGOmaiJaei4waSLaeGOmaiJae8xSde2aaSbaaWqaamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbaceaGae4xgHafabeaaliabcIcaOiabdsha0jabcMcaPiabgkHiTiabikdaYiabc2faDbaakiabcYcaSaaa@5136@
where F(t) is an arbitrary function. This approach has the advantage that no assumption needs to be made about the t dependence. The first indications from measurements of αℙ(t) in exclusive J/ψ photoproduction 822 are that αℙ(0) is larger and α′ℙ
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciqb=f7aHzaafaWaaSbaaSqaamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbaceaGae4xgHafabeaaaaa@3783@ is smaller than those of the above soft Pomeron trajectory.
3 Experimental set-up
The present measurement is based on data taken with the ZEUS detector during two running periods of the HERA ep collider. During 1996–1997, protons with energy 820 GeV collided with 27.5 GeV positrons, while during 1998–2000, 920 GeV protons collided with 27.5 GeV electrons or positrons. The sample used for this study corresponds to an integrated luminosity of 118.9 pb-1, consisting of 37.2 pb-1 e+ p sample from 1996–1997 and 81.7 pb-1 from the 1998–2000 sample (16.7 pb-1 e- and 65.0 pb-1 e+)1.
A detailed description of the ZEUS detector can be found elsewhere 2324. A brief outline of the components that are most relevant for this analysis is given below.
Charged particles are tracked in the central tracking detector (CTD) 252627. The CTD consists of 72 cylindrical drift chamber layers, organised in nine superlayers covering the polar-angle2 region 15° <θ <164°. The CTD operates in a magnetic field of 1.43 T provided by a thin solenoid. The transverse-momentum resolution for full-length tracks is σ(pT)/pT = 0.0058pT ⊕ 0.0065 ⊕ 0.0014/pT, with pT in GeV.
The high-resolution uranium-scintillator calorimeter (CAL) 28293031 covers 99.7% of the total solid angle and consists of three parts: the forward (FCAL), the barrel (BCAL) and the rear (RCAL) calorimeters. Each part is subdivided transversely into towers and longitudinally into one electromagnetic section (EMC) and either one (in RCAL) or two (in BCAL and FCAL) hadronic sections. The CAL energy resolutions, as measured under test-beam conditions, are σ(E)/E = 0.18/E
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamaakaaabaGaemyrauealeqaaaaa@2C23@ for electrons and σ(E)/E = 0.35/E
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamaakaaabaGaemyrauealeqaaaaa@2C23@ for hadrons, with E in GeV.
The position of the scattered electron was determined by combining information from the CAL, the small-angle rear tracking detector 32 and the hadron-electron separator 33.
In 1998, the forward plug calorimeter (FPC) 34 was installed in the 20 × 20 cm2 beam hole of the FCAL with a small hole of radius 3.15 cm in the centre to accommodate the beam pipe. The FPC increased the forward calorimeter coverage by about one unit in pseudorapidity to η ≤ 5.
The leading-proton spectrometer (LPS) 35 detected positively charged particles scattered at small angles and carrying a substantial fraction, xL, of the incoming proton momentum; these particles remained in the beam-pipe and their trajectories were measured by a system of silicon microstrip detectors, located between 23.8 m and 90.0 m from the interaction point. The particle deflections induced by the magnets of the proton beam-line allowed a momentum analysis of the scattered proton.
During the 1996–1997 data taking, a proton-remnant tagger (PRT1) was used to tag events in which the proton dissociates. It consisted of two layers of scintillation counters perpendicular to the beam at Z = 5.15 m. The two layers were separated by a 2 mm-thick lead absorber. The pseudorapidity range covered by the PRT1 was 4.3 <η < 5.8.
The luminosity was measured from the rate of the bremsstrahlung process ep → eγp. The photon was measured in a lead-scintillator calorimeter 363738 placed in the HERA tunnel at Z = -107 m.
4 Data selection and reconstruction
The following kinematic variables are used to describe exclusive ρ0 production and its subsequent decay into a π+π- pair:
• the four-momenta of the incident electron (k), scattered electron (k'), incident proton (P), scattered proton (P') and virtual photon (q);
• Q2 = -q2 = -(k - k')2, the negative squared four-momentum of the virtual photon;
• W2 = (q + P)2, the squared centre-of-mass energy of the photon-proton system;
• y = (P·q)/(P·k), the fraction of the electron energy transferred to the proton in its rest frame;
• Mππ, the invariant mass of the two decay pions;
• t = (P - P')2, the squared four-momentum transfer at the proton vertex;
• three helicity angles, Φh, θh and φh (see Section 9).
The kinematic variables were reconstructed using the so-called "constrained" method 1039, which uses the momenta of the decay particles measured in the CTD and the reconstructed polar and azimuthal angles of the scattered electron.
The online event selection required an electron candidate in the CAL, along with the detection of at least one and not more than six tracks in the CTD.
In the offline selection, the following further requirements were imposed:
• the presence of a scattered electron, with energy in the CAL greater than 10 GeV and with an impact point on the face of the RCAL outside a rectangular area of 26.4 × 16 cm2;
• E - PZ > 45 GeV, where E - PZ = ∑i(Ei - pZi
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdchaWnaaBaaaleaacqWGAbGwdaWgaaadbaGaemyAaKgabeaaaSqabaaaaa@2F5A@) and the summation is over the energies and longitudinal momenta of the final-state electron and pions, was imposed. This cut excludes events with high energy photons radiated in the initial state;
• the Z coordinate of the interaction vertex within ± 50 cm of the nominal interaction point;
• in addition to the scattered electron, exactly two oppositely charged tracks, each associated with the reconstructed vertex, and each having pseudorapidity |η| less than 1.75 and transverse momentum greater than 150 MeV; this excluded regions of low reconstruction efficiency and poor momentum resolution in the CTD. These tracks were treated in the following analysis as a π+π- pair;
• events with any energy deposit larger than 300 MeV in the CAL and not associated with the pion tracks (so-called 'unmatched islands') were rejected 404142.
In addition, the following requirements were applied to select kinematic regions of high acceptance:
• the analysis was restricted to the kinematic regions 2 <Q2 < 80 GeV2 and 32 <W < 160 GeV in the 1996–1997 data and 2 <Q2 < 160 GeV2 and 32 <W < 180 GeV in the 1998–2000 sample;
• only events in the π+π- mass interval 0.65 <Mππ < 1.1 GeV and with |t| < 1 GeV2 were taken. The mass interval is slightly narrower than that used previously 10, in order to reduce the effect of the background from non-resonant π+π- production. In the selected Mππ range, the resonant contribution is ≈ 100% (see Section 8).
The above selection yielded 22,400 events in the 1996–1997 sample and 49,300 events in the 1998–2000 sample, giving a total of 71,700 events for this analysis.
5 Monte Carlo simulation
The relevant Monte Carlo (MC) generators have been described in detail previously 10. Here their main features are summarised.
The program ZEUSVM 43 interfaced to HERACLES4.4 44 was used. The effective Q2, W and t dependences of the cross section were parameterised to reproduce the data 42.
The decay angular distributions were generated uniformly and the MC events were then iteratively reweighted using the results of the present analysis for the 15 combinations of matrix elements rik04
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdkhaYnaaDaaaleaacqWGPbqAcqWGRbWAaeaacqaIWaamcqaI0aanaaaaaa@312D@, rikα
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdkhaYnaaDaaaleaacqWGPbqAcqWGRbWAaeaaiiGacqWFXoqyaaaaaa@30EF@ (see Section 9).
The contribution of the proton-dissociative process was studied with the EPSOFT 45 generator for the 1996–1997 data and with PYTHIA 46 for the 1998–2000 data. The Q2, W and t dependences were parameterised to reproduce the control samples in the data. The decay angular distributions were generated as in the ZEUSVM sample.
The generated events were processed through the same chain of selection and reconstruction procedures as the data, thus accounting for trigger as well as detector acceptance and smearing effects. For both MC sets, the number of simulated events after reconstruction was about a factor of seven greater than the number of reconstructed data events.
All measured distributions are well described by the MC simulations. Some examples are shown in Fig. 1, for the W, Q2, t variables, and the three helicity angles, θh, φh, and Φh, and in Fig. 2 for the transverse momentum pT of the pions, for different Q2 bins.
<p>Figure 1</p>
Comparison between the data and the ZEUSVM MC distributions for (a) W, (b) Q2, (c) |t|, (d) cosθh, (e) φh and (f) Φh for events with 0.65 <Mππ < 1.1 GeV and |t| < 1.0 GeV2
Comparison between the data and the ZEUSVM MC distributions for (a) W, (b) Q2, (c) |t|, (d) cosθh, (e) φh and (f) Φh for events with 0.65 <Mππ < 1.1 GeV and |t| < 1.0 GeV2. The MC distributions are normalised to the data.
![]()
<p>Figure 2</p>
Comparison between the data and the ZEUSVM MC distributions for the transverse momentum, pT, of π+ and π- particles, for different ranges of Q2, as indicated in the figure
Comparison between the data and the ZEUSVM MC distributions for the transverse momentum, pT, of π+ and π- particles, for different ranges of Q2, as indicated in the figure. The events are selected to be within 0.65 <Mππ < 1.1 GeV and |t| < 1.0 GeV2. The MC distributions are normalised to the data.
![]()
6 Systematics
The systematic uncertainties of the cross section were evaluated by varying the selection cuts and the MC simulation parameters. The following selection cuts were varied:
• the E - PZ cut was changed within the appropriate resolution of ±3 GeV;
• the pT of the pion tracks (default 0.15 GeV) was increased to 0.2 GeV;
• the distance of closest approach of the extrapolated track to the matched island in the CAL was changed from 30 cm to 20 cm;
• the π+π--mass window was changed to 0.65–1.2 GeV;
• the Z vertex cut was varied by ±10 cm;
• the rectangular area of the electron impact point on the CAL was increased by 0.5 cm in X and Y ;
• the energy of an unmatched island was lowered to 0.25 GeV and then raised to 0.35 GeV.
The dependence of the results on the precision with which the MC reproduces the performance of the detector and the data was checked by varying the following inputs within their estimated uncertainty:
• the reconstructed position of the electron was shifted with respect to the MC by ±1 mm;
• the electron-position resolution was varied by ±10% in the MC;
• the Wδ-dependence in the MC was changed by varying δ by ±0.03;
• the exponential t-distribution in the MC was reweighted by changing the nominal slope parameter b by ±0.5 GeV-2;
• the angular distributions in the MC were reweighted assuming SCHC;
• the Q2-distribution in the MC was reweighted by (Q2 + Mρ2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabd2eannaaDaaaleaaiiGacqWFbpGCaeaacqaIYaGmaaaaaa@2EFE@)k, where k = ±0.05.
The largest uncertainty of about ± 4% originated from the variation of the energy of the unmatched islands. All the other checks resulted on average in a 0.5% change in the measured cross sections. All the systematic uncertainties were added in quadrature. In addition, the cross-section measurements have an overall normalisation uncertainty of ±2% due to the luminosity measurement.
7 Proton dissociation
The production of ρ0 mesons may be accompanied by the proton-dissociation process, γ*p → ρ0N. For low masses MN of the dissociative system N, the hadronisation products may remain inside the beam-pipe, leaving no signals in the main detector. The contribution of these events to the exclusive ρ0 cross section was estimated from MC generators for proton-dissociative processes.
A class of proton dissociative events for which the final-state particles leave observed signals in the surrounding detectors was used to tune the MN and the t distribution in the MC. In the 1998–2000 running period, these events were selected by requiring a signal in the FPC detector with energy above 1 GeV. The comparison of the data with PYTHIA expectations for the energy distribution in the FPC is shown in Fig. 3(a). The same procedure was repeated with a sample of ρ0 events for which the FPC energy was less than 1 GeV and a leading proton was measured in the LPS detector, with the fraction of the incoming proton momentum xL < 0.95. The comparison between the xL distribution measured in the data and that expected from PYTHIA is shown in Fig. 3(b), where the elastic peak in the data (xL > 0.95) is also observed. Also shown in Fig. 3(c–e) is the fraction of proton-dissociative events expected in the selected ρ0 sample as a function of Q2, W and t. The fraction is at the level of 19%, independent of Q2 and W, but increasing with increasing |t|. The combined use of the FPC and LPS methods leads to an estimate of the proton dissociative contribution for |t| < 1 GeV2 of 0.19 ± 0.02(stat.) ± 0.03(syst.). The systematic uncertainty was estimated by varying the parameters of the MN distribution and by changing the FPC cut.
<p>Figure 3</p>
(a) The energy distribution in the FPC
(a) The energy distribution in the FPC. The data (full dots) are compared to the expectations from the PYTHIA MC, normalised to the data. (b) The xL distribution in the LPS. The data (open circles) are compared to the expectations from the PYTHIA MC, normalised to the data for xL < 0.95. The extracted fraction of proton-dissociation events, from the FPC data (dots) and from the LPS data (open circles), as a function of (c) Q2, (d) W and (e) |t|. All events were selected in the ρ0 mass window (0.65–1.1 GeV). The dotted line in (c) and (d) represents a fit of a constant to the proton-dissociation fraction.
![]()
In the 1996–1997 data-taking period, a similar procedure was applied, after tuning the EPSOFT MC to reproduce events with hits in the PRT1 or energy deposits in the FCAL. The proton-dissociative contribution for |t| < 1 GeV2 was determined to be 0.07 ± 0.02 after rejecting events with hits in the PRT1 or energy deposits in the FCAL. This number is consistent with that determined from the LPS and FPC because of the different angular coverage of the PRT1.
After subtraction of the proton-dissociative contribution, a good agreement between the cross sections derived from the two data-taking periods was found. For all the quoted cross sections integrated over t, the overall normalisation uncertainty due to the subtraction of the proton-dissociative contributions was estimated to be ± 4% and was not included in the systematic uncertainty. The proton-dissociative contribution was statistically subtracted in each analysed bin, unless stated otherwise.
8 Mass distributions
The π+π--invariant-mass distribution is presented in Fig. 4. A clear enhancement in the ρ0 region is observed. Background coming from the decay φ → K+ K-, where the kaons are misidentified as pions, is expected 42 in the region Mππ < 0.55 GeV. That coming from ω events in the decay channel ω → π+π-π0, where the π0 remains undetected, contributes 42 in the region Mππ < 0.65 GeV. Therefore defining the selected ρ0 events to be in the window 0.65 <Mππ < 1.1 GeV ensures no background from these two channels.
<p>Figure 4</p>
The π+π- acceptance-corrected invariant-mass distribution
The π+π- acceptance-corrected invariant-mass distribution. The line represent the best fit of the Söding form to the data in the range 0.65 <Mππ < 1.1 GeV. The vertical lines indicate the range of masses used for the analysis. The dashed line is the shape of a relativistic Breit-Wigner with the fitted parameters given in the figure. The dotted line is the interference term between the non-resonant background (dash-dotted line) and the ρ0 signal.
![]()
In order to estimate the non-resonant π+π- background under the ρ0, the Söding parameterisation 47 was fitted to the data, with results shown in the figure. The resulting mass and width values are in agreement with those given in the Particle Data Group 48 compilation. The integrated non-resonant background is of the order of 1% and is thus neglected.
The π+π- mass distributions in different regions of Q2 and t are shown in Fig. 5 and Fig. 6, respectively. The shape of the mass distribution changes neither with Q2 nor with t. The results of the fit to the Söding parameterisation are also shown. Note that the interference term decreases with Q2 as expected but is independent of t, indicating that the non-exclusive background is negligible.
<p>Figure 5</p>
The π+π- acceptance-corrected invariant-mass distribution, for different Q2 intervals, with mean values as indicated in the figure
The π+π- acceptance-corrected invariant-mass distribution, for different Q2 intervals, with mean values as indicated in the figure. The lines are defined in the caption of Fig. 4.
![]()
<p>Figure 6</p>
The π+π- acceptance-corrected invariant-mass distribution, for different t intervals, with mean values as indicated in the figure
The π+π- acceptance-corrected invariant-mass distribution, for different t intervals, with mean values as indicated in the figure. The lines are defined in the caption of Fig. 4.
![]()
9 Angular distributions and decay-matrix density
The exclusive electroproduction and decay of ρ0 mesons is described, at fixed W, Q2, Mππ and t, by three helicity angles: Φh is the angle between the ρ0 production plane and the electron scattering plane in the γ*p centre-of-mass frame; θh and φh are the polar and azimuthal angles of the positively charged decay pion in the s-channel helicity frame. In this frame, the spin-quantisation axis is defined as the direction opposite to the momentum of the final-state proton in the ρ0 rest frame. In the γ*p centre-of-mass system, φh is the angle between the decay plane and the ρ0 production plane. The angular distribution as a function of these three angles, W(cos θh, φh, Φh), is parameterised by the ρ0 spin-density matrix elements, ρikα
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciab=f8aYnaaDaaaleaacqWGPbqAcqWGRbWAaeaacqWFXoqyaaaaaa@313D@, where i, k = -1, 0, 1 and by convention α = 0, 1, 2, 4, 5, 6 for an unpolarised charged-lepton beam 49. The superscript denotes the decomposition of the spin-density matrix into contributions from the following photon-polarisation states: unpolarised transverse photons (0); linearly polarised transverse photons (1,2); longitudinally polarised photons (4); and from the interference of the longitudinal and transverse amplitudes (5,6).
The decay angular distribution can be expressed in terms of combinations, rik04
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdkhaYnaaDaaaleaacqWGPbqAcqWGRbWAaeaacqaIWaamcqaI0aanaaaaaa@312D@ and rikα
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdkhaYnaaDaaaleaacqWGPbqAcqWGRbWAaeaaiiGacqWFXoqyaaaaaa@30EF@, of the density matrix elements
r
i
k
04
=
ρ
i
k
0
+
ε
R
ρ
i
k
4
1
+
ε
R
,
r
i
k
α
=
{
ρ
i
k
α
1
+
ε
R
,
α
=
1
,
2
R
ρ
i
k
α
1
+
ε
R
,
α
=
5
,
6
,
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqadeGadaaabaGaemOCai3aa0baaSqaaiabdMgaPjabdUgaRbqaaiabicdaWiabisda0aaaaOqaaiabg2da9aqcfayaamaalaaabaacciGae8xWdi3aa0baaeaacqWGPbqAcqWGRbWAaeaacqaIWaamaaGaey4kaSIae8xTduMaemOuaiLae8xWdi3aa0baaeaacqWGPbqAcqWGRbWAaeaacqaI0aanaaaabaGaeGymaeJaey4kaSIae8xTduMaemOuaifaaiabcYcaSaGcbaGaemOCai3aa0baaSqaaiabdMgaPjabdUgaRbqaaiab=f7aHbaaaOqaaiabg2da9aqaamaaceqabaqbaeaabiGaaaqaaKqbaoaalaaabaGae8xWdi3aa0baaeaacqWGPbqAcqWGRbWAaeaacqWFXoqyaaaabaGaeGymaeJaey4kaSIae8xTduMaemOuaifaaOGaeiilaWcabaGae8xSdeMaeyypa0JaeGymaeJaeiilaWIaeGOmaidabaqcfa4aaSaaaeaadaGcaaqaaiabdkfasbqabaGae8xWdi3aa0baaeaacqWGPbqAcqWGRbWAaeaacqWFXoqyaaaabaGaeGymaeJaey4kaSIae8xTduMaemOuaifaaOGaeiilaWcabaGae8xSdeMaeyypa0JaeGynauJaeiilaWIaeGOnayJaeiilaWcaaaGaay5Eaaaaaaaa@7476@
where ε is the ratio of the longitudinal- to transverse-photon fluxes and R = σL/σT, with σL and σT the cross sections for exclusive ρ0 production from longitudinal and transverse virtual photons, respectively. In the kinematic range of this analysis, the value of ε varies between 0.96 and 1 with an average value of 0.996; hence ρik0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciab=f8aYnaaDaaaleaacqWGPbqAcqWGRbWAaeaacqaIWaamaaaaaa@3091@ and ρik4
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciab=f8aYnaaDaaaleaacqWGPbqAcqWGRbWAaeaacqaI0aanaaaaaa@3099@ cannot be distinguished.
The Hermitian nature of the spin-density matrix and the requirement of parity conservation reduces the number of independent parameters to 15 49. A 15-parameter fit was performed to the data and the obtained results are listed in Table 1 and shown in Fig. 7 as a function of Q2. The published ZEUS results 50 at lower Q2 values and the expectations of SCHC, when relevant, are also included. The observed Q2 dependence, expected in some calculations 51 and previously reported by H1 52, is driven by the R dependence on Q2 under the assumption of helicity conservation and natural parity exchange. The significant deviation of r005
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdkhaYnaaDaaaleaacqaIWaamcqaIWaamaeaacqaI1aqnaaaaaa@2F63@ from zero shows that SCHC does not hold 51 as was observed previously 5052.
<p>Table 1</p>
Spin density matrix elements for electroproduction of ρ0, for different intervals of Q2. The first uncertainty is statistical, the second systematic.
Element
2 <Q2 < 3 GeV2
3 <Q2 < 4 GeV2
4 <Q2 < 6 GeV2
6 <Q2 < 10 GeV2
10 <Q2 < 100 GeV2
r
00
04
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdkhaYnaaDaaaleaacqaIWaamcqaIWaamaeaacqaIWaamcqaI0aanaaaaaa@304F@
0.590
±
0.006
−
0.010
+
0.012
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabiwda1iabiMda5iabicdaWiabgglaXkabicdaWiabc6caUiabicdaWiabicdaWiabiAda2maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaIWaamaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaIYaGmaaaaaa@417F@
0.659
±
0.008
−
0.015
+
0.009
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabiAda2iabiwda1iabiMda5iabgglaXkabicdaWiabc6caUiabicdaWiabicdaWiabiIda4maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaI1aqnaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIWaamcqaI5aqoaaaaaa@41A5@
0.725
±
0.008
−
0.008
+
0.014
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabiEda3iabikdaYiabiwda1iabgglaXkabicdaWiabc6caUiabicdaWiabicdaWiabiIda4maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIWaamcqaI4aaoaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaI0aanaaaaaa@4195@
0.752
±
0.008
−
0.008
+
0.011
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabiEda3iabiwda1iabikdaYiabgglaXkabicdaWiabc6caUiabicdaWiabicdaWiabiIda4maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIWaamcqaI4aaoaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaIXaqmaaaaaa@418F@
0.814
±
0.010
−
0.019
+
0.008
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabiIda4iabigdaXiabisda0iabgglaXkabicdaWiabc6caUiabicdaWiabigdaXiabicdaWmaaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaI5aqoaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIWaamcqaI4aaoaaaaaa@418F@
Re(r1004
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdkhaYnaaDaaaleaacqaIXaqmcqaIWaamaeaacqaIWaamcqaI0aanaaaaaa@3051@)
0.024
±
0.005
−
0.009
+
0.003
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0.005
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1
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1
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdkhaYnaaDaaaleaacqaIXaqmcqGHsislcqaIXaqmaeaacqaIWaamcqaI0aanaaaaaa@3140@
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0.012
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0.015
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabicdaWiabicdaWiabicdaWiabgglaXkabicdaWiabc6caUiabicdaWiabicdaWiabiEda3maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIWaamcqaI2aGnaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaI1aqnaaaaaa@4175@
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1
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabgkHiTiabicdaWiabc6caUiabicdaWiabicdaWiabiIda4iabgglaXkabicdaWiabc6caUiabicdaWiabicdaWiabiEda3maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaI5aqoaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIWaamcqaI2aGnaaaaaa@427A@
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0.015
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabgkHiTiabicdaWiabc6caUiabicdaWiabigdaXiabisda0iabgglaXkabicdaWiabc6caUiabicdaWiabikdaYiabiAda2maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaI1aqnaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaI0aancqaI2aGnaaaaaa@4276@
0.020
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0.028
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0.013
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0.072
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabicdaWiabikdaYiabicdaWiabgglaXkabicdaWiabc6caUiabicdaWiabikdaYiabiIda4maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaIZaWmaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaI3aWncqaIYaGmaaaaaa@4181@
0.019
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0.030
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0.060
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0.008
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0.006
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0.003
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabgkHiTiabicdaWiabc6caUiabicdaWiabicdaWiabiAda2iabgglaXkabicdaWiabc6caUiabicdaWiabigdaXiabigdaXmaaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIZaWmcqaIWaamaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIWaamcqaIZaWmaaaaaa@4258@
−
0.042
±
0.016
−
0.009
+
0.029
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabgkHiTiabicdaWiabc6caUiabicdaWiabisda0iabikdaYiabgglaXkabicdaWiabc6caUiabicdaWiabigdaXiabiAda2maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIWaamcqaI5aqoaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIYaGmcqaI5aqoaaaaaa@427E@
r
1
−
1
1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdkhaYnaaDaaaleaacqaIXaqmcqGHsislcqaIXaqmaeaacqaIXaqmaaaaaa@304C@
0.195
±
0.009
−
0.019
+
0.012
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabigdaXiabiMda5iabiwda1iabgglaXkabicdaWiabc6caUiabicdaWiabicdaWiabiMda5maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaI5aqoaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaIYaGmaaaaaa@4199@
0.151
±
0.011
−
0.011
+
0.014
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabigdaXiabiwda1iabigdaXiabgglaXkabicdaWiabc6caUiabicdaWiabigdaXiabigdaXmaaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaIXaqmaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaI0aanaaaaaa@416F@
0.121
±
0.011
−
0.011
+
0.016
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabigdaXiabikdaYiabigdaXiabgglaXkabicdaWiabc6caUiabicdaWiabigdaXiabigdaXmaaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaIXaqmaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaI2aGnaaaaaa@416D@
0.095
±
0.011
−
0.029
+
0.006
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabicdaWiabiMda5iabiwda1iabgglaXkabicdaWiabc6caUiabicdaWiabigdaXiabigdaXmaaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIYaGmcqaI5aqoaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIWaamcqaI2aGnaaaaaa@4191@
0.100
±
0.016
−
0.032
+
0.023
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabigdaXiabicdaWiabicdaWiabgglaXkabicdaWiabc6caUiabicdaWiabigdaXiabiAda2maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIZaWmcqaIYaGmaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIYaGmcqaIZaWmaaaaaa@4173@
Im(r102
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdkhaYnaaDaaaleaacqaIXaqmcqaIWaamaeaacqaIYaGmaaaaaa@2F5F@)
0.040
±
0.007
−
0.020
+
0.010
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabicdaWiabisda0iabicdaWiabgglaXkabicdaWiabc6caUiabicdaWiabicdaWiabiEda3maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIYaGmcqaIWaamaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaIWaamaaaaaa@416B@
0.024
±
0.009
−
0.020
+
0.005
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabicdaWiabikdaYiabisda0iabgglaXkabicdaWiabc6caUiabicdaWiabicdaWiabiMda5maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIYaGmcqaIWaamaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIWaamcqaI1aqnaaaaaa@417B@
0.029
±
0.009
−
0.011
+
0.012
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabicdaWiabikdaYiabiMda5iabgglaXkabicdaWiabc6caUiabicdaWiabicdaWiabiMda5maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaIXaqmaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaIYaGmaaaaaa@4181@
0.031
±
0.009
−
0.012
+
0.016
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabicdaWiabiodaZiabigdaXiabgglaXkabicdaWiabc6caUiabicdaWiabicdaWiabiMda5maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaIYaGmaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaI2aGnaaaaaa@417D@
0.026
±
0.015
−
0.005
+
0.028
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabicdaWiabikdaYiabiAda2iabgglaXkabicdaWiabc6caUiabicdaWiabigdaXiabiwda1maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIWaamcqaI1aqnaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIYaGmcqaI4aaoaaaaaa@4189@
Im(r1−12
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdkhaYnaaDaaaleaacqaIXaqmcqGHsislcqaIXaqmaeaacqaIYaGmaaaaaa@304E@)
−
0.186
±
0.009
−
0.024
+
0.009
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabgkHiTiabicdaWiabc6caUiabigdaXiabiIda4iabiAda2iabgglaXkabicdaWiabc6caUiabicdaWiabicdaWiabiMda5maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIYaGmcqaI0aanaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIWaamcqaI5aqoaaaaaa@428A@
−
0.148
±
0.011
−
0.015
+
0.019
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabgkHiTiabicdaWiabc6caUiabigdaXiabisda0iabiIda4iabgglaXkabicdaWiabc6caUiabicdaWiabigdaXiabigdaXmaaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaI1aqnaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaI5aqoaaaaaa@427A@
−
0.124
±
0.012
−
0.013
+
0.029
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabgkHiTiabicdaWiabc6caUiabigdaXiabikdaYiabisda0iabgglaXkabicdaWiabc6caUiabicdaWiabigdaXiabikdaYmaaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaIZaWmaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIYaGmcqaI5aqoaaaaaa@426E@