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33751160253
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By the Riemann mapping theorem, this conformal transformation is unique up to the choice of t0 and an overall phase See e.g..
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By the Riemann mapping theorem, this conformal transformation is unique up to the choice of t0 and an overall phase See e.g..
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14
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33751175726
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The quantity A defined in is normalized to the leading OPE prediction: A≡(32π2mb2/3)a02AF+2.
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The quantity A defined in is normalized to the leading OPE prediction: A≡(32π2mb2/3)a02AF+2.
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16
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33751194746
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At Q2=2GeV2, the result for χ0 is consistent with a related study in, where however uncertainties due to the poor convergence of the perturbation series, and the effects of power corrections, were not considered.For Q2 m^ΛQCD, AF0∼1, with dominant contributions from t∼m^ΛQCD.
-
At Q2=2GeV2, the result for χ0 is consistent with a related study in, where however uncertainties due to the poor convergence of the perturbation series, and the effects of power corrections, were not considered.
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26
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33751180216
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For heavy-quark systems the choice of in 8 14 also ensures that AF does not scale as some power of the heavy-quark mass.
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For heavy-quark systems the choice of in 8 14 also ensures that AF does not scale as some power of the heavy-quark mass.
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27
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33751177425
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This value of t0 is insensitive to the precise form factor shape, e.g. whether F+ is assumed constant, or is input from experimental data. Since Ref. measures a distribution in transverse momentum, t π=mK2+mπ2-2mKmπ2+p 2, the partial rates dΓ/dt π involve a smearing over different t values. Taking this into account, the correlations remain very insensitive to the form factor shape, and are minimized in the ideal case for t0 ≈0.37t- (at Q2=2GeV2).
-
This value of t0 is insensitive to the precise form factor shape, e.g. whether F+ is assumed constant, or is input from experimental data. Since Ref. measures a distribution in transverse momentum, t π=mK2+mπ2-2mKmπ2+p 2, the partial rates dΓ/dt π involve a smearing over different t values. Taking this into account, the correlations remain very insensitive to the form factor shape, and are minimized in the ideal case for t0 ≈ 0.37t- (at Q2=2GeV2).
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28
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33751200573
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Another commonly used parametrization, the pole model, suffers from similar difficulties. To avoid biases, this model would need to be generalized. A pedestrian, but rigorous, approach is to break apart the dispersive integral into the sum of effective poles. Although there is no analogue of the OPE here, we can establish a bound on the coefficients of the effective poles using the τ-decay data in (and a negligible perturbative contribution) to obtain (1/π)∫t+∞dt′|F+(t′)|/t′ 2.3±0.4.
-
Another commonly used parametrization, the pole model, suffers from similar difficulties. To avoid biases, this model would need to be generalized. A pedestrian, but rigorous, approach is to break apart the dispersive integral into the sum of effective poles. Although there is no analogue of the OPE here, we can establish a bound on the coefficients of the effective poles using the τ-decay data in (and a negligible perturbative contribution) to obtain (1/π)∫t+∞dt′|F+(t′)|/t′ 2.3±0.4.
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0039819887
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33751184065
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Semileptonic data combined with the bound 17 yields a more precise value for the slope than is obtained from combining unitarity with the Callan-Treiman point in the absence of data, cf.
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Semileptonic data combined with the bound 17 yields a more precise value for the slope than is obtained from combining unitarity with the Callan-Treiman point in the absence of data, cf.
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31
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33749820673
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For some recent analysis along these lines, see: PRVDAQ 0556-2821 10.1103/PhysRevD.74.074009
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