Study of systematic errors of bound-state parameters in SVZ sum rules
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ELEMENTARY PARTICLES AND FIELDS Theory
Study of Systematic Errors of Bound-State Parameters in SVZ Sum Rules* W. Lucha1), D. I. Melikhov1), 2) , and S. Simula3) Received July 23, 2007; in final form, January 23, 2008
Abstract—We study systematic errors of the ground-state parameters obtained from Shifman– Vainshtein–Zakharov sum rules, making use of the harmonic-oscillator potential model as an example. In this case, one knows the exact solution for the polarization operator, which allows one to obtain both the OPE to any order and the parameters (masses and decay constants) of the bound states. We determine the parameters of the ground state making use of the standard procedures of the method of sum rules and compare the obtained results with the known exact values. We show that, in the situation when the continuum contribution to the polarization operator is not known and is modeled by an effective continuum, the method of sum rules does not allow one to control the systematic uncertainties of the extracted groundstate parameters. PACS numbers: 11.55.Hx, 12.38.Lg, 03.65.Ge DOI: 10.1134/S1063778808080188
1. INTRODUCTION A QCD sum-rule calculation of hadron parameters [1] involves two steps: (i) one calculates the operator product expansion (OPE) series for a relevant correlator and obtains the sum rule which relates this OPE to the sum over hadronic states, and (ii) one extracts the parameters of the ground state by a numerical procedure. Each of these steps leads to uncertainties in the final result. The first step lies fully within QCD and allows a rigorous treatment of the uncertainties: the correlator in QCD is not known precisely (because of uncertainties in quark masses, condensates, αs , radiative corrections, etc.), but the corresponding errors in the correlator may be systematically controlled (at least in principle). The second step lies beyond QCD and is more cumbersome: even if several terms of the OPE for the correlator were known precisely, the hadronic parameters might be extracted by a sum rule only within some error, which may be treated as a systematic error of the method. It is useful to recall that a successful extraction of the hadronic parameters by a sum rule is not guaranteed: as already noted in the ∗
The text was submitted by the authors in English. Institute for High Energy Physics, Austrian Academy of Sciences, Vienna, Austria. 2) Institute of Nuclear Physics, Moscow State University, Moscow, Russia. 3) INFN, Sez. di Roma III, Roma, Italy. 1)
classical papers [1, 2], the method may work in some cases and fail in others; moreover, error estimates (in the mathematical sense) for the numbers obtained by sum rules may not be easily provided—e.g., according to [1], any value obtained by varying the parameters in the sum-rule stability region has equal probability. However, for many applications of sum rules, especially in flavor physics, one needs rigorous error estimates of the theoretical results for comparing theoretical predictions with the experimental data. Systematic errors of the sum-r
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