Leptonic Decay of theϒ(1S)Meson at Third Order in QCD

Abstract

We present the complete next-to-next-to-next-to-leading order short-distance and bound-state QCD correction to the leptonic decay rate Γ(Υ(1S) → ℓ + ℓ − ) of the lowest-lying spin-1 bottomonium state. The perturbative QCD prediction is compared to the measurement Γ(Υ(1S) → e + e − ) = 1.340(18) keV.PACS numbers: 13.20. Gd, 12.38.Bx Bound states of a heavy quark and antiquark provide an ideal laboratory to study non-relativistic quantum chromodynamics (NRQCD). The bound-state dynamics is characterized by three scales, the mass of the heavy quark (hard scale), m, its typical momentum (soft scale), mv, and energy (ultrasoft scale), mv 2 . Here v ∼ α s (mv) is the velocity of the quark in the bound state and α s the strong coupling. The theoretical description of heavyquark bound states uses the fact that the different scales are well-separated since the velocity is small. This allows to construct a series of effective theories by integrating out the larger scales. Starting from QCD, the first step is to integrate out the hard modes to obtain NRQCD [1][2][3]. The second step is to integrate out potential and soft gluons and soft light quarks, leading to potential NRQCD (PNRQCD) [4]. PNRQCD contains only potential heavy quarks, whose energy and momentum are of order mv 2 and mv, respectively, and ultrasoft gluons and light quarks.A "classical" application of NRQCD is the prediction of the decay rate of heavy-quark bound states into leptons. The simplest such system is the Υ(1S) meson, the lowest-lying spin-triplet bound state of a bottom quark and antiquark. To next-to-next-to-next-to-leading order accuracy (N 3 LO) the decay rate can be computed with the help of the formula [5]with α being the fine structure constant and m b the bottom-quark pole mass. c v and d v are matching constants of leading and sub-leading bb currents in NRQCD, and ψ 1 (0) is the wave function of the (bb) system at the origin, which at leading order is given byThe mass of the Υ(1S) is M Υ(1S) = 2m b + E 1 , and the perturbative part of the binding energy E 1 is given at leading order by E p,LO 1 = −(4m b α 2 s )/9. In the following we assume that the bound-state dynamics of the Υ(1S) state is governed by weak coupling, which formally requires that the ultrasoft scale m b v 2 is large compared to the strong interaction scale Λ. It is generally believed that this is a reasonable assumption for the lowest-lying 1S state, but not for the higher states, which, though more non-relativistic, are too large to be considered as bound states dominated by the colour-Coulomb interaction. Even for the 1S state the assumption m b v 2 ≫ Λ is questionable. In fact, the leptonic decay that we consider in this Letter should be considered as one of the crucial tests of perturbative QCD bound-state dynamics, when all three scales (hard, soft, ultrasoft) are relevant to the problem. The more recent analyses of the leptonic Υ(1S) decay are based on next-to-leading order QCD together with nonperturbative condensate corrections [6], or second-order QCD without non...

Bottom quark mass from $ \varUpsilon $ sum rules to $ \mathcal{O}\left( {\alpha_s^3} \right) $

“…The third-order term for general n has only been evaluated in the logarithmic approximation [5,6,37]. Recently the total third-order correction for n = 1 has been published in a numerical form [7]. Below we present the result for the excited states.…”

“…Though significant nonperturbative effects are expected in the QCD analysis of the Υ-resonance mass and width, it is instructive to figure out how perturbative QCD results [4,7] reproduce the experimental data for the lowest Υ(1S) state, where the nonperturbative contribution is minimal. From the result of the previous section we get the following numerical series for the ground state mass and width for m c = 0 …”

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Y and ψ leptonic decays as probes of solutions to the R(D (*)) puzzle

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