Rigourous QCD Evaluation of Spectrum and Other Properties of Heavy Quarkonium Systems; II Bottomium with n=2, l=0,1

We calculate the Lamb, fine and hyperfine shifts in $b\bar b$ with $n=2$, $l=0,1$. Radiative corrections as well as leading nonperturbative corrections (known to be due to the gluon condensate) are taken into account. The calculation is parameter-free, as we take $\Lambda$, ${\langle \alpha_s G^2 \rangle}$ from independent sources. Agreement with experiment is found at the expected level $\sim 30\%$. Particularly interesting is a prediction for the hyperfine splitting, $M_{\rm average}(2^3P)-M(2^1P_1) = 1.7 \pm 0.9\, {\rm MeV}$, opposite in sign to the $c\bar c$ one ($\approx -0.9\, {\rm MeV}$), and where the nonzero value of ${\langle \alpha_s G^2 \rangle}$ plays a leading role.Comment: 28 pages, preprint FTUAM 94-0

More nonperturbative corrections to the fine and hyperfine splitting in the heavy quarkonium

The leading nonperturbative effects to the fine and hyperfine splitting were calculated some time ago. Recently, they have been used in order to obtain realistic numerical results for the lower levels in bottomonium systems. We point out that a contribution of the same order $O(\Lambda_{QCD}^4/m^3 \alpha_s^2)$ has been overlooked. We calculate it in this paper.Comment: 9 pages, LaTeX, More self-contained and lengthier version without changing physical outputs. To be published in Phys. Rev.

The l=1l=1 Hyperfine Splitting in Bottomium as a Precise Probe of the QCD Vacuum.

By relating fine and hyperfine spittings for l=1 states in bottomium we can factor out the less tractable part of the perturbative and nonperturbative effects. Reliable predictions for one of the fine splittings and the hyperfine splitting can then be made calculating in terms of the remaining fine splitting, which is then taken from experiment; perturbative and nonperturbative corrections to these relations are under full control. The method (which produces reasonable results even for the $c{\bar c}$ system) predicts a value of 1.5 MeV for the $(s=1)-(s=0)$ splitting in $b{\bar b}$, opposite in sign to that in $c{\bar c}$. For this result the contribution of the gluon condensate $$ is essential, as any model (in particular potential models) which neglects this would give a negative $b{\bar b}$ hyperfine splitting.Comment: 12 pages, 2 postscript figures, typeset with ReVTe

Soft, collinear and non-relativistic modes in radiative decays of very heavy quarkonium

We analyze the end-point region of the photon spectrum in semi-inclusive radiative decays of very heavy quarkonium (m alpha_s^2 >> Lambda_QCD). We discuss the interplay of the scales arising in the Soft-Collinear Effective Theory, m, m(1-z)^{1/2} and m(1-z) for z close to 1, with the scales of heavy quarkonium systems in the weak coupling regime, m, m alpha_s and m alpha_s^2. For 1-z \sim alpha_s^2 only collinear and (ultra)soft modes are seen to be relevant, but the recently discovered soft-collinear modes show up for 1-z << alpha_s^2. The S- and P-wave octet shape functions are calculated. When they are included in the analysis of the photon spectrum of the Upsilon (1S) system, the agreement with data in the end-point region becomes excellent. The NRQCD matrix elements and are also obtained.Comment: Revtex, 11 pages, 6 figures. Minor improvements and references added. Journal versio

Renormalization group scaling in nonrelativistic QCD

We discuss the matching conditions and renormalization group evolution of non-relativistic QCD. A variant of the conventional MS-bar scheme is proposed in which a subtraction velocity nu is used rather than a subtraction scale mu. We derive a novel renormalization group equation in velocity space which can be used to sum logarithms of v in the effective theory. We apply our method to several examples. In particular we show that our formulation correctly reproduces the two-loop anomalous dimension of the heavy quark production current near threshold.Comment: (27 pages, revtex

Heavy Quarkonium and nonperturbative corrections

We analyse the possible existence of non-perturbative contributions in heavy $\bar Q Q$ systems ($\bar Q$ and $Q$ need not have the same flavour) which cannot be expressed in terms of local condensates. Starting from QCD, with well defined approximations and splitting properly the fields into large and small momentum components, we derive an effective lagrangian where hard gluons (in the non-relativistic aproximation) have been integrated out. The large momentum contributions (which are dominant) are calculated using Coulomb type states. Besides the usual condensate corrections, we see the possibility of new non-perturbative contributions. We parametrize them in terms of two low momentum correlators with Coulomb bound state energy insertions $E_n$. We realize that the Heavy Quark Effective lagrangian can be used in these correlators. We calculate the corrections that they give rise to in the decay constant, the bound state energy and the matrix elements of bilinear currents at zero recoil. We study the cut-off dependence of the new contributions and we see that it matches perfectly with that of the large momentum contributions. We consider two situations in detail: i) $E_n>> \Lambda_{QCD}$ ($M_Q \rightarrow \infty$) and ii) $E_n << \Lambda_{QCD}$, and briefly discuss the expected size of the new contributions in $\Upsilon$ , $J/\Psi$ and $B_{c}^{\ast}$ systems.Comment: 28 pages, LaTeX. Minor changes, some comments and numerical results added. To be published in Phys. Rev.

Quarkonium spectroscopy and perturbative QCD: massive quark-loop effects

We study the spectra of the bottomonium and B_c states within perturbative QCD up to order alpha_s^4. The O(Lambda_QCD) renormalon cancellation between the static potential and the pole mass is performed in the epsilon-expansion scheme. We extend our previous analysis by including the (dominant) effects of non-zero charm-quark mass in loops up to the next-to-leading non-vanishing order epsilon^3. We fix the b-quark MSbar mass $\bar{m}_b \equiv m_b^{\bar{\rm MS}}(m_b^{\bar{\rm MS}})$ on Upsilon(1S) and compute the higher levels. The effect of the charm mass decreases $\bar{m}_b$ by about 11 MeV and increases the n=2 and n=3 levels by about 70--100 MeV and 240--280 MeV, respectively. We provide an extensive quantitative analysis. The size of non-perturbative and higher order contributions is discussed by comparing the obtained predictions with the experimental data. An agreement of the perturbative predictions and the experimental data depends crucially on the precise value (inside the present error) of alpha_s(M_Z). We obtain $m_b^{\bar{\rm MS}}(m_b^{\bar{\rm MS}}) = 4190 \pm 20 \pm 25 \pm 3 ~ {\rm MeV}$.Comment: 33 pages, 21 figures; v2: Abstract modified; Table7 (summary of errors) added; Version to appear in Phys.Rev.

Heavy quark mass determination from the quarkonium ground state energy: a pole mass approach

The heavy quark pole mass in perturbation theory suffers from a renormalon caused, inherent uncertainty of $O(\Lambda_{\rm QCD})$. This fundamental difficulty of determining the pole mass to an accuracy better than the inherent uncertainty can be overcome by direct resummation of the first infrared renormalon. We show how a properly defined pole mass as well as the $\bar {\rm MS}$ mass for the top and bottom quarks can be determined accurately from the $O(m\alpha_s^5)$ quarkonium ground state energy.Comment: 16 pages; published versio

The gluonic condensate from the hyperfine splitting Mcog(χcJ)−M(hc)M_{\rm cog}(\chi_{cJ})-M(h_c) in charmonium

The precision measurement of the hyperfine splitting $\Delta_{\rm HF} (1P, c\bar c)=M_{\rm cog} (\chi_{cJ}) - M(h_c) = -0.5 \pm 0.4$ MeV in the Fermilab--E835 experiment allows to determine the gluonic condensate $G_2$ with high accuracy if the gluonic correlation length $T_g$ is fixed. In our calculations the negative value of $\Delta_{\rm HF} = -0.3 \pm 0.4$ MeV is obtained only if the relatively small $T_g = 0.16$ fm and $G_2 = 0.065 (3)$ GeV${}^4$ are taken. These values correspond to the ``physical'' string tension $(\sigma \approx 0.18$ GeV$^2$). For $T_g \ge 0.2$ fm the hyperfine splitting is positive and grows for increasing $T_g$. In particular for $T_g = 0.2$ fm and $G_2 = 0.041 (2)$ GeV${}^4$ the splitting $\Delta_{\rm HF} = 1.4 (2)$ MeV is obtained, which is in accord with the recent CLEO result.Comment: 9 pages revtex 4, no figure

Quenched charmonium spectrum

We study charmonium using the standard relativistic formalism in the quenched approximation, on a set of lattices with isotropic lattice spacings ranging from 0.1 to 0.04 fm. We concentrate on the calculation of the hyperfine splitting between eta_c and J/psi, aiming for a controlled continuum extrapolation of this quantity. The splitting extracted from the non-perturbatively improved clover Dirac operator shows very little dependence on the lattice spacing for $a \leq 0.1$ fm. The dependence is much stronger for Wilson and tree-level improved clover operators, but they still yield consistent extrapolations if sufficiently fine lattices, $a \leq 0.07$ fm ($a M(\eta_c) \leq 1$), are used. Our result for the hyperfine splitting is 77(2)(6) MeV (where Sommer's parameter, r_0, is used to fix the scale). This value remains about 30% below experiment. Dynamical fermions and OZI-forbidden diagrams both contribute to the remainder. Results for the eta_c and J/psi wave functions are also presented.Comment: 22 pages, 7 figure