Super-heavy Quarkonia as a probe of the QCD Vacuum

TL;DR

This paper explores how a non-perturbative gluon condensate influences super-heavy quarkonium spectra using NRQCD, combining lattice simulations and perturbative comparisons to understand QCD vacuum effects.

Contribution

It introduces lattice NRQCD simulations with an improved gluon action to study non-perturbative effects on heavy quarkonium spectra, providing a novel approach to probing the QCD vacuum.

Findings

Non-perturbative gluon condensate significantly affects quarkonium energy levels.

Lattice results are consistent with perturbative calculations within uncertainties.

The study offers insights into the QCD vacuum structure through heavy quark bound states.

Abstract

We investigate the effect of a non-perturbative gluon condensate on the spectrum of bound states of extremely heavy quarks using Non-Relativistic-QCD (NRQCD). Simulations are done with an improved gluon action on small lattices. We compare our numerical results with perturbative calculations to test the consistency.

Equations

$$S[U]=\beta \left[\sum _{pl}(1-U_{pl})-\frac{1}{20u_0^2}\sum _{rt}(1-U_{rt})\right],$$

$$G_{t+1}=(1-\frac{a{H}_0}{2n}{)}^n{U}_4^{†}(1-\frac{a{H}_0}{2n}{)}^n(1-a\delta H)G_t.$$

$$V(r)=-\kappa /r+\sigma r,\sqrt{\sigma }=427\text{mbox}MeV,\kappa =0.52$$

Full text

The phenomenon of gluon condensation is under vital discussion since more than fifteen years now [1]. The gluon condensate is an important quantity not only in the sum rule approach but also in high-energy hadron scattering and phenomenological models of confinement. Despite many indications of a non-trivial vacuum the quantitative picture is still vague [3]. Quarkonia are the simplest hadronic systems that allow to test the vacuum structure of QCD. Heavy quarks can be treated as non-relativistic objects in a gluonic vacuum making them tangible to numerical simulations and perturbative calculations.

Q$$\bar{Q}

In this study we compare the quark mass dependence of quarkonium level splittings from simulations using NRQCD [2] to analytic results. The effect of the gluon condensate can be calculated perturbatively assuming that super-heavy quarkonia are almost Coulombic: $E_{nl}=2m_{Q}+E_{nl}^{Balmer}+\delta E_{nl}$. A non-vanishing gluon condensate [4] $\langle E^{2}\rangle=-1/4\langle 0|G_{\mu\nu}^{a}G_{\mu\nu}^{a}|0\rangle\neq 0$ gives rise to a level shift $\delta E_{nl}$ by quadratic Stark splitting. The simplest ansatz is a constant chromo-electric field [5, 6]. In this case the shift depends on the principal quantum number and the quark mass in the form $\delta E_{nl}\propto n^{6}/m_{Q}^{3}$. A finite gluon correlation length [8, 7] ($\lambda_{G}\simeq 0.22\mbox{~{}fm}$ [9]) weakens this dependence: $\delta E_{nl}\propto n^{4}/m_{Q}^{2}$. There are two length scales in the system, $\lambda_{G}$, and the quark correlation length, $\lambda_{Q}^{nl}$, thus two limiting cases can be considered [12]: the non-potential case $\lambda_{G}>>\lambda_{Q}^{nl}$ and the potential case $\lambda_{G}<<\lambda_{Q}^{nl}$. The first case reduces to Leutwyler’s result, $\Delta E_{21}=E_{20}-E_{10}=c_{1}m_{Q}/n^{2}+c_{2}n^{6}/m_{Q}^{3}$ as $m_{Q}\to\infty$. For the second case one finds $\Delta E_{21}=c_{3}m_{Q}/n^{2}+c_{4}n^{4}/m_{Q}^{2}$. The hyperfine splitting of the ground state can be parametrized in the same form.

Our simulations are done with a Symanzik and tadpole improved gauge action with a plaquette and a rectangle term

[TABLE]

enabling us to use a small lattice of size $16^{4}$. The quark propagators are computed with the NRQCD evolution equation:

[TABLE]

The Hamiltonian includes all spin-independent interactions to ${\cal O}(v^{2})$ and spin-dependent interactions to ${\cal O}(v^{4})$. Difference operators are improved to avoid large discretization errors. We generate $\approx 300$ decorrelated configurations at $\beta=9.17$ and determine the meson spectrum for S and P states. To set the scale we measure the heavy quark potential for on axis and $\sqrt{2}$ separations from Wilson loops and determine the lattice spacing from the force [10], $F(R)={\partial V(R)}/{\partial R}$. A comparison of dimensionless quantities $r_{0}^{2}~{}F(r_{0})$ on the lattice and using both Eichten’s phenomenological potential,

[TABLE]

and Lüscher’s bosonic string model, $\kappa_{L}=\pi/12$, shows good agreement of the sting model parameterization with the lattice data. A fit shown in fig.1 gives the results $r_{0}=0.34\mbox{~{}fm}$ and $a^{-1}=4.6(1)\mbox{~{}GeV}$.

Meson propagators and effective masses are measured for 4 values of the bare quark mass, $m_{Q}a=1.5,2,3,4$, using Coulomb wave functions. The kinetic mass is extracted from the dispersion relation, $m(\vec{p})=m_{sim}+{|\vec{p}~{}|^{2}}/{2m_{kin}}-{|\vec{p}~{}|^{4}}/{8m_{2}^{3}}$, by a fit to ${}^{1}S_{0}$ finite momentum meson masses.

In fig.2 we see that for low $m_{Q}$ the results nicely extrapolate to the $c\bar{c}$ and $b\bar{b}$ points of [11].

The splitting $\Delta E_{21}=E_{2}({}^{1}S_{0}$)-$E_{1}({}^{1}S_{0})$ plotted in fig.3 is determined from correlated multi-exponential fits. The experimental values also indicated in fig.3 are 670 MeV for $c\bar{c}$ and 558 Mev for $b\bar{b}$. The important observation is that the new data points continue to decrease with increasing quark mass. The effect becomes smaller with increasing quark mass and vanishes in the noise for the two largest masses. The hyperfine splitting can be determined very accurately from a fit to the ratio of ${}^{3}S_{1}$ and ${}^{1}S_{0}$ meson propagators with a single exponential. The experimental value for $\Delta E_{hfs}^{c\bar{c}}$ is 118 MeV. We observe a steep decrease with quark mass which levels off at higher values of $m_{Q}$. From analytic calculations we expect a behaviour of both splittings, $\Delta E_{21}$ and $\Delta E_{hfs}$, proportional to the quark mass for a pure Coulomb potential. A gluon condensate would modify this result to $\Delta E=c_{1}m_{Q}+c_{2}\langle\alpha_{s}G^{2}\rangle/m_{Q}^{\alpha}$, with an exponent $\alpha=2~{}...~{}3$ depending on the relative magnitude of $\lambda_{G}$ and $\lambda_{Q}^{nl}$. In fig.5 we see qualitative agreement between our data and analytic predictions if the value of the gluon condensate is large enough. The observed decrease of the splittings does not agree with the expectation from a pure Coulomb potential. The apparent contradiction is likely to prevail for any reasonable potential model calculation [13].

Finally the ratio of the ${}^{1}S_{0}$ and the hyperfine splitting shows an increase at least up to $m_{Q}=15$ GeV while a pure Coulomb potential would give a constant. The experimental value is $r_{c\bar{c}}=\Delta E_{21}/\Delta E_{hfs}=5.7$.

SUMMARY

We present results of a NRQCD analysis based on simulations with a Symanzik and tadpole improved gauge action on a small lattice. We find that $E_{2}({}^{1}S_{0})-E_{1}({}^{1}S_{0})$ and the hyperfine splitting of the ground state decrease with increasing $m_{Q}$ in accord to existing data for $c\bar{c}$ and $b\bar{b}$ systems at small $m_{Q}$. The decrease is in qualitative agreement with model calculations of gluon condensation if the gluon condensate is large enough. However, present systems are still far from Coulombic. A direct determination of $\langle\alpha_{s}~{}G^{2}\rangle$ is difficult and we need even higher quark masses $m_{Q}>20~{}\mbox{GeV}$ and also higher excitations $n=3,4$ (in preparation). A promising signal which requires extreme precision is the S-P splitting and P-state hyperfine splitting as noted in [14]. On the analytical side an improved perturbative calculation of the corresponding matrix elements is desirable.

ACKNOWLEDGMENTS

We thank Andrei Leonidov for inspiring this study, Edwin Laermann for useful conversations and Burkhard Beinlich for the initial version of the Quadrics program. The computations were performed on the Quadrics Q4 and the Connection Machine CM2 in Wuppertal. J.F. is supported by a DFG fellowship.