Effective-field theories for heavy quarkonium

This article reviews recent theoretical developments in heavy-quarkonium physics from the point of view of effective-field theories of QCD. We discuss nonrelativistic QCD and concentrate on potential nonrelativistic QCD. The main goal will be to derive Schrödinger equations based on QCD that govern heavy-quarkonium physics in the weak- and strong-coupling regimes. Finally, the review discusses a selected set of applications, which include spectroscopy, inclusive decays, and electromagnetic threshold production.

Figure 1

Relevant one-loop diagrams for the matching of the four-fermion operators at order $\mathcal{O}(1∕m^2)$ for the case of unequal masses. The incoming and outgoing particles are on shell and exactly at rest.

Figure 2

Relevant diagrams for the matching of the four-fermion operators at order $\mathcal{O}(1∕m^2)$ and one loop that only appear for the equal-mass case. The incoming and outgoing particles are on shell and exactly at rest.

Figure 3

(Color online) Radial and orbital splittings in the ${\rm Y}$ system from lattice QCD in the quenched approximation (open circles) and including dynamical $u$, $d$, and $s$ quarks (filled circles). The lattice spacing has been fixed by the radial splitting between ${\rm Y}\left(2S\right)$ and ${\rm Y}\left(1S\right)$. The $b$-quark mass has been tuned to get the ${\rm Y}$ mass correct. From 200.

Figure 4

(Color online) Mass gap between the singlet and hybrid fields. From 23.

Figure 5

Propagators and vertices of the pNRQCD Lagrangian (60). Dashed lines represent longitudinal gluons and curly lines transverse gluons. $P^{\mu }$ represents the gluon incoming momentum.

Figure 6

Matching between NRQCD and pNRQCD without considering pinch singularities. The dashed and curly lines represent the longitudinal and transverse gluon exchanges, respectively.

Figure 7

Matching between NRQCD and pNRQCD taking into account pinch singularities. In pNRQCD the loop regulates the pinch singularity.

Figure 8

A graphical representation of the static Wilson loop. We adopt the convention that the time direction is from the left to the right. Therefore the quark trajectories are represented by horizontal lines and the equal-time end-point Wilson lines by shorter vertical lines.

Figure 9

The matching of $V_s^{\left(0\right)}$ and $Z_s^{\left(0\right)}$ at next-to-leading order (NLO) in the multipole expansion. On the left-hand side is the Wilson loop in NRQCD, on the right-hand side are the pNRQCD propagators. The first and second diagrams on the right-hand side symbolically represent the first and second terms in Eq. (125).

Figure 10

The matching of $V_o^{\left(0\right)}$ and $Z_o^{\left(0\right)}$ at NLO in the multipole expansion. On the left-hand side is the Wilson loop in NRQCD with color matrix insertions, on the right-hand side are the pNRQCD propagators.

Figure 11

Octet self-energy graph proportional to $V_B^2$.

Figure 12

One possible contribution to the running of $D_{d,s}^{\left(2\right)}$ at next-to-leading-logarithm order (NLL). The first picture represents the calculation in terms of the free quark-antiquark propagator $G_c^{\left(0\right)}$ and the potentials (the small rectangles). The picture on the right is the representation within a more standard diagrammatic interpretation in terms of quarks and antiquarks. The delta potentials are displayed as local interactions and the Coulomb potential as an extended object in space (but not in time).

Figure 13

(Color online) Different hybrid potentials (169) at a lattice spacing $a_{\sigma }\approx 0.2\mathrm{fm}\approx 0.4r_0$, where $r_0\approx 0.5\mathrm{fm}$ is the scale for which ${\phantom{\mid }-r_0^{2}dV∕dr\mid }_{r=r_0}=1.65$ (285), in comparison with the gluelump spectrum (133) (circles, left-most data points). The gluelump spectrum has been shifted by a constant to adjust the $1^{+-}$ state with the ${\Pi }_u$ and ${\Sigma }_u^{-}$ potentials at short distance. In addition, we include the sum of the ground-state $\left({\Sigma }_g^{+}\right)$ potential and the scalar glueball mass $m_{0^{++}}$ (22; 236; 203). The lines are drawn to guide the eye. From 20.

Figure 14

(Color online) Plot of $r_0[V_{\mathrm{RS}}\left(r\right)-V_{\mathrm{RS}}\left(r^{\prime }\right)+E_{\mathrm{latt.}}\left(r^{\prime }\right)]$ vs $r$ at tree (dashed line), one-loop (dash-dotted line), two-loop (dotted line), and three-loop level (estimate) plus the renormalization-group (RG) expression for the ultrasoft (US) logarithms (solid line) compared with the lattice simulations $E_{\mathrm{latt.}}\left(r\right)$ (238). For the scale of ${\alpha }_s\left(\nu \right)$, we set $\nu =1∕r$. Further, ${\nu }_f={\nu }_{\mathit{us}}=2.5r_0^{-1}$, ${\Lambda }_{\overline{\mathrm{MS}}}=0.602r_0^{-1}$ (93), and $r^{\prime }=0.15399r_0$. From 264.

Figure 15

(Color online) Splitting between the ${\Sigma }_u^{-}$ and ${\Pi }_u$ potentials extrapolated to the continuum limit and comparison with a quadratic fit to the $r\lesssim 0.5r_0$ data points $(r_0^{-1}\approx 0.4\mathrm{GeV})$. The big circles correspond to the data obtained by 169. The errors in this case are smaller than the symbols. The smaller circles correspond to the data obtained by 20. From 20.

Figure 16

The incoming energy $E$ is of order ${\Lambda }_{\mathrm{QCD}}$, $p$, and $q$ of order $mv$. The vertex describes the interaction with an external potential $V$.

Figure 17

The Coulomb-exchange graph contributing to the leading mixing interaction between semihard and potential fields.

Figure 18

(a) The spin-orbit potential $-V_1^{\prime }$ with the fit $\sigma +h∕r^2$ and (b) the potential $V_d$ together with the curve $-\sigma ∕9r$. The lattice simulations are quenched. The fitting parameters are $\sigma \approx {\left(468\mathrm{MeV}\right)}^2$ and $h\approx 0.067$. From 21.

Figure 19

(Color online) The spin ratio as a function of the renormalization scale $\nu$ in $\mathrm{LO}\equiv \mathrm{LL}$ (dotted line), NLO (short-dashed line), NNLO (long-dashed line), NLL (dot-dashed line), and NNLL (solid line) approximation. For the NNLL result the band reflects the errors due to ${\alpha }_s\left(M_Z\right)=0.118\pm 0.003$. (a) The bottomonium ground-state case for which ${\nu }_h=m_b$. (b) The charmonium ground-state case for which ${\nu }_h=m_c$. In the charmonium case, the upper band represents the experimental error of the ratio (121), where the central value is given by the horizontal solid line. From 246.

Figure 20

(Color online) Plot of the one-loop RG-improved expression for $\mathcal{E}$ vs $\mu :\mathcal{E}\left(\mu \right)=\mathcal{E}\left(m\right)+(24N_c{C}_F∕{\beta }_0)\ln [{\alpha }_s\left(m\right)∕{\alpha }_s\left(\mu \right)]$. $\mathcal{E}\left(m\right)$ has been extracted from charmonium $P$-wave data. The error band accounts only for the uncertainties inherited from the charmonium data. From 246.

Figure 21

(Color online) The left-hand side of Eqs. (341, 342) plotted vs $\mu$. We have taken ${\mathcal{E}}_3$ from Fig. 20. The LO and NLO bands refer to the Wilson coefficients at LO and NLO, respectively. From 297.

Figure 22

(Color online) (a) The results for $e_t^2{R}^v$ ($e_t=2∕3$ is the top-quark electric charge) with $m_t=175\mathrm{GeV}$ (the $1S$ threshold mass is used, see Sec. 5) and ${\Gamma }_t=1.43\mathrm{GeV}$ in fixed-order perturbation theory at LO (dotted lines), NLO (dashed lines), and NNLO (solid lines). (b) The results for $e_t^2{R}^v$ with the same parameters in RG-improved perturbation theory at LL (dotted lines), NLL (dashed lines), and (partial) NNLL (solid lines) order. For each order, curves are plotted for ${\nu }_p∕m_t=0.15$, 0.20, and 0.3. From 156.

Figure 23

(Color online) End-point region of the photon spectrum in semi-inclusive ${\rm Y}$ decay. The points are CLEO data (239), the dashed line is the (best) curve obtained by 128, and the solid and dot-dashed lines are the results of 135 (the solid line is the central value and the dot-dashed lines are obtained by a $2^{\pm 1}$ variation of the relevant scale). From 135.