1. Y. Sumino (Tohoku Univ.) Basics of potential-NRQCD and Quarkonium Spectroscopy

2. Heavy Quarkonium: ? For, bound-state theory based on pert. QCD is valid. gluons with decouples Appropriate EFT NRQCD potential NRQCD (pNRQCD) velocity NRQCD (vNRQCD) Caswell, Lepage Pineda, Soto Manohar, Stewart

3. Virtue of EFT Principle sym. & small parameter Constrain possible interactions Systematic expansion Matching to full theory OPE (separating pert. vs non-pert. effects) Wilson coeffs.Matrix elements of operators cf. Chiral PT

4. Relevant d.o.f. and interacting via potential(s) + IR gluons (ultrasoft gluons) To undestand systematic expansion, consider hydrogen-like atom (pNRQED) Bohr radius Bound-stateIR photon All other (short-dist.) modes are “integrated out” Effects incorporated into Wilson coefficients=“potentials”. Expansion parameters:

5. Lagrangian where should be expanded in. Field redefinition gauge singlet Expansion parameters:

6. Lagrangian gauge inv. at each order of -expansion : electric field at LO Lagrangian dipole interaction … US photon e.g. Lamb shift with Propagator of hydrogen-like atom in QM:

7. Systematic expansion in “Potential”=Wilson coeff.: function of Non-Local in Local in Integrating out

8. Potential-NRQCD and OPE of QCD potential QCD potential: as : OPE: expansion in Brambilla, Pineda, Soto, Vairo IR contr.UV contr. singlet octet singlet US gluon

9. OPE: expansion in Brambilla, Pineda, Soto, Vairo IR contr.UV contr. singlet octet singlet US gluon

10. OPE: expansion in Brambilla, Pineda, Soto, Vairo IR contr.UV contr. singlet octet singlet US gluon : Wilson coeff.,, perturbative : non-pert. matrix element + Potential NRQCD is valid below cut-off scale, where. prediction of OPE and pert. QCD non-pert. matrix element Folklore: pert., non-pert. Wrong at r < 0.5 fm !

11. Y.S. Accuracy of perturbative prediction of improved drastically around 1998, due to discovery of renormalon cancellation in. Hoang, Smith, Steltzer, Willenbrock; Beneke

12. Application to quarkonium spectroscopy and determination of. Global level structure of bottomonium is reproduced. Brambilla, Sumino, Vairo Determination of bottom and charm quark MS masses: Brambilla, Sumino, Vairo Fine and hyperfine splittings of charmonium/bottomonium reproduced. In particular, mass of is predicted correctly. Recksiegel, Y.S. However, mass of disagrees: Relation between lattice and MS is accurately measured (quenched approx.) Y.S. Recksiegel, Y.S (prediction) (exp.09)

13. D.o.f. that are integrated out in pNRQCD: hard (, + gluon): soft (, + gluon): potential (gluon): No other scales are known to play roles in Coulombic bound-states. Beneke, Smirnov potential (, ): ultrasoft (gluon): Dynamical d.o.f. in pNRQCD:

14. 3. QCD Potential: OPE Analysis at §

15. QCD Potential: Pert. QCD

16. Lattice QCD: Determination of

17. OPE of Wilson coeff. non-pert. contr. cancel