Y. Sumino (Tohoku Univ.)

1. Modern View of Perturbative QCD and Application to Heavy QuarkoniumSystem （現在の視点から見る摂動ＱＣＤ及び 重いクォーコニウム系への応用） Y. Sumino (Tohoku Univ.)

2. ☆Plan of Talk Review of Pert. QCD (Round 1, Quick overview) • What’s Pert. QCD? • Today’s computational technologies 2. Review of Pert. QCD (Round 2, Some details) 3. Application to Heavy QuarkoniumSystem • physics in the heavy quark mass and interquark force (4. More details of specific interests, upon request)

3. Review of Pert. QCD (Round 1, Quick overview) What’s Pert. QCD? 3 types of so-called “pert. QCD predictions” : (Confusing without properly distinguishing between them.) (i) Predict observable in series expansion in IR safe obs., intrinsic uncertainties (ii) Predict observable in the framework of Wilsonian EFT OPE as expansion in , uncertainties of (i) replaced by non-pert. matrix elements Do not add these non-pert. corr. to (i). (iii) Predict observable assisted by model predictions Many obs in high-energy experiments depend on hadronization models, PDFs. Necessary (in MC) to compare with experimental data Systematic uncertainties difficult to control, O(10%) accuracy at LHC

4. Remarkable progress of computational technologies in the last 10-20 years (i) Higher-loop corrections Resolution of singularities in multi-loop integrals Numerical and analytical methods Cross-over with frontiers of mathematics (ii) Lower-order (NLO/NLL) corrections to complicated processes Cope with proliferation of diagrams and many kinematical variables Motivated by LHC physics (iii) Factorization of scales in loop corrections Provide powerful and precise foundation for constructing Wilsonian EFT Dim. reg.: common theoretical basis Essentially analytic continuation of loop integrals Contrasting/complementary to cut-off reg.

5. Comment on Impacts on Physics Insights: new interpretations, viewpoints, concepts, … To date, scattered over specialized fields, yet to frame a general overview Examples: • Various EFTs triggered new paradigms, such as HQET for b-physics, SCET for jets • physics in the heavy quark mass and interquark force cannot appear in series expansion in?

6. 2. Review of Pert. QCD (Round 2, Some details) 3 types of so-called “pert. QCD predictions” : (i) Predict observable in series expansion in (ii) Predict observable in the framework of Wilsonian EFT (iii) Predict observable assisted by model predictions

7. 2. Review of Pert. QCD (Round 2, Some details) 3 types of so-called “pert. QCD predictions” : (i) Predict observable in series expansion in (ii) Predict observable in the framework of Wilsonian EFT (iii) Predict observable assisted by model predictions

8. 2. Review of Pert. QCD (Round 2, Some details) 3 types of so-called “pert. QCD predictions” : (i) Predict observable in series expansion in (ii) Predict observable in the framework of Wilsonian EFT (iii) Predict observable assisted by model predictions

9. renormalization scale Pert. QCD Theory of quarks and gluons Same input parameters as full QCD. Systematic: has its own way of estimating errors. (Dependence on is used to estimate errors.) Differs froma model Predictable observables testable hypothesis (i) Inclusive observables (hadronic inclusive) insensitive to hadronization • -ratio: • Inclusive decay widths • Distributions of non-colored particles, (ii) Observables of heavy quarkonium states (the only individual hadronic states) • spectrum, leptonic decay width, transition rates

10. IR sensitivity at higher-order Renormalon uncertainty -ratio:

12. Infinite sum

13. Consequence Renormalon uncertainty ~ Asymptotic series (Empirically good estimate of true corr.) Limited accuracy

14. 2. Review of Pert. QCD (Round 2, Some details) 3 types of so-called “pert. QCD predictions” : (i) Predict observable in series expansion in (ii) Predict observable in the framework of Wilsonian EFT (iii) Predict observable assisted by model predictions

15. integrate out Wilsonian EFT in terms of light quarks and IR gluons EFT less d.o.f. Determine Wilson coeffssuch that physics at is unchanged, viapert. QCD: Matching Asymptotic expansion of diagrams include only UV contr. Free from IR renormalon uncertainties

16. OPE in Wilsonian EFTmultipole expansion integrate out Observable which includes a high scale light quarks and IR gluons replace renormalons

17. Remarkable progress of computational technologies in the last 10-20 years (i) Higher-loop corrections Resolution of singularities in multi-loop integrals Numerical and analytical methods Cross-over with frontiers of mathematics (ii) Lower-order (NLO/NLL) corrections to complicated processes Cope with proliferation of diagrams and many kinematical variables Motivated by LHC physics (iii) Factorization of scales in loop corrections Provide powerful and precise foundation for constructing Wilsonian EFT Dim. reg.: common theoretical basis Essentially analytic continuation of loop integrals Contrasting/complementary to cut-off reg.

18. Remarkable progress of computational technologies in the last 10-20 years (i) Higher-loop corrections Resolution of singularities in multi-loop integrals Numerical and analytical methods Cross-over with frontiers of mathematics (ii) Lower-order (NLO/NLL) corrections to complicated processes Cope with proliferation of diagrams and many kinematical variables Motivated by LHC physics (iii) Factorization of scales in loop corrections Provide powerful and precise foundation for constructing Wilsonian EFT Dim. reg.: common theoretical basis Essentially analytic continuation of loop integrals Contrasting/complementary to cut-off reg.

19. Dim. reg. Advantages • Preserves important symmetries (Lorentz sym, gauge sym) • In a single step, all loop integrals are rendered finite; both UV and IR. • (cf. Pauli-Villars reg.) • Many useful computational techniques Disadvantages • Not defined as a quantum field theory (cf. lattice reg.) • Nevertheless, well-definedand uniquely defined in pert. computations. • Difficult to interpret physically Does represent IR or UV divergence? Unphysical equalities? Is only UV part of the theory modified? (I can give an argument why I believe dim. reg. leads to correct predictions.)

20. Integration-by-parts (IBP) Identities Chetyrkin, Tkachov Most powerful application of Dim. Reg. ; Standard technology used to reduce a large number of loop integrals to a small set of integrals (master integrals). Example:

21. Integration-by-parts (IBP) Identities Chetyrkin, Tkachov Most powerful application of Dim. Reg. ; Standard technology used to reduce a large number of loop integrals to a small set of integrals (master integrals). Example:

22. Remarkable progress of computational technologies in the last 10-20 years (i) Higher-loop corrections Resolution of singularities in higher-loop integrals cross-over with frontiers of mathematics (ii) Lower-order (NLO/NNLO/NLL) corrections to complicated processes Cope with proliferation of diagrams and many variables Strongly motivated by LHC physics (iii) Factorization of scales in loop corrections Provide powerful and precise foundation for constructing Wilsonian EFT Dim. reg. as the common theoretical basis to all of them Essentially analytic continuation of loop integrals Contrasting to cut-off reg.

23. Asymptotic Expansion of Diagrams Simplified example:

24. Asymptotic expansion of a diagram and Wilson coeffs in EFT in the case

25. Asymptotic expansion of a diagram and Wilson coeffs in EFT L H H L L H in the case = = = = = Vertices and Wilson coeffs in EFT

26. Remarkable progress of computational technologies in the last 10-20 years (i) Higher-loop corrections Resolution of singularities in higher-loop integrals cross-over with frontiers of mathematics Theory of Multiple Zeta Values (MZV) (ii) Lower-order (NLO/NNLO/NLL) corrections to complicated processes Cope with proliferation of diagrams and many variables Strongly motivated by LHC physics (iii) Factorizing and separating scales in loop corrections Provide solid and precise foundation for constructing Wilsonian EFT Dim. reg. as the common theoretical basis to all of them Essentially analytic continuation of loop integrals Contrasting to cut-off reg.

27. Example: Anomalous magnetic moment of electron () terms omitted Li4 ln

28. ☆ Generalized Multiple Zeta Value (MZV) Given as a nested sum Can also be written in a nested integral form e.g. Li4 ln

29. MZVs can be expressed by a small set of basis (vector space over ) weight For : e.g. Dimension=1 at weight 3: . weight dim #(MZVs) Shuffle relations are powerful in reducing MZVs. (Probably sufficient for .) New relations for:Anzai,YS MZV as a period of cohomology, motives

30. Relation between topology of a Feynman diagram and MZVs? What kind of MZVs are contained in a diagram? Which s ?

31. Singularities in Feynman Diagrams ☆ Classes of singularities in a Feynman diagram • IR singularity • UV singularity • Mass singularity • Threshold singularity Complex -plane cuts also log singularity at

32. What kind of MZVs are contained in a diagram? Which s ? Singularities map In simple cases all square-roots can be eliminated by (successive) Euler transf. Integrals convertible to MZVs

33. Summary of Overview Pert. QCD Higher-order computations IRrenormalons increase of at IR

34. Summary of Overview Pert. QCD OPE in WilsonianEFT Higher-order computations IRrenormalons Separation of UV & IR contr. Wilson coeffsvs. non-pert. matrix elements

35. Summary of Overview Pert. QCD OPE in WilsonianEFT only UV Higher-order computations IRrenormalons Separation of UV & IR contr. Wilson coeffsvs. non-pert. matrix elements replaced

36. Summary of Overview Pert. QCD OPE in WilsonianEFT only UV Higher-order computations IRrenormalons Separation of UV & IR contr. Wilson coeffsvs. non-pert. matrix elements replaced scale separation using analyticity Dim. reg. Asymptotic expansion integration by region

37. Summary of Overview Pert. QCD OPE in WilsonianEFT only UV Higher-order computations IRrenormalons Separation of UV & IR contr. Wilson coeffsvs. non-pert. matrix elements replaced scale separation using analyticity Dim. reg. Asymptotic expansion integration by regions Reduction by IBP identities Resolution of singularities

38. Summary of Overview Pert. QCD OPE in WilsonianEFT only UV Higher-order computations IRrenormalons Separation of UV & IR contr. Wilson coeffsvs. non-pert. matrix elements replaced scale separation using analyticity Dim. reg. Asymptotic expansion integration by region Reduction by IBP identities Resolution of singularities tough intermediate comp. Singularities Topology of a diagram MZVs final results very simple short-cut ?

39. Pert. QCD: Today’s benchmarks More than 10 digits! Universality Precisions 0.6% accuracy 0.8% accuracy 2% accuracy 3% accuracy (0.06% at ILC)

40. 3. Application to Heavy QuarkoniumSystem • physics in the heavy quark mass and interquark force • IR renormalization of Wilson coeffs in EFT

41. Static QCD Potential 3-loop pert. QCDvs. lattice comp. Anzai, Kiyo, YS

42. Consider (naively) a “short-distance expansion” at According to renormalon analysis in pert. QCD, constant and term contain uncertainties IR renormalon in is canceled in the total energy by the MS mass (). if we express the quark pole mass () Drastic improvement of convergence of pert. series

43. at IR renormalon in is canceled in the total energy by the MS mass (). if we express the quark pole mass () Drastic improvement of convergence of pert. series

44. at IR renormalon in is canceled in the total energy by the MS mass (). if we express the quark pole mass () Drastic improvement of convergence of pert. series [GeV-1] N=3 N=0 N=0 N=3 [GeV-1] Exact pert. potential up to 3 loops

45. General feature of gauge theory . Couples to total charge as

46. General feature of gauge theory . Couples to total charge as

47. What are UV contributions? at IR contributions cancel against OPE of QCD potential in Potential-NRQCD EFT Uncetainty in replaced by a non-local gluon condensate within pNRQCD US gluon singlet octet singlet

48. A ‘Coulomb+Linear potential’ is obtained by resummation of logs in pert. QCD: YS IR contributions at UV contributions

49. A ‘Coulomb+Linear potential’ is obtained by resummation of logs in pert. QCD: YS UV contributions Expressed by param. of pert. QCD

50. Formulas for Define via then In the LL case Coulombic pot. with log corr. at short-dist. Coefficient of linear potential (at short-dist.)