Non-Relativistic Quantum Chromo Dynamics (NRQCD)

1. Non-Relativistic Quantum Chromo Dynamics (NRQCD) Heavy quark systems as a test of non-perturbative effects in the Standard Model Victor Haverkort en Tom Boot, 21 oktober 2009

2. Topics of Today • Motivation for NRQCD • NRQCD • Philosophy • Energy scales in heavy quark systems • Non-Relativistic version of the QCD Lagrangian • Components • Power counting; relative importance of components • Origin of the correction terms • Application of NRQCD: Annihilation • Use NRQCD to describe annihilation of heavy quarkonia (charmonium)

3. : 4 component spinor 1. Motivation • Lagrangian density of QCD • Symmetry group: SU(3) • Looks simple! Don’t forget

4. 1. Motivation • It´s not! Hmm, maybe not so simple…

5. 1. Motivation • Standard way of calculating probabilities: Feynman Diagrams • Relies on perturbation theory: expansion in orders of the coupling constant • Very long and difficult calculations if many diagrams have to be taken into account • Method for calculations: Lattice QCD

6. 1. Motivation • Solution: choose a particular energy region and select only relevant degrees of freedom • Effective Field Theory (EFT) • Is this allowed? Compare results with lattice QCD • NRQCD selects an energy scale at which relativistic degrees of freedom do not appear in leading order terms • No expansion in the coupling constant so all diagrams are included • Therefore we look for non-perturbative effects in the Standard Model

7. 2a. NRQCD Philosophy • Heavy Quark systems • Bound state of quark-antiquark • For example: Charmonium (or Bottomonium) • What is the scale parameter that selects relevant degrees of freedom? From comparison of hadron masses From the charmonium level scheme

9. 2a. NRQCD Philosophy • Heavy Quark systems • Bound state of quark-antiquark • For example: Charmonium • What is the scale parameter that selects relevant degrees of freedom? From comparison of hadron masses From the charmonium level scheme

10. 2b. Energy scales in heavy quark systems • M: heavy quark mass; rest energy • Mv: momentum of the charm quark • Mv2: kinetic energy of the charm quark • Because v<1: Mv2 < M v < M • Now we will discuss these scales in more detail

11. 2b. Energy scales in heavy quark systems • M: heavy quark mass; rest energy • Processes which happen above this energy M: • Well described by perturbation theory (Why?) • Example: Formation of high energy jets and asymptotically free quarks strong coupling constant vs. energy

12. 2b. Energy scales in heavy quark systems • Leading order terms in the Lagrangian will have an energy ~ kinetic energy of the bound state • This value is obtained by looking at the splitting between radial excitations • C.f. harmonic oscillator

13. 2b. Energy scales in heavy quark systems • Momentum • Sets size of the bound state • Heisenberg uncertainty principle

14. 2b. Energy scales in heavy quark systems • Assume scales to be well separated

15. 3. Non-Relativistic Version of the QCD Lagrangian • Recipe: • Introduce UV-cut off Λ to separate energy region > M • Excludes explicitly relativistic heavy quarks and gluons and light quarks of order M • Non-relativistic region: • decoupling of quarks-antiquarks • Covariant derivative splits up in time component and spatial component • Result:

16. 3a. Non-Relativistic Version of the QCD LagrangianLight quarks and gluons Gluon Field Strength Tensor This describes the free gluon field and the free light quark fields

17. Creates heavy antiquark 2 component spinor Kinetic term Annihilates heavy quark 2 component spinor are the time and space components of 3a. Non-Relativistic Version of the QCD LagrangianHeavy quarks-antiquarks This is just a Schrődinger field theory Reproduce relativistic effects with correction terms

18. electric color field magnetic color field spin operator 3a. Non-Relativistic Version of the QCD LagrangianCorrection terms • And last but not least • These terms are allowed under the symmetries of QCD • First we will explain the ordering of the Lagrangian • Then we will explain the exact origin of the terms

19. 3b. Power CountingWavefunction • Dimensionless (probability) • Use Heisenberg to relate momentum to position • So the quark annihilation field scales according to

20. 3b. Power CountingTime and spatial derivatives • Recall that gives an expectation value for the kinetic energy • And then • From the field equations:

21. 3b. Power CountingScalar, electric, magnetic field • For the scalar field, the color electric field and the color magnetic field:

22. 3b. Power CountingExample: 2nd correction term What order is this? How does it compare to the leading order terms?

23. 3b. Power CountingConclusion • The correction terms are of order and are suppressed by a factor of with respect to the leading order terms • Correction terms are all possible terms but have a more fundamental origin

24. 3c. Origin of the correction termsKinetic energy correction • First correction term • This is a correction to the energy

25. 3c. Origin of the correction terms Field interaction corrections • Second and third correction term • Correction to the interaction of a quark with a scalar field • Fourth correction term • Correction to the interaction of a quark with a vector field

26. Summary • QCD calculations using perturbation theory are hard • For heavy quark systems degrees of freedom can be separated to make calculations simpler • Diagrams up to every order in g are included so we can test non-perturbative effects • We have to add correction terms to maintain correspondence to the full theory

27. After the break • Annihilation: a process we can describe using an extended version of NRQCD and which can be compared to measurements

28. Annihilation

29. Conclusions before the break Until some cut-off energy  we can use NRQCD to describe strong interaction Now can we apply NRQCD to annihilation processes of heavy quarkonia in order to check the theory with experiment?

30. Overview Goal: Use NRQCD to desribeannihilation of heavy quarkonia (charmonium) Describeannihilation of heavy quarkonia Arguethat we canuse NRQCD Find the contribution order of annihilation Comparewith experiment Conclusions

31. J/Ψ to light hadrons We need at least 3 gluons Different light hadrons can form Complicated process Example of annihilation light hadrons

32. Annihilation of heavy quarkonia Process of heavy quarks going into light quarks Light quark - heavy quarks interaction Lagrangian is separated We need an extra correction

33. What does this correction look like? Can it be nonrelativistic? … this is quite relativistic Annihilation of heavy quarkonia

34. Annihilation of heavy quarkonia What do we do? Use nice trick, optical theorem: Γ: decay rate, H: heavy hadrons, LH: light hadrons If we know the scattering amplitude of we get the annihilation decay rate of HLH! (1)

35. Optical theorem from the literature: σ: cross section, k: wavenumber, f: scattering amplitude, f(0) means forward scattering usc: scattered wave ui: incident wave uf: final wave r: distance to scattering centre Optical theorem

36. Proof: Start with scattering amplitude: l = number of partial wave, Pl = Legendre polynomial al: effect on l’th partial wave, 0 ≤ ηl ≤ 1, amplitude, δl = phase shift ηl=1: elastic, no change in amplitude ηl<1: inelastic We are going to make use of this Optical theorem (2)

37. Optical theorem We want to calculate the total cross section Differential cross section: For the elastic cross section: using: with δ the delta function

38. Optical theorem Analogue for the inelastic part: In total: (3)

39. Optical theorem If we fill in for the scattering amplitude (2), θ=0 (so Pl(1)=1) and take imaginary part: We can identify this with (3): (2) Optical theorem!

40. Optical theorem We have: If we now use: and (follows from dimension analysis) We get: This corresponds to (1): λ = wavelength Γ=annihilation rate

41. How do we evaluatewithin NRQCD? First look at annihilation process: Scattering At whatlengthscale does this happen?

42. Scattering Annihilation is a localprocess (1/M) pgluon = M Trace back the interaction vertex Uncertainty principle tells us:

43. Scattering Because annihilation is local we need local scattering interactions: 4-fermion operators These have the form:

44. Scattering On(Λ): local 4-fermion operator fn(Λ): coef. of local operator dn: massscalingdimension n: rank of color tensor Λ: energyscale • Extra correction term: • Scattering is described by • We are interested in the order of contributions • General form:

45. Scattering gives M6v6so d=6 O has contributions in powers of M and v Mass dimension compensates Example: So dL is proportional to M4 note: Lagrangian density

46. Scattering Ordering of local operators can be done in mass dimension Lowest order: d=6, all terms allowed are:

47. Scattering All terms scale as v3 so v compressed wrt Lheavy Similar for d=8 terms: v3 compressed

48. Scattering This seems more important than Lcorrection But now: Coefficients fn Calculated by setting perturbative QCD equal to NRQCD Have imaginary parts for d=6 and d=8 terms: αs2

49. Compare to experiment So in theory: Energy splittings (from Lheavy) are order Mv2 Relative contribution of annihilation

50. For ηc: Γ=27MeV ΔE: 400MeV Γ/ ΔE = 0.07