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

2/52 Topics of Today 1.Motivation for NRQCD 2.NRQCD a.Philosophy b.Energy scales in heavy quark systems 3.Non-Relativistic version of the QCD Lagrangian a.Components b.Power counting; relative importance of components c.Origin of the correction terms 4.Application of NRQCD: Annihilation –Use NRQCD to describe annihilation of heavy quarkonia (charmonium)

3/52 1. Motivation Lagrangian density of QCD –Symmetry group: SU(3) Looks simple! Don’t forget : 4 component spinor

4/52 1. Motivation It´s not! Hmm, maybe not so simple…

5/52 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

6/52 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

7/52 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

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9/52 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

10/52 2b. Energy scales in heavy quark systems 1.M: heavy quark mass; rest energy 2.Mv: momentum of the charm quark 3.Mv 2 : kinetic energy of the charm quark Because v<1: Mv 2 < M v < M Now we will discuss these scales in more detail

11/52 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

12/52 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

13/52 2b. Energy scales in heavy quark systems Momentum Sets size of the bound state –Heisenberg uncertainty principle

14/52 2b. Energy scales in heavy quark systems Assume scales to be well separated

15/52 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:

16/52 3a. Non-Relativistic Version of the QCD Lagrangian Light quarks and gluons Gluon Field Strength Tensor This describes the free gluon field and the free light quark fields

17/52 3a. Non-Relativistic Version of the QCD Lagrangian Heavy quarks-antiquarks Annihilates heavy quark 2 component spinor Creates heavy antiquark 2 component spinor Kinetic term This is just a Schrődinger field theory Reproduce relativistic effects with correction terms are the time and space components of

18/52 3a. Non-Relativistic Version of the QCD Lagrangian Correction 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 electric color field magnetic color field spin operator

19/52 3b. Power Counting Wavefunction Dimensionless (probability) Use Heisenberg to relate momentum to position So the quark annihilation field scales according to

20/52 3b. Power Counting Time and spatial derivatives Recall that gives an expectation value for the kinetic energy And then From the field equations:

21/52 3b. Power Counting Scalar, electric, magnetic field For the scalar field, the color electric field and the color magnetic field:

22/52 3b. Power Counting Example: 2nd correction term What order is this? How does it compare to the leading order terms?

23/52 3b. Power Counting Conclusion 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

24/52 3c. Origin of the correction terms Kinetic energy correction First correction term This is a correction to the energy

25/52 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

26/52 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

27/52 After the break Annihilation: a process we can describe using an extended version of NRQCD and which can be compared to measurements

28/52 Annihilation

29/52 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?

30/52 Overview Goal: Use NRQCD to desribe annihilation of heavy quarkonia (charmonium) 1.Describe annihilation of heavy quarkonia 2.Argue that we can use NRQCD 3.Find the contribution order of annihilation 4.Compare with experiment 5.Conclusions

31/52 J/Ψ to light hadrons We need at least 3 gluons Different light hadrons can form Complicated process Example of annihilation J/Ψgluon P S11 light hadrons

32/52 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

33/52 What does this correction look like? Can it be nonrelativistic? … this is quite relativistic Annihilation of heavy quarkonia quarkmass u/d MeV c1.27 GeV

34/52 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)

35/52 Optical theorem from the literature: σ: cross section, k: wavenumber, f: scattering amplitude, f(0) means forward scattering u sc : scattered wave u i : incident wave u f : final wave r: distance to scattering centre Optical theorem

36/52 Proof: Start with scattering amplitude: l = number of partial wave, P l = Legendre polynomial a l : effect on l’th partial wave, 0 ≤ η l ≤ 1, amplitude, δ l = phase shift η l =1: elastic, no change in amplitude η l <1: inelastic We are going to make use of this (2) Optical theorem

37/52 Optical theorem We want to calculate the total cross section Differential cross section: For the elastic cross section: using: with δ the delta function

38/52 Optical theorem Analogue for the inelastic part: In total: (3)

39/52 Optical theorem If we fill in for the scattering amplitude (2), θ=0 (so P l (1)=1) and take imaginary part: We can identify this with (3): Optical theorem! (2)

40/52 Optical theorem We have: If we now use: and (follows from dimension analysis) We get: This corresponds to (1): λ = wavelength Γ=annihilation rate

41/52 How do we evaluate within NRQCD? First look at annihilation process: Scattering At what length scale does this happen?

42/52 Scattering p gluon = M Trace back the interaction vertex Uncertainty principle tells us: Annihilation is a local process (1/M)

43/52 Scattering Because annihilation is local we need local scattering interactions: 4-fermion operators These have the form:

44/52 Scattering Extra correction term: Scattering is described by We are interested in the order of contributions General form: O n (Λ): local 4-fermion operator f n (Λ): coef. of local operator d n : mass scaling dimension n: rank of color tensor Λ: energy scale

45/52 Scattering O has contributions in powers of M and v Mass dimension compensates Example: So dL is proportional to M 4 note: Lagrangian density gives M 6 v 6 so d=6

46/52 Scattering Ordering of local operators can be done in mass dimension Lowest order: d=6, all terms allowed are:

47/52 Scattering All terms scale as v 3 so v compressed wrt L heavy Similar for d=8 terms: v 3 compressed

48/52 Scattering This seems more important than L correction But now: Coefficients f n Calculated by setting perturbative QCD equal to NRQCD Have imaginary parts for d=6 and d=8 terms: α s 2

49/52 Compare to experiment So in theory: Energy splittings (from L heavy ) are order Mv 2 Relative contribution of annihilation

50/52 For η c : Γ=27MeV ΔE: 400MeV Γ/ ΔE = 0.07

51/52 Summary In order to describe annihilation of heavy quarkonia we need an extra correction term to NRQCD lagrangian Because the interaction is local we can use the optical theorem which says we can use local scattering operators The contribution of this extra correction term for annihilation agrees with experiment We can use NRQCD to obtain physical predictions

52/52 Literature “Rigorous QCD Analysis of Inclusive Annihilation and Production of Heavy Quarkonium” Bodwin, Braaten, Lepage arxiv: hep-ph/ v2 (1997) “Improved Nonrelativistic QCD for Heavy Quark Physics” Lepage, Magnea, Nakhleh arxiv: hep-lat/ v1 (1992) - “IHEP-Physics-Report-BES-III v1” Different contributors; editors: Kuang-Ta Chao and Yifang Wang Particle data group