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

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Topics of Today1. Motivation for NRQCD2. NRQCD

a. Philosophyb. Energy scales in heavy quark systems

3. Non-Relativistic version of the QCD Lagrangiana. Componentsb. Power counting; relative importance of componentsc. Origin of the correction terms

4. Application of NRQCD: Annihilation– Use NRQCD to describe annihilation of heavy quarkonia

(charmonium)

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1. Motivation• Lagrangian density of QCD

– Symmetry group: SU(3)

• Looks simple!

Don’t forget

: 4 component spinor

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1. Motivation

• It´s not!

Hmm, maybe not so simple…

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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

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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

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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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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

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2b. Energy scales in heavy quark systems

1. M: heavy quark mass; rest energy2. Mv: momentum of the charm quark3. Mv2: kinetic energy of the charm quark

• Because v<1: Mv2 < M v < M

• Now we will discuss these scales in more detail

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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

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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

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2b. Energy scales in heavy quark systems

• Momentum

• Sets size of the bound state – Heisenberg uncertainty principle

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2b. Energy scales in heavy quark systems

• Assume scales to be well separated

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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:

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3a. Non-Relativistic Version of the QCD Lagrangian

Light quarks and gluonsGluon Field Strength Tensor

This describes the free gluon field and the free light quark fields

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3a. Non-Relativistic Version of the QCD Lagrangian

Heavy quarks-antiquarks

Annihilates heavy quark2 component spinor

Creates heavy antiquark2 component spinorKinetic term

This is just a Schrődinger field theoryReproduce relativistic effects with correction terms

are the time and space components of

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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

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3b. Power CountingWavefunction

• Dimensionless (probability)

• Use Heisenberg to relate momentum to position

• So the quark annihilation field scales according to

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3b. Power CountingTime and spatial derivatives

• Recall thatgives an expectation value for the kinetic energy

• And then• From the field equations:

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3b. Power CountingScalar, electric, magnetic field

• For the scalar field, the color electric field and the color magnetic field:

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3b. Power CountingExample: 2nd correction term

What order is this?How does it compareto the leading orderterms?

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3b. Power CountingConclusion

• The correction terms are of orderand are suppressed by a factor ofwith respect to the leading order terms

• Correction terms are all possible termsbut have a more fundamental origin

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3c. Origin of the correction termsKinetic energy correction

• First correction term

• This is a correction to the energy

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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

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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

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After the break

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

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Annihilation

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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?

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Overview• Goal: Use NRQCD to desribe annihilation of

heavy quarkonia (charmonium)

1.Describe annihilation of heavy quarkonia2.Argue that we can use NRQCD3.Find the contribution order of annihilation4.Compare with experiment5.Conclusions

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• J/Ψ to light hadrons

• We need at least 3 gluons• Different light hadrons can form• Complicated process

Example of annihilation

J/Ψ gluonP -1 -1S 1 1

light

hadrons

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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

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• What does this correction look like?• Can it be nonrelativistic?

• … this is quite relativistic

Annihilation of heavy quarkonia

..9999.02/

2

2/2

/

duc

duc

du

MMcMM

cv

quark mass

u/d 1.5-3.3 MeV

c 1.27 GeV

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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 HLH!

HscatteringHLHH Im2)(

QQQQ

(1)

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• Optical theorem from the literature:

σ: cross section, k: wavenumber, f: scattering amplitude, f(0) means forward scattering

usc: scattered waveui: incident waveuf: final waver: distance to scattering centre

Optical theorem

)0(Im4 fk

),()exp( frikruuu ifsc

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• 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: inelasticWe are going to make use of this

0

)(cos)1)(12(21)(

lll Pal

ikf (2)

lill ea 2

Optical theorem

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Optical theorem

• We want to calculate the total cross section

• Differential cross section:

• For the elastic cross section:

2

02

2 )(cos2

1)12(1)()( l

l

l Pi

alk

fdd

0

2

2

2 1)12()(l

lel alk

df

inelel

124 '

' ldPP llll

using:

with δ the delta function

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Optical theorem• Analogue for the inelastic part:

• In total:

0

22 )1)(12(l

linel alk

02

0

22

2

)Re1)(12(2

)11)(12(

ll

lll

alk

aalk

(3)

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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):

0

)Re1)(12(21)0(Im

llal

kf

)0(Im4 fk Optical theorem!

0

)(cos)1)(12(21)(

lll Pal

ikf

02

0

22

2

)Re1)(12(2

)11)(12(

ll

lll

alk

aalk

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Optical theorem

• We have:

• If we now use:and (follows from dimension analysis)

• We get: • This corresponds to (1):

)0(Im4 fk

2

k λ = wavelength

Γ=annihilation rate

)0(Im2 f

HscatteringHLHH Im2)(

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• How do we evaluatewithin NRQCD?

• First look at annihilation process:

Scattering

HscatteringH

At what length scale does this happen?

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Scattering

• pgluon = M• Trace back the interaction vertex• Uncertainty principle tells us: fm

Mdx 1.01

fm1.0charmonium of size

Annihilation is a local process (1/M)

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Scattering

• Because annihilation is local we need local scattering interactions:

4-fermion operators

These have the form:

antiquark ofcreation quark ofon annihilati

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Scattering

• Extra correction term:

• Scattering is described by• We are interested in the order of contributions• General form:

fermioncorrectionlightheavy 4LLLLL

fermion4L

HH fermion4L

n

ndn

fermion OMf

n)()(

44L

On(Λ): local 4-fermion operatorfn(Λ): coef. of local operatordn: mass scaling dimensionn: rank of color tensor Λ: energy scale

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Scattering

• O has contributions in powers of M and v• Mass dimension compensates• Example:

So dL is proportional to M4

• note: Lagrangian density

gives M6v6 so d=6

n

ndn

fermion OMf

n)()(

44L

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n

ndn

fermion OMf

n)()(

44L

Scattering

• Ordering of local operators can be done in mass dimension

• Lowest order: d=6, all terms allowed are:

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Scattering

• All terms scale as v3 so v compressed wrt Lheavy

• Similar for d=8 terms: v3

compressed

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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: αs

2

n

ndn

fermion OMf

n)()(

44L

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Compare to experiment

• So in theory: • Energy splittings (from Lheavy) are order Mv2

• Relative contribution of annihilation06.0)3.01(55.024.0)()( 2322 vMavMa

SS

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For ηc: Γ=27MeVΔE: 400MeVΓ/ ΔE = 0.07

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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

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Literature

• “Rigorous QCD Analysis of Inclusive Annihilation and Production of Heavy Quarkonium” Bodwin, Braaten, Lepage arxiv: hep-ph/9407339v2 (1997)

• “Improved Nonrelativistic QCD for Heavy Quark Physics”Lepage, Magnea, Nakhleharxiv: hep-lat/9205007v1 (1992)

• - “IHEP-Physics-Report-BES-III-2008-001-v1”Different contributors; editors: Kuang-Ta Chao and Yifang Wanghttp://arxiv.org/abs/0809.1869v1

• Particle data grouphttp://pdg.lbl.gov/