Transcript Non-Relativistic Quantum Chromo Dynamics (NRQCD)

Non-Relativistic
Quantum Chromo Dynamics
(NRQCD)
Heavy quark systems as a test of nonperturbative effects in the Standard
Model
Victor Haverkort en Tom Boot, 21 oktober 2009
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)
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1. Motivation
• Lagrangian density of QCD
– Symmetry group: SU(3)
Don’t forget
: 4 component spinor
• Looks simple!
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1. Motivation
• It´s not!
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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
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Standard Model
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:
2. Mv:
3. Mv2:
heavy quark mass; rest energy
momentum of the charm quark
kinetic energy of the charm quark
Mv2 < M v < M
•
Because v<1:
•
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:
strong coupling constant vs. energy
– Well described by perturbation
theory (Why?)
– Example: Formation of high
energy jets and asymptotically
free quarks
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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 gluons
Gluon 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
Kinetic term
Annihilates heavy quark
2 component spinor
Creates heavy antiquark
2 component spinor
are the time and space
components of
This is just a Schrődinger field theory
Reproduce relativistic effects with correction terms
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3a. Non-Relativistic Version of the QCD
Lagrangian
Correction terms
• And last but not least
electric color field
magnetic color field
spin operator
• 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 18/52
3b. Power Counting
Wavefunction
• Dimensionless (probability)
• Use Heisenberg to relate momentum to
position
• So the quark annihilation field scales
according to
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3b. Power Counting
Time and spatial derivatives
• Recall that
gives an expectation value for the kinetic
energy
• And then
• From the field equations:
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3b. Power Counting
Scalar, electric, magnetic field
• For the scalar field, the color electric field and
the color magnetic field:
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3b. Power Counting
Example: 2nd correction term
What order is this?
How does it compare
to the leading order
terms?
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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
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3c. Origin of the correction terms
Kinetic 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.
2.
3.
4.
5.
Describe annihilation of heavy quarkonia
Argue that we can use NRQCD
Find the contribution order of annihilation
Compare with experiment
Conclusions
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Example of annihilation
• J/Ψ to light hadrons
light
hadrons
J/Ψ gluon
P -1
-1
S 1
1
• We need at least 3 gluons
• Different light hadrons can form
• Complicated process
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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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Annihilation of heavy quarkonia
• What does this correction look like?
• Can it be nonrelativistic?
quark
mass
u/d
1.5-3.3 MeV
c
1.27 GeV
Mu/d
M  2
vu / d
c

 0.9999..
2
2
c
Mc  Mu/d
2
c
• … this is quite relativistic
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Annihilation of heavy quarkonia
• What do we do?
• Use nice trick, optical theorem:
( H  LH )  2 Im H scattering H
(1)
Γ: decay rate, H: heavy hadrons, LH: light hadrons
If we know the scattering amplitude of
QQ  QQ we get the annihilation decay rate of
HLH!
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Optical theorem
• Optical theorem from the literature:
4
  k Im f (0)
σ: cross section, k: wavenumber, f: scattering amplitude,
f(0) means forward scattering
exp( ikr )
u sc  u f  ui 
f ( ,  )
r
usc: scattered wave
ui: incident wave
uf: final wave
r: distance to scattering centre
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Optical theorem
• Proof:
• Start with scattering amplitude:
1 
f ( ) 
(2l  1)(al  1) Pl (cos )

2ik l 0
(2)
l = number of partial wave, Pl = Legendre polynomial
al  l e 2il
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
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Optical theorem
• We want to calculate the total cross section
   el   inel
• Differential cross section:
d ( )
1
2
 f ( )  2
d
k
 al  1 
(
2
l

1
)

Pl (cos  )

 2i 
l 0

• For the elastic cross section:
 el   f ( ) d 
2

k
2

 (2l  1) 1  a
l 0
2
l
2
using:
 Pl Pl 'd 
4 ll '
2l  1
with δ the
delta function
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Optical theorem
• Analogue for the inelastic part:
 inel 

k2

 (2l  1)(1  al )
2
l 0
• In total:


k
2
2
 2
k

 (2l  1)(1  a
l 0
l
2
 1  al )
2

 (2l  1)(1  Re a )
l 0
l
(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:

1 
1
f ( )  Im
 1()2Pll (cos
f((20l )1)(al 
 1)(1) Re al )

2ik l 0
2k l 0
• We can identify this with (3):
4

Im f (0)
k

Optical theorem!

k2
2
 2
k

 (2l  1)(1  al  1  al )
2
2
l 0

 (2l  1)(1  Re a )
l 0
l
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Optical theorem
• We have:

4
Im f (0)
k
• If we now use: k  2
λ = wavelength

and    Γ=annihilation rate
(follows from dimension analysis)
• We get:   2 Im f (0)
• This corresponds to (1):
( H  LH )  2 Im H scattering H
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Scattering
• How do we evaluate
H scattering H
within NRQCD?
• First look at annihilation process:
At what length scale
does this happen?
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Scattering
• pgluon = M
• Trace back the interaction vertex
1
• Uncertainty principle tells us:
dx 
 0.1 fm
M
size of charmonium 0.1 fm
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:
  
 annihilation of quark
 creationof antiquark


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Scattering
• Extra correction term: L4 fermion
L  Lheavy  Llight  Lcorrection  L4 fermion
• Scattering is described by
H L4 fermion H
• We are interested in the order of contributions
• General form:
L4 fermion  
n
f n ()
On ()
d n 4
M
On(Λ): local 4-fermion operator
fn(Λ): coef. of local operator
dn: mass scaling dimension
n: rank of color tensor
Λ: energy scale
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Scattering
L4 fermion
f n ()
  d n 4 On ()
n M
• O has contributions in powers of M and v
• Mass dimension compensates
• Example:
gives M6v6 so d=6
So dL is proportional to M4
• note: Lagrangian density
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Scattering
L4 fermion
f n ()
  d n 4 On ()
n M
• 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:
f ()
L

O ()
M
n
4  fermion
d n 4
n
n
• Coefficients fn
• Calculated by setting perturbative QCD equal
to NRQCD
• Have imaginary parts
• for d=6 and d=8 terms: αs2
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Compare to experiment
• So in theory:
• Energy splittings (from Lheavy) are order Mv2
• Relative contribution of annihilation
aS2 (M ) v  a2S (M ) v3  0.242  0.55(1  0.3)  0.06
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For ηc: Γ=27MeV
ΔE: 400MeV
Γ/ 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, Nakhleh
arxiv: hep-lat/9205007v1 (1992)
•
- “IHEP-Physics-Report-BES-III-2008-001-v1”
Different contributors; editors: Kuang-Ta Chao and Yifang Wang
http://arxiv.org/abs/0809.1869v1
•
Particle data group
http://pdg.lbl.gov/
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