Transcript Lattice QCD - KVI - Center for Advanced Radiation Technology

Lattice QCD
By Arjen van Vliet
Outline
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Introduction QCD
Lattice QCD basics
Scalar field calculation
Monte Carlo Method
Wilson loops and Wilson action
Quenched Approximation
Results
QCD
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Quantum field theory of quarks and gluons
Based on symmetry group SU(3)
Complex because of gluon-gluon interactions
At high energies:
- small coupling constant
- perturbation theory applies
- very good quantitative predictions
At low energies:
- large coupling constant
- perturbation theory does not apply
- no good quantitative predictions
QCD Lagrangian
1  a




[
i
q

L QCD
G a G   f
D  m f ] q f
4
f
with the gluon field strength tensor


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

 



g
f
Ga  Aa  Aa
Ab Ac
bc
a
and the gauge covariant derivative
g  a
D    i Aa 
2

where A is the gluon field, g is the strong coupling constant
and f denotes the quark flavor. Looks very similar to QED,
except for the last term in the second equation.


a
Perturbation Theory
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Calculate Feynman
diagrams.
Stop at certain order.
Order corresponds to
number of vertices.
Proportional to coupling
constant, only applicable
for small coupling
constant.
I. Allison, “Matching the Bare and MS Charm Quark
Mass using Weak Coupling Simulations”, presentation
at Lattice 2008
Intrinsic QCD Scale
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Running coupling constant.
Intrinsic QCD scale  in
the order of 1 GeV.
Scale below which the
coupling constant becomes
so large that standard
perturbation theory no
longer applies.
Many unresolved question
about low-energy QCD.
This is where Lattice QCD
comes in!
QCD
 s ( ) 
R. Timmermans, D. Bettoni and K. Peters, “Strong
interaction studies with antiprotons”
g s2 (  )
4
Lattice QCD
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Proposed by Wilson, 1974.
Nonperturbative low-energy solution of QCD.
E.o.m. discretized on 4d Euclidean space-time
lattice.
Quarks and gluons can only exist on lattice
points and travel over connection lines.
Solved by large scale numerical simulations on
supercomputers.
Set up LQCD action
http://globe-meta.ifh.de:8080/lenya/hpc/live/APE/physics/lattice.html
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From continuum to discretized lattice:
d
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4
xa
4

n
n four-vector that labels the lattice site, a lattice constant
Check, take an appropriate continuum limit (a→0) to get back
the continuum theory.
Scalar field action
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Scalar field (x), action of continuum field theory in
Euclidean space:
S ()   d 4 x[ 12 (  ) 2  12 m 2 2  4  4 ]
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Discretize to a lattice:
4
4
d
x

a


n
( x )   n
  ( x)  1a ( n  ˆ   n )
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Result:
S ()  {
n
2
a
2
4
 ( nˆ   n )  a (
2
 1
4 m2
2
 2n  4  4n )}
Expectation value calculation
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Feynman path integral formalism
Expectation value of an operator
 0 | O( n1 ,  n2 ,...,  nl ) | 0  Z1   [d n ]O( n1 ,  n2 ,...,  nl )e  S (  )
n
where
Z    [d n ]e  S (  )
n
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Rescale fields: 'n    n
Lattice action becomes: S ()  1 S ' ( ' )
S ()  {
n
2
a
2
4
 ( nˆ   n )  a (
2
 1
4 m2
2
 2n  4  4n )}
Statistical Mechanics
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Rescaled expectation value
 0 | O( 'n  'n ... 'n ) | 0 
1
2
l
1
Z'
 [d
'
'
'
'
'
1
]
O
(


...

)
exp{

S
(

)}

n
n
n
n
'
n
1
2
l
Z '    [d 'n ] exp{ 1 S ' ( ' )}
n
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Recognizable?
Statistical mechanics partition function with
1

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   kT1
Similar for fermion fields
R. Gupta, “Introduction to Lattice QCD”, arXiv:hep-lat/9807028
Monte Carlo Method
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Method from statistical mechanics to calculate
expectation value numerically.
Generate random distribution.
Calculate expectation value for this distribution.
Repeat this process very many times.
Average over results.
Results have statistical errors.
A lot of computational power needed!
Supercomputers
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Largest computing power in
Japan, especially for LQCD
Combination of Hitachi
SR11000 model K1 (peak
performance 2.15 TFlops)
and IBM Blue Gene Solution
(peak performance 57.3
TFlops)
IBM-Blue Gene/L in
Groningen: peak
performance 27.5 TFlops
http://www.kek.jp/intrae/press/2006/supercomputer_e.html
Wilson Loops
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Closed paths on the 4d Euclidean space-time lattice
U x ,  matrices defined on the links that connect the
neighboring sites x and x  aˆ
Traces of product of such matrices along Wilson loops
are gauge invariant
Plaquette: the elementary building block of the lattice,
the 1 x 1 lattice square
R. Gupta, “Introduction to Lattice QCD”, arXiv:hep-lat/9807028
Wilson action
SW 
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1
g2
Re
Tr

 
x, 
1
2
(1  U x ,U x  aˆ , U
†
x  aˆ , 
†
x,ν
U )
Simplest discretized action of the Yang-Mills part of the
a
( x)
QCD action SYM   14  d 4 xGa ( x)G
Agrees with the QCD action to order O(a2).
Proportional to the gauge-invariant trace of the sum
over all plaquettes.
From Wilson to Yang Mills
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Matrices U given by U  ( x)  exp( iagA ( x  2ˆ ))
The simplest Wilson loop, the 1x1 plaquette given by
1x1
W
 U  ( x)U ( x  ˆ )U † ( x ˆ)U† ( x)
 exp( iag[ A ( x  2ˆ )  A ( x  ˆ  2ˆ )  A ( x ˆ  2ˆ )  A ( x  2ˆ )])
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Expanding about x  ˆ 2ˆ gives
 exp[ ia g (  A   A ) 
2
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( 3 A  3 A )  ...]
The Taylor series of the exponent gives
 1  ia gF 
2

ia4 g
12
a4 g 2
2
F F   O(a 6 )  ...
From this we derive
Re Tr (1  W ) 
1x1
a4 g 2
2
F F   ...
Method of operation
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Six unknown input parameters, coupling constant and
the masses of the up, down, strange, charm and bottom
quark.
Top quark too short lived to form bound states at the
energies we are looking at.
Fix in terms of six precisely measured masses of
hadrons.
Masses and properties of all the other hadrons can be
obtained this way.
They should agree with experiment.
Lattice constant
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Lattice constant a should be small to approach
continuum limit, but not too small or the
computation time becomes too long.
Size nucleon in the order of 1 Fermi (1 Fermi =
1.0x10–15 m).
a between 0.05 and 0.2 Fermi
Results also have systematic errors due to this
lattice discretization.
Quenched Approximation
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Quarks fully dynamical
degrees of freedom that can
be produced and annihilated.
In the quenched
approximation vacuum
polarization effects of quark
loops are turned off.
Very popular approximation,
reduces computation time by
a factor of about 103-105.
R. Gupta, “Introduction to Lattice QCD”, arXiv:heplat/9807028
Consequence of QQCD
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Difference with QCD for
large distances.
In QCD separated quarks
split up by forming a quark
anti-quark pair.
At smaller distances a
reasonable but not great
approximation, as can be
seen from this picture.
R. Timmermans, D. Bettoni and K. Peters, “Strong
interaction studies with antiprotons”
Goals of LQCD
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Test whether QCD is the correct theory of
strong interactions in the nonperturbative
regime.
Improve understanding of low-energy aspects of
QCD.
Determine quark masses and the value of the
strong coupling constant in this energy range.
Determine hadron spectra and masses.
Results glueball spectrum
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LQCD glueball
spectrum.
Glueball: strongly
interacting particle
without any valence
quarks.
Entirely composed of
gluons and quarkantiquark pairs.
R. Timmermans, D. Bettoni and K. Peters, “Strong
interaction studies with antiprotons”
Multiple QQCD results
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QQCD
predictions for
the charmonium,
the glueball, and
the spin-exotic
cc-glue hybrids
spectrum.
R. Timmermans, D. Bettoni and K. Peters, “Strong interaction
studies with antiprotons”
Results
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The lattice QCD
prediction of the mass
of the Bc meson.
Approaching the
precision of the value
measured at Fermilab.
Fermilab Today, “Precise Prediction of Particle Mass,
Confirmed by Experiment”, May 11, 2005
Thanks for listening!
Any question left?