1
Lattice Quantum Chromodynamics
1- Literature : Lattice QCD , C. Davis Hep-ph/02051812- Burcham and Jobes
By
Leila Joulaeizadeh
19 Oct. 2005
2
Outline
- Introduction
- Hamilton principle
- Local gauge invariance and QED
- Local gauge invariance and QCD
- Lattice QCD calculations
- Some results
- Conclusion
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What is Quantum Chromodynamics and why LQCD?
- Strong interaction between coloured quarks by exchange of coloured gluon
- Gluons carry colour so they have self interaction
- Self interaction of gluons , nonabelian group SU(3)
- QCD is a nonlinear theory so there is no analytical solution and we should use numerical methods
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Euler Lagrange Equation
0y
f
dx
d
y
f
0)y(x andn integratio partial gsinU
dx)y)dx
d(
y
f y
y
f(dx)y
y
f y
y
f(
y dx
d y 0)y(x)y(x
0 dx)y,y(fI I minimisingfor dx)y,y(fI
'
'
x
x
''
x
x
'10
x
x
'x
x
'
1
0
1
0
1
0
1
0
5
For motion of a point like particle with mass m in a central potential:
Physical systems will evolve in such a way to minimize the action
0q
L
dt
d
q
L minimized is dt.LS Action
tiongeneraliza
0x
L
dt
d
x
L Pxm
x
L F
x
V
x
L
)r(V)zyx(m2
1L
ii
t
t
xx
222
1
0
Hamilton Principle
6
In Quantum Field Theory
),t
(x
coordinate time-space ly varyingcontinuous :x
field)(x
1,2,3,...i 0))(
L(
L
)x
,L( :Density Lagrangian
ii
7
Examples
Scalar field (spin 0 particle)
Spinor field(spin 1/2 particle)
2222 m2
1))((g
2
1m
2
1))((
2
1L
Eq.Gordon -Klein 0m2
eq. Dirac 0 mi miL____
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Local Gauge Invariance and QED
qj
AjLqAmimDiL
)x(- )x(A )x(A , D)]x(iqexp[D , )x(iqAD
:derivativecovariant Gauge
Lmqi
)x()]x(iqexp[)x()]x(iqexp[m))x()]x(iq(exp[)x()]x(iqexp[iL
:rmation transfogauge local After
miL :m mass of particle Dirac free a For
)x()]x(iqexp[ , )x()]x(iqexp[
(x): xof function a is parameter tion Transforma
free
'
''
9
Massless vector field(spin 1)
equation Maxwell of formCovariant
jF AjFF4
1L
: ermenergy t kinetic add We
AAF FF4
1AjLL μν
free
L) A)(A(m2
1L AAm
2
1L :masslessnot ephoton wer theIf 2'2
Example
10
aaa
ajkasjkjk
aaj
qjq
kqjk
j
aqaqsqqqqqqqqqq
aaaa
as
'
jq
jq
kq
jq
q
cabcba
q'
aasq'
qqaasq'
q
BBB
1,2,...,8a B)T(ig)D(
BB4
1m)D(iL : termkinetic add We
B)T(gmimDiL
)x(- )x(B )x(B , BTigD
LL:rmation transfogauge local After
miL :m mass of particle Dirac free a For
generators group SU(3) TifT,T
)x(]T)x(igexp[ , )x(]T)x(igexp[
Non-Abelian nature of SU(3)
Gluon self interaction term
cbabcs BBfg
Local Gauge Invariance and QCD
)x(B)x(fg cbabcs
BDBDB
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Diagrams representing propagation of free quark and gluon and their interaction
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O : operator whose expectation value we want to calculate
Lattice QCD
))n(m2
1
a2
)1n()1n(
2
1(aS :action Lattice
m2
1)(
2
1L lagrangian theory fieldScalar
points ebetween th spacing:a a)na,(nt)(x, , axd :tion discretisa After
xLdS
edA]d[d
e]A,,[OdA]d[d0O0
22
24
1n
4
222
tin
44
4
S
S
13
Lattice gauge theory for gluons
n!calibratio requires 0.1(fm)a6
g
6 TrUS :action plaquette Wilson
)BB(Tr4g
1xd :action QCD continum of piece gluonic Purely
)x(U)1x(U)...x(U fieldgluon of string
(x))U1(x)U1(x(x)UU(x) Ugluon of loop closed
matrixation transformGauge:)x(G
1GG (x)G (x)(x) (x)G(x)(x) )1x(G)x(U)x(G)x(U
ikgB1e Ulatticein field Gluon
B continuumin field Gluon
2platt
24
221
jjiijip
gg)g(
ikgB
b
x x X+1X+1
)x(UU)1x( U )x(U 1-
1x
2x_
x
14
Lattice gauge theory for gluons
integral the toonscontributi large with ionsconfigurat choose : sampling Importance
ionconfiguratin that operator O of valuee:
latticein link each for one matrices Uofset :
00 : tion discretiza
00 : integralpath
thO
aU
eU
eOUOAfter
edU
OedUOFeynman
S
S
S
S
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Fermion doubling problem of quarks on the lattice
ma
apsini)p(G : propagator inverse atticel
:L ofon ansformatiFourier tr
ma2
aS
mpi)p(G : propagator inverse Continuum
:L ofon ansformatiFourier tr
)m(xdS
1-naivelatt,
f
4
1
_
xx1x1x
x4
n
naivelatt,f
1-cont
f
_4
f
: fermions free offlavor single afor action Continuum
:ion dicretizat lattice Naive
p
0a
a
0
!! 1 of instead fermions 24
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Solutions of Fermion doubling problem
quarks Wilson
naivef
wf
4
`12
1xx1x
xx
5naivef
wf
S S 0a
errors!tion discretizaLarger :Problem
parameterWilson r )a
2(a
2
rSS
quarks Staggered
/a)!p to0p (from scattering changingFlavour :Problem
flavoursdiffrent as doublers of tionInterpreta
17
!problems! numerical big causes Mdet
nformulatioquark on the depends and massesquark dynamical ofmatrix :M
e M)(det dUe]d[ddUS gg S)MS(
ionapproximat Quenched
quarks sea of dynamics about the Forget
ionextrapolat chiral
d andu like quarkslight ofion extrapolat and quarksheavier Work with
Action with quarks
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Relating lattice results to physics
0 T
massesHadron quarks ofDensity obabilityPr)rx( (r)(x) .
.
:(r)on distributi spatial relative
vector)( )(
arpseudoscal )( )(
spin with mesons
)( )(
Ti0i
T505
T0
Make the correlators of quarks by using matrices
r
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1- choose the lattice spacing - close to the continuum - computation costs
2- Choose a quark formulation and number of quark flavors
3- generating an ensemble of gluon configurations - Try to go near small masses - computation costs
4- calculation of quark propagators on each gluon configuration
5- combination of quark propagators to form hadron correlators
6- Determination of lattice spacing in Gev(lattice calibration)
7- extrapolation of hadron masses as a function of bare quark masses
8- repeat the calculation using several lattice spacing to compare with physical results at the limit of a 0
9- compare with experiment or give a prediction for experiment
Steps of typical lattice calculation
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Some results of lattice QCD calculations
The spectrum of light mesons and baryons in the quenched approximation
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The ratio of inverse lattice spacing
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spacing lattice offunction a as mesons K and of masses The *
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Charmonium spectrum in quenched approximation
c
JPC
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Summary
- Photons don’t carry any colour charge, so QED is analytically solvable.
- Gluons do carry colour charge,so to solve the QCD theory, approximations are proposed
(e.g. Lattice calculation method ).
- There is a fermion doubling problem in lattice which can be solved by various methods.
- In order to obtain light quark properties, we need bigger computers and the
calculation costs will be increased.
- Quenched approximation is reasonable in order to decrease the computation costs.
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