Transcript Euler Lagrange Equation

Lattice Quantum Chromodynamics
By
Leila Joulaeizadeh
19 Oct. 2005
1- Literature : Lattice QCD , C. Davis
Hep-ph/0205181
2- Burcham and Jobes
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Outline
- Introduction
- Hamilton principle
- Local gauge invariance and QED
- Local gauge invariance and QCD
- Lattice QCD calculations
- Some results
- Conclusion
2
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
x1
I

x1
f ( y, y ' )dx
for minimisingI

I  f ( y, y ' )dx  0
x0
x0
y(x0 )  y(x1 )  0
x1

x0
(
f
f
y  ' y ' )dx 
y
y
y' 
x1

x0
(
d
y
dx
f
f d
y  ' ( y))dx
y
y dx
U sin g partialintegration and y(x)  0
f d f

0
'
y dx y
4
For motion of a point like particle with mass m in a central potential:
L
1
m( x 2  y 2  z 2 )  V (r )
2
L
V

 Fx
x
x
L
 mx  Px
x

L d L

0
x dt x
 generalization
t1

Action S  L.dt is minimized
t0
L d L

0

q i dt q i
Hamilton Principle
Physical systems will evolve in such a way to
minimize the action
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In Quantum Field Theory

LagrangianDensity: L(,
)
x 
L
L

 (
)0
i
 (   i )
i  1,2,3,...
(x  )  field
x  : continuous
ly varyingspace- time coordinate
 


 ( ,)
x 
t
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Examples
Scalar field (spin 0 particle)
L
1
1
1
1
(  )(  )  m 2  2  g  (  )(  )  m 2  2
2
2
2
2
      m2  0
Klein- GordonEq.
Spinor field(spin 1/2 particle)
_

_
_

_
L  i       m    i     m   0 Diraceq.
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Local Gauge Invariance and QED
Transformation parameter  is a function of x : (x)
   '  exp[iq ( x )] ( x )
,
   '  exp[iq ( x )] ( x )
For a free Dirac particle of mass m : L  i        m 
After local gauge transformation :
L'  i exp[iq ( x )] ( x )     (exp[iq ( x )] ( x ))  m exp[iq ( x )] ( x ) exp[iq ( x )] ( x )
 i        q        m   L
Gaugecovariantderivative:
D      iqA ( x ) , D    exp[iq ( x )]D  
,
A  ( x )  A  ( x ) -   ( x )
L  i    D    m   i        m   qA      L free  j A 
j  q    
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We add kinetic energy term :
L  Lfree  j A  
1
F Fμν
4
If the photon were not massless: L  
F    A     A
1 2
1
m  A A  L'   m2 (A    )(A     )  L 
2
2
Example
Massless vector field(spin 1)
1
L   F F  j A     F  j
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Covariantform of Maxwellequation
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Local Gauge Invariance and QCD
 q   ' q  exp[igs  a ( x )Ta ] q ( x )
Ta , Tb   ifabcTc
,
 q   ' q  exp[igs  a ( x )Ta ] ' q ( x )
SU(3) group generators
For a free Dirac particle of mass m : L 
i 
j
q
     qk 
m  
q
q
j
q
j
q
q
After local gauge transformation : L'  L
D      igs Ta Ba
,
Ba ( x )  Ba ( x ) -    a ( x ) 
g s f abc  b (x)Bc (x)
Non-Abelian nature of SU(3)
L  i  q   D   q  m q  q  q  i  q      q  m q  q  q  g s (  q   Ta  q )Ba
We add kinetic term : L 

j
i  q   (D  ) jk  qk 
q
(D  ) jk   jk    igs (Ta ) jk Ba

q
m q  qj  qj 
1 a 
B B a
4
a  1,2,...,8
b c
a
B
   B a    Ba  g s f abc B B
Gluon self interaction term
B  D  B  D  B
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Diagrams representing propagation of free quark and gluon and their interaction
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Lattice QCD
O : operator whose expectation value we want to calculate
0O0 
S




d

[
d

]
dA
O
[

,

,
A
]
e


 d[d  ]dA e


After discretisation: d 4 x 


n
S
a4
,
(x, t)  (n i a, n t a)
a : spacingbetween the points
n
Scalar field theorylagrangianL 
Lattice action: S 

S  Ld4 x
1
1
(  ) 2  m 2  2
2
2
2
 (n  1 )  (n  1 ) 
1
1 2 2
a4(

  m  (n ))
2  1 
2a
2

4

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Lattice gauge theory for gluons
Gluonfield in continuum B b
x
X+1
U ( x)
U e
Gluonfield in lattice
U (g ) ( x )
ikgB
 G ( x ) U  ( x )G ( x  1)
X+1
U - (x  1)  U1  U (x)
 1  ikgB
g

x
 (x)  G(x) (x)
g
 (x)   (x) G  (x)
G G  1
G ( x ) : Gaugetransformation matrix

closedloopof gluon
stringof gluonfield
U p (x)  U i (x)U j (x  1i )U i (x  1 j )U j (x)
U  ( x1 )...U  ( x 2  1 ) U  ( x 2 )
P urelygluonicpieceof continumQCD action:
 TrU 
Wilsonplaquetteaction: Slatt  
  6  a  0.1(fm)
p
requirescalibration!

d4x

1
4g2
6
x2
_

x1
x
Tr (B B )
g2
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Lattice gauge theory for gluons
Feynm anpat hintegral : 0 O 0
S


dU
Oe


S


dU
e

 U  O e S

After discretization
: 0O0 

U  e


 S
U  : a set of U matricesone for each link in lat tice
O : the value of O operatorin thatconfiguration
Importancesampling: choose configurations with large contributons
i to theintegral
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Fermion doubling problem of
quarks on the lattice
Continuumactionfor a singleflavor of free fermions :

_
S f  d x  (      m )
4
Fourier transformation of L f :
1
Continuuminversepropagator: G -cont
( p )  i  p   m
0
Naive lattice dicretizat ion :


4
_




x 1
x 1
naive
4
Slatt,

a



m


 x

x x
f
2
a
n
  1



Fourier transformation of L f :


1

latticeinversepropagator: G -latt,
naive ( p)  i
sin p  a
a
m
p


a
0

a
2 4 fermions insteadof 1 !!
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Solutions of Fermion doubling
problem
Wilsonquarks
Sfw
 Sfnaive
r
 a5
2
4
  (
 x 1  2 x   x 1
x
x
 1`
a
2
)
r  Wilsonparameter
P roblem: Largerdiscretization errors!
a  0  Sfw  Sfnaive
Staggeredquarks
Interpretationof doublersas diffrent flavours
P roblem: Flavourchangingscattering(from p  0 to p  /a)!
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Action with quarks
S
 dUd[d ] e
(Sg   M )

 dU (det M) e
Sg
M : matrix of dynamicalquark masses and dependson thequark formulation
det M causesbig numericalproblems!!
Quenched approximation
Forget about thedynamicsof sea quarks
chiral extrapolation
Work withheavierquarksand extrapolation of lightquarkslikeu and d
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Relating lattice results to physics
Make the correlators of quarks by using  matrices

r

0
(  ) 0
T

(  ) T
mesons with spin
( 5  ) 0

( 5  ) T pseudoscalar
( i  ) 0

( i  ) T vector
relativespatialdistribution (r) :
.
. (x) (r)    ( x  r )  P robabilityDensityof quarks Hadron masses
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Steps of typical lattice calculation
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
19
Some results of lattice QCD
calculations
The spectrum of light mesons and baryons in the quenched approximation
20
The ratio of inverse lattice spacing
21
The masses of  and K* mesons as a functionof latticespacing
22
c 

JPC
Charmonium spectrum in quenched approximation
23
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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