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

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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

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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

qq

kqjk

j

qq

aqaqsqqqqqqqqqq

aaaa

as

'

jq

jq

qq

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

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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

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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

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!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.