# Simulating QCD

In this se tion, we briey outlinethe idea behind aLatti e QCD simulation

(with any kindof fermions) and des ribe the most widely used algorithmof

generating gauge eld ongurations  the Hybrid MonteCarlo algorithm.

3.1.1 General idea

simulation onsistsinapproximatelyevaluatinganintegraloftheform(1.16)

by aMonteCarlomethod. Thisisa high-dimensionalintegraloverall

possi-blegaugeeldandfermioneld ongurations. Fortunately,the dependen e

ontheGrassmann-valuedfermionelds an alwaysbeeliminated,leavingan

integral overonlythe gaugeelds, weighted by the Boltzmannfa tor

,

where

### S ef f

is some ee tive a tiondependent on the algorithmused.

More-over, foramajority ofgauge eld ongurations the a tion isverylarge and

hen e theirweight isnegligiblysmall. Therefore, one shouldperform

impor-tan e sampling, i.e. use analgorithm that ee tively hooses ongurations

that have ahigh Boltzmannfa tor. Thus, having alarge number(of the

or-der of several thousand) of su h ongurations, one an ompute the Monte

Carlo average of anobservable

### O

, whi h wewill denoteby

:

(3.1)

where

### O[U i ]

denotes theobservable

### O

omputedinaba kgroundgaugeeld

### U i

belongingtothe Markov hain of generated ongurations. Ifthe simula-tion is performed orre tly, inthe limit

### N → ∞

the Monte Carlo average

### O ¯

will orrespondtothea tualensembleaverage

### hOi

. The onditionsthathave

tobesatisedinordertoobtainthe orre taveragearemeasurepreservation

and detailed balan e. The latter reads:

(3.2)

where

### P (U → U ′ )

denotesthe probabilityof transitionfrom onguration

to

### U ′

.

Let us now onsider the partition fun tion (1.17). After integrating out

the fermion elds, one obtains:

where

### det( ˆ D i [U])

isthedeterminantoftheDira operatormatrixforfermion

avour

### i

. Su hformoftheintegrandimpliesthattheprobabilitydistribution that hastobesimulateddepends onahighlynon-lo alfermiondeterminant.

The ost of al ulating this determinant(s) is by far the highest ost in a

Monte Carlo simulation. However, the rst approximation to the partition

fun tion ouldbetonegle tthefermiondeterminant,i.e. setittoa onstant.

Su h approximation is alled the quen hed approximation and it physi ally

onsists in negle ting the fermion loops. As su h, it is very rude. Still,

for many years it was very mu h used in simulations, sin e the

omputa-tional ost relatedtothe determinantwasjusttoohigh forthe generationof

omputers then available.

However, the omputational power has been in reasing for many years

and presently itis possibletoperformfully dynami alsimulations(i.e. with

the determinant in luded 1

),whi h isalsodue tomany algorithmi

improve-ments. In the next subse tion we des ribe the algorithmof hoi e for most

simulations with dynami alfermions the HybridMonte Carlo algorithm.

3.1.2 Hybrid Monte Carlo

The HybridMonteCarlo(HMC)algorithmwasoriginallyintrodu edby

Du-ane,Kennedy,PendletonandRoweth[70℄. It ombinesamole ulardynami s

update of gaugeelds witha Metropolisa ept/reje t step. Here we outline

the basi steps that need tobe performed inan HMCsimulation[71℄.

Given the a tiontosimulate

### S(U)

, rst one onstru ts the Hamiltonian:

### H(π, U) = 1

Thedeterminantis not omputedexpli itlyoneusuallyrepresentsitinanindire t

way,e.g. byasetofpseudofermionelds, tobedis ussedlater.

where

### π x,µa

is a omponent of a momentum eld:

### π x,µ = π x,µa t a

(3.5)

onjugate to ea h latti e link

### U x,µ

. In this way, the integral one wants to

evaluate:

### R DU O[U] exp(−S(U))

an be written in the equivalent form:

### R DUDπ O[U] exp(−H(π, U))

,sin e the additionalintegrationover momen-tum elds

### π

yields just a Gaussian integral and hen e produ es a onstant

fa tor.

Thus, one obtains a lassi al Hamiltoniansystem. The evolution of this

system in a titiousMonteCarlo time

### τ

an be al ulated fromthe

Hamil-ton's equations:

(3.6)

### U ˙ x,µ = π x,µ U x,µ ,

(3.7)

wherethedotdenotesdierentiationwithrespe ttothe  titioustime

and

the for e

is given by2:

### .

(3.8)

Solving the above system of dierential equations, one obtains a traje tory

in phase spa e,i.e. the values of

and

### π x,µ (τ )

for every value of

### τ

.

The steps inthe HMCalgorithmare the following:

1. Randomly generate the initial(

) momentum eld

### π x,µ (0)

a ord-ing tothe distribution

### x,µ π x,µa π x,µa )

.

2. Numeri allyintegrate(e.g. bytheso- alledleap-frogalgorithm)

Hamil-ton's dierential equations (3.6)-(3.7) to obtain

and

### π x,µ (τ )

fromtheir initialvalues

and

### π x,µ (0)

, respe tively. By onstru -tion, su h evolution preserves the value of the Hamiltonian up to a

numeri al integration error.

3. A ept the new ongurationwith probability:

min

(3.9)

where

### ∆H(τ) = H(π(τ), U(τ)) − H(π(0), U(0))

is ingeneral non-zero,

whi hisdueonlytothenumeri alintegrationerror. Ifthe onguration

is reje ted, then

### U(τ ) = U(0)

, i.e. the initial onguration does not hange.

2

variable. The derivative with respe t to an SU(3) element an be formally dened as

### ω a (x,µ)=0

.

tions)as isdesired. The initialvaluesof the variables

and

for step

2oftraje tory

### N

arethevaluesofthesevariablesattheendofstep3of

the pre edingtraje tory

, i.e.:

traje tory

traje tory

,

traje tory

traje tory

### N −1 (τ )

.

The above algorithm ould in prin iple be used to simulate QCD with

dynami al quarks. However, to make su h simulations pra ti al, one has to

over ome the omputational problem of ee tively al ulating the fermion

determinant. This is usually done with the pseudo-fermion method, whi h

onsists in repla ing the fermion elds by auxiliary bosoni elds