Simulating QCD - Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University

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

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

generating gauge eld ongurations the Hybrid MonteCarlo algorithm.

3.1.1 General idea

AsalreadystatedinSe tion1.1, omputinganyobservableinaLatti eQCD

simulation onsistsinapproximatelyevaluatinganintegraloftheform(1.16)

by aMonteCarlomethod. Thisisa high-dimensionalintegraloverall

possi-blegaugeeldandfermioneld ongurations. Fortunately,the dependen e

ontheGrassmann-valuedfermionelds an alwaysbeeliminated,leavingan

integral overonlythe gaugeelds, weighted by the Boltzmannfa tor

e ^−S ^{ef f}

where

S ef f

^is ^some ^ee
tive ^a
tion^dependent ^on ^the ^algorithm^used.

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

hen e theirweight isnegligiblysmall. Therefore, one shouldperform

impor-tan e sampling, i.e. use analgorithm that ee tively hooses ongurations

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

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

Carlo average of anobservable

O

^, ^whi
h ^we^will ^denote^by

O ¯

O = ¯ 1

N

X

i=1

O[U ⁱ ],

^(3.1)

where

O[U ⁱ ]

^denotes ^the^observable

O

^omputedⁱⁿ^a^ba
kground^gauge^eld

U i

^belonging^to^the ^Markov ^hain ^of ^generated ongurations. Ifthe simula-tion is performed orre tly, inthe limit

N → ∞

^the ^Monte ^Carlo ^average

O ¯

will orrespondtothea tualensembleaverage

hOi

^. ^The^onditions^that^have

tobesatisedinordertoobtainthe orre taveragearemeasurepreservation

and detailed balan e. The latter reads:

e ^{−S[U ]} P (U → U ^′ ) = e ^−S[U ^′ ^] P (U ^′ → U),

^(3.2)

where

P (U → U ^′ )

^denotes^the probabilityof transitionfrom onguration

U

U ^′

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

the fermion elds, one obtains:

Z =

where

det( ˆ D _i [U])

^is^thedeterminantoftheDira operatormatrixforfermion

avour

i

^. ^Su
h^form^of^the^integrand^implies^that^theprobabilitydistribution 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 gaugeelds 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 itlyoneusuallyrepresentsitinanindire t

way,e.g. byasetofpseudofermionelds, tobedis ussedlater.

where

π _x,µ ^a

^is ^a^omponent ^of ^a ^momentum ^eld:

π x,µ = π _x,µ ^a 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 ^from^the

Hamil-ton's equations:

˙π x,µ = −F ^x,µ ,

^(3.6)

U ˙ x,µ = π x,µ U x,µ ,

^(3.7)

wherethedotdenotesdierentiationwithrespe ttothe titioustime

τ

^and

the for e

F _x,µ

^is ^given ^by²^:

F _x,µ = ∂S(U)

∂U x,µ

.

^(3.8)

Solving the above system of dierential equations, one obtains a traje tory

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

U x,µ (τ )

^and

π x,µ (τ )

^for ^every ^value ^of

τ

The steps inthe HMCalgorithmare the following:

1. Randomly generate the initial(

τ = 0

⁾ ^momentum ^eld

π x,µ (0)

a ord-ing tothe distribution

exp(− ¹ ₂ P

x,µ π _x,µ ^a π _x,µ ^a )

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

Hamil-ton's dierential equations (3.6)-(3.7) to obtain

U _x,µ (τ )

^and

π _x,µ (τ )

fromtheir initialvalues

U x,µ (0)

^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 ongurationwith probability:

P =

^min

1, e ^{−∆H(τ )} ,

^(3.9)

where

∆H(τ) = H(π(τ), U(τ)) − H(π(0), U(0))

^is ⁱⁿ^general ^non-zero,

whi hisdueonlytothenumeri alintegrationerror. Ifthe onguration

is reje ted, then

U(τ ) = U(0)

^, ^i.e. ^the ^initial onguration does not hange.

In(3.8)weusesymboli notationforaderivativeofthea tionwithrespe ttoalink

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

∂S(exp(ω ^a (x,µ)t ^a )U(x,µ))

∂ω ^a (x,µ)

ω ^a (x,µ)=0

tions)as isdesired. The initialvaluesof the variables

U

^and

π

^for ^step

2oftraje tory

N

^are^the^values^of^these^variables^at^the^end^of^step³^of

the pre edingtraje tory

N − 1

^, ^i.e.:

U

^traje
tory

^N (0) = U

^traje
tory

^{N −1} (τ )

π

^traje
tory

^N (0) = π

^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 ee 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

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 62-65)

Simulating QCD

e −S ef f

S ef f

O

O ¯

O = ¯ 1

N

N

X

i=1

O[U i ],

O[U i ]

O

U i

N → ∞

O ¯

hOi

e −S[U ] P (U → U ′ ) = e −S[U ′ ] P (U ′ → U),

P (U → U ′ )

U

U ′

Z =

det( ˆ D i [U])

i

S(U)

H(π, U) = 1

π x,µ a

π x,µ = π x,µ a t a

U x,µ

R DU O[U] exp(−S(U))

R DUDπ O[U] exp(−H(π, U))

π

τ

˙π x,µ = −F x,µ ,

U ˙ x,µ = π x,µ U x,µ ,

τ

F x,µ

F x,µ = ∂S(U)

∂U x,µ

.

U x,µ (τ )

π x,µ (τ )

τ

τ = 0

π x,µ (0)

exp(− 1 2 P

x,µ π x,µ a π x,µ a )

U x,µ (τ )

π x,µ (τ )

U x,µ (0)

π x,µ (0)

P =

1, e −∆H(τ ) ,

∆H(τ) = H(π(τ), U(τ)) − H(π(0), U(0))

U(τ ) = U(0)

∂S(exp(ω a (x,µ)t a )U(x,µ))

∂ω a (x,µ)

ω a (x,µ)=0

U

π

N

N − 1

U

N (0) = U

N −1 (τ )

π

N (0) = π

N −1 (τ )

e ^−S ^{ef f}

O[U ⁱ ],

O[U ⁱ ]

e ^{−S[U ]} P (U → U ^′ ) = e ^−S[U ^′ ^] P (U ^′ → U),

P (U → U ^′ )

U ^′

det( ˆ D _i [U])

π _x,µ ^a

π x,µ = π _x,µ ^a t ^a

˙π x,µ = −F ^x,µ ,

F _x,µ

F _x,µ = ∂S(U)

exp(− ¹ ₂ P

x,µ π _x,µ ^a π _x,µ ^a )

U _x,µ (τ )

π _x,µ (τ )

1, e ^{−∆H(τ )} ,

∂S(exp(ω ^a (x,µ)t ^a )U(x,µ))

∂ω ^a (x,µ)

ω ^a (x,µ)=0

^N (0) = U

^{N −1} (τ )

^N (0) = π

^{N −1} (τ )