The QCD Lagrangian

Quantum ChromoDynami s (QCD) is a gauge theory of strong nu lear

in-tera tions between the onstituents of hadrons. The hadrons are a lass of

parti les in luding baryons (e.g. the nu leon) and mesons (e.g. the pion).

The theory is based on the prin iple of lo al gauge invarian e with a

non-Abelian SU(3) gauge group [3, 4℄. The fundamental degrees of freedom of

the theory are quarks and gluons. The Lagrangian density of QCD an be

written as:

L ^QCD = L ^quark + L ^gluon + L ^int ,

^(1.1)

where

L ^quark

^{is the purely fermioni (quark) part,}

L ^gluon

^{the purely bosoni}

(gluon) part and

L ^int

^the intera tion part that ouples quarksand gluons.

Let us now onsider the dierent parts that onstitute the QCD

La-grangian. The quark term is 1

:

L quark =

N f

X

f =1

ψ ¯ _f (x)(iγ ^µ ∂ _µ − m f )ψ _f (x),

^(1.2)

where

N _f

^{is the number of avours2,}

ψ _f (x)

^{is the quark (spinor) eld}

or-responding to avour

f

^and

m f

^{is the}

f

^{-avour bare quark mass and the}

1

Throughoutthethesis,weemploytheEinsteinsummation onventionforDira indi es

(denoted byGreekletters)andSU(3)-groupgeneratorindi es(denotedbyLatinletters).

2

The Standard Model in orporates 6 avours of quarks (up, down, strange, harm,

bottom, top). However, investigating the low-energy properties of QCD with Latti e

QCDmethods, oneusuallyrestri tsoneselftothelightest2,3or4avours.

{γ ^µ , γ ^ν } = 2η ^µν ,

^(1.3)

where

η ^µν =

^diag

(1, −1, −1, −1)

^{isthe metri tensor.}

The gluon part reads:

L ^gluon = − 1

4 F _µν ^a (x)F _a ^µν (x),

^(1.4)

where

F _µν ^a (x)

^{is the eld strength tensor, whi h is relatedto the gluon eld}

omponents

A ^a _µ (x)

^:

F _µν ^a (x) = ∂ µ A ^a _ν (x) − ∂ ^ν A ^a _µ (x) − gf ^abc A ^b _µ (x)A ^c _ν (x),

^(1.5)

where

g

^{isthe bare oupling onstantand}

f ^abc

^{are the stru ture onstantsof}

SU(3), satisfying the ommutation relations:

[t ^a , t ^b ] = if ^abc t ^c ,

^(1.6)

where

t ^a

^{are the generators of the group SU(3).}

The purely bosoni part of the Lagrangian is invariant with respe t to

the lo al gauge transformation. If we want the fermioni part to obey the

lo algauge symmetry aswell, we haveto introdu e a term that ouples the

fermions and bosons, i.e. des ribes the intera tion between them. This is

the basi buildingprin ipleof alllo algaugetheories. It wasrst dis overed

in the ase of the ele tromagneti intera tion, where a term that ouples

ele trons and photons is ne essary to guarantee the lo al gauge invarian e.

In the ase ofQCD, the sum

L quark + L gluon

^{is not invariant withrespe t to}

the lo al SU(3) transformation and the way to guarantee this invarian e is

to introdu e the intera tion term

L ^int

^{that ouples the quark elds}

ψ

^and

gluon elds

A µ

^:

L ^int = g

N f

X

f =1

ψ ¯ f (x)γ ^µ A µ (x)ψ f (x),

^(1.7)

where the gluon eld

A _µ

^{is related toits omponentsin the followingway:}

A µ (x) = t ^a A ^a _µ (x).

^(1.8)

Conventionally,onewritestheterms

L ^quark

^and

L ^int

^together,introdu ing the ovariant derivative

D _µ

^:

D µ (x) = ∂ µ − igA ^µ (x).

^(1.9)

L ^QCD =

N f

X

f =1

ψ ¯ f (x)(iγ ^µ D µ − m ^f )ψ f (x) − 1

4 F _µν ^a (x)F _a ^µν (x).

^(1.10)

Letusalsodenethe ( lassi al)QCDa tion,whi his the integralof the

Lagrangian density overspa e-time:

S QCD = Z

d ⁴ x L ^QCD .

^(1.11)

Anelegant(and relevantfrom the point of viewof Latti e QCD) way to

quantize a lassi al theory, like the one given by the lassi al QCD a tion

(1.11), is to use the Feynman path integral formalism [5℄. The expe tation

value of any observable

O

^{is given by:}

hOi = 1 Z

Z

D ¯ ψDψDA O[ψ, ¯ ψ, A] e ^{iS QCD [ψ, ¯ ψ,A]} ,

^(1.12)

with the partition fun tion:

Z = Z

D ¯ ψDψDA e ^{iS QCD [ψ, ¯ ψ,A]} .

^(1.13)

Itisworthtoemphasizethatalleldsinthepathintegralare lassi al. Su h

path integral an not beevaluated analyti ally (ex ept for few spe ial ases

mu h simpler than QCD) and one has to swit h to approximate methods.

Formany theories, likeQuantumEle troDynami s (QED), averysu essful

method is perturbation theory. It onsists in expanding the path integral

with respe t to a small parameter (e.g. the ne stru ture onstant

α ≈ 1/137.036

^{inQED)anddroppingtermsbeyondsomeorder. Forexample,the}

most re ent al ulation of the anomalous magneti moment of the ele tron

(usually parametrized in terms of the so- alled

g

^{-fa tor) up to} fourth-order in

α

^{agrees with experiment up to 10 signi ant digits, making it one of}

the most pre isely veried predi tion of physi s  the ele tron

g

^{-fa tor is}

g e = 2a e +2

^,wherethetheoreti alvalue:

a ^th _e = 1 159 652 182.79(7.71)×10 ⁻¹²

andtheexperimentalone:

a ^exp _e = 1 159 652 180.73(0.28)×10 ⁻¹²

^{[6℄. However,}

for perturbative methods to work, there has to be a small parameter with

respe t to whi h one expands the path integral. In the ase of QCD, the

oupling onstant of the olour intera tion depends on energy and one has

to onsider two regimes. For high energy or large momentum transfer, the

QCD oupling onstantissmallenoughforperturbativemethodstowork. In

this regime,theintera tionof quarksand gluons anbearbitrarilyweakand

by Gross, Politzer and Wil zek. However, in the ase of low energy orsmall

momentum transfer, this oupling onstant be omes of the order of unity

and perturbation theory is bound to fail  the strong intera tions be ome

strongindeed. Quantitatively,theenergys alewhenithappens

Λ

^strong

≈ 250

MeV, where the value is not pre isely dened and depends on the hosen

observable. Anyway, its approximate value implies that a vast number of

relevant phenomena in QCD, su h asthe onnement of quarks and gluons

into hadrons, happen in the non-perturbative regime. Thus, one needs

non-perturbativemethods,su hasLatti eQCD,whi histheonlyknownmethod

of extra ting quantitative predi tions about the low-energy regimeof QCD.

This approa h onsists in dis retizing the QCD path integral. In this way,

one obtainsafullyregularized andwell-dened theory, whi h an bestudied

numeri ally, but also analyti ally  the dis retized version of QCD enabled

manyrelevant on eptualdevelopmentsandledtoimportantinsightintothe

nature of strong intera tions.

However, the os illating exponential

e ^{iS QCD [ψ, ¯ ψ,A]}

^{renders the numeri al}

evaluation of the QCD path integral unfeasible from the pra ti al point of

view. Fortunately, integrals like (1.12) are tra table, if one swit hes from

Minkowski spa e-time with metri tensor

η ^µν

^{with signature e.g.}

(+ − −−)

toEu lideanspa e-timewithsignature

(++++)

^{. Thisisa hievedbyanalyti}

ontinuation (Wi k rotation of the time dire tion:

t → −iτ

^{). In order that}

the Eu lidean formulation an be ontinued ba k to physi al (Minkowski)

spa e,the Eu lidean orrelationfun tionshavetosatisfya ertain ondition,

alled the Osterwalder-S hrader ree tion positivity [7, 8℄. This ondition

ensures that the transition probabilities between gauge-invariant states are

non-negative and the quantum me hani al Hamiltonian has only real and

positiveeigenvalues [9℄.

The QCDLagrangian density inEu lidean spa e reads [10℄:

L ^E QCD =

N f

X

f =1

ψ ¯ f (x)(γ _µ ^E D µ + m f )ψ f (x) − 1

4 F _µν ^a (x)F _a ^µν (x)

^(1.14)

and the Eu lidean gammamatri es satisfy:

{γ ^µ , γ ^ν } = 2δ ^µν ,

^(1.15)

where

δ ^µν =

^diag

(1, 1, 1, 1)

^{isthe Eu lidean metri tensor. The}expe tation value of any observable

O

^{is then given by:}

hOi = 1 Z ^E

Z

D ¯ ψDψDA O[ψ, ¯ ψ, A] e ^{−S QCD E [ψ, ¯ ψ,A]} ,

^(1.16)

where

S _QCD ^E = R d ⁴ xL ^E QCD

^{is the Eu lidean a tion and the Eu lidean}

parti-tion fun tion reads:

Z ^E = Z

D ¯ ψDψDA e ^{−S QCD E [ψ, ¯ ψ,A]} .

^(1.17)

The os illating exponential in (1.12) is repla ed by the well-behaved fa tor

e ^{−S QCD E}

^{and thus the} multi-dimensional integral (1.16) an beevaluated nu-meri ally,atleastinprin iple,e.g. withMonteCarlomethods. Formally,the

quantum eld theory dened by the partition fun tion (1.17) an be

inter-preted as a statisti al me hani al system and the exponential

e ^{−S E QCD}

^plays

the role of aBoltzmann fa tor.

Fromnowon, wewillworkonlywiththe Eu lideanformulationof SU(3)

non-Abelian gauge theory (QCD) and hen e we drop the supers ript

E

^and

the subs ript

QCD

^{thatremind us of it.}

Now,wewilldis ussafewimportantfeaturesof ontinuumQCDthatare

relevant from the point of view of further onsiderations, espe iallythe role

of hiral symmetry and spontaneous hiral symmetry breaking[10, 11,4℄.

Tobespe i , letusrestri tourselvestotwoavours ofquarks (

u

^and

d

quarks). The lassi alQCD Lagrangian an be rewritten as:

L = ¯uγ ^µ D µ u + ¯ dγ µ D µ d + ¯ um u u + ¯ dm d d − 1

4 F _µν ^a F _a ^µν ≡

≡ L ^u + L ^d + L ^{m u} + L ^m d + L ^gluon ,

^(1.18)

where

u ≡ ψ ^u

^and

d ≡ ψ ^d

^{are the} orresponding spinors and we have sep-arated the mass terms in the fermioni Lagrangian. We an de ompose the

quark Lagrangian further by dening left-handed and right-handed quark

spinor elds:

q R ≡ P ⁺ q, q L ≡ P ⁻ q, q = u, d,

^(1.19)

where:

P ^± = 1 ± γ ⁵

2 .

^(1.20)

Eq. (1.19) impliesfor the onjugate spinor elds:

¯

q R = ¯ qP ⁻ , q ¯ L = ¯ qP ⁺ .

^(1.21)

Thus, the rst twoterms in Lagrangian (1.18)be ome:

L ^u + L ^d = ¯ u L γ µ D µ u L + ¯ u R γ µ D µ u R + ¯ d L γ µ D µ d L + ¯ d R γ µ D µ d R =

^(1.22)

terms we obtain:

i.e. the mass terms oupleelds of opposite hiralities.

Let us now onsider the massless terms

L ^u

^and

L ^d

^{in the} Lagrangian.

They are invariant with respe t to the following transformations,

respe -tively:

where

L

^and

R

^areunitary

2×2

^{matri es,i.e. elementsofthe(avour)group}

U(2). This means that the Lagrangian

L u + L d

^{is invariant with respe t to}

the group U(2)

L ×

^U(2)

^R

^.

Let us take a loser look at the possible forms of transformations. The

masslessquark Lagrangianisinvariantunder fourSU(2)

×

^{U(1)ve tor}

trans-formations: (ve tor) Noether urrents

j _i ^µ

^{asso iated with these 4} transformations and hen e 4 onserved harges

Q i = R d ³ xj _i ⁰

^{ the baryon number (}

i = 0

^{) and}

the isospin (

i = 1, 2, 3

^).

Inaddition,therearetransformationsinvolving

γ 5

^{, alled hiralrotations:}

u

Togetherwithtransformations(1.25),themasslessquarkLagrangian

L u +L d

is invariant underthe symmetry groupSU(2)

R ×

^SU(2)

^L ×

^U(1)

^V ×

^U(1)

^A

^.

However, it an be shown that the fermion integration measure in the

quantized theory is not invariant under the transformation (1.26) for

i = 0

^,

whi hredu es the fullsymmetry to SU(2)

R ×

^SU(2)

^L ×

^U(1)

^V

^{. This isthe}

so- alled axial anomaly and it has important onsequen es e.g. for the meson

spe trum  the hiral avour singlet symmetry an not be broken

sponta-neously and hen e there isno Goldstone boson asso iatedwith spontaneous

breaking of this symmetry. This implies that the mass of the avour

sin-glet

η ^′

^{meson does not vanish in the limit of vanishing quark masses (as}

opposed to the mass of the

η

^{meson, whi h is one of the} pseudo-Goldstone bosons),but it isrelatedto topologi alu tuationsof the QCD va uumvia

the Witten-Veneziano formula[12, 13℄:

f _π ²

2N _f m ² _η + m ² _η ′ − 2m ² K
= χ top ,

^(1.27)

where

f π

^{is the pion de ay onstant,}

m x

^{the mass of the}

x

^{meson and}

χ top

the topologi al sus eptibility, whi h willbe dened later.

Letus now onsider the mass terms of the QCD Lagrangian

L m u + L m d

^.

They areinvariantwithrespe t tothe transformation(1.25)for

i = 0

^,sothe

baryon number is onserved also in the massive theory. For

i = 1, 2, 3

^the

transformation(1.25)isasymmetryonlyif thequarkmassesare equal

m _u = m d

^{. Hen e, theisospinis onserved inthemassivetheory,butonlyfor}

mass-degenerate quarks. However, the mass terms

L ^{m u} + L ^m d

^{are not invariant}

under hiralrotations(1.26),whi his ausedbythefa tthattheexponential

in (1.26) is the same for the spinor

(u d) ^T

^{and the onjugate spinor}

u ¯ ¯ d

,

whi h is, in turn, due to the anti ommutation relation

{γ ^µ , γ 5 } = 0

^{. Thus,}

the symmetry ofthe quantum QCDLagrangianisbroken toSU(2)

V ×

^U(1)

^V

in the mass-degenerate ase and to U(1)

V ×

^U(1)

^V

^if

m u 6= m ^d

^.

In the ase of arbitrary number

N f

^{of quark avours, the analysis is}

easilygeneralized (the matri es

u i

^{are nowthe}

N f × N ^f

^{identity matrix and}

N _f ² − 1

^{generatorsofthe avour groupSU(}

N f

^{))andthefullsymmetry ofthe}

quantized massless QCD Lagrangian is SU(

N f

⁾

R ×

^SU(

N f

⁾

L ×

^U(1)

^V

^{, whi h}

is redu ed to SU(

N f

⁾

V ×

^U(1)

^V

^{in the} mass-degenerate ase and further to U(1)

V × . . . ×

^U(1)

^V

^(with

N f

^{fa tors U(1)}

V

^{) in the ase of dierent quark}

masses. Thus, in the latter ase, the only exa t symmetry is the baryon

number onservation.

However,sin etheisospinsymmetryisonlyslightlybrokenforthelightest

twoquarks, itisoftentreatedasexa t 3

,while theheavierquarksare treated

separately. Moreover, sin e the up and down quarks are so light, ompared

to the heavier quarks (

m u ≈ m ^d ≈

^{a few MeV, whereas already}

m s ≈ 100

MeV), the full symmetry of the massless Lagrangian with

N f = 2

^avours

SU(2)

R ×

^SU(2)

L ×

^U(1)

V

^{remains an important} approximate symmetry and is the basis of

N f = 2

^hiral perturbation theory (

χ

^{PT ). At low energy,}

the quarks and gluons are onned into hadrons and hen e one an dene

an ee tive eld theory, in whi h the fundamental degrees of freedom are

not quarksand gluons,but lighthadrons. Two-avour

χ

^{PT wasformulated}

by Gasser and Leutwyler [14℄. The Lagrangian of this theory is onstru ted

from elds des ribing the pions (

π ^±

^,

π ⁰

^{) in a way whi h is onsistent with}

3

InLatti eQCDoneusuallysimulatesthelightesttwoquarksasmass-degenerate.

ganized in terms of expansion parameters

p/Λ χ

^and

m π /Λ χ

^{, where}

p

^{is the}

momentum,

m π

^{thepion massand}

Λ χ = (4πf ) ²

^{thetypi alhadroni s ale}

≈

1 GeV,with

f

^{the pion de ay onstantin the hirallimit. Thereare many}

appli ations of

χ

^{PT in the analysis of the low-energy regime of QCD, e.g.}

pion s attering experiments. Moreover, it is also essential in the analysis of

Latti eQCDdata, sin emost of ontemporaryLatti eQCDsimulationsare

performed at unphysi al values of the pion mass 4

 hen e an extrapolation

to the physi al point (physi al pion mass) is ne essary and is performed by

tting

χ

^{PT formulas. What is more, even thoughthe strange quarkmass is}

mu h larger than the mass of the up and down quarks, it is still relatively

small ompared to the typi al QCD s ale of

≈

^{1 GeV and the symmetry}

SU(3)

R ×

^SU(3)

^L ×

^U(1)

^V

^{of the massless}

N f = 3

^{Lagrangian is also an}

ap-proximate symmetry and forms the basis of

N _f = 3

^hiral perturbation the-ory,whi hisalsoof use inthe analysis oflow-energyQCDexperiments,e.g.

in luding the kaons (also in kaon physi s from Latti e QCD). Three-avour

χ

^{PT was also introdu ed by Gasser and Leutwyler [15℄ as a} generalization of the two-avour ase to in lude the strange quark. The three-avour

La-grangianin ludes,besidesthepionelds,alsootherlightpseudos alarmeson

elds (of the remaining pseudo-Goldstone bosons  the kaons

K ^±

^,

K ⁰

^,

K ¯ ⁰

and the

η

^meson). Quantitatively, the expli it breaking of hiral symmetry by the quark masses an be expressed by the ratios

m ² _π /(4πf ) ² ≈ 0.007

^and

m ² _K /(4πf ) ² ≈ 0.09

^{. In this sense, the expli itbreaking by the strangequark}

mass isroughlya10%ee t, whileforthe lightestquarksitisa<1% ee t.

Obviously,itisnot possibletotreat the

N f = 4

^symmetryasapproximately valid,sin ethe harmquarkisalreadyheavy(

m c ≈ 1.3

^{GeV)andthemesons}

ontaining itare mu hheavier than the s ale

Λ χ

^.

However, if hiralsymmetrywasbroken onlyexpli itly,wewouldobserve

degenerate multiplets of hadrons  e.g. there should be s alar mesons with

massesverysimilartothepseudos alarones. Also,inthis aseoneshouldnot

expe t su h big dieren e between the masses of the pions and kaons. The

explanation of these phenomena an beprovided by anassumption that the

hiral symmetry of QCD is not only expli itly broken by the quark masses,

butalsospontaneouslybroken. Wespeakofspontaneoussymmetrybreaking

if asymmetrywhi hispresentatthe Lagrangianlevelisabsentinthe

phys-i al ground state 5

. If a ontinuous symmetry is broken spontaneously, then

4

Some ollaborationshavere entlystartedorarepreparingsimulationsatthephysi al

pion mass.

5

A learexampleis provided byferromagnets. EventhoughtheHamiltonian ofsu h

system is invariantwith respe t to a simultaneous ip of allspins, in an experimentall

spins are aligned, i.e. only one of two degenerate ground states must be hosen  the

interpreted asthe wouldbe-Goldstonebosons of hiralsymmetrybreaking,

wheretheprex wouldbe-referstothefa tthattheyare notmassless,but

have asmallmass ( ompared tothe masses of other hadrons) that isdue to

(small) expli it breaking of hiral symmetry by the quark masses.

Also, spontaneous breaking of hiral symmetry an be observed in the

mass dieren e of parti les that are hiral partners and should have the

same mass,if hiralsymmetry wasexa t. Sin e hiralsymmetry isexpli itly

broken bythe quark masses,the experimental massvalues of hiral partners

should not be equal, but they should be lose to ea h other, be ause the

masses of the lightquarks are so small. This is not observed. Forexample,

the ve tor mesons

ρ

^and

a 1

^{have massesequalto,}respe tively, 770 and 1260 MeV,whi hisamu hlargerdieren ethanone wouldexpe tfromthesmall

expli itbreakingof hiralsymmetry[16℄. Anotherexampleisthenu leonand

itsnegative-paritypartner,usuallydenotedby

N ^∗

^{[11,17℄. The}experimental value of the nu leon mass is

m N ≈ 940

^{MeV, while}

m N ^∗ ≈ 1535

^MeV.

Spontaneous hiralsymmetrybreakingissignalledbyanon-zerovalueof

the hiral ondensate

h0|¯uu|0i

^,where

|0i

^{is theva uumstate. This quantity}

emerges in hiral perturbation theory as an important low-energy onstant

B ₀

^:

B 0 = −f ⁻² h0|¯uu|0i,

^(1.28)

where the tree-level pion de ay onstant

f

^{is another low-energy onstant.}

A well-known relationthat involves the hiral ondensate is the Gell-Mann,

Oakes, Renner (GMOR) relation[18℄:

f ² m ² _π = −(m u + m _d )h0|¯uu|0i,

^(1.29)

whi h an be derived in

χP T

^{. As su h, it is desirable to assess the value}

of the hiral ondensate from experiment  thus the value of

B 0

^{would be}

known. It has been argued that the best estimate an be obtained from

the low-energypion-pions attering [19, 20℄. However, the al ulation of the

ondensatefromempiri aldatarequiressomemodelassumptions,i.e. one in

fa t has to assumethat spontaneous hiral symmetry breaking takes pla e.

Therefore,animportant he k would beto al ulatethe ondensate

non-perturbativelyfromrstprin iples,withoutanyadditionalassumptions. One

su hway isprovidedby Latti eQCD.Indeed, Latti eQCDsimulations

on-rmthatitisnon-zeroatzerotemperature(areviewofresultsonthistopi is

provided e.g. in. [21℄). However, thereexists atemperaturewherethe hiral

ondensate vanishes,thus signalling hiral symmetry restoration. Moreover,

spin-ip symmetryisspontaneouslybroken.

ment temperature, i.e. the temperature at whi h the quark-gluon plasma

formsand quarksand gluons areno longer onned intohadrons. Up tothe

presentday,thisissuehasnotbeenresolved ompletely,butitisastronghint

that Latti eQCD al ulations point to the fa t that both temperatures are

equal, up tostatisti alerror. This strongly suggests that spontaneous hiral

symmetry breaking isrelated to onnement and onrms that

understand-ing hiral symmetry and spontaneous hiral symmetry breaking is essential

to fully omprehend QCD. However, mu h more pre ise results are needed

to unambiguously resolve this question. In Latti e QCD investigations of

these phenomena it is therefore essential to take hiral symmetry properly

into a ount,i.e. fermionswith good hiralpropertieshave tobeused. This

is one of the motivations for employing overlap fermions, whi h will be the

main subje t of this thesis.

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 12-21)