Observables in Latti e QCD

In this se tion, we show how one an extra t hadron properties from

Lat-ti e QCD simulations, on entratingon the quantities of interestfor further

onsiderations.

Letus onsider a generalzero-momentumtwo-point orrelation fun tion

of the form

C(t) ≡ h0|O i (t) ¯ O _j (0)|0i

^{, where}

O _i (t)

^{is some} interpolating op-erator orresponding to the state with quantum numbers of the hadron we

want to analyze. Sin e:

O _i (t) = e ^Ht O _i e ^−Ht ,

^(1.82)

insertinga ompleteset of energy eigenstatesinto

C(t)

^{yields(wetake}

i = j

where

E n

^{is the energy of the state}

n

⁽

1/2E n

^{is a} normalization fa tor for energy eigenstates).

One an immediatelysee that in the limitof large Eu lidean time

t

^{, the}

above expression isdominated by the lowest energy state

|1i

^:

C(t) −−−→ ^t→∞ |h0|O ⁱ |1i| ²

2E ₁ e ^{−E 1 t} ,

^(1.84)

where

E 1

^{isthe energy of this state, i.e. the mass of the lightest parti le. In}

thisway,one anextra tthismassbytting(insomeinterval

t ∈ [t min , t _max ]

⁾

the orrelation fun tion with an exponential fun tion

A exp(−m ¹ t)

^{, where}

A

^,

m 1

^{are tting} parameters, whi h provide estimates for the parti le mass

E ₁ = m ₁

^{and thematrix element}

|h0|O i |1i| ² = 2Am ₁

^{. Tond the tinterval}

[t min , t max ]

^{,one usually omputes the so- alledee tive mass:}

m

^e

(t) ≡ log

 C(t) C(t + 1)



(1.85)

and plotsittond theplateauregion,i.e. the regionwherethe ontribution

of the ex ited states is negligible and the ee tive mass is stable, up to

statisti al u tuations.

Sin e a latti e omputation is usually performed with a nite latti e

ex-tent

T

^{in the temporal dire tion11 with e.g. periodi boundary onditions}

in time, the large-time form of the orrelation fun tion is modied in the

followingway:

In su h ase,the ee tive massattime

t

^{an beextra ted bysolving}

numer-i ally the equation

C(t)/C(t + 1) = cosh E ₁ t − ^T ₂
/ cosh E 1 t + 1 − ^T ₂

.

11

However, it is sometimespossible to onsider latti es with innite time extent. An

examplewillbegivenin thenext hapter.

Wenow on entrateonmeson orrelatorsinthe aseof

N f = 2

^degenerate

quarks. The generalform of aninterpolating operatorfor mesons is:

O i (~x, t) = ¯ ψ(~x, t)Γ ⁱ ψ(~x, t),

^(1.87)

where

Γ

^{denotes any Dira matrix (an identity matrix, a gamma matrix or}

a ombinationof gammamatri es).

Expli itly introdu ing Dira (

µ

^,

ν

^,

ρ

^,

σ

^{) and olour (}

a

^,

b

^{) indi es, the}

orrelation fun tion an be writtenas:

C(t) = X

~ x

h0| ¯ ψ _µ ^a (~x, t)Γ ⁱ _µν ψ _ν ^a (~x, t) ¯ ψ _ρ ^b (~0, 0)Γ ^j _ρσ ψ _σ ^b (~0, 0)|0i,

^(1.88)

where the sum over

~x

^Fourier-transforms the orrelation fun tion to zero momentum. Contra tingfermion elds pairwise intofermion propagators:

h0|ψ µ ^a (~x, t) ¯ ψ _ν ^b (0, 0)|0i = S µν ^ab (~x, t;~0, 0),

^(1.89)

a ording to Wi k's theorem, one nds that there are two possible

on-tra tions (

ψ _ν ^a (~x, t) ↔ ¯ ψ _µ ^a (~x, t)

^,

ψ _σ ^b (~0, 0) ↔ ¯ ψ ^b _ρ (~0, 0)

^and

ψ ^a _ν (~x, t) ↔ ¯ ψ _ρ ^b (~0, 0)

^,

ψ ^b _σ (~0, 0) ↔ ¯ ψ ^a _µ (~x, t)

^{),whi h lead to:}

C(t) = X

~ x

Tr

(S(~x, t; ~x, t) Γ ⁱ )

^Tr

(S(~0, 0;~0, 0) Γ ^j ) +

− X

~ x

Tr

(S(~x, t;~0, 0) Γ ⁱ S(~0, 0; ~x, t) Γ ^j ),

^(1.90)

where the tra e is overspin and olour.

The rst term in the above expression an be represented by a

dis on-ne ted diagramand ontributesonlytoavour singletmesons. Lateron, we

willbeinterested onlyinavournon-singletmesons,i.e. ones thatare

repre-sentedby onne teddiagrams, orrespondingtothese ondtermintheabove

expression. Hen e, we now drop the rst term and use the

γ ₅

-hermiti ity property of the propagator:

S(~0, 0; ~x, t) = γ 5 S ^† (~x, t;~0, 0)γ 5

^{to rewrite:}

C(t) = − X

~ x

Tr

(S(~x, t;~0, 0) Γ ⁱ γ 5 S ^† (~x, t;~0, 0) γ 5 Γ ^j ).

^(1.91)

Inthisway,toevaluatethis orrelatoritisenoughto omputethepropagator

from a given sour e (lo atedat the origin (

~0, 0

^{) in the above formula) to all}

possiblesinks(alllatti esites(

~x, t

^{)). Su hpropagatoris alleda}point-to-all propagator. This an be done by solving the following matrix equation:

Dψ ˆ _µa = η _µa

^(1.92)

Table 1.1: Meson interpolatingoperators.

J ^{P C}

lassi ationdenotes parti le spin

J

^{, parity}

P

^{and harge} onjugation

C

^[63℄.

hannel

J ^{P C} Γ

pseudos alar

0 ⁻⁺ γ 5

^,

γ 0 γ 5

s alar

0 ⁺⁺

^1,

γ 0

ve tor

1 ⁻⁻ γ i

^,

γ 0 γ i

axialve tor

1 ⁺⁺ γ _i γ ₅

tensor

1 ⁺⁻ γ i γ j

12 times for ea h spin- olour ombination

µa

^{, with a point sour e}

η µa

^{, i.e.}

a ve tor

(0 . . . 010 . . . 0) ^T

^{, where the only non-zero number is pla ed in one}

of the rst 12 entries, orrespondingto12 spin- olour omponentsatlatti e

site

(0, 0, 0, 0)

^.

The solutionof this equation:

ψ _µa = ˆ D ⁻¹ η _µa

^(1.93)

is the point-to-all quarkpropagator, denoted by

S(~x, t;~0, 0)

^{ineq. (1.91), in}

whi hthe spin- olour indi esare suppressed.

Obviously, the Dira equation (1.92) does not have to be solved with a

point sour e lo ated at the origin. Other hoi es of the sour e an be e.g.

pointsour es with randomlo ationof the sour eor sto hasti sour es. The

latter areof spe ial relevan e fromthepointofviewof thiswork and willbe

dis ussed later.

Table1.1summarizesthemost ommonlyusedmesoninterpolating

oper-ators. The names of dierent hannels ome from the transformation

prop-erties of parti les with respe t to spin and parity. Here we have assumed

that the

Γ

^{matrix at the sour e (denoted by}

Γ ^j

^{in eq. (1.91)) and at the}

sink (

Γ i

^{) are thesame. However, itis alsopossibleto onstru t mesonswith}

Γ i 6= Γ ^j

^{, e.g.}

Γ i = γ 5

^,

Γ j = γ 0 γ 5

^{, whi h belongs to the} pseudos alar hannel and hen eit analsobeusedtoextra t themass ofthepseudos alarmeson.

From the point of view of further onsiderations, the most important

meson hannel will be the pseudos alar one. The PP orrelation fun tion

(

Γ i = γ 5 ≡ P

^,

Γ j = γ 5 ≡ P

^{) is the simplest} orrelation fun tion that an be onstru ted. Puttingits gammamatrix stru ture ineq. (1.91),one obtains:

C P P (t) = − X

~ x

Tr

(S(~x, t;~0, 0) S ^† (~x, t;~0, 0)).

^(1.94)

son (pion)

m π

^{from the de ay of the PP orrelator and also the pion de ay}

onstant

f π

^{from the matrix element}

|h0|P |πi|

^:

f _π = 2m

m ² _π |h0|P |πi|,

^(1.95)

where

m

^{is the bare quark mass.}

Anequivalent denition of the pion de ay onstant reads:

f π = Z A

m π |h0|A ⁰ |πi|,

^(1.96)

where

Z A

^isthe renormalization onstant ofthe axial urrent and

|h0|A ⁰ |πi|

the matrix element of this urrent.

Forthe ase of overlapfermions, the

O(a)

^-improved interpolating opera-tors formesons are onstru ted inthe following way [64℄:

O _i ^ov (~x, t) = ¯ ψ(~x, t)Γ ⁱ 1 − a ˆ D ov (0) 2

!

ψ(~x, t) = 1

1 − ^am ₂ ψ(~x, t)Γ ¯ ⁱ ψ(~x, t),

(1.97)

where the last equality holds for orrelation fun tions at non-zero physi al

distan e.

We also give here the expressions for baryon interpolating operators 

for the proton

p

^{(uud), the neutron}

n

^{(udd) and the deltas}

∆ ⁺⁺

^(uuu),

∆ ⁺

(uud),

∆ ⁰

^(udd),

∆ ⁻

^{(ddd) [65, 66℄, i.e. the o tet and the de uplet baryons}

that ontain only lightquarks (up and down, no strangequarks).

J p = ǫ abc u ^T _a Cγ 5 d b
u c ,

^(1.98)

J n = ǫ abc d ^T _a Cγ 5 u b
d c ,

^(1.99)

J _∆ ^µ ++ = ǫ _abc u ^T _a Cγ ^µ u _b
u _c ,

^(1.100)

J _∆ ^µ + = 1

√ 3 ǫ abc 2 u ^T _a Cγ ^µ d b
u c + u ^T _a Cγ ^µ u b
d c  ,

^(1.101)

J _∆ ^µ 0 = 1

√ 3 ǫ abc 2 d ^T _a Cγ ^µ u b
d c + d ^T _a Cγ ^µ d b
u c  ,

^(1.102)

J _∆ ^µ − = ǫ _abc d ^T _a Cγ ^µ d _b
d _c ,

^(1.103)

where

C = γ 4 γ 2

^{isthe harge} onjugation matrix.

The two-point orrelation fun tionfor baryon

B

^reads:

C B (t) = 1

2 Tr(1 ± γ ⁴ ) X

~ x

hJ ^B (~x, t) ¯ J B (~0, 0)i,

^(1.104)

where

(1 ± γ ⁴ )/2

^{isthe parityproje tor. Forexample,the physi alprotonis}

des ribed by the orrelation fun tion

C p (t)

^{with proje tiontopositiveparity}

and the negativeparityproje tion orresponds tothebaryon

N ^∗

^{, mentioned}

earlier inthe ontext of spontaneous hiral symmetry breaking.

We will be interested in light baryon masses, whi h are evaluated in an

analogouswayasinthe aseofmesons,i.e. fromtheexponentialfall-oofthe

orresponding orrelation fun tion. The ee tive masses are thus extra ted

numeri ally fromthe ratiosof the orrelationfun tionsof the form(1.86) at

two subsequent timesli es.

Wenishbyshortlydis ussingthedegenera iesbetweenthelightbaryons

in the ase of fermions that preserve isospin symmetry (e.g. overlap) and

violate it (e.g. twisted mass). In the overlap ase, the proton

p

^{and neutron}

n

^are degenerate,aswellasalldeltabaryons. Forthe twisted mass ase, the degenera y isredu ed, but stillholds between

p

^

n

^,

∆ ⁺⁺

^

∆ ⁻

^and

∆ ⁺

^

∆ ⁰

^,

whi h isdue to

γ 5

-hermiti ity. Therefore, we willalways refer tothe proton and neutron asthenu leon

N

^{, butwe will}distinguish between

∆ ⁺⁺

^and

∆ ⁺

in the twisted mass ase.

Tree-level s aling test

In this hapter we willshow the results of tree-level s aling tests of overlap,

twistedmassand Creutzfermionsand thusexpli itlydemonstratethe

O(a)

-improvementintheobservables[67,55,56℄. Wewill onsiderthreequantities

 the pseudos alar meson mass and de ay onstant and the pseudos alar

orrelationfun tionataxedphysi aldistan e. Wewillalsoanalyzethe ase

when the pseudos alar orrelation fun tion is onstru ted with propagators

orresponding to two dierent fermion dis retizations.

2.1 Fermion propagators

Thetree-leveltestofdierentkindsoflatti efermions onsistsinanalyti ally

evaluating the momentum-spa e fermion propagator and then using it to

onstru t the relevant orrelation fun tion, from whi h the observables of

interest an beextra ted.

2.1.1 Overlap fermions

The startingpointforthe evaluationofthe tree-leveloverlap fermion

propa-gatoristhe freemasslessoverlapDira operatorinmomentumspa e 1

,whi h

was given by Lüs her [29℄:

a ˆ D

^ov

(p) = 1 − 

1 − iaγ µ ˚ p _µ − a ² 2 p ˆ ² 

1 + a ⁴ 2

X

µ<ν

ˆ

p ² _µ p ˆ ² _ν  −1/2

.

^(2.1)

The massive operator is,a ording to (1.73):

a ˆ D ov (p, m) = 

1 − am 2



a ˆ D ov (p) + am,

^(2.2)

1

An expli itderivationof thisoperatorisgivenin AppendixB.

where

m

^{is the bare overlap quark mass.}

The expression for the quarkpropagator in momentum spa e

S ov (p)

^an

befound by omputing the inverse of the above Dira operator

a ˆ D ov (p, m)

^:

S ^ov (p) = −i(1 − ^ma ₂ )F (p) ^−1/2 ˚ p µ γ µ + M(p)

¹

(1 − ^ma 2 ) ² F (p) ⁻¹ P

µ ˚ p ² _µ + M(p) ²

^(2.3)

where 1 is the identity matrix in Dira spa e and we have introdu ed the

fun tions:

The propagator has a matrix stru ture in Dira spa e and for later

onve-nien e we write ithere in terms of its omponents:

S ^ov (p) = S _µ ^ov (p)γ µ + S ₀ ^ov (p)

¹

.

^(2.6)

2.1.2 Wilson twisted mass fermions

The twisted mass fermion propagator an be found as an inverse of the

followingDira operator inmomentum spa e:

D ˆ

^tm

(p) = i˚ p µ γ µ

¹

f + ar

2 p ˆ ² _µ

¹¹

f + m

¹¹

f + iµγ 5 τ 3 ,

^(2.7)

where the relevant notation has been introdu ed in Se tion 1.3.3 and we

show here expli itly the matrix stru ture in avour spa e. The rst three

terms have a trivial stru ture in avour spa e (1

f

^{is the identity matrix in}

this spa e), but the twisted mass term

iµγ ₅ τ ₃

^{breaks the isospin symmetry}

between up and down quarks and hen e it modies the expression for the

tree-levelWilsonpropagatorinmomentumspa e(1.51)inthefollowingway:

S ˆ

^tm

(p) = −i˚ p µ γ µ

¹

f + ( ^ar ₂ P

µ p ˆ ² _µ + m)

¹¹

f − iµγ ⁵ τ 3

P

µ ˚ p ² _µ + ( ^ar ₂ P

µ p ˆ ² _µ + m) ² + µ ² .

^(2.8)

The propagator has a matrix stru ture in Dira and avour spa e and we

again write it here in terms of its omponents, expli itly distinguishing

be-tween up and down quark propagators:

S ^tm,u (p) = S _µ ^tm (p)γ µ + S ₅ ^tm (p)γ 5 + S ₀ ^tm (p)

¹

,

^(2.9)

S ^tm,d (p) = S _µ ^tm (p)γ µ − S 5 ^tm (p)γ 5 + S ₀ ^tm (p)

¹

,

^(2.10)

where the propagators of the two avours dier only in the sign of the

γ 5

-matrix oe ient.

Toobtainautomati

O(a)

-improvement(maximaltwist), inthe free the-ory it is enough to set the bare untwisted quark mass

m

^{to0. Su h variant}

of twisted mass fermions is usually referred to as maximally twisted mass

(MTM) fermions.

2.1.3 Creutz fermions

It an beshown [55℄ that the momentum spa e tree-level Dira operatorfor

Creutz fermions an be writtenas:

D Creutz (p) = i X

All notationused inthis subse tion is explainedinAppendix B. This yields

the following formof the fermion propagator:

S Creutz (p) = −i P

The tree-level Dira operator for the variant suggested by Bori i is:

D Borici (p) = i X

where we haveagain introdu ed anauxiliaryfun tion

G _µ (p)

^.

The matrix stru ture of both Creutz and Bori i fermions is of the same

form asin the ase of overlap (2.6).

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 36-44)