GWC ode point sour es

For the test of subtra tion routines, the rst step was to expli tly ompute

the zero modes. The numberof zero modes in the free-eld ase is equal to

N c N d

^, ^i.e. ^there ^are ¹² ^zero ^modes ⁱⁿ ^our ^ase ^of ^interest, ⁶ ⁱⁿ ^the ^positive

and 6 in the negative hirality se tor.

steps:

1. Read in allzero modes.

2. Compute the propagator

Ψ ⁰

^oming ^only ^from ^the ^zero ^modes, ^using

formula4.22,i.e. taking intoa ountthesour e. This sour ehastobe

exa tly the same asthe one used for fullinversion (with allmodes).

3. Compute (or read in, if omputed before) the full propagator

Ψ

^(with

all modes) with the same point sour e asinthe previous step.

4. Constru t the non-zero modes propagator

Ψ ^N = Ψ − Ψ ⁰

5. Use the GWC ontra tion ode to ompute the PP and SS orrelation

fun tions from

Ψ ^N

^. ^This ^gives ^the ^part

C N N (t)

^of ^these orrelators.

The result for the orrelation fun tions with no ontribution from the zero

modes is:

t C_PP(t)

0 +3.2070498264e+00

1 +5.3875529993e-02

2 +1.1851597433e-02

3 +4.3685038347e-03

4 +2.5747032913e-03

5 +4.3685038347e-03

6 +1.1851597433e-02

7 +5.3875529993e-02

t C_SS(t)

0 -2.8998469255e+00

1 +5.3495732753e-02

2 +9.7986124350e-03

3 +2.3495604466e-04

4 -2.4074086241e-03

5 +2.3495604466e-04

6 +9.7986124350e-03

7 +5.3495732753e-02

These numbers are exa tly the same as ones obtained with the analyti al

formulainthe previous se tion.

Wefollowan analogous pro edurein the ase of sto hasti sour es:

1. Read in allzero modes.

2. Read in sample

r

^of ^sto
hasti ^sour
e.

3. Compute the propagator

Ψ ⁰ _r

^oming ^only ^from ^the ^zero ^modes, ^using

formula 4.22 with sample

r

^of ^the ^sour
e

4. Compute (or readin,if omputed before) the fullpropagator

Ψ r

^(with

all modes) with the same sampleof the sour e

r

5. Constru t the non-zero modes propagator

Ψ ^N _r = Ψ r − Ψ ⁰ r

6. Usethe light ontra tion ode to ompute the PPandSS orrelation

fun tions from

Ψ ^N _r

Su h pro edure is then repeated

N _r

^times ^for^dierent ^samples^of ^sto
hasti

noise. Ea hsample ofthe sour eleads toa orrelation fun tion

C N N (t)

^. ^W^e

have used

N r = 600

^samples ^and ^nally ^averaged ^the orrelation fun tions to obtain:

t C_PP(t) dC_PP(t)

0 3.207395e+00 6.036502e-04

1 5.341456e-02 4.634707e-04

2 1.163657e-02 2.135815e-04

3 4.276079e-03 9.325929e-05

4 2.518227e-03 5.732913e-05

5 4.276079e-03 9.325929e-05

6 1.163657e-02 2.135815e-04

7 5.341456e-02 4.634707e-04

t C_SS(t) dC_SS(t)

0 2.899527e+00 3.025145e-04

1 -5.303857e-02 4.597752e-04

2 -9.629567e-03 1.669140e-04

3 -2.358282e-04 3.102437e-06

4 2.351388e-03 5.740869e-05

5 -2.358282e-04 3.102437e-06

6 -9.629567e-03 1.669140e-04

7 -5.303857e-02 4.597752e-04

the ones from the analyti al formula and from the GWC ode with point

sour es, we on lude thatallresultsare onsistent,up tothe statisti alerror

forthe aseofsto hasti sour es. Thelight ontra tion odeusesadierent

sign onvention for the s alar orrelator and hen e the sign of

C SS (t)

^is

alwaysoppositetotheonefromtheGWC ontra tion odeandtheanalyti al

formula. Hen e, with the light ontra tion ode the ontribution of the

zero modes is exa tly an elled in the dieren e

C _{P P} − C SS

^. ^Therefore, ^for

omputations inthe intera ting ase we always use

C P P − C ^SS

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simu-lations with lover-improved Wilson Fermions, Nu l.Phys. B659, 299;

hep-lat/0211042.

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with multiple time s ale integration and mass pre onditioning,

Com-put.Phys.Commun. 174, 87; hep-lat/0506011.

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hybrid MonteCarlo algorithm,Nu l.Phys. B380, 665.

[131℄ Omelyan I.P., Mryglod I.M., Folk R. (2003), Symple ti analyti ally

integrablede ompositionalgorithms: lassi ation,derivation,and

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sim-ulations, Comp.Phys.Comm. 151, 272.

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for Dynami al Fermion Computations with Non-Lo al A tions,

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[135℄ Clark M.A. (2006), The Rational Hybrid Monte Carlo Algorithm,

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[136℄ Clark M.A., Kennedy A.D. (2007), A elerating Dynami al Fermion

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a domain de omposition method, Comp.Phys.Comm. 156, 209;

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JHEP 0712, 011; 0710.5417 (hep-lat).

1.1 Mesoninterpolatingoperators.

J ^{P C}

lassi ationdenotes par-ti le spin

J

^,^parity

P

^and ^harge onjugation

C

^[63℄. ^. ^. ^. ^. ^. ³⁸

2.1 Simulationparameters for the tree-levels aling test. . . 46

2.2 Fitting oe ients for the pion mass eq. (2.20). . . 47

2.3 Fitting oe ients for the pion de ay onstant eq. (2.21). . . 49

2.4 Fitting oe ientsforthepseudos alar orrelationfun tionat

a xed physi aldistan e

t/N = 4

^eq. ^(2.22). ^. ^. ^. ^. ^. ^. ^. ^. ^. ⁴⁹

1.1 Continuum limit s aling in xed nite volume for

r 0 f P S

^at

xed values of

r ₀ m _{P S}

^(a) ^and ^for

(r ₀ m _{P S} ) ²

^at^xed ^values ^of

renormalized quark mass

r 0 µ R

^(b). ^In ^(b) ^data ^at

β = 4.2

(

(a/r 0 ) ² = 0.0144

⁾ ^are ^not ^in
luded, ^due ^to^the ^missing ^value

of the renormalizationfa tor

Z _P

^. ^Sour
e: ^[33℄. ^. ^. ^. ^. ^. ^. ^. ^. ^. ²⁸

2.1 Continuum limits alingof the pion mass for overlap, twisted

mass and Creutz fermions. . . 47

2.2 Continuumlimits alingofthepionde ay onstantforoverlap,

twisted mass and Creutz fermions. . . 48

2.3 Continuum limits aling of the pseudos alar orrelation

fun -tion ata xed physi al distan e

t/N = 4

^for ^overlap, ^twisted

mass and Creutz fermions. . . 50

2.4 Continuum limits alingof the pion mass foroverlap-overlap,

MTM-MTM and overlap-MTM quarks. . . 51

2.5 Continuumlimits alingofthepionde ay onstantfor

overlap-overlap, MTM-MTM and overlap-MTM quarks. . . 52

2.6 Continuum limits aling of the pseudos alar orrelation

fun -tion at a xed physi al distan e

t = 4N

^for overlap-overlap, MTM-MTM and overlap-MTM quarks. . . 53

2.7 Continuum limits aling of the pion mass at a xed physi al

distan e

t/N = 4

^for ^twisted ^mass ^and ^overlap ^fermions. ^The

quark masses are mat hed up to

O(1/N ² )

^. ^The ^lower ^plot ^is

a zoomof the upperone forlarge values of

N

^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ⁵⁵

2.8 Continuum limits aling of the pion de ay onstant ata xed

physi aldistan e

t/N = 4

^for^twisted^mass^and^overlap^fermions.

Thequarkmassesaremat hedupto

O(1/N ² )

^. ^The^lower^plot

is azoom of the upperone for large values of

N

^. ^. ^. ^. ^. ^. ^. ^. ^. ⁵⁶

2.9 The mat hing of MTMand overlap quark masses. . . 57

2.10 Themismat hbetween theMTMandoverlappionde ay

on-stants atthe mat hing point

Nm ^MTM _π = Nm ^overlap _π

^. ^. ^. ^. ^. ^. ^. ⁵⁸

tions (at a xed physi al distan e) at the mat hing point

Nm ^MTM _π = Nm ^overlap _π

^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ⁵⁸

2.12 Thedieren ebetween the MTMand overlappionde ay

on-stants atthe mat hing point

Nm ^MTM _π = Nm ^overlap _π

^, ^as^a

fun -tion of

1/N ²

^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ⁵⁹

2.13 Thedieren ebetweentheMTMandoverlap orrelation

fun -tions (at a xed physi al distan e) at the mat hing point

Nm ^MTM _π = Nm ^overlap _π

^, ^as^a ^fun
tion ^of

1/N ²

^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ⁶⁰

2.14 The dieren e between the MTM and overlap quark mass at

the mat hing point

Nm ^MTM _π = Nm ^overlap _π

^, ^as^a^fun
tion^of

1/N ²

^. ⁶⁰

3.1 5 lowest eigenvalues and the highest eigenvalue for various

gauge eld ensembles. The latti e spa ing is

a ≈ 0.079

^fm

(

β = 3.9

⁾^for^upper^plots,

a ≈ 0.063

^fm ⁽

β = 4.05

⁾^for^bottom

left and

a ≈ 0.051

^fm ⁽

β = 4.2

⁾ ^for ^bottom ^right^plot. ^. ^. ^. ^. ⁶⁷

4.1 Maximal norm of the overlap operator in logarithmi s ale.

The linear t orresponds to the value of

s

^whi
h ^yields ^the

maximal de ay rate. Parameters:

β = 3.9

L/a = 16

^. ^. ^. ^. ^. ^. ⁸¹

4.2 Thedependen eoftheoverlapDira operatornormde ayrate

ρ

^on ^the ^parameter

s

^for ^gauge ^eld ongurations with and withoutHYP smearing. Parameters:

β = 3.9

L/a = 16

^. ^. ^. ^. ⁸¹

4.3 The ontinuum limits aling ofthe overlap operatorde ay rate. 82

4.4 The ontinuum limits aling of the ratio of the pion mass (at

the mat hing mass) and the overlap operator de ay rate. . . . 82

4.5 Ee tive pion mass plateau for the ensemble

16 ³ × 32

a ≈ 0.079

^fm⁽

β = 3.9

^),

aµ = 0.004

^. ^The ^bare ^valen
e ^quark^mass

am q = 0.04

^. ^F^or ^ea
h ^timesli
e ³ ^values ^of ^the ^pion ^mass

are omputed, orresponding to dierent kinds of smearing

of the sour es (des ribed in Se tion 3.4.1). The horizontal

band orresponds to a simultaneous t of the LL, LF and

FF pseudos alar orrelation fun tions, whi h yields a value

0.2884(17).. . . 83

4.6 Mat hingthe pionmassforthreevalues ofthelatti espa ing,

orresponding to

β = 3.9

^, ^4.05 ^and ^4.2. ^The ^horizontal^bands

are unitary MTM (maximally twisted mass) values and the

urves show the bare quark mass dependen e of the overlap

pion mass. . . 84

quark mass. The dashed lines orrespond to the mat hing

quark masses

a ˆ m

^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ⁸⁵

4.8 Continuumlimits alingofthe overlap pionde ay onstantat

the mat hing mass and two other referen e values of

r 0 m π

^. ^. ^. ⁸⁶

4.9 Continuum limit s aling of the MTM pion de ay onstant at

the mat hing mass. . . 87

4.10 Continuum limit s aling of the dieren e of the overlap and

MTM pion de ay onstant at the mat hing mass. . . 88

4.11 The omparison of the quark mass dependen e of the pion

mass extra ted from PP and PP-SS orrelators for

β = 3.9

ensemble. . . 92

4.12 The omparison of the quark mass dependen e of the pion

de ay onstant extra ted from PP and PP-SS orrelators for

β = 3.9

^ensemble. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ⁹³

4.13 Ensembleaveragesforthefollowing orrelationfun tions:

pseu-dos alar (PP), s alar (SS), the dieren e of PP and SS

(PP-SS). The inset shows the PP and PP-SS orrelation fun tions

on a single onguration. Parameters:

β = 3.9

L/a = 16

aµ = 0.004

am _q = 0.004

^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ⁹⁴

4.14 Ensembleaveragesforthefollowing orrelationfun tions:

pseu-dos alar (PP), s alar (SS), the dieren e of PP and SS

(PP-SS). Parameters:

β = 3.9

L/a = 16

aµ = 0.004

am _q = 0.04

(mu h largervalen equark mass than in Fig. 4.13). . . 95

4.15 Mat hingthepionmass(extra tedfromthePP-SS orrelator)

for three values of the latti e spa ing, orresponding to

β = 3.9

^,^4.05 ^and ^4.2. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ⁹⁷

4.16 Thedependen eofthepionde ay onstantonthebareoverlap

quark mass. The dashed lines orrespond to the mat hing

quark masses

a ˆ m

^(from^PP-SS orrelator). The solid verti al lines (left of the dashed lines) show the dieren e of

f _π ^overlap

and

f _π ^{M T M}

^(at ^the ^mat
hing ^mass) ^extra
ted ^from ^the ^PP

orrelator. . . 98

4.17 Continuum limit s aling of the overlap pion de ay onstant

(extra ted from the PP-SS orrelator) at the mat hing mass

and twoother referen e values of

r 0 m π

^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ⁹⁹

4.18 Continuumlimits alingof the dieren eof the overlap (from

the PP-SS orrelator) and MTM pion de ay onstant at the

mat hing mass. . . 100

4.19 Mat hing the pion mass for3 dierent volumesata xed

lat-ti e spa ing

a ≈ 0.079

^fm. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ¹⁰²

dierentvolumes ata xed latti e spa ing

a ≈ 0.079

^fm. ^. ^. ^. ¹⁰³

4.21 The relative dieren e between the overlap and MTM pion

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 154-0)

GWC ode  point sour es

N c N d

Ψ 0

Ψ

Ψ N = Ψ − Ψ 0

Ψ N

C N N (t)

r

Ψ 0 r

r

Ψ r

r

Ψ N r = Ψ r − Ψ 0 r

Ψ N r

N r

C N N (t)

N r = 600

C SS (t)

C P P − C SS

C P P − C SS

SU 3 × SU 3

h¯qqi

π − π

O(a)

3

L s = 16

Z 2

H W

I = 2 ππ

I = 2 ππ

ǫ

J P C

J

P

C

t/N = 4

r 0 f P S

r 0 m P S

(r 0 m P S ) 2

r 0 µ R

β = 4.2

(a/r 0 ) 2 = 0.0144

Z P

t/N = 4

t = 4N

t/N = 4

O(1/N 2 )

N

t/N = 4

O(1/N 2 )

N

Nm MTM π = Nm overlap π

Nm MTM π = Nm overlap π

Nm MTM π = Nm overlap π

1/N 2

Nm MTM π = Nm overlap π

1/N 2

Nm MTM π = Nm overlap π

1/N 2

a ≈ 0.079

β = 3.9

a ≈ 0.063

β = 4.05

a ≈ 0.051

β = 4.2

s

β = 3.9

L/a = 16

ρ

s

β = 3.9

L/a = 16

16 3 × 32

a ≈ 0.079

β = 3.9

aµ = 0.004

am q = 0.04

β = 3.9

a ˆ m

r 0 m π

Ψ ⁰

Ψ ^N = Ψ − Ψ ⁰

Ψ ^N

Ψ ⁰ _r

Ψ ^N _r = Ψ r − Ψ ⁰ r

Ψ ^N _r

N _r

C _{P P} − C SS

C P P − C ^SS

SU 3 × SU ³

J ^{P C}

r ₀ m _{P S}

(r ₀ m _{P S} ) ²

(a/r 0 ) ² = 0.0144

Z _P

O(1/N ² )

O(1/N ² )

Nm ^MTM _π = Nm ^overlap _π

Nm ^MTM _π = Nm ^overlap _π

Nm ^MTM _π = Nm ^overlap _π

1/N ²

Nm ^MTM _π = Nm ^overlap _π

1/N ²

Nm ^MTM _π = Nm ^overlap _π

1/N ²

16 ³ × 32

am _q = 0.004

am _q = 0.04

f _π ^overlap

f _π ^{M T M}