# 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:

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:

### r

of sto hasti sour e.

3. Compute the propagator

### Ψ 0r

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 Nr = Ψ r − Ψ 0 r

.

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

fun tions from

### Ψ Nr

.

Su h pro edure is then repeated

### N r

times fordierent samplesof sto hasti

noise. Ea hsample ofthe sour eleads toa orrelation fun tion

. We

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. Thelight ontra tion odeusesadierent

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 dieren 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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1.1 Mesoninterpolatingoperators.

### J P C

lassi ationdenotes par-ti le spin

,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

at

xed values of

(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

(

### (a/r 0 ) 2 = 0.0144

) are not in luded, due tothe 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

fortwistedmassandoverlapfermions.

Thequarkmassesaremat hedupto

### O(1/N 2 )

. Thelowerplot

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 MTMπ = Nm overlapπ

. . . . . . . 58

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

### Nm MTMπ = Nm overlapπ

. . . . . . . . . . . . . . . . . . . . . . . . 58

2.12 Thedieren ebetween the MTMand overlappionde ay

on-stants atthe mat hing point

, asa

fun -tion of

### 1/N 2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.13 Thedieren ebetweentheMTMandoverlap orrelation

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

### Nm MTMπ = Nm overlapπ

, asa fun tion of

### 1/N 2

. . . . . . . . . . . 60

2.14 The dieren e between the MTM and overlap quark mass at

the mat hing point

, asafun tionof

### 1/N 2

. 60

3.1 5 lowest eigenvalues and the highest eigenvalue for various

gauge eld ensembles. The latti e spa ing is

fm

(

)forupperplots,

fm (

)forbottom

left and

fm (

### β = 4.2

) for bottom rightplot. . . . . 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:

,

### L/a = 16

. . . . . . 81

4.2 Thedependen eoftheoverlapDira operatornormde ayrate

on the parameter

### s

for gauge eld ongurations with and withoutHYP smearing. Parameters:

,

### 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 Ee tive pion mass plateau for the ensemble

,

fm(

),

### aµ = 0.004

. The bare valen e quarkmass

is

### am q = 0.04

. For ea h timesli e 3 values of the pion mass

are omputed, orresponding to dierent 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 horizontalbands

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 dieren 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 dieren e of PP and SS

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

on a single onguration. Parameters:

,

,

,

### am q = 0.004

. . . . . . . . . . . . . . . . . . . . . . 94

4.14 Ensembleaveragesforthefollowing orrelationfun tions:

pseu-dos alar (PP), s alar (SS), the dieren e of PP and SS

(PP-SS). Parameters:

,

,

,

### 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

(fromPP-SS orrelator). The solid verti al lines (left of the dashed lines) show the dieren e of

and

### f πM 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 dieren 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 dierent volumesata xed

lat-ti e spa ing

### a ≈ 0.079

fm. . . . . . . . . . . . . . . . . . . . . 102

dierentvolumes ata xed latti e spa ing

### a ≈ 0.079

fm. . . . 103

4.21 The relative dieren e between the overlap and MTM pion

4.21 The relative dieren e between the overlap and MTM pion