Comparison of overlap, twisted mass and Creutz fermions

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 46-53)

Creutz fermions

In this se tion, we present the results of a tree-level s aling test of dierent

kinds of fermions:

N N 4 m

or

µ t = 4N

4 256 0.125000 16

8 512 0.062500 32

12 768 0.041667 48

16 1024 0.031250 64

20 1280 0.025000 80

24 1536 0.020833 96

28 1792 0.017857 112

32 2048 0.015625 128

36 2304 0.013889 144

40 2560 0.012500 160

44 2816 0.011364 176

48 3072 0.010417 192

52 3328 0.009615 208

56 3584 0.008929 224

64 4096 0.007813 256

overlap fermions,

Wilson twistedmass fermions at maximaltwist (MTM),

Creutz fermions with

C = 3/ √ 10

,

Creutz fermions with

C = 3/ √ 14

,

Bori i fermions.

.

First,we onsiderthepionmass,whi hisdepi tedinFig. 2.1. Thepoints

intheplotshowtheresultextra ted fromthe orrelationfun tion(2.16)and

the orrespondinglines are ts of the followingformula:

Nm π = a m + b m

1 N 2 + c m

1

N 4 .

(2.20)

Inall ases,wendtheexpe tedbehaviouri.e.

O(a 2 )

(

O(1/N 2 )

)s aling

violations. It is worth to emphasize here again that in the ase of overlap

and Creutz fermions this results dire tly from hiral symmetry and in the

ase of twisted mass fermions from automati

O(a)

-improvement, whi h is a hieved only atmaximal twist, i.e. for bare untwisted quark mass set to 0.

0.9998 0.99985 0.9999 0.99995 1 1.00005 1.0001 1.00015 1.0002

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 Nm π

1/N 2

Nm=0.5 Nµ=0.5

N=16 N=32

N=64

cont. limit = 1.0 MTM

Overlap Borici Creutz C=3/10 1/2 Creutz C=3/14 1/2

Figure 2.1: Continuum limit s aling of the pion mass for overlap, twisted

mass and Creutz fermions.

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

fermion

a m b m c m

MTM 1.0 -0.0104167 0.000296044

Overlap 1.0 0.0208333 0.000783869

Bori i 1.0 -0.0494792 0.00564291

Creutz

C = 3/ √

10

1.0 -0.0078125 -0.0101045

Creutz

C = 3/ √

14

1.0 -0.0488282 0.00282578

The ontinuum limit (the oe ient

a m

) is the same for all kinds of

fermions (and equal to the expe ted value

Nm π = 2Nm

(overlap, Creutz

fermions) and

Nm π = 2Nµ

(twisted mass fermions)). This is a ne essary ondition that ea h fermion a tion has to fulll the ontinuum limit of all

physi al observables has to be the same. This is ensured if the ontinuum

limitofthefermionpropagatorforthedis retizationofinterestisequaltothe

3.4638 3.464 3.4642 3.4644 3.4646 3.4648 3.465 3.4652 3.4654

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 Nf π

1/N 2

Nm=0.5 Nµ=0.5

N=16 N=32

N=64

cont. limit = 3.4641 MTM

Overlap Borici Creutz C=3/10 1/2 Creutz C=3/14 1/2

Figure 2.2: Continuum limit s aling of the pion de ay onstant for overlap,

twistedmass and Creutz fermions.

ontinuum fermion propagator. In other words, various fermion

dis retiza-tions dier inthe way the latti e artefa ts are introdu ed.

Itisalsointerestingto omparethe magnitudeof

O(1/N 2 )

dis retization errors( oe ient

b m

Tab. 2.2)forthis observable. Theyare thelargestfor Bori i and Creutz (

C = 3/ √

14

) fermions, around twi e smaller for overlap

fermions, a further fa tor of two smaller for twisted mass fermions and the

smallest for Creutz (

C = 3/ √

10

) fermions. Moreover, omputations for dierent xed values of

Nm

onrm that this behaviour is universal for a

wide range of values of

Nm

.

Furthermore,thevalueofthe oe ient

c m

that hara terizesthe

O(1/N 4 )

dis retization errors is in general smaller than

b m

, indi ating that the

or-re tions to the

O(1/N 2 )

behaviour are small. However, there are some

ex- eptions tothis rule (e.g. the Creutz

C = 3/ √

10

ase), where

c m

isslightly

larger than

b m

, but still rathersmall.

Asthe se ond observable, we onsider the pionde ay onstant,shown in

Fig. 2.2. The points in the plot show the result omputed from eq. (2.19)

fermion

a f b f c f

MTM 3.4641 0.0541266 -0.000811859

Overlap 3.4641 0.108253 0.00553143

Bori i 3.4641 -0.0676584 -0.00527683

Creutz

C = 3/ √

10

3.4641 0.293186 -0.0746106

Creutz

C = 3/ √

14

3.4641 -0.00789431 -0.0379067

and the orrespondinglines are ts of the followingformula:

Nf π = a f + b f 1

N 2 + c f 1

N 4 .

(2.21)

Inall ases,weobserveagain

O(1/N 2 )

leadingdis retizationerrors. How-ever, the oe ients

b f

(Tab. 2.3)leadtodierent on lusionsregarding the sizeoftheseee tsforthekindsoffermionsunderanalysis. Thelargest

oef- ientis observed forCreutz (

C = 3/ √

10

)fermions, whi h hadthe smallest

dis retizationerrorinthepionmass (

b m

). A ordingly,Creutz (

C = 3/ √ 14

)

fermions had the se ond largest oe ient

b m

, but the oe ient

b f

is the

smallest amongall dis retizations.

Generalizing,thismeansthatthe sizeofdis retizationee tsdepends on

the hoi e of the observable, i.e. that small

O(a 2 )

ee ts in one observable

do not mean that for otherobservables one an expe t the same.

Table 2.4: Fitting oe ients for the pseudos alar orrelation fun tion at a

xed physi aldistan e

t/N = 4

eq. (2.22).

fermion

a C b C c C

MTM 0.109894 0.00457891 -0.0000333779

Overlap 0.109894 0.00457891 0.000181293

Bori i 0.109894 0.00114472 -0.0013941

Creutz

C = 3/ √

10

0.109894 0.0194604 -0.00269918 Creutz

C = 3/ √

14

0.109894 0.00486504 -0.00300215

This is onrmed by the result for the third observable  the orrelation

fun tion at a xed physi al distan e

t = 4N

, shown in Fig. 2.3. Again, the

pointsin the plot orrespond tothe orrelation fun tion omputed fromeq.

0.10989 0.1099 0.10991 0.10992 0.10993 0.10994 0.10995

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 N 3 C PP (t/N=4)

1/N 2

Nm=0.5 Nµ=0.5

N=16 N=32

N=64

t/N=4

cont. limit = 0.109894 MTM

Overlap Borici Creutz C=3/10 1/2 Creutz C=3/14 1/2

Figure2.3: Continuum limits aling ofthe pseudos alar orrelationfun tion

at a xed physi al distan e

t/N = 4

for overlap, twisted mass and Creutz

fermions.

(2.16) and the lines are ts of the followingformula:

N 3 C P P (t = 4N) = a C + b C

1 N 2 + c C

1

N 4 .

(2.22)

The oe ient

b C

is again the largest for Creutz (

C = 3/ √

10

) fermions

and the smallest for Bori i fermions. As a oin iden e,

b C

for overlap and

twistedmass fermions isthe same, whi h isnot true for other values of

Nm

(for

Nm < 0.5

the value foroverlap islarger, for

Nm > 0.5

itis smaller).

Other interesting quantities to ompute are the mixed orrelators. In

the meson ase they orrespond to taking the two quarks dis retized with

dierent a tions. This is relevant from the point of view of mixed a tion

simulations in the intera ting theory, where it is possible to build a meson

from two valen e quarks, two sea quarks or one valen e and one sea quark.

If one imposes a mat hing ondition that the valen e-valen e pion and the

sea-seapionhavethe samemass,themixed valen e-seapioningeneralhasa

dierentmassandtheobtainedmassdieren equantiesunitarityviolations

0.99994 0.99996 0.99998 1 1.00002 1.00004 1.00006 1.00008 1.0001

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 Nm π

1/N 2 Nm=0.5 Nµ=0.5

N=16 N=32

N=64

cont. limit = 1.0 MTM

Overlap Mixed

Figure 2.4: Continuum limit s aling of the pion mass for overlap-overlap,

MTM-MTM and overlap-MTM quarks.

inthe mixeda tionsetup. Itis,however, worth toemphasizethatthis ee t

is onlya latti e artefa t with nophysi alsigni an e.

The way to onstru t the mixed pion attree-levelis to use two dierent

propagators in formula (2.16) for the pseudos alar orrelation fun tion. We

willshowanexampleofoverlap-MTMmixed orrelator,i.e. wewilltakeone

of the propagators to be the overlap fermion propagator and the other one

to be the MTM fermion propagator. The results of the s aling test for su h

mixed ase (with

Nm = 0.5

and

Nµ = 0.5

) are shown in Figs. 2.4, 2.5 and

2.6.

The mixed pion mass, de ay onstant and orrelator at a xed physi al

distan e allshowleading

O(a 2 )

ut-o ee ts. Furthermore,inall ases, the mixed meson lineis situatedexa tlyhalfway between the overlapand MTM

lines, whi h implies that the tting oe ients

b m

,

b f

and

b C

are always

arithmeti averages of the orresponding oe ients for the overlap and the

MTM ase. The onsequen eofthisisalsothatattree-levelitisnotpossible

to observe a splitting between the mixed pion mass and the overlap/MTM

pion masses, if the latter are mat hed. This results from the fa t that at

3.46405 3.4641 3.46415 3.4642 3.46425 3.4643 3.46435 3.4644 3.46445 3.4645 3.46455

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 Nf π

1/N 2 Nm=0.5 Nµ=0.5

N=16 N=32

N=64

cont. limit = 3.4641 MTM

Overlap Mixed

Figure 2.5: Continuum limits aling of the pion de ay onstant for

overlap-overlap, MTM-MTM and overlap-MTMquarks.

tree-level there are no unitarity violations  their sour e is a dierent Dira

operator used to generate the gauge eld ongurations and a dierent one

for the valen equarks and in the free ase su h situationdoes not o ur.

To summarize, there are no denite on lusions from the tree-level test.

It an not bededu ed that one type of fermionsexhibits the smallestor the

largest dis retization errors  this depends on the observable and of ourse

in the intera ting theory one should expe t the same. A general on lusion

fromthetest isthatallfermionsexhibit

O(a 2 )

s alingviolations. Thisagain should hold inthe intera ting theory, but it has tobeexpli itlytested. The

results of su h test for overlap fermions willbepresented in Chapter 4.

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 46-53)