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
orµ 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 withC = 3/ √ 10
,•
Creutz fermions withC = 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 alingviolations. 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 -0.0101045Creutz
C = 3/ √
14
1.0 -0.0488282 0.00282578The ontinuum limit (the oe ient
a m
) is the same for all kinds offermions (and equal to the expe ted value
Nm π = 2Nm
(overlap, Creutzfermions) 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 allphysi 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 ientb m
Tab. 2.2)forthis observable. Theyare thelargestfor Bori i and Creutz (C = 3/ √
14
) fermions, around twi e smaller for overlapfermions, 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 ofNm
onrm that this behaviour is universal for awide range of values of
Nm
.Furthermore,thevalueofthe oe ient
c m
that hara terizestheO(1/N 4 )
dis retization errors is in general smaller than
b m
, indi ating that theor-re tions to the
O(1/N 2 )
behaviour are small. However, there are someex- eptions tothis rule (e.g. the Creutz
C = 3/ √
10
ase), wherec m
isslightlylarger 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 -0.0746106Creutz
C = 3/ √
14
3.4641 -0.00789431 -0.0379067and 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 ientsb f
(Tab. 2.3)leadtodierent on lusionsregarding the sizeoftheseee tsforthekindsoffermionsunderanalysis. Thelargestoef- ientis observed forCreutz (
C = 3/ √
10
)fermions, whi h hadthe smallestdis retizationerrorinthepionmass (
b m
). A ordingly,Creutz (C = 3/ √ 14
)fermions had the se ond largest oe ient
b m
, but the oe ientb f
is thesmallest 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 observabledo 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 CreutzC = 3/ √
14
0.109894 0.00486504 -0.00300215This 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, thepointsin 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 Creutzfermions.
(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
) fermionsand the smallest for Bori i fermions. As a oin iden e,
b C
for overlap andtwistedmass fermions isthe same, whi h isnot true for other values of
Nm
(for
Nm < 0.5
the value foroverlap islarger, forNm > 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
andNµ = 0.5
) are shown in Figs. 2.4, 2.5 and2.6.
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 MTMlines, whi h implies that the tting oe ients
b m
,b f
andb C
are alwaysarithmeti 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. Theresults of su h test for overlap fermions willbepresented in Chapter 4.