4.8 Expli it subtra tion of zero modes
4.8.2 Ee ts of expli it zero modes subtra tion
An ee t of the subtra tion pro edureon the pionmass and de ay onstant
(with respe t to the PP ase) an be observed in Figs. 4.27 and 4.28. For
omparison, alsothe urves orresponding tothe PP-SS ase are plotted.
Thepionmassextra tedfromthePP-SS orrelatorandthePP orrelator
with subtra ted zero modes (PPsubtr.) agree forsmallquark masses,while
the behaviour of the pion de ay onstant is very dierent. At the level of
orrelationfun tions,this resultsfromthefa tthat thePP subtr. orrelator
has a very similar slope to the one of the PP-SS orrelator, but its matrix
element is signi antlylower.
For larger quark masses (larger than
r 0 m q ≈ 0.08
), the pion massesex-tra tedfromthePP-SSandthePPsubtr. orrelatorsarenot onsistentwith
ea h other the PP subtr. urve hanges slope and deviates the morefrom
thePP-SS urvethelargerthequarkmass. Thisisinapparent ontradi tion
withtheexpe tationthatexpli itsubtra tionofzeromodesremovesthe
on-tributionof thesemodes,sin e atlarger quarkmass values this ontribution
tends to zero and the PP subtr. urve should onverge to the PP (and
PP-SS) urve. Su h behaviour of the pion mass from the PP subtr. orrelator
provides a warning about the expli it subtra tion method. It was observed
beforein quen hed studieswith the xed point Dira operator, whi h is
an-other variant of a hirally improved latti e Dira operator. The studies by
Hauswirth [123℄ and Gattringer etal. [124℄ obtained asimilar pi ture the
pion mass at small quark mass is approximately the same from the PP-SS
0 0.3 0.6 0.9 1.2 1.5 1.8 2.1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
(r 0 m π ) 2
r 0 m q β=3.9 L/a=16 aµ=0.004 MTM
Overlap PP subtr.
Overlap PP-SS Overlap PP
Figure4.27: The omparisonofthequarkmass dependen e ofthe pionmass
extra ted fromPP,PP-SSandPPwithexpli itlysubtra tedzeromodes(PP
subtr.) orrelators for
β = 3.9
ensemble.0.2 0.25 0.3 0.35 0.4 0.45
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
r 0 f π
r 0 m q β=3.9 L/a=16 aµ=0.004 MTM
Overlap PP Overlap PP-SS Overlap PP subtr.
Figure4.28: The omparisonofthequarkmassdependen eofthepionde ay
onstant extra ted from PP, PP-SS and PP with expli itlysubtra ted zero
modes (PPsubtr.) orrelators for
β = 3.9
ensemble.0.14 β=3.9, L/a=16, aµ=0.004, am q =0.011 PP subtr. correlator
Figure 4.29: Ee tive mass plateaus for PP subtr. orrelation fun tions.
Parameters:
β = 3.9
,L/a = 16
,aµ = 0.004
. Upper plot:am q = 0.011
.Lowerplot:
am q = 0.04
.and the PP subtr. orrelator, while at larger quark masses the PP subtr.
orrelation fun tion leads to mu h smaller pion masses than ones obtained
fromthe PPandPP-SS orrelators (whi htendtoagreeatquarkmassesfor
whi hthe ee ts of zero modes are negligible).
This on lusion isfurther on rmed by Fig. 4.29, whi hshows the
ee -tivepion mass plateausfromthe PPsubtr. orrelator,for two quarkmasses
the mat hing mass (upper plot) and a signi antly heavier mass (lower
plot). The plateau observed for the mat hing mass looks rather normal,
fromthespe tralde omposition(1.83). However, fortheheavierquarkmass,
there is no plateau(this plot an be ompared to Fig. 4.5, whi h shows the
samequarkmass,butthe ee tivemassisextra ted fromthe(unsubtra ted)
PP orrelator). This implies that the PP subtr. orrelation fun tion might
not be a sum of exponential fun tions, but rather a sum of power
fun -tions. This results from the fa t that expli it subtra tionof zero modes isa
non-lo al pro edure, i.e. it an modify the simulated theory in a non-lo al
way, thus leading to unphysi al ee ts in the orrelation fun tions, whi h
auses that the spe tral de omposition (1.83) is not valid. It also implies
that the pion mass values obtained from the PP subtr. orrelator are not
meaningful at high values of the quark mass. However, sin e there is no
fundamental reason why dierent quark masses should lead to qualitatively
dierentbehaviourof the orrelationfun tions,we an notbesure thateven
atarelativelysmallquarkmass(su hasthemat hingmass)thesubtra tion
pro edure isvalid.
The above dis ussion leads to a on lusion that expli it subtra tion of
zero modes is adangerous hand-made pro edure, whi h may lead to
un on-trollable unphysi al ee ts in the extra ted observables. However, sin e we
have observed the onsisten y between the pion masses fromthe PP-SS and
PP subtr. orrelators (for relatively small quark masses), we may assume
here as a working hypothesis that at the mat hing mass the expli it
sub-tra tionpro edure isvalid,i.e. that the unphysi alee ts ofsubtra tion are
small. This is justi ed by the fa t that the ee tive mass plateau for the
mat hing mass does not show the pathology observed at the larger quark
mass.
To on lude this subse tion, we show in Fig. 4.30 the s alar orrelation
fun tion with expli itly subtra ted zero modes (SS subtr.) for two values
of the valen e quark mass the lightest onsidered mass and the mat hing
mass. Theplotshowsthatthedominant ontributiontothefullSS orrelator
omes from the zero modes (the SS subtr. urve for
am q = 0.004
should beomparedtothefullSS urveatthesamequarkmassFig. 4.13). Moreover,
after the zero modes are subtra ted, the s alar orrelator is negative, whi h
may be attributed to the unitarity violation ee t dis ussed in Se tion 4.1.
We have alsohypothesized in Se tion4.4that this ee t in uen es the pion
de ay onstantextra tedfromthePP-SS orrelator. Thiswouldalsoexplain
thedieren ein
f π
extra tedfromthePP-SSandPPsubtr. orrelatorsthe latter does not have the enhan ed unitarity violation ee t from the s alarorrelator. This ee t willbe investigated further inthe next hapter.
In the next subse tion we will use the PP subtr. orrelation fun tion to
extra tthe pionde ay onstantandperformits ontinuumlimits alingtest.
-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05
0 2 4 6 8 10 12 14 16
C(t)
t
β=3.9, L/a=16, aµ=0.004 SS subtr. am q =0.004
am q =0.011
Figure4.30: TheSSsubtr. orrelationfun tion(SSwithexpli itlysubtra ted
zero modes). Parameters:
β = 3.9
,L/a = 16
,aµ = 0.004
, 2 valen e quarkmasses:
am q = 0.004
,am q = 0.011
(mat hing mass).The results will not be ontaminated by the zero modes ontribution, but
we again emphasize that they have to be interpreted with aution, due to
the fa tthatthesubtra tionpro edureisnot leanfromthe eld-theoreti al
pointof view.
4.8.3 Pion de ay onstant s aling test PP subtr.
orrelator
Weagainbeginby ndingthe mat hingmass forea h ensemble. Theresults
ofthe mat hingpro edureareshown inFig. 4.31 andthebareoverlapquark
masses that lead to the same pion mass as inthe unitary setup are:
• β = 3.9
a ˆ m = 0.0115(15)
,• β = 4.05
a ˆ m = 0.0065(15)
,• β = 4.2
a ˆ m = 0.0055(15)
.These mat hing masses are onsistent with the ones obtained from the
PP-SS orrelationfun tion, on rming againthe on lusionthat these methods
give onsistent results for smallquark masses.
0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
(r 0 m π ) 2
r 0 m q β=4.2 L/a=24 aµ=0.002 MTM
Overlap
Figure 4.31: Mat hing the pion mass (extra ted from the PP subtr.
orre-lator) for three values of the latti e spa ing, orresponding to
β = 3.9
, 4.05and 4.2.
The urves that showthe quark mass dependen e of the pionde ay
on-stantextra tedfromthePP orrelatorwithexpli itlysubtra tedzeromodes
(PP subtr.) liewell belowthe orresponding urves for the PP and the
PP-SS ase (Fig. 4.32). This ee t has already been dis ussed in the previous
subse tion. In omparison with the PP ase, the values of the pion de ay
onstantatthemat hingmass areverymu hredu ed,whi hisshown inthe
plots by verti al lines the dashed ones orrespond to the dieren e in the
pion de ay onstantinthe PPsubtr. ase and thesolid onestothe PP ase.
0.15 0.2 0.25 0.3 0.35 0.4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 r 0 f π
r 0 m q β=3.9 L/a=16 Overlap
β=4.05 L/a=20 Overlap β=4.2 L/a=24 Overlap
MTM aµ=0.004 MTM aµ=0.003 MTM aµ=0.002
Figure4.32: The dependen e of the pion de ay onstantonthe bare overlap
quark mass. The dashedlines orrespond to the mat hing quark masses
a ˆ m
(fromPPsubtr. orrelator). Thesolidverti allines(leftofthedashedlines)
show the dieren e of
f π overlap
andf π M T M
(at the mat hing mass) extra tedfrom the PP orrelator.
Again, the in uen e of zero modes an ellation is relatively the largest for
β = 4.2
.Fig. 4.33showsthe ontinuumlimits alingofthepionde ay onstantfor
threereferen e valuesof
r 0 m π ≈ 1.3
(herewetakethe highestvalueavailablefor the
β = 3.9
ensemble),r 0 m π ≈ 1.0
and the one that orresponds to the mat hing riterionr 0 m π ≈ 0.85
. As in the previous ases of the PP andthe PP-SS orrelation fun tions, we observe good s aling behaviour for all
analyzed values of
r 0 m π
, withO(a 2 )
leading ut-o ee ts.Moreover, the ontinuum limitofthe pionde ay onstantextra tedfrom
the PP subtr. orrelation fun tion agrees with the MTM ontinuum limit,
whi h is shown in Fig. 4.34. Therefore, both methods of subtra ting the
zero modes lead to a onsistent ontinuum limit value, whi h is the one of
the unitary approa h. This is a strong hint that the zero modes are indeed
responsible for the observed behaviourof the pion de ay onstant extra ted
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 r 0 (f π overlap -f π MTM )
(a/r 0 ) 2 cont. limit = 0.223(14)
cont. limit = 0.311(12)
cont. limit = 0.173(13)
L ≈ 1.3 fm
β=4.2 β=4.05 β=3.9
matching mass r 0 m π ≈ 1.0 r 0 m π ≈ 1.3
Figure 4.33: Continuum limit s aling of the overlap pion de ay onstant
(extra ted from the PP subtr. orrelator) at the mat hing mass and two
other referen e values of
r 0 m π
.from the PP orrelator, i.e. for the wrong ontinuum limit value of this
observable.
However, wehavetoemphasize againthat the resultsof theexpli it zero
modes subtra tion pro edure have to be treated with aution. The
onsis-ten y between both methods is a hint that the pathologi al ee ts of the
hand-made subtra tion pro edure are not very large at the mat hing mass,
but this method of an elling the zero modes ontribution is not
re om-mended,sin ethereisnosystemati wayto ontrolthepotentialnon-physi al
ee ts.
A learly safer pro edure to remove the ontribution of the zero modes
is the one with the PP-SS orrelationfun tion. Thismethoddoes not suer
from the aforementioned ee ts, sin e no modi ation atthe levelof
propa-gators is made. In this way, both the PP and SS orrelation fun tions have
theproperspe tralde ompositionand,aswehaveshown,the ontributionof
the zero modes is exa tly an elled inthe orrelator dieren e. The pri e to
pay, however, isthat the s alar orrelatormay introdu e enhan ed unitarity
violations oming from the double pole ontributionspe i to non-unitary
approa hes (quen hed, partially quen hed and mixed a tion theories). On
the other hand, su h ee ts are
O(a 2 )
latti e artefa ts and hen e should-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 r 0 (f π overlap -f π MTM )
(a/r 0 ) 2 L ≈ 1.3 fm
β=4.2 β=4.05 β=3.9
matching mass r 0 m π overlap =r 0 m π MTM
Figure 4.34: Continuum limit s aling of the dieren e of the overlap (from
the PP subtr. orrelator) and MTM pion de ay onstant at the mat hing
mass.
not ae t extrapolations to the ontinuum limit. This ee t will be further
analyzed in the next hapter.
Various further results
In the previous hapter we have performed an analysis of the ontinuum
limit s aling of the pion de ay onstant. We have dis ussed the role of the
zero modesinamixed a tionsetupof hirally-symmetri valen equarksand
non- hirally-symmetri sea quarks. To an el the non-physi al ontribution
of the zero modes we haveused the PP-SS orrelationfun tion and we have
hypothesized that while this orrelator orre tly removes the zero modes
ontribution,italsointrodu esenhan edunitarityviolations. Inthis hapter
we will analyze this ee t. We will also present some additional results
regarding the ontinuumlimits aling ofbaryon (nu leonand delta)masses,
as well assome topologi alaspe ts.