5.1 Unitarity violations
5.1.2 Small volume analysis
For onvenien e, werewrite here the formulafor the s alar orrelation
fun -tion at the mat hing mass:
C SS (t) −−−→ − t→∞ B 0 2 2L 3
e −2M V V t
M V V 3 (γ V V + γ SS − 2γ V S ) a 2 t.(5.1)
The low-energy onstants
γ V V = γ V S = 0, due to exa t hiral symmetry
in the valen e se tor [110℄, but
γ SSis non-vanishing, sin e the sea Dira operator is not hirally-symmetri .
Formula(5.1)impliesthatthes alar orrelationfun tionforthemat hing
quark mass an be ome negative at large times (provided that
γ SS > 0).
However, the shape of this orrelator is basi ally the one observed in Fig.
4.13, whi h shows the
β = 3.9ase at a quark mass below the mat hing
mass. Clearly, this orrelation fun tion does not be ome negative, sin e
it has a large positive ontribution from the zero modes and the unitarity
violation ee t is obs ured.
In order to analyze the ee t predi ted by eq. (5.1), we would have to
removethezeromode ontributionfromthes alar orrelatororworkatlarge
The latterrequiresavery omputer-timeintensive omputationandishen e
beyond the s ope of the urrent proje t. However, su h analysis is planned
for the future and would provide the leanest way of testing the predi tion
of eq. (5.1).
Inthe urrentproje t,wethereforehavetorestri tourselvestotheformer
method, i.e. to remove the zero mode ontribution from the SS orrelator.
This is possible by following the pro edure of expli it subtra tion of zero
modes at the level of propagators. As we have shown in the previous
hap-ter, this is a dangerous pro edure with hard to ontrol systemati ee ts.
Therefore, the results of this analysis have to be interpreted with aution
and treated as an outlook on this kind of analysis, whi h will be later
per-formed in a lean setup of large volume and large enough quark mass, so
that the zero mode ee ts willbe negligible. Analternativeapproa h ould
onsistinusingonly ongurationsinthetrivialtopologi alse tor,whi hare
ongurations in this se tor is too small to allow for meaningful ts of eq.
Fig. 4.30 shows that after the zero mode ontribution is removed, the
s alar orrelator at the mat hing mass be omes negative indeed. We have
also he ked that the SS orrelator on topologi ally trivial ongurations is
negative at large time (the error bands are too large to perform ts of eq.
(5.1), however, the on lusion about the sign of the orrelator is
unambigu-ous), whi h onrms that the unitarity ee t is really present in our mixed
a tion setup.
parameters are given inSe tion4.2.1 and expli itlysubtra t the zero modes
at the level of propagators, as des ribed in Se tion 4.8.1. In this way, we
obtain for ea h ensemble the SS subtr. orrelator at the mat hing mass.
Then, we teq. (5.1) to the latti edata.
Spe i ally, we write this equationas:
C SS (t) t→∞ = −γ t e −2M V V t .(5.2)
where we havedened aparameter
γ ≡ B 0 2 γ SS
2(M V V L) 3 a 2 ≡ ˜γa 2 .(5.3)
Sin e the temporal extent of the latti e is nite and equals
Tfor ea h
en-semble(with periodi boundary onditions intime), the ttingformulathat
we use reads:
C SS (t) tlarge
= −γ t e −2M V V t + (T − t) e −2M V V (T −t) .(5.4)
The parameters that we t are
γand the pion mass
M V V. The denition
of the parameter
γshould have a quadrati dependen e on
the latti e spa ing,sin e
γ SSare low-energy onstantsand
M V V Lis
approximatelythe same for ea hensemble.
Thetfortheensembleatthe oarsest latti espa ing(
β = 3.9)isshown
inFig. 5.3. Thettingintervalis
t ∈ [9, 23]and inthis intervalthet
repre-sentsaverygooddes ription oflatti edata. Qualitativelysimilarbehaviour
is observed also inthe
β = 4.05and
β = 4.2ases.
Oneof the ttingparametersisthe pion mass
M V V. Its valuesextra ted
fromthets anbe omparedwithvaluesofthe mat hingpionmass (known
pre isely from the maximallytwisted mass PP orrelator). This provides a
onsisten y he kforthets. Inall asesthettedvaluesof
M V Varearound
-0.1 -0.08 -0.06 -0.04 -0.02 0
0 5 10 15 20 25 30
C SS (t)
β=3.9, L/a=16, aµ=0.004, am q =0.011 SS subtr.
Figure 5.3: The SS subtr. orrelation fun tion at the mat hing mass. The
solid line represents the t of eq. (5.4).
2 standard deviations below the mat hing pion mass. This is a reasonable
agreement,taking intoa ount the unknown systemati ee trelated tothe
subtra tion pro edure.
γhas a few sour es of un ertainties, related to:
1. statisti alerrors in
C SS (t)(errorbars in Fig. 5.3),
2. the hoi e of the tting interval to estimate it we have performed
several ts with dierenttting intervals,
3. errors of the mat hing pro edure to estimate it we have performed
ts not only forthe mat hing quarkmasses, but alsofor quarkmasses
diering in latti e units by
±0.001(whi h orresponds to the error in the mat hing mass),
4. the fa t that the produ t
M V V Lis not exa tlymat hed for all
5. unknownvaluesoftherenormalization onstant
Z Softhes alar urrent
Z Sis equalfor all ensembles,
6. anunknownsystemati errorintrodu edbythezeromodessubtra tion
-0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
(a/r 0 ) 2 SS subtr. correlator
L ≈ 1.3 fm
Figure 5.4: Continuum limit s aling of the tting parameter
γ, dened by
Fig. 5.4 shows the tted values of
γ. The errors on ea h value in lude
sour es 1-4 from the above list of un ertainties. The error related to the
unknown value of
Z Sshould be rather small ompared to the overall size of
the error from sour es 1-4. We have not tried to estimate the error related
to the zero modes subtra tion pro edure.
We observe good s aling of the parameter
O(a 2 )ut-o
dependen e. The value of this parameter extrapolated to the ontinuum is
onsistent with zero. This result is in very good agreement with the
hy-pothesis that the s alar orrelator is inuen ed by the unitarity violation
ee t predi ted and analyzed in [107, 108, 109, 110℄. It also provides an
explanation for the behaviour des ribed in the previous subse tion, i.e. the
seemingly in onsistent with the hypothesisaboutthe role of the zero modes
la k of ee t of subtra ting the SS orrelator at
β = 3.9. The unitarity
vio-lations analysis suggests that there are indeed two ompeting ee ts in the
PP-SS orrelatorthezero mode ontributionisremoved,but the orrelator
is ontaminatedbyaunitarityviolationoriginatingfromanenhan eddouble
pole ontribution. These two ee ts are roughly balan ed at
β = 3.9, but at
β = 4.05and
β = 4.2, thenite volume ee t ofthe zero modes an ellation dominates overthe
O(a 2 )unitarityviolationee t, whi hissmalleratthese
latti e spa ings.
ay onstant extra ted from the PP-SS and the PP subtr. orrelator, sin e
the latter does not have the double pole ontribution of the SS orrelator.
the unphysi al zeromodessubtra tion pro edureand itmay suer from
un-predi table ee ts. Therefore, this analysis has to be treated with aution.
It providesaplausibleexplanationofthe observed ee ts. However, inorder
to quantitatively analyze the ee t of unitarity violations in the s alar
or-relator and reliably extra t the low-energy onstant
γ SS, a simulation with
large enough volume and quark mass would have to be performed in order
to have a negligible ontributionfrom the zero modes to the full s alar
or-relator (without expli it subtra tion pro edure). In addition, eq. (5.1) was
derived for an innite volume and hen e it would be very advantageous to
have large volume data for the SS orrelation fun tion in order to use the
tting ansatz of this formulain anappropriate way.