Lattice Flavourdynamics

0.12 MB·English

by  Chris T Sachrajda

SHEP0602
Lattice Flavourdynamics 1
Chris T Sachrajda
SchoolofPhysicsandAstronomy,UniversityofSouthampton,SouthamptonSO171BJ,UK
6
0
0
Abstract. Ipresenta selectionofrecentlatticeresultsinflavourdynamics,includingthestatusof
2
thecalculationofquarkmassesandavarietyofweakmatrixelementsrelevantforthedetermination

n ofCKMmatrixelements.Recentimprovementsinthemomentumresolutionoflatticecomputations
a andprogresstowardsprecisecomputationsofK→pp decayamplitudesarealsoreviewed.
J

0
1
INTRODUCTION

1
v
OneofthemainapproachestotestingtheStandardModelofParticlePhysicsandsearch-
4
1 ing for signatures of new physics is to study a large number of physical processes to
0
obtain information about the unitarity triangle and to check its consistency. The preci-
1
0 sion with which this check can be accomplished is limited by non-perturbative QCD
6 effectsandlatticeQCDprovidestheopportunitytoquantifytheseeffectswithoutmodel
0
assumptions. Of course, lattice computations themselves have a number of sources of
/
t systematicuncertainty, and much of our current effort is being devoted to reducing and
a
l controllingtheseerrors. In this talk I briefly discussthe evaluationof quark masses and
-
p weak matrixelementsusinglatticesimulations.
e
h Formostlatticecalculationsofphysicalquantities,theprincipalsourceofsystematic
: uncertainty is the chiral extrapolation, i.e. the extrapolation of results obtained with
v
i unphysicallylargeuandd quarkmasses.Ideallywewouldliketoperformcomputations
X
with140MeVpionsandhencewithm /m ofabout1/25(wherem (m )istheaverage
q s q s
r
a light quark mass (strange quark mass)). In practice values m /m ≥ 1/2 are fairly
q s
typical, so that the MILC Collaboration’s simulation with m /m ≃ 1/8 is particularly
q s
impressive[1] and provides a challenge to the rest of the community to reach similarly
low masses. Its configurations have been widely used to determine physical quantities
withsmallquotederrors.
The MILC collaboration uses the staggered formulation of lattice fermions and for a
varietyofreasonsitisveryimportanttoverifytheresultsusingotherformulations.With
staggeredfermionseachmesoncomesin16tastesandtheunphysicalonesareremoved
bytakingthefourthrootofthefermiondeterminant.Althoughthereisnodemonstration
that this procedure is wrong, there is also no proof that it correctly yields QCD in the
continuumlimit[2].Thepresenceofunphysicaltastesleadstomanyparameterstobefit
1 Plenarylecturepresentedatthe2005ParticlesandNucleiInternationalConference(PANIC05),Santa
Fe,NewMexico,USA,Oct.24–28th2005.
in staggered chiral perturbation theory (typically many tens of parameters) and to date
the renormalization has only been performed using perturbation theory. It is therefore
pleasing to observe that the challenge of reaching lower masses is being taken up by
groupsusingotherformulationsoflatticefermions (seee.g. ref.[3]).
In this talk I will discuss a selection of issues and results in lattice flavourdynamics.
I start by describing some new thoughts on improving the momentum resolution in
simulations, by varying the boundary conditions on the quark fields. I then review the
status of lattice calculations of quark masses, K decays (for which computations have
ℓ3
onlyrecently began)and B . Thisisfollowedby adiscussionofsomeofthekeyissues
K
inthecomputationofK →pp decays and inheavy-quark physics.
Improving the Momentum Resolution on the Lattice
Numerical simulations of lattice QCD are necessarily performed on a finite spatial
volume, V = L3. Providing that V is sufficiently large, we are free to choose any
consistentboundaryconditionsforthefieldsf (~x,t),anditisconventionaltouseperiodic
boundary conditions, f (x +L) = f (x) (i = 1,2 or 3). This implies that components
i i
of momenta are quantized to take integer values of 2p /L. Taking a typical example
of a lattice with 24 points in each spatial direction, L = 24a, with a lattice spacing
a = 0.1fm so that a−1 ≃ 2GeV, we have 2p /L = .52GeV. The available momenta
for phenomenological studies (e.g. in the evaluation of form-factors) are therefore very
limited,withtheallowedvaluesofeach component p separated by about1/2GeV. The
i
momentumresolutioninsuch simulationsisvery poor.
Bedaque[4]hasadvocatedtheuseoftwisted boundaryconditionsforthequarkfields
q(~x)e.g.
2p q
q(x +L)=eiq iq(x) withmomentumspectrum p =n + i , (1)
i i i i
L L
with integer n . Modifying the boundary conditions changes the finite-volume effects,
i
however, for quantities which do not involve Final State Interactions (e.g. hadronic
masses,decayconstants,form-factors)theseerrorsremainexponentiallysmallalsowith
twisted boundary conditions [5]. Since we usually neglect such errors when using peri-
odicboundaryconditions,wecan usetwistedboundaryconditionswiththesamepreci-
sion.Moreoverthefinite-volumeerrorsarealsoexponentiallysmallforpartiallytwisted
boundary conditions in which the sea quarks satisfy periodic boundary conditions but
thevalencequarkssatisfytwistedboundaryconditions[5,6].Thisisofsignificantprac-
ticalimportance,implyingthatwedonotneedtogeneratenewgluonconfigurationsfor
everychoiceoftwistingangle{q }.
i
The use of partially twisted boundary conditions opens up many interesting phe-
nomenological applications, solving the problem of poor momentum resolution. It also
appearstoworknumerically.Considerforexample,theplotsinfig.1,obtainedusingan
unquenched (2 flavours of sea quarks) UKQCD simulationon a 163×32 lattice, with a
spacing of about 0.1fm. The plots correspond to a value for the light-quark masses for
which mp /mr = 0.7 [7]. The lower (upper) left-hand plot shows the energy of the p
0.4
1
0.35
0.8 0.3
2) 0.25
) 0.6 are
qE( afb 0.2
a
( 0.4 0.15
0.1
0.2
0.05
0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
(~pL)2 (~pL)2
FIGURE 1. Plots of the Dispersion Relation (left) and Decay Constants (right) as a function of the
momentum~pofthemesons.Inbothcasesthetop(bottom)plotcorrespondstother -meson(p -meson).
(r )asafunctionofthemomentumofthemeson,andtheright-handplotshowsthebare
values of the leptonic decay constants fp and fr . The x-axis denotes (|~p|L)2. The re-
sultsarebeautifullyconsistentwithexpectations(particularlyfor pL≤2p wherelattice
artifacts are small); thepredicted dispersion relation is satisfied and the extracted decay
constantsareindependentofthemomenta.Usingperiodicboundaryconditionsonlythe
resultsat valuesof~p indicatedby thedashedlinesareaccessible. Withpartiallytwisted
boundaryconditionsallmomentaare reachable.
QUARK MASSES
Quark Masses are fundamental parameters of the Standard Model, but unlike leptons,
quarks are confined inside hadrons and are not observed as physical particles. Quark
massesthereforecannotbemeasureddirectly,buthavetobeobtainedindirectlythrough
theirinfluenceonhadronicquantitiesandthisfrequentlyinvolvesnon-perturbativeQCD
effects.Latticesimulationsprovetobeveryusefulinthedeterminationofquarkmasses;
particularlyforthelightquarks(u,d ands)forwhichperturbationtheoryisinapplicable.
In order to determine the quark masses we computea convenient and appropriate set
of physical quantities (frequently a set of hadronic masses) and vary the input masses
until the computed values correctly reproduce the set of physical quantities being used
forcalibration.Inthiswayweobtainthephysicalvaluesofthebarequarkmasses,from
whichbyusingperturbationtheory,orpreferablynon-perturbativerenormalization,the
resultsinstandardcontinuumrenormalizationschemes can bedetermined.
My current best estimates for the values of the quark masses as determined from
latticesimulationsare presented intable1.
The relatively large error on the mass of the charm quark is a reflection of the fact
that the most detailed study to date was performed in the quenched approximation [8],
whose authors find m¯ = 1.301(34)GeV. I have added a conservative 15% error as an
c
estimate of quenching effects. Current and future calculations will be dominated by
unquenched simulationsso that the error will decrease very significantly. Indeed a very
TABLE1. Mysummaryofthestatusoflatticedeterminationsofquarkmasses(intheMSrenormal-
ization scheme). For mˆ ≡(m +m )/2 and m the results are presented at 2GeV and for m ,m , the
u d s c b
results are presented at the mass itself (m¯ ≡m¯(m¯)). For comparisonthe values quoted by the PDG in
2004,usingorexcludinglatticesimulations,arealsopresented.
Flavour BestLatticeValues PDG2004(Lattice) PDG2004(Non-Lattice)
mˆ(2GeV) (3.8±0.8)MeV (4.2±1.0)MeV (1.5<m (2GeV)<5)MeV
u
(5<m (2GeV)<9)MeV
d
m (2GeV) (95±20)MeV (105±25)MeV 80–155MeV
s
m¯ (1.26±0.13±0.20)GeV (1.30±0.03±0.20)GeV 1–1.4GeV
c
m¯ (4.2±0.1±0.1)GeV (4.26±0.15±0.15)GeV 4–4.5GeV
b
recent unquenchedcalculationfinds m¯ =1.22(9)GeV [9].
c
The relative error on m is small because what is actually calculated is m −m .
b B b
The calculations are performed in the Heavy Quark Effective Theory and the major
sourceofsystematicerror is the subtractionofO(1/(aL ))terms. Using stochastic
QCD
perturbation theory, Di Renzo and Scorzato have performed this calculation to 3-loop
order[10].Theseconderroronm¯ intable1ismyconservativeestimateofthefactthat
b
thesimulationshavebeen performed withtwoflavours ofseaquarks.
SELECTED TOPICS IN KAON PHYSICS
K Decays
ℓ3
Anewareaofinvestigationforlatticesimulationsistheevaluationofnon-perturbative
QCD effects in K → p ℓn decays, from which the CKM matrix element V can be
ℓ us
determined. TheQCD contributionto the amplitudeis contained in two invariant form-
factors f0(q2) and f+(q2)defined by
hp (pp )|s¯g m u|K(pK)i= f0(q2)MK2q−2Mp2qm + f+(q2)(cid:20)(pp +pK)m −MK2q−2Mp2qm (cid:21),
whereq=pK−pp .(ParityInvarianceimpliesthatonlythevectorcurrentfromtheV−A
chargedcurrentcontributestothedecay.)Ausefulreferencevaluefor f+(0)comesfrom
the20-yearoldpredictionofLeutwylerandRoos, f+(0)=1+ f + f +···=0.961(8)
2 4
where f =O(M2 ). f =−0.023iswelldetermined,whereasthehigherorderterms
n K,p ,h 2
inthechiralexpansionrequiremodelassumptions.
To be useful in extracting V from experimental measurements we need to be able
us
to evaluate f0(0) = f+(0) to better than about 1% precision. This would seem to be
impossible until one notes that it is possible to compute 1− f+(0), so that an error of
1% on f+(0) is actually an error of O(25%) on 1− f+(0). The calculation follows a
similarstrategytothatproposedinref. [11]fortheform-factorsofB→Dsemileptonic
decays (which in the heavy quark limitare also close to 1), starting with a computation
JLQCD(1997)
CP-PACS(2001)
Leeetal. (2004)
SPQR(2004)
Alpha(2005)
RBC(2005)
Berrutoetal. (2004)
DeGrand(2004)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
FIGURE2. AcompilationofrecentquenchedresultsforBMS(2GeV).
K
ofdoubleratiossuchas
hhpp ||sl¯¯gg00ll||pKiihhKK||s¯l¯gg00ss||Kp ii =(cid:2)f0(q2max)(cid:3)2 (m4Km+Kmmpp )2, (2)
whereallthemesonsareat rest and q2max =(MK−Mp )2.
Followingaquenched calculationby theSPQR collaborationlastyear [13], in which
the strategy for determining the form-factors was presented, there have been 3 very
recent unquenched(albeit largelypreliminary)results:
RBC [14] f+(0)=0.955(12)
JLQCD [15] f+(0)=0.952(6)
FNAL/MILC/HPQCD[16] f+(0)=0.962(6)(9)
ingoodagreement withtheresultofLeutwylerand Roos [12].
B
K
B , the parameter which contains the non-perturbativeQCD effects in K0−K¯0 mix-
K
ing,hasbeen computedin latticesimulationsby manygroups.Itis defined by
8
hK¯0|(s¯g m (1−g 5)d)(s¯g m (1−g 5)d)|K0i= 3MK2 fK2BK. (3)
B depends on therenormalization schemeand scaleand is conventionallygiven in the
K
NDR, MS scheme at m =2GeV or as the RGI parameter Bˆ (the relation between the
K
twoisBˆ ≃1.4BMS(2GeV)).
K K
A compilation of recent results for B obtained in the quenched approximation is
K
presented in fig.2. From such results recent reviewers have summarised the status of
quenchedcalculationsas:
BMS(2GeV)=0.58(4)[17] and BMS(2GeV)=0.58(3)[18]. (4)
K K
The dashed lines in fig.2 correspond to B =.58(3), which I am happy to take as the
K
current bestestimate.
Thechallengenowistoobtainreliableunquenchedresults;suchcomputationsareun-
derwaybyseveralgroupsbutsofartheresultsareverypreliminary.Wewillhavetowait
a year or two for precise results, but I mention in passing C.Dawson’s guesstimate[18]
(stressing that it is only a guesstimate), based on a comparison of quenched and un-
quenchedresultsat similarmassesand latticespacings,ofBMS(2GeV)=0.58(3)(6).
K
pp
K → Decays
Aquantitativeunderstandingofnon-perturbativeeffectsinK →pp decayswillbean
importantfuturemilestoneforlatticeQCD. Twoparticularlyinterestingchallengesare:
i) an understanding of the empirical D I =1/2 rule, which states that the amplitude for
decays inwhichthetwo-pionfinal statehas isospinI=0 islarger byafactor ofabout22
thanthatin whichthefinal statehas I =2;
ii) a calculation of e ′/e , whose experimental measurement with a non-zero value,
(17.2±1.8)×10−4, was thefirst observationofdirect CP-violation.
The two challenges require the computationof the matrix elements of the D S=1 oper-
atorswhich appearintheeffectiveWeak Hamiltonian.
About 4 years ago, two collaborations published some very interesting quenched
resultsforthesequantities:
Collaboration(s) Re A /Re A e ′/e
0 2
RBC [19] 25.3±1.8 −(4.0±2.3)×10−4
CP-PACS[20] 9–12 (-7–-2)×10−4
Experiments 22.2 (17.2±1.8)×10−4
Both collaborations obtain a considerable octet enhancement (significantly drivenhow-
ever,bythechiralextrapolation)ande ′/e withthewrongsign.Aparticularlyimpressive
feature of these calculations was that the collaborations were able to perform the sub-
tractionoftheunphysicaltermswhichdivergeaspowersoftheultra-violetcut-off(a−1,
whereaisthelatticespacing).Theresultsareveryinterestingandwillprovidevaluable
benchmarksforfuturecalculations,howeverthelimitationsofthecalculationsshouldbe
noted,inparticulartheuseofchiralperturbationtheory(c PT)onlyatlowestorder.This
has the practical advantage that K → pp matrix elements do not have to be evaluated
directly, it is sufficient at lowest order to study the mass dependence of the matrix ele-
mentshM|O |Mi and h0|O |Mi, where M isa pseudoscalarmeson and theO are the
i i i
D S = 1 operators appearing in the effective Hamiltonian, to determine the low-energy
constants and hence the amplitudes. It is not very easy to estimatethe errors due to this
approximation, but they should be at least of O(m2/L 2 ). Since for e ′/e the dom-
K QCD
inant contributions appear to be from the QCD and electroweak penguin operators O
6
and O , which are comparable in magnitude but come with opposite signs, it is not to-
8
tally surprising that the prediction for e ′/e at lowest order in c PT has the wrong sign.
It should also be noted that in the simulationsdescribed in ref. [19, 20] the light quarks
masseswerelarge(thepionswereheavierthanabout400MeV)andsoonecanquestion
thevalidityof c PT intherangeofmassesused(about 400-800MeV).
To improvethe precision, apart from performing unquenched simulationsand reduc-
ingthemassesofthelightquarks,oneneeds togobeyondlowestorder c PT (forexam-
ple by going to NLO [21, 22]) and, in general, this requires the evaluation of K →pp
matrixelementsandnotjustM →M ones.Thetreatmentoftwo-hadronstatesinlattice
computations has a new set of theoretical issues, most notably the fact that the finite-
volume effects decrease only as powers of the volume and not exponentially. Starting
with the pioneering work of Lüscher [23], the theory of finite-volume effects for two-
hadron states in the elastic regime is now fully understood, both in the centre-of-mass
andmovingframes, [23]–[28]and Iwillnowbriefly discussthis.
Considerthetwo-hadron correlationfunctionrepresented by thediagram
p
E
where the shaded circles represent two-particle irreducible contributions in the s-
channel. For simplicity let us take the two-hadron system to be in the centre-of mass
frame and assume that only the s-wave phase-shift is significant (the discussion can be
extendedtoincludehigherpartialwaves).Considertheloopintegration/summationover
p (see the figure). Performing the p integration by contours, we obtain a summation
0
overthespatialmomentaoftheform:
1 (cid:229) f(p2)
(5)
L3 p2−k2
~p
wheretherelativemomentumkisrelatedtotheenergybyE2=4(m2+k2),thefunction
f(p2) is non-singular and (for periodic boundary conditions) the summation is over
momenta~p=(2p /L)~nwhere~nisavectorofintegers.Ininfinitevolumethesummation
in eq.(5) is replaced by an integral and it is the difference between the summation
and integration which gives the finite-volume corrections. The relation between finite-
volume sums and infinite-volume integrals is the Poisson Summation Formula, which
(in1-dimension)is:
¥
1(cid:229) g(p)= (cid:229) dpeilLpg(p). (6)
L Z 2p
p l=−¥
Ifthefunctiong(p)isnon-singular,theoscillatingfactorsontheright-handsideensures
that only the term with l =0 contributes, up to terms which vanish exponentially with
L. The summand in eq.(5) on the other hand is singular (there is a pole at p2 = k2)
and this is the reason why the finite-volume corrections only decrease as powers of L.
The detailed derivation of the formulae for the finite-volume corrections can be found
in refs. [23]–[28] and is beyond the scope of this talk. The results hold not only for
K →pp decays, butalsoforp -nucleonand nucleon-nucleonsystems.
For decays in which the two-pions have isospin 2, we now have all the necessary
techniques to calculate the matrix elements with good precision and such computations
are underway. For decays into two-pion states with isospin 0 there are also no barriers
inprinciple.However,in thiscase, purelygluonicintermediatestatescontributeandwe
need to learn how to calculate the corresponding disconnected diagrams with sufficient
precision.Inadditionthesubtractionofpower-likeultravioletdivergencesrequireslarge
datasets (as demonstrated in refs.[19, 20] in quenched QCD). For these reasons it will
take a longer time for some of the D I = 1/2 matrix elements to be computed than
D I =3/2 ones.
HEAVY QUARK PHYSICS
Latticesimulationsareplayinganimportantroleinthedeterminationofphysicalquanti-
tiesinheavyquarkphysicsincludingdecayconstants(f , f , f , f ),theB-parameters
B Bs D Ds
ofB−B¯ mixing(from whichtheCKM matrixelementsV andV can bedetermined),
td ts
form-factors of semileptonic decays (which giveVcb andVub), the gBB∗p coupling con-
stantofheavy-mesonchiralperturbationtheory andthelifetimesofbeautyhadrons.
Thetypicallatticespacingin current simulationsa≃0.1fm is largerthan theComp-
ton wavelength of the b-quark and comparable to that of the c-quark. The simulations
aretherefore generally performedusingeffectivetheories, suchas theHeavy Quark Ef-
fectiveTheoryorNon-RelativisticQCD.Anotherinterestingapproachwasproposedby
the Fermilab group [29], in which the action is improved to the extent that, in principle
at least, artefacts of O((m a)n) are eliminated for all n, where m is the mass of the
Q Q
heavy quark Q. Determining the coefficients of the operators in these actions requires
matching with QCD, and this matching is almost always performed using perturbation
theory (most often at one-loop order). This is a significant source of uncertainty and
providesthemotivationforattemptstodevelopnon-perturbativematchingtechniques.
I only have time here to consider very briefly a single topic, semileptonic B-decays.
For B→p decays, the pion’s momentum has to be small in order to avoid large lattice
artefacts, so that q2 =(pB−pp )2 is large (q2 >15GeV2 or so). There continues to be
aconsiderableeffortinextrapolatingtheseresultsoverthewholeq2 range. Recently,as
experimental results begin to be presented in q2 bins, it has become possible to com-
bine the lattice results at large q2 with the binned experimental results and theoretical
constraintsto obtainV withgoodprecision [30].
ub
As an exampleI present a recent result, obtained using theMILC gaugefield config-
urationswithstaggered lightquarksand theFermilabaction fortheb-quark [31]
|V |=3.48(29)(38)(47)×10−3. (7)
ub
I mention that other semileptonic decays of heavy mesons are also being studied, in-
cludingB→D(∗) decays(arecentresultis|V |=3.9(1)(3)×10−2[16])andD→p ,K
cb
decays.
SUMMARY AND CONCLUSIONS
Lattice QCD simulations, in partnership with experiments and theory, play a central
rôle in the determination of the fundamental parameters of the Standard Model (e.g.
quark masses, CKM matrix elements) and in searches for signatures of new physics
andultimatelyperhapswillhelptounravelitsstructure.Withtheadventofunquenched
simulations,a major source of uncontrolled systematicuncertainty has been eliminated
and the main aim now is to control the chiral extrapolation and reduce other systematic
uncertainties. We continue to extend the range of applicability of lattice simulations to
moreprocessesandphysicalquantities.InthistalkIhaveonlybeenabletogiveasmall
selection of recent results and developments; a more complete set can be found on the
web-siteofthe2005internationalsymposiumonlatticefield theory[32].
REFERENCES
1. C. Bernard et al. [MILC Collaboration], PoS LAT2005 (2005) 025 [arXiv:hep-lat/0509137] and
referencestherein.
2. S.Durr,PoSLAT2005(2005)021[arXiv:hep-lat/0509026].
3. M. Luscher, PoS LAT2005 (2005) 002 [arXiv:hep-lat/0509152]; L. Del Debbio, L. Giusti,
M.Luscher,R.PetronzioandN.Tantalo,arXiv:hep-lat/0512021.
4. P.F.Bedaque,Phys.Lett.B59382(2004)[arXiv:nucl-th/0402051].
5. C.T.SachrajdaandG.Villadoro,Phys.Lett.B60973(2005)[arXiv:hep-lat/0411033].
6. P.F.BedaqueandJ.W.Chen,Phys.Lett.B616208(2005)[arXiv:hep-lat/0412023].
7. J.M.Flynn,A.JuttnerandC.T.Sachrajda[UKQCDCollaboration],Phys.Lett.B632313(2006)
[arXiv:hep-lat/0506016];J.Flynn,A.Juttner,C.SachrajdaandG.Villadoro,PoSLAT2005(2005)
352[arXiv:hep-lat/0509093].
8. J.RolfandS.Sint[ALPHACollaboration],JHEP0212007(2002)[arXiv:hep-ph/0209255].
9. M.NobesandH.Trottier,PoSLAT2005(2005)209[arXiv:hep-lat/0509128].
10. F.DiRenzoandL.Scorzato,Nucl.Phys.Proc.Suppl.140473(2005)[arXiv:hep-lat/0409151].
11. S. Hashimoto, A. X. El-Khadra, A. S. Kronfeld, P. B. Mackenzie, S. M. Ryan and J. N. Simone,
Phys.Rev.D61014502(2000)[arXiv:hep-ph/9906376].
12. H.LeutwylerandM.Roos,Z.Phys.C2591(1984).
13. D.Becirevicetal.,Nucl.Phys.B705339(2005)[arXiv:hep-ph/0403217].
14. C.Dawsonet.al.,PoSLAT2005(2005)337[arXiv:hep-lat/0510018].
15. N.Tsutsuietal.[JLQCDCollaboration],PoSLAT2005(2005)357[arXiv:hep-lat/0510068].
16. M.Okamoto[FermilabLatticeCollaboration],arXiv:hep-lat/0412044.
17. S.Hashimoto,Int.J.Mod.Phys.A205133(2005)[arXiv:hep-ph/0411126].
18. C.Dawson,http://www.maths.tcd.ie/lat05/plenary-talks/Dawson.pdf
19. T.Blumetal.[RBCCollaboration],Phys.Rev.D68114506(2003)[arXiv:hep-lat/0110075].
20. J.I.Noakietal.[CP-PACSCollaboration],Phys.Rev.D68014501(2003)[arXiv:hep-lat/0108013].
21. C. J. D. Lin et.al., Nucl. Phys. B 650 301 (2003) [arXiv:hep-lat/0208007]; P. Boucaud et al. [The
SPQ(CD)RCollaboration],Nucl.Phys.Proc.Suppl.106329(2002)[arXiv:hep-lat/0110206].
22. J. Laiho and A. Soni, Phys. Rev. D 65 114020 (2002) [arXiv:hep-ph/0203106]; Phys. Rev. D 71
014021(2005)[arXiv:hep-lat/0306035].
23. M. Luscher, Commun. Math. Phys. 104 177 (1986);Commun. Math. Phys. 105 153 (1986);Nucl.
Phys.B354531(1991);Nucl.Phys.B364237(1991).
24. K.RummukainenandS.A.Gottlieb,Nucl.Phys.B450397(1995)[arXiv:hep-lat/9503028].
25. L.LellouchandM.Luscher,Commun.Math.Phys.21931(2001)[arXiv:hep-lat/0003023].
26. C. J. D. Lin, G. Martinelli, C. T. Sachrajda and M. Testa, Nucl. Phys. B 619 467 (2001)
[arXiv:hep-lat/0104006].
27. C.h.Kim,C.T.SachrajdaandS.R.Sharpe,Nucl.Phys.B727(2005)218[arXiv:hep-lat/0507006];
PoSLAT2005(2005)359[arXiv:hep-lat/0510022].
28. N.H.Christ,C.KimandT.Yamazaki,Phys.Rev.D72114506(2005)[arXiv:hep-lat/0507009].
29. A. X. El-Khadra, A. S. Kronfeld and P. B. Mackenzie, Phys. Rev. D 55 3933 (1997)
[arXiv:hep-lat/9604004].
30. T.BecherandR.J.Hill,arXiv:hep-ph/0509090.
31. M.Okamoto,arXiv:hep-ph/0505190.
32. www.maths.tcd.ie/lat05