Non-perturbative renormalization of kaon four-quark operators with nf=2+1 Domain Wall fermions

0.2 MB·English

by  Peter Boyle

Non-perturbative renormalization of kaon four-quark
operators with n = 2+1 Domain Wall fermions
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Edinburgh2011/05

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a Peter Boyle, Nicolas Garron∗for the RBC-UKQCD collaborations.
J
SchoolofPhysicsandAstronomy,UniversityofEdinburgh,EdinburghEH93JZ,U.K.
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2 E-mail: [email protected]

]
t
a We present our strategy and some preliminary results for the renormalization of four-quark op-
l
- erators relevant for kaon physics. We follow the non-perturbative Rome-Southampton method,
p
e with both exceptional and non-exceptional kinematics. We also implement momentum sources
h
and twisted boundary conditions. We use an (almost) unitary setup: Domain-Wall valence on
[
nf =2+1Domain-WallseaandIwasakigaugeaction,attwovaluesofthelatticespacingcorres-
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v pondingtoapproximately0.086fmand0.114fm. Thechiralpropertiesofthesefermionsplaya
9 crucialroleinthiscompuationandarestudiedindetailinthiswork.
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TheXXVIIIInternationalSymposiumonLatticeFieldTheory,Lattice2010
June14-19,2010
Villasimius,Italy
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
NicolasGarron
1. Introduction
Kaonphysicshasbeenextensivelystudiedthoughlatticesimulationformorethanthirtyyears,
andthankstotherecentalgorithmsandhardwaredevelopmentsonecannowachievetheprecision
requiredtoconstraintthestandardmodelandhopefullyrevealtheeffectofnewphysics. Recently,
B ,thequantitywhichparametrizesneutralkaonmixinginthestandardmodelhasbeencomputed
K
with an accuracy of a few percents [1]. Nevertheless, other kaon matrix elements are still poorly
determinedalthoughtheycanhaveagreatimpactinthesearchfornewphysicsorimplystrongcon-
straintsonbeyondthestandardmodel(BSM)theories. Forexample,thenon-perturbativecontribu-
tionstoneutralkaonmixingbeyondthestandardmodelhavebeencomputedonlyinthequenched
approximation [2, 3], although computation with dynamical fermions are currently underway and
some preliminary studies have been already presented [4, 5]. Probably even more importantly, a
completecomputationofK →ππ decayswithdynamicalquarksisstillmissing. Becausetheex-
perimentalparametersofCPviolationsareverywellmeasured(|ε|=(2.228±0.011)×10−3 and
Re(ε(cid:48)/ε)=(1.65±0.26)×10−3 [6]), a precise and realistic computation of the relevant matrix
elements would provide important constraints on the CKM matrix. One of the difficulties for the
lattice implementation comes from the two-body final state. In the past this problem was usually
circumventbyinvokingthesoftpiontheoremtorelatethetwo-pionstatetoaone-pionstate. But,
asithasbeenshownin[7],thisapproachisnotreliableforaprecisecomputation,mainlybecause
of the poor convergence of chiral perturbation theory at masses around those of the kaon. A very
importantstepforwardhasbeenmadewhenitwasrealizedhowtheenergyshiftofatwo-particle
state can be computed on the lattice [8]. The RBC-UKQCD collaborations have started the com-
putation of this decay along this line, and the first results for the matrix element of the ∆I =3/2
operatorshavebeenpresentedatthisconference[9]. Analternativemethodhasbeenrecentlypre-
sented in [10]. In this proceeding we present our strategy and some preliminary results for the
renormalization of some of the relevant four-quark operators. It has become traditional to use a
non-perturbativerenormalizationschemeliketheSchrödingerfunctionalortheRI-MOMscheme.
Here we use a modified version of the latter, following what was done recently for B [1], but
K
generalizedtootherfour-quarkoperators. Inthenextsectionweexplainwhataretheoperatorswe
consider in this work. In the third section, we give more details about the numerical techniques,
andpreliminaryresultsarepresentedinthefourthsection.
2. GeneralFramework
2.1 Kaondecay
In the standard model at an energy scale below the charm quark mass, the dominant non
perturbative contributions to the effective ∆s = 1, ∆d = −1 Hamiltonian can be described by a
linearcombinationoftenfour-quarkoperators: twocurrent-current,fourQCDpenguins,andfour
electroweak penguins. Among these 10 operators, only seven are actually independent and it is
useful to classify them according to their chiral and ispospin properties. Hence one notices that
theyfallintothreedifferentrepresentationsofSU(3) ×SU(3) whichare(27,1),(8,1)and(8,8),
L R
andtheycancontributetotwoispospinchannels∆I=3/2and∆I=1/2(seeforexample[11,12]).
We will use the seven-operator renormalization basis defined in [12], in which the operators have
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NicolasGarron
s d
d
d
Figure1: ExampleofeyediagramwhichcontributetoK→ππ decayinthe∆I=1/2channel.
thefollowingproperties:
Operator SU(3) ×SU(3) ∆I
L R
Q(cid:48) (27,1) 1/2, 3/2
1 (2.1)
Q(cid:48),Q(cid:48),Q(cid:48),Q(cid:48) (8,1) 1/2
2 3 5 6
Q(cid:48),Q(cid:48) (8,8) 1/2, 3/2
7 8
If chiral symmetry was exact, the operators of different chirality would not mix under renormali-
zation,andinthebasisdescribedabovetherenormalizationmatrixwouldtaketheblockdiagonal
form:
 
Z∆s=1
11
 Z∆s=1 Z∆s=1 Z∆s=1 Z∆s=1 
 22 23 25 26 
 Z∆s=1 Z∆s=1 Z∆s=1 Z∆s=1 
 32 33 35 36 
Z∆s=1= Z∆s=1 Z∆s=1 Z∆s=1 Z∆s=1  . (2.2)
 52 53 55 56 
 Z∆s=1 Z∆s=1 Z∆s=1 Z∆s=1 
 62 63 65 66 
 Z∆s=1 Z∆s=1
 77 78 
Z∆s=1 Z∆s=1
87 88
Furthermore,sinceispospinisanexactsymmetryinthechirallimit,forthe(27,1)and(8,8)opera-
tors,itisenoughtoconsideronlythe∆I=3/2parts. Thisisnumericallyadvantageousbecausethe
“eyediagram”(seefig.1)whicharedifficulttocomputecanonlycontributeto∆I=1/2processes.
3. Neutralkaonmixing
Inthestandardmodel,neutralkaonmixingisdominatedbyboxdiagramsliketheoneshown
in figure 2. The non-perturbative contributions are given by (cid:104)K0|O∆s=2 |K0(cid:105), where O∆s=2 is
VV+AA VV+AA
theparityconservingpartof(sγLd)(sγLd). Beyondthestandardmodel,otheroperatorscontribute
µ µ
andtheyareusuallygivenintheso-calledSUSYbasis
O∆s=2 = (s γ (1−γ )d )(s γ (1−γ )d ), (3.1)
1 α µ 5 α β µ 5 β
O∆s=2 = (s (1−γ )d )(s (1−γ )d ), (3.2)
2 α 5 α β 5 β
O∆s=2 = (s (1−γ )d )(s (1−γ )d ), (3.3)
3 α 5 β β 5 α
O∆s=2 = (s (1−γ )d )(s (1+γ )d ), (3.4)
4 α 5 α β 5 β
O∆s=2 = (s (1−γ )d )(s (1+γ )d ). (3.5)
5 α 5 β β 5 α
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W
s d
t t
d W s
Figure2: ExampleofboxdiagramcontributingtoK−K mixingintheStandardmodel.
InthisbasisO∆s=2 isthestandardmodeloperatorandO∆s=2,i>1aretheBSMones. IntheSU(3)
1 i
flavor limit, It is straightforward to relate O∆s=2 and O∆s=2 to the ∆I = 3/2 components of the
4 5
electroweak penguins Q(cid:48) and Q(cid:48), which transform under (8,8). As one can find out from their
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flavorstructures,theremainingoperatorsO∆s=2andO∆s=2transformunder(6,6) 1. Thus,ifchiral
2 3
symmetryisrespected,O∆s=2 renormalizesmultiplicatively,O∆s=2 andO∆s=2 mixtogether,andso
1 2 3
doO∆s=2 andO∆s=2. Tosimplifythenumericalimplementationweworkinthefollowingbasis:
4 5
Q∆s=2 = (s γ d )(s γ d )+(s γ γ d )(s γ γ d ), (3.6)
1 α µ α β µ β α µ 5 α β µ 5 β
Q∆s=2 = (s γ d )(s γ d )−(s γ γ d )(s γ γ d ), (3.7)
2 α µ α β µ β α µ 5 α β µ 5 β
Q∆s=2 = (s d )(s d )−(s γ d )(s γ d ), (3.8)
3 α α β β α 5 α β 5 β
Q∆s=2 = (s d )(s d )+(s γ d )(s γ d ), (3.9)
4 α α β β α 5 α β 5 β
1
Q∆s=2 = (s σ d )(s σ d ), σ = [γ ,γ ]. (3.10)
5 α µν β α µν β µν 2 µ ν
Theparityconservingpart(denotedbyasuperscript“+”)oftheoperators(3.1)-(3.5)canbewritten
intermsoftheoperators(3.6)-(3.10)
(27,1) (cid:2)O∆s=2(cid:3)+=Q∆s=2 (3.11)
1 1
(cid:40)(cid:2)O∆s=2(cid:3)+ = Q∆s=2 (cid:40)(cid:2)O∆s=2(cid:3)+ = Q∆s=2
(6,6) 2 4 (8,8) 4 3
(cid:2)O∆s=2(cid:3)+ = −1(Q∆s=2−Q∆s=2) (cid:2)O∆s=2(cid:3)+ = −1Q∆s=2
3 2 4 5 5 2 2
(3.12) (3.13)
It follows from the above considerations that Q∆s=2 renormalizes multiplicatively, Q∆s=2 mixes
1 2
with Q∆s=2 and Q∆s=2 mixes with Q∆s=2. We denote the renormalization factors computed in this
3 4 5
basisbyZ∆s=2. Moreover, intheSU(3)flavorlimittheyaresomerelationsbetweentherenorma-
ij
lization factors of the ∆s=2 operators (3.6)-(3.10) and those of the ∆s=1 operators (2.1). For
example O∆s=2 and Q(cid:48) have the same renormalization factor, and the two by two renormalization
1 1
matrixof(Q ,Q )isrelatedtotheoneof(Q∆s=2,Q∆s=2)inthefollowingway
7 8 2 3
Z∆s=1 = Z∆s=2 Z∆s=1 = −1Z∆s=2
77 22 87 2 32 (3.14)
Z∆s=1 = −2Z∆s=2 Z∆s=1 = Z∆s=2
87 32 33 88
1Intheliterature,wesometimesfindthenotationVLLforO ,SLLfortheset(O ,O )andLRfortheset(O ,O ).
1 2 3 4 5
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NicolasGarron
4. Numericalimplementationandpreliminaryresults
The numerical setup of this computation is the same as the one presented in details in two
recentpublications[1,13]. Weusen =2+1flavorsofdomainwallfermiononaIwasakigauge
f
action at two values of the lattice spacing a∼0.086 fm and a∼0.114 fm , corresponding to the
lattice volumes 32×64×16 and 24×64×16, respectively. On each ensemble the strange sea
quark mass is fixed, while several values of the light sea quark masses have been considered (the
corresponding unitary pion mass varies in the range 290−420 MeV). In this work we consider
onlythelightvalencequarkmasseswhichhavethesamevaluesastheircorrespondingseaquarks,
andperformthechiralextrapolationslinearlyinthequarkmass. Thiscomputationwasdonewith
20 configurations, and 100 bootstrap samples. In addition to the standard RI-MOM scheme, we
implementalsoaschemewithnon-exceptional(andsymmetric)kinematic,whichexhibitsabetter
infrared behavior 2 [14, 15]. We also employ momentum sources [16] in order to obtain small
statisticalerrorsdespitetheexpensivecostofthequarkdiscretization. Furthermore,weusetwisted
periodicboundaryconditions,whichallowsustochangesmoothlythemagnitudeofthemomentum
without changing its direction (and thus control the O(4) discretization effects) [17]. This setup
has been used recently used for the computation of Z [1]. Here we generalize this computation
BK
to the operators relevant for neutral kaon mixing beyond the standard model, and to the ∆I =3/2
partofK →ππ decay. Asdescribedin[18],theZmatrixisessentiallytheinvertofΛ =P{O},
ij j i
where O is a four-quark operator which belong to the basis given in eqs (3.6)-(3.10). P projects
i j
onto the Dirac and color structure of the operator O (and a given flavor structure which depends
j
onthechoiceofexternalstates).
Infigures3and 4weshowthenormalizedGreenvertexfunctionΛ extrapolatedtothechiral
ij
limit. The range of simulated momenta corresponds to 1.98 GeV≤ p≤3.29 GeV on the finest
latticeandto1.80GeV≤ p≤2.64GeVonthecoarserone. Asexpectedtheuseofthemomentum
sourcesgiveusaccesstoaveryhighstatisticalprecision: atagivenmomentumthestatisticalerror
is below the permille (∼10−4 for Z ). Thanks to the twisted boundary conditions, the Z factors
BK
are smooth functions of the momentum (no scatter coming from the O(4) discretization effects is
visible). Finallytheeffectofthenon-exceptionalkinematicisclearlyvisibleinfigure4. Thematrix
elementsshownintheseplotsshouldbezeroifchiralsymmetrywasexact. WiththeDomain-Wall
fermions, this is true only in the limit L →∞, so in practice one has to check whether the effects
s
ofchiralsymmetrybreakingcanbeseenwithinthenumericalprecision. Forthenon-perturbative
renormalization,thiscanbecomplicatedbythepresenceofsomeGoldstonepoles,whichcanaffect
the vertex functions. These poles are suppressed by the use of a non-exceptional kinematic. We
confirmhereaneffectalreadyseenin [4],thatthegoodpropertiesoftheDomainWallactioncan
beobscuredbyapoorchoiceofkinematic.
2Thenon-exceptionalschemeimplementedhereiscalledSMOM(γµ,γµ)in[1]
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2 2
Λ /Λ2 Λ /Λ2
11 AV 11 AV
1.5 Λ /Λ2 1.5 Λ /Λ2
22 AV 22 AV
Λ /Λ2 Λ /Λ2
23 AV 23 AV
Λ /Λ2 Λ /Λ2
1 32 AV 1 32 AV
Λ /Λ2 Λ /Λ2
33 AV 33 AV
Λ /Λ2 Λ /Λ2
44 AV 44 AV
0.5 Λ /Λ2 0.5 Λ /Λ2
45 AV 45 AV
Λ /Λ2 Λ /Λ2
54 AV 54 AV
0 Λ55/Λ2AV 0 Λ55/Λ2AV
-0.5 -0.5
1 1.5 2 2.5 1 1.5 2 2.5
(ap)2 (ap)2
Figure3: PhysicalGreenvertexfunctionsforthenonexceptionalkinematic(SMOMscheme),inthechiral
limit. Ontheleftpanelweshowtheresultsforthe243 lattice(a∼0.114fm)andontherightforthe323
lattice(a∼0.086fm). Theerrorbarsaresmallerthanthesymbols.
Λ /Λ2 Λ /Λ2
4 12 AV 12 AV
Λ /Λ2 Λ /Λ2
13 AV 0.004 13 AV
Λ /Λ2 Λ /Λ2
14 AV 14 AV
2 Λ /Λ2 Λ /Λ2
Λ15/Λ2AV 0.002 Λ15/Λ2AV
21 AV 21 AV
Λ /Λ2 Λ /Λ2
0 Λ24/Λ2AV 0 Λ24/Λ2AV
25 AV 25 AV
Λ /Λ2 Λ /Λ2
Λ31/Λ2AV -0.002 Λ31/Λ2AV
-2 34 AV 34 AV
Λ /Λ2 Λ /Λ2
35 AV 35 AV
Λ /Λ2 -0.004 Λ /Λ2
41 AV 41 AV
-4 Λ42/Λ2AV Λ42/Λ2AV
Λ /Λ2 -0.006 Λ /Λ2
43 AV 43 AV
Λ /Λ2 Λ /Λ2
51 AV 51 AV
-6 1 1.5 2 Λ252./5Λ2AV -0.008 1 1.5 2 Λ252./5Λ2AV
(ap)2 Λ /Λ2 (ap)2 Λ /Λ2
53 AV 53 AV
Figure4: ChiralydisallowedGreenvertexfunctionsonthefinestlatticefortheexceptional(left)andnon-
exceptional(right)kinematic. Weseethatfortheexceptionalkinematicsomeofthesematrixelementsare
suppressed onlyat highmomenta, whilein the nonexceptional casethey are allzero withinthe statistical
precision.
5. Conclusionsandoutlook
Bycombiningmomentumsourcewithtwistedboundaryconditionsandnonexceptionalkine-
matic we can obtain the renormalization factors of kaon four-quark operators with a very good
handle on the different kinds or errors: the statistical errors are tiny (below the permille), most of
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NicolasGarron
theunwantedinfraredeffectsaresuppressedandtheusualscattercomingfromtheO(4)discretiza-
tionerrorsisabsent. Evenwiththisprecision, whenanon-exceptionalkinematicisimplemented,
the non-physical mixing of the four-quark operators is compatible with zero, thanks to the good
chiralpropertiesoftheDomainWallaction. Wearecurrentlyextendingourcomputationtoalarger
physicalvolumediscussedin[9],wherethesimulatedpionmassissignificantlysmaller(downto
180MeV).Weplantousethestepscalingmethodintroducedin[17]inordertoenlargetheRome-
Southampton window. We have also started a computation of the eye diagrams, with the use of
stochasticsources.
WethankourRBC/UKQCDcolleaguesformanydiscussionsandcontributionstothiswork.
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