Delta I = 3/2, K to Pi Pi Decays with a Nearly Physical Pion Mass

0.11 MB·English

by  Elaine J Goode

D pp
I = 3/2, K Decays with a Nearly Physical Pion
→
Mass
1
1
0
2

n
a
J
Elaine JGoode
3
1 UniversityofSouthampton,SchoolofPhysicsandAstronomy,Highfield,Southampton,SO17
1BJ,UnitedKingdom
]
t E-mail: [email protected]
a
l
- MatthewLightman
p ∗
e DepartmentofPhysics,ColumbiaUniversity,NewYork,NY10027USA
h DepartmentofPhysics,WashingtonUniversity,St. Louis,MO63130USA
[
E-mail: [email protected]

1
v
3 TheD I=3/2K pp decayamplitudeiscalculatedonRBC/UKQCD323 64,L =32dynam-
s
7 → ×
icallatticeswith2+1flavorsofdomainwallfermionsusingtheDSDRandIwasakigaugeaction.
4
2 Thecalculationisperformedwithasinglepionmass(mp =141.9(2.3)MeV,partiallyquenched)
.
1 andkaonmass(m =507.4(8.5)MeV)whicharenearlyphysical,andwithnearlyenergycon-
K
0
servingkinematics. Antiperiodicboundaryconditionsin two spatialdirectionsare used to give
1
1 thetwopionsnon-zerogroundstatemomentum.Resultsfortimeseparationsof20,24,28and32
:
v betweenthekaonandtwo-pionsourcesarecomputedandanerrorweightedaverageisperformed
i
X toreducetheerror.WefindprelimenaryresultsforRe(A )=1.396(081) (160) 10 8GeV
2 stat sys −
×
r andIm(A )= 8.46(45) (1.95) 10 13GeV.
a 2 − stat sys× −
TheXXVIIIInternationalSymposiumonLatticeFieldTheory,Lattice2010
June14-19,2010
Villasimius,Italy
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
D I=3/2,K pp DecayswithaNearlyPhysicalPionMass MatthewLightman
→
1. Introduction
Preciselattice calculations ofK pp decays willprovide quantitative insight intotheorigin
→
of the D I = 1/2 rule and direct CP violation in kaon decays. Previous calculations have relied
on the quenched approximation or have attempted to use chiral perturbation theory to extrapolate
from heavy quark masses down to physical masses, or both [1, 2, 3, 4, 5, 6, 7]. The calculation
presented here avoids both these sources of error by computing the K pp amplitude directly
→
using dynamical lattices with 2+1 flavours of domain wall fermions (DWF) at near physical pion
mass. We use RBC/UKQCD 323 64, L = 32 lattices which use the Dislocation Suppressing
s
×
Determinant Ratio(DSDR)plusIwasaki gaugeaction withinverse lattice spacing a 1 1.4 GeV
−
≈
(b =1.75) and domain wall height M =1.8. We use ensembles generated with amsea =0.001,
5 l
amssea =0.045,corresponding toaunitary pionmassofmp 180MeV.
≈
2. Four-Quark Operators andThe Effective Hamiltonian
The weak interactions and the effects of heavier quarks can be included in the lattice QCD
simulation by evaluating matrix elements of an effective Hamiltonian [8, 9]. In particular the
conventions of [3]are used. Wecalculate matrix elements offour-quark operators between K and
pp states. In this paper the amplitude A is calculated, which requires the evaluation of matrix
2
elements ofthreeoperators. Theseoperators areclassified byhowtheytransform underSU(3)
L
×
SU(3) : Q ,Q andQ . Progressincalculating A isdescribed in[10].
R (27,1) (8,8) (8,8)mx 0
3. Boundary Conditions
Wewish to simulate the K pp decay at physical kinematics, which requires the final state
→
pions to have a non-zero momentum. This is achieved by imposing antiperiodic boundary condi-
tions on the quark fields in one ormore spatial directions. Theallowed momenta of the quark are
thengivenby p =(p +2p n)/L,whereListhespatialextentofthelattice.
n
Werelatethephysical p +p 0 QD I=3/2 K+ matrixelementtotheunphysical matrixelement
D Iz=1/2| i
hp +p +|QDD IIz==33//22|K+i using(cid:10)the W(cid:12)(cid:12)igner-Eckart theorem. This simplifies the operators and allows
us to use periodic boundary conditions on the up- and strange-quarks while using antiperiodic
boundaryconditionsonlyonthedown-quark,thusgivingthetwopionsmomentumwhilethekaon
remainsatrest.
If antiperiodic boundary conditions are imposed on the d-quark in only the x direction with
periodic boundary conditions in the y and z directions then we can have a two-pion ground state
p p
in which one pion has momentum p = and the other pion has momentum p = . The
x L x −L
antiperiodic boundary conditions allow us to extract non-zero momentum pions without the need
to fit to an excited state, which would have been necessary had we imposed periodic boundary
conditions onallthequarkfields. Inprinciple wecanimposeantiperiodic boundary conditions on
thed-quarkinone,two,orallthreespatialdirectionscorrespondingtoindividualgroundstatepion
momentaof p=p /L,√2p /Land√3p /Lrespectively.
4. Detailsofthe Calculation
The calculation was carried out on 62 configurations of dynamical 323 64 lattices using
×
DSDR+IwasakigaugeactionanddomainwallfermionswithL =32,generatedonBG/Pmachines
s
2
D I=3/2,K pp DecayswithaNearlyPhysicalPionMass MatthewLightman
→
atArgonneNationalLaboratory. Furtherdetailsoftheensemblegeneration aregivenin [11]. The
inverse lattice spacing is a 1 = 1.365(22) GeV, the physical volume is (4.62fm)3 and we set
−
the light and strange valence quark masses to am =0.0001 and am =0.049 respectively. This
l s
corresponds toapionmassofmp =141.9(2.3) MeVandakaonmassofmK =507.4(8.5) MeV.
We combined propagators with periodic and antiperodic boundary conditions in the time di-
rection in order to double the effective time extent of the lattice. The meson correlation functions
contained propagators which were computed with a source at t =0 (corresponding to (P+A) bc)
andt=64(corresponding to(P-A)bc). Wealsogeneratedstrange-quark propagatorswithsources
att =20,24,28,32,36,40and44inordertocalculate K pp correlators withkaonsources at
K
→
these times,whilethetwo-pion sources remainedateithert =0ort =64. Thuswecouldachieve
timeseparations betweenthekaonandtwopionsof20,24,28and32intwodifferent wayswhich
doubled the statistics. These separations were chosen so that the signals from the kaon and two
pionsdidnotdecaytonoisebeforereachingthefour-quark operator Q.
For the kaon and pions with zero momentum we use propagators with Coulomb gauge-fixed
wallsources. Forthetwopionswithnon-zeromomentumweusethesametypeofpropagators for
theuquarkbutusedpropagatorswithantiperodic spatialboundaryconditions forthed-quarkwith
Coulombgauge-fixedmomentumwallsourcesofthe“cosine”type
s (x)=cos(p x)cos(p y)cos(p z). (4.1)
p,cos x y z
Weusethesamecosinesourceforeachd-quark,whichintroducesacrosstermthatcouplesto
two-pion states withnon-zero totalmomentum. Forexample, ifweconsider giving momentum in
onlythexdirection theproductofthesources ofthetwod-quarks is
p p
s (x )s (x )=cos x cos x
p,cos 1 p,cos 2 1 2
L L
(4.2)
= 1 e(cid:16)ipLx1e(cid:17)ipLx2+(cid:16)eipLx1e(cid:17)−ipLx2+e−ipLx1eipLx2+e−ipLx1e−ipLx2 .
4
(cid:16) (cid:17)
p p
Werequirethetwopionstohaveindividualmomentump = xˆ andp = xˆ,butthefirstandlast
1 L 2 −L
p p
terms of equation (4.2) couple to two-pion states with total momentum 2 and 2 respectively.
L − L
Weeliminate theunwanted termsinthetwo-pioncorrelator byusingpureexponential momentum
sinks which constrain the final state to have zero total momentum. In the K pp correlator, the
→
zeromomentumkaonhasasimilareffectonthecosinesourcesofthetwo-pions. Hadweusedthe
moreconventional momentumsource
s (x)=eipx (4.3)
p ·
we would have needed to perform two separate d-quark inversions with momentum +p for one
and p for the other. The cosine source eliminates one of these inversions. In practice we only
−
compute d-quark propagators with antiperiodic boundary conditions in 0 or 2 spatial directions,
correspondingtopionswithgroundstatemomenta p=0and p=√2p /L. Thischoiceismotivated
by the expectation that for our choice of quark masses, p= √2p /L will correspond to on-shell
kinematics.
5. AnalysisandResults
WeextracttheK pp matrixelementM byfittingaconstant tothelefthandsideof(5.1)
→
3
D I=3/2,K pp DecayswithaNearlyPhysicalPionMass MatthewLightman
→
Ci (t) M
Kpp = i . (5.1)
CK(tK t)Cpp (t) ZKZpp
−
Ci istheK pp correlator withakaon source att , ilabels the four-quark operator Q which
Kpp → K i
isinsertedattimet,andZK andZpp arecalculated fromthekaonandtwo-pioncorrelators respec-
tively,whosesourcesareatt=0. Thelefthandsideofequation(5.1)isplottedinfigure1foreach
of the three operators. The figure demonstrates that sufficiently far from the kaon and two-pion
sourceswearejustifiedinfittingtoaconstant. ThefitresultsforMi/(ZKZpp )areindicatedonthe
plot.
7x 10−17 14x 10−16 4.5x 10−15
ors6 ors12 ors 4
elat5 elat10 elat3.5
orr orr orr 3
nt of C4 nt of C 8 nt of C2.5
otie3 otie 6 otie 2
u u u
Q2 Q 4 Q1.5
10 5 10 15 20 25 20 5 10 15 20 25 10 5 10 15 20 25
t t t
(a) (27,1)operator (b) (8,8)operator (c) (8,8)mxoperator
Figure1: K pp quotientplotsfor p=√2p /L. Thetwopionsourceisatt=0whilethekaonsourceis
→
att=24. Thedashedlineshowstheerroronthefit
The finite volume matrix elements are related to the infinite volume amplitudes A using the
i
Lellouch-Lüscher factor[12,13]. Inparticular wehave
n /4 ¶f ¶d 1
Ai="pp qp s¶ qp +¶ qp #√n √mKEpp Mi (5.2)
wherethequantityinsquarebrackets(denotedbyLLintable2)containstheeffectsoftheLellouch-
Lüscher factor beyond the free field normalization. Epp is the energy of the two-pion state, d is
the s-wave phase shift, n is a factor counting the free-field degenerate states, qp is a dimension-
less quantity related to the individual pion momentum kp via qp =kp L/2p and f is a kinematic
function defined in [12]. Once qp is known, d can be calculated using the Lüscher quantisation
condition [14].
np =d (kp )+f (qp ). (5.3)
Epp isfoundbyfittingthequotientofcorrelatorsCpp /(Cp )2 Ae−D Et toextractD E =Epp
∼ −
2Ep . We then get the two-pion energy by calculating D E+2Ep , where in the case of p = 0,
Ep =mp whilefor p=√2p /L,Ep isfoundfroma2parameterfittothepioncorrelation function
whichalsohas p=√2p /L. Thismethodispreferred todirectlyextracting Epp fromthetwo-pion
correlatorbecausethequotientCpp /(Cp )2cancelssomecommonfluctuationsinthenumeratorand
denominator andreducestheerror.
Thepionmomentumkp inthetwo-pionstateisdeterminedfromthetwo-pionenergyusingthe
dispersion relation Epp =2 m2p +kp2. It differs from p=0,√2p /L due to interactions between
the two pions. Results for Epp , kp , qp and d are presented in table 1. ¶f /¶ q can be calculated
p
4
D I=3/2,K pp DecayswithaNearlyPhysicalPionMass MatthewLightman
→
analytically sotheonlyunknowninequation(5.2)is¶d /¶ q. Theresultsforthephaseshiftcanbe
plotted against kp and compared with experiment [15, 16]. This is done in figure 2(a) and we see
goodagreementwithexperiment. For p=0wemaketheapproximation thatd islinearwithkp in
order to calculate ¶d /¶ qp (see figure 2(b)). For p=√2p /L weuse the phenomenological curve
[17] shown in figure 2(a) to calculate the derivative of the phase shift at the corresponding value
of qp . Thederivative of the phase shift is found to be a small factor in comparison with ¶f /¶ qp .
Resultsfor¶f /¶ qp and¶d /¶ qp areshownintable2.
Table1: Twopionenergyands-wavephaseshift
p Epp (MeV) kp (MeV) qp d (degrees)
0 285.9(4.6) 17.55(61) 0.0655(21) -0.306(29)
√2p /L 489.2(8.1) 199.2(3.8) 0.743(11) -10.4(3.3)
Table2: ContributionstoLellouch-Lüscherfactor
p ¶f /¶ qp ¶d /¶ qp LL
0 0.239(14) -0.0815(50) 0.9636(22)
√2p /L 5.039(35) -0.2927(52) 0.933(11)
0
0
s)-0.1
e
e
grees) Degr-0.2
Phase Shift (De-10 THHhooioosgg Cllaaannlcddu eeltta..t iaaoll..n ((AB)) Phase Shift (---000...453 TPhheisn oCmal.c (uSlacthieonnk)
Losty et. al. Scattering Length
-20
Phenom. (Schenk)
0 10 20 30
0 100 200 300 400 k (MeV)
kPi (MeV) Pi
(a) Comparison of calculated phase shift with (b) Zoomof2(a)showingthatforsmallkd is
experimentalresults[15,16,17]. approximately linear. The scattering length is
calculatedusingchiralperturbationtheory[18].
Figure 2: Plot of I =2 two-pion s-wave phase shift against momentum kp . The results from p=0 and
p=√2p /Lareshowninred.
Theamplitudes A arerelatedtothephysicaldecayamplitudeA via
i 2
A =a 3 3G V V (cid:229) C(m )Z (m )A , (5.4)
2 − 2 F ud u∗s i ij j
r i,j
where C are the Wilson Coefficients and Z are the renormalization constants, calculated using
i ij
non-perturbativerenormalization(NPR).Thefactor 3/2isneededtoconvertfromtheunphysical
K+ p +p + amplitudes backtothephysical K+ p +p 0 amplitudes. Atpresent onlyZ has
→ →p (27,1)
been calculated; for the (8,8) and (8,8) operators (which mix under renormalization) wemake
mx
the approximation Z = 0.9Z2d . A full calculation of Z for the (8,8) and (8,8) operators
ij q ij ij mx
5
D I=3/2,K pp DecayswithaNearlyPhysicalPionMass MatthewLightman
→
on the lattice is currently under way [19]. Comparing Epp for p=√2p /L with the kaon mass
m =507.4(8.5) MeV we see that the decay is nearly energy conserving, so we use the results
K
from p=√2p /LtocomputeA . ResultsforRe(A )andIm(A )forthefourdifferentkaonsource
2 2 2
times are shown in table 3. Our final result for Re(A ) and Im(A ) is an error weighted average
2 2
(EWA)overthefourkaonsourcetimes.
Table3: FinalresultsforA . Theerrorsquotedarestatisticalerrorsonly.
2
t Re(A )(unitsof10 8 GeV) Im(A )(unitsof10 13 GeV)
K 2 − 2 −
20 1.33(11) -8.11(52)
24 1.44(11) -8.77(60)
28 1.53(13) -8.58(58)
32 1.20(16) -9.01(75)
EWA 1.396(81) -8.46(45)
6. SystematicError
Themajor sources of systematic error inthe determination of A are scaling violations, finite
2
volume effects, partialquenching, uncertainty in¶d /¶ q, andthefactthatthemassesandmomen-
tumareslightly differentfromtheirphysicalvalues. Furthermore, theapproximation madeforthe
renormalization constants for the (8,8) operators introduces a large systematic error into Im(A )
2
whichwewillestimateas20%. A isverysensitivetoscalingviolations because itisproportional
2
to a 3. Weestimate this systematic by calculating Re(A ) with a lattice spacing determined from
− 2
fK, mW , and r0 respectively, and find afluctuation of 8.5% among the three values. Forfinite vol-
umeeffectsweestimate7%forthesystematic errorusing finitevolume chiralperturbation theory
for the K pp matrix elements [20, 21]. One expects that for D I =3/2 decays partial quench-
→
ing will introduce small errors and in [22] the use of partial quenching has been shown to affect
Re(A )byabout2%. Avalueof¶d /¶ qthatisratherlargerinmagnitudeisobtainedjustbyputting
2
astraightlinethroughthetwo-pionphaseshiftdatapointsfromthiscalculation infigure2(a);this
value yields a result for Re(A ) that differs by 2% which we use as our conservative estimate of
2
this systematic. Finally, a K pp calculation on 243 quenched lattices was done for a variety of
→
mesonmassesandtwo-pion energies [23],andshowsthatthedeviations oftheseparameters from
their physical values in the present calculation causes an 1.2% difference in Re(A ). Adding all
2
errorsinquadratureresultsinapreliminaryestimateof11%forthesystematicerrorinRe(A )and
2
23%forthesystematic errorinIm(A ).
2
7. Conclusions
Wehave presented preliminary results forthe D I=3/2 K pp decay amplitude on323 lat-
→
ticeswith2+1flavoursofDWFandtheIwasaki-DSDRgaugeaction. Wefindmp =141.9(2.3)MeV,
mK =507.4(8.5)MeVandEpp =489.2(8.1)MeV.ThemaincontributiontoRe(A2)isexpectedto
befromthe(27,1)operator, andourresult1.396(081) (160) 10 8 GeVcanbecomparedto
stat sys −
×
theexperimentalresultof1.5 10 8 GeVandisfoundtoagreewithinerror. Thisisthefirsttimea
−
×
calculation ofthis type hasbeen achieved. Im(A )isdominated by theoperators inthe (8,8) rep-
2
resentation, so we expect there to be a large systematic error on Im(A ) due to the approximation
2
6
D I=3/2,K pp DecayswithaNearlyPhysicalPionMass MatthewLightman
→
made for Z . This is reflected in our final answer Im(A )= 8.46(45) (1.95) 10 13 GeV.
ij 2 stat sys −
− ×
Thissourceofsystematic errorwillbeeliminatedoncetheNPRcalculation hasbeencompleted.
WethankallofourcolleaguesintheRBCandUKQCDcollaborations forhelpfuldiscussions
and the development and support of the QCDOChardware and software infrastructure which was
essentialtothiswork. InadditionweacknowledgeColumbiaUniversity,RIKEN,BNL,ANL,and
the U.S. DOE for providing the facilities on which this work was performed. This research used
resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which
issupportedbytheOfficeofScienceoftheU.S.DOEundercontractDE-AC02-06CH11357. This
work was supported in part by U.S. DOE grant number DE-FG02-92ER40699. Elaine Goode is
supported byanSTFCstudentship andgrantST/G000557/1andbyEUcontract MRTN-CT-2006-
03542(Flavianet).
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