D-Meson Mixing in 2+1-Flavor Lattice QCD

0.49 MB·

by  Chia Cheng Chang

DDD-Meson Mixing in 2+1-Flavor Lattice QCD
Chia Cheng Changabc , C. M. Bouchardc†, A. X. El-Khadraab, E. Freelandd, E. Gámize,
∗
A. S. Kronfeld‡bf, J. W. Laihog, E. T. Neilhi, J. N. Simoneb, and R. S. Van de Waterb
aDepartmentofPhysics,UniversityofIllinois,Urbana,Illinois,61801,USA
7
bFermiNationalAcceleratorLaboratory,Batavia,Illinois,60510,USA
1
0 cPhysicsDepartment,TheCollegeofWilliamandMary,Williamsburg,Virginia,23185,USA
2 dLiberalArtsDepartment,SchooloftheArtInstituteofChicago,Chicago,Illinois,60603,USA

n eCAFPEandDepartamentodeFísicaTeóricaydelCosmos,UniversidaddeGranada,
a
18071Granada,Spain
J
fInstituteforAdvancedStudy,TechnischeUniversitätMünchen,85748Garching,Germany
0
2 gDepartmentofPhysics,SyracuseUniversity,Syracuse,NewYork13244,USA
hDepartmentofPhysics,UniversityofColorado,Boulder,Colorado80309,USA
]
t iRIKEN-BNLResearchCenter,BrookhavenNationalLaboratory,Upton,NewYork11973,USA
a
l E-mail: [email protected],[email protected]
-
p
Fermilab Lattice and MILC Collaborations
e
h
[
WepresentresultsforneutralD-mesonmixingin2+1-flavorlatticeQCD.Wecomputethematrix

1
elementsforallfiveoperatorsthatcontributetoDmixingatshortdistances,includingthosethat
v
6 only arise beyond the Standard Model. Our results have an uncertainty similar to those of the
1
ETM collaboration (with 2 and with 2+1+1 flavors). This work shares many features with a
9
5 recentpublicationonBmixingandwithongoingworkonheavy-lightdecayconstantsfromthe
0
FermilabLatticeandMILCCollaborations.
.
1
0
7
1
:
v
i
X
r
a
34thannualInternationalSymposiumonLatticeFieldTheory
24-30July2016
UniversityofSouthampton,UK
Presentaddress:LawrenceBerkeleyNationalLaboratory,Berkeley,California,94720,USA
∗
†Presentaddress:SchoolofPhysicsandAstronomy,UniversityofGlasgow,GlasgowG128QQ,UK
‡Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
D-MesonMixing A.S.Kronfeld
1. Introduction
TheseproceedingscontainastatusupdateofanongoingcalculationofD0-D¯0 mixingmatrix
elements [1], similar to our published work on B0-B¯0 mixing [2]. We present nearly final results
for all five matrix elements, sufficient to describe D0-D¯0 mixing not only in the Standard Model,
butalsoinanyhigh-energyextensionthatmodifiesonlythelocal∆C=2interaction.
IntheStandardModel,neutral-mesonmixingismediatedbyone-loop,GIM-suppressedpro-
cesses, shown in Fig. 1. In extensions of the Standard Model, other particles could appear in the
boxes; there could even be tree-level flavor-changing neutral currents. Mixing has been observed
in all four neutral-meson systems—K0, D0, B0, and B0—but the pattern of internal quark masses
s
andCKMfactorsexplainswhythephenomenologydifferssogreatlyfromonesystemtoanother.
BecausetheW bosonsandbquarkshavemasseswellabovetheQCDscale,mixingcanbere-
expressedasstemmingbothfromalocal∆C=2interactionandtwo∆C=1interactionsseparated
by a distance of order 1/Λ . From degenerate perturbation theory, the off-diagonal term in the
QCD
mass-widthmatrixis[3]
i D0 L∆C=1 n n L∆C=1 D¯0
M Γ ∝ D0 L∆C=2 D¯0 +∑(cid:104) | | (cid:105)(cid:104) | | (cid:105). (1.1)
12−2 12 (cid:104) | | (cid:105) M E +i0+
n D− n
Thesecondtermisverydifficulttoestimate. ForD0 mesonsitisalsonotnegligible,unlikeforB0
andB0,wheret,c,anduquarksappearinthebox. (Forkaons,thesecondtermisimportantbutnot
s
dominant.) Onecanrelatethemeasuredmassandwidthdifferences, ∆M and∆Γ, to M , Γ ,
12 12
| | | |
and the relative phase arg(Γ /M ) [4]. In some extensions of the Standard Model, only the first
12 12
termand,thus,M isaltered[5].
12
The effective Lagrangian L∆C=2 (at energies below the b-quark mass) is built out of the fol-
lowingoperators(andtheirWilsoncoefficients)[6,7,8]:
O =c¯γµLuc¯γ Lu, O˜ =c¯γµRuc¯γ Ru, (1.2)
1 µ 1 µ
O =c¯Luc¯Lu, O˜ =c¯Ruc¯Ru, (1.3)
2 2
O =c¯αLuβc¯βLuα, O˜ =c¯αRuβc¯βRuα, (1.4)
3 3
O =c¯Luc¯Ru, (1.5)
4
O =c¯αLuβc¯βRuα, (1.6)
5
where L (R) denotes a left-(right-)handed projector on the Dirac indices, and α and β are color
indices. By parity conservation in QCD, D0 O˜ D¯0 = D0 O D¯0 , i = 1, 2, 3. Thus, the five
i i
(cid:104) | | (cid:105) (cid:104) | | (cid:105)
matrix elements D0 O D¯0 , i=1,...5, suffice to describe the short-distance part of all ∆C =2
i
(cid:104) | | (cid:105)
processes,whethertheiroriginisW-bboxorsomethingelse. Intheseproceedings,wereportona
calculationofallfivematrixelementsusinglatticeQCDwith2+1flavorsofseaquarks.
W∓ d,s,b
u c u c
d¯,s¯,¯b
W W
d,s,b ± ∓
c¯ W u¯ c¯ d¯,s¯,¯b u¯
±
Figure1: BoxdiagramsmediatingD0-D¯0mixingintheStandardModel.
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D-MesonMixing A.S.Kronfeld
2. Lattice-QCDcalculation
Our D-meson calculations have much in common with our published B-meson work [2]. We
use the same ensembles (generated by the MILC collaboration) with 2+1 flavors of sea quark [9].
The light quarks (valence and sea) are based on the staggered asqtad action; the heavy c (or b)
quark on the Fermilab interpretation of the clover action. The lattice spacings for the ensembles
satisfya 0.045fm, 0.06fm, 0.09fm,and 0.12fm. Thesea-quarkmassesyieldpionswith
≈ ≈ ≈ ≈
177MeV(cid:46) M (cid:46)555MeV, (2.1)
π
257MeV(cid:46) Mrms (cid:46)670MeV, (2.2)
π
Theensemblescontain600–2200gauge-fieldconfigurations,andweuse4or8sources/config.
Tocarryoutthechiral-continuumextrapolation,wetakeintoaccountthesubtlewayinwhich
spin emerges for staggered fermions with staggered-Wilson four-fermion lattice operators. The
three-point correlation function, it turns out, contains contributions not only from the continuum-
limitoperatorofdesiredspin,butalsosomeofthewrongspin[10]. Becauseonlythefiveoperators
in Eqs. (1.2)–(1.6) can arise, we automatically have the information needed to disentangle this
effect. Weusetheone-loopchiral-perturbation-theoryformulasofRef.[10]toremovethewrong-
spincontributioninthecourseofourchiral-continuumfit.
The operators in Eqs. (1.2)–(1.6) require renormalization for any ultraviolet regulator. We
carry out the renormalization of the lattice operators corresponding to Eqs. (1.2)–(1.6) together
withmatchingtoMSschemesincontinuumQCD.Weuseamostlynonperturbativemethodtohan-
dle the largest lattice-to-continuum matching corrections [11, 12], supplemented with a one-loop
calculation of the remaining, small renormalization parts [13, 2]. We choose the renormalization
scaleforD-mesonmatrixelementstobe3GeV,whilewechosem forB mesons.
b (s)
ThemaindifferencebetweenourworkonDvs.B mesonsistheanalysisofthecorrelation
(s)
functions. The signal-to-noise ratio is much better for D-meson correlators. For the two-point
correlators, the optimal time ranget (cid:46)t (cid:46)t differs: t 0.7(0.2) fm,t 3.0(2.4) fm
min max min max
≈ ≈
forD(B )mesons. Thedifferenceforthethree-pointcorrelatorsismorestriking. Wefixthefour-
(s)
quark operators att =0 andthe mesoncreation (annihilation) operator attimet <0 (t >0). As
x y
0 30
5 25
10
20
15 or]
tyty 15err ty
20 %
10[
25
30 5
35 0
0 5 10 15 20 25 30 35
|t |
|t|t|| x
xx
Figure2: Fittingrangesforthree-pointcorrelators: triangular(green)and-orfan-shaped(magenta)regions
for B mixing (left); two-strip diagonal region (green) for D mixing (right). Background color shows the
signal-to-noiseratiofromgood(blue)tobad(red).
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D-MesonMixing A.S.Kronfeld
showninFig.2,weuseatriangularand-orfan-shapedregioninthe t -t planeforB mesons[2],
x y (s)
| |
whileweusealongdiagonalofwidth2forDmixing, t =t t =t +1 . Thelongdiagonal
x y x y
{| | }∪{| | }
makesiteasiertodisentanglethelowest-lying state, ifthesignalpersiststhatfar. Asimultaneous
fittotwo-andthree-pointfunctionsisusedtoextractthematrixelements O D0 O D¯0 .
i i
(cid:104) (cid:105)≡(cid:104) | | (cid:105)
3. Chiral-continuumextrapolation
Tocarryoutthechiral-continuumextrapolation,wedevelopafitfunctionbasedonchiralper-
turbationtheory(χPT),Symanzikeffectivefieldtheory,andheavy-quarkeffectivetheory(HQET).
Ittakestheform
F =Flogs+Fanalytic+FHQdisc+Fαsa2 gen+Frenorm+Fκ, (3.1)
i i i i i i i
whereFlogsdenotesthenext-to-leadingorderdescriptionfromheavy-mesonrootedstaggeredχPT,
i
withnonanalytictermsincludingthosethatdisentanglethewrong-spincontributions[10];Fanalytic
i
is a polynomial of various terms that arise in χPT at next-to-leading or higher order; FHQdisc
i
describesheavy-quarkdiscretizationeffectsusingHQETasatheoryofcutoffeffects[12];Fαsa2 gen
i
parametrizesgenericcutoffeffectsoflightquarksandgluons,àlaSymanzik;andFrenormallowsthe
i
fittobesensitivetohigherordersinα formatchingandrenormalization. Finally,Fκ incorporates
s i
acorrectionfortuningthecharm-quarkhoppingparameterκ,basedonextrarunsata 0.12fm.
≈
Figure 3: Stability of the chiral-continuum extrapolation for several variants of the fit function F:
i
O (left),minimizedχ2 /dof(right). Stabilityplotsfortheother O looksimilar.
(cid:104) 1(cid:105) aug (cid:104) i(cid:105)
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D-MesonMixing A.S.Kronfeld
BBGLN[16] O /M (cid:0)GeV3(cid:1) f2 B (cid:0)GeV2(cid:1)
(cid:104) i(cid:105) D Bq Bq
q=d q=s
O 0.0432(29)(9) 0.0342(29)(7) 0.0498(30)(10)
1
O 0.0833(38)(17) 0.0303(27)(6) 0.0449(29)(9)
2
−
O 0.0248(16)(5) 0.0399(77)(8) 0.0571(77)(11)
3
O 0.1469(69)(30) 0.0390(28)(8) 0.0534(30)(11)
4
O 0.0554(38)(11) 0.0361(35)(7) 0.0493(36)(10)
5
µ 3GeV m m
b b
Table1: ResultsforD[thiswork]andB[2]mixingintherenormalizationschemeofRef.[16].
Both the renormalization and wrong-spin effects mix operators 1, 2, and 3 with each other,
and also 4 and 5 with each other. It is thus natural to fit the matrix elements in each sector si-
multaneously. Some ingredients in Flogs are common for all i, such as masses, f , light-meson
i π
χPT constants [14], and the D -D-π coupling. We introduce these external inputs with Gaussian
∗
priors, for example g =0.53 0.8. Because of these common ingredients, we choose to fit
D Dπ
∗ ±
allfivematrixelementssimultaneously. Weforma χ2 functionfromF O¯ andthesampleco-
i i
−(cid:104) (cid:105)
variancematrixofthe O¯ ,whereO¯ denotestherenormalizedlatticeoperators(whichdifferfrom
i i
(cid:104) (cid:105)
the continuum O by discretization effects and higher-order matching effects). We then augment
i
this χ2 withGaussianpriorsforthefitparametersimpliedinEq.(3.1),choosingacentralvalueof
0 and width of 1 in natural units for χPT and HQET [15] and minimize the resulting χ2 . We
aug
±
reconstitutethefitfunctionatzerolatticespacingandphysicalquarkmassestoobtainourestimate
ofthe O andtheiruncertainty.
i
(cid:104) (cid:105)
We have 510 data points for O¯ , ranging over the ensembles, valence-quark masses, and
i
(cid:104) (cid:105)
five operators. In our base version of F, there are 127 parameters. To check whether the final
i
results are robust, we repeat the procedure with several variants of F, as illustrated in Fig. 3.
i
We express the χPT with f instead of f ; we choose different orders of α in Frenorm and even
K π s i
replacethemostlynonperturbative(mNPR)matchingwithafullyperturbative(PT)one;wecheck
various alternatives for the polynomial Fanalytic (NLO, NNLO, N3LO); we check what happens
i
when the χPT prior widths in Fanalytic are doubled; we check alternatives for the heavy-quark
i
discretization errors; we substitute infinite-volume one-loop integrals for the finite-volume sums
in one-loop χPT; we omit the data from the coarsest or finest lattice spacing; we fit each matrix
element separately, thereby ignoring data constraints on wrong-spin contributions. As one can
see from Fig. 3, the results for the O are very stable, so we take these variations in the fit as
1
(cid:104) (cid:105)
crosschecks. Thesameappliestotheother O . Thelargestdeviationsare 1σ andcomefrom
i
(cid:104) (cid:105) ∼
fits that omit important information. Our nearly final results for D mixing are given in Table 1,
together with published results for B mixing from Ref. [2]. These matrix elements (as noted
(s)
above) depend on the renormalization scheme; the tabulated results are in the MS scheme with
naive (fully commuting) γ5 and the evanescent-operator basis used by Beneke, Buchalla, Greub,
Lenz,andNierste(BBGLN)[16].
The MILC asqtad ensembles omit the charmed-quark sea. As in Ref. [2], we assign an ad-
ditional 2% uncertainty to account for this omission. This uncertainty is given separately, in the
secondsetofparentheses,inTable1.
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D-MesonMixing A.S.Kronfeld
4. Outlook
Ourresultsagreewellwithandhavesimilaruncertaintyaspreviouslattice-QCDresultsfrom
theETMcollaboration,with2[17]or2+1+1[18]flavorsinthesea. Thecomparisonoftheseresults
testsnotonlytheflavor-dependenceofthematrixelementsbutalsothesensitivitytolatticefermion
formulation: ETM employs twisted-mass Wilson fermions, while we employ staggered fermions.
Allthesecalculationsuseseverallatticespacingsandtakethecontinuumlimit. References[17,18]
report the so-called “bag factors” often used in phenomenology [7]; a detailed comparison would
require choices of quark masses and decay constants (and their uncertainties) that would obscure
the error budget of one or the other set of results. We have a set of calculations underway [19]
tocomputetheD-andB -mesondecayconstantsonthesameensemblesandwillreportthebag
(s)
factorsthen.
EstimatesofthecontributiontoM ofthesecondterminEq.(1.1)rangeover(10 3–10 2)Γ
12 − −
[20], where Γ is the total width of the neutral D meson. It turns out, however, that all Standard-
Model phases appearing in Eq. (1.1) are small. Thus, in a TeV-scale model that might produce a
large phase in M , the results for the O can be used to constrain the model’s parameters. Fur-
12 i
(cid:104) (cid:105)
thermore,untilamethodisdevelopedtotamethesecondterminEq.(1.1),theaccuracyachieved
inthisworkandRefs.[17,18]shouldsufficeforthispurpose.
Acknowledgments
We thank our collaborators in the Fermilab Lattice and MILC Collaborations. Computations
forthisworkwerecarriedoutwithresourcesprovidedbytheUSQCDCollaboration,theNational
Energy Research Scientific Computing Center, and the Argonne Leadership Computing Facility,
which are funded by the Office of Science of the U.S. Department of Energy; and with resources
providedbytheNationalInstituteforComputationalScienceandtheTexasAdvancedComputing
Center, which are funded through the National Science Foundation’s Teragrid/XSEDE Program.
This work was supported in part by the U.S. Department of Energy under grants No. DE-FG02-
13ER42001(C.C.C.,A.X.K.),No.DE-SC0015655(A.X.K.),No.DE-SC0010005(E.T.N.);bythe
U.S.NationalScienceFoundationundergrantPHY14-17805(J.L.);bytheFermilabFellowshipin
Theoretical Physics (C.M.B., C.C.C.); by the URA Visiting Scholars’ program (C.M.B., C.C.C.,
A.X.K.); by the MICINN (Spain) under grant FPA2010-16696 (E.G.); by the Junta de Andalucía
(Spain) under Grants No. FQM-101 and No. FQM-6552 (E.G.); by the European Commission
(EC)underGrantNo.PCIG10-GA-2011-303781(E.G.); bytheGermanExcellenceInitiativeand
the European Union Seventh Framework Program under grant agreement No. 291763 as well as
the European Union’s Marie Curie COFUND program (A.S.K.). Fermilab is operated by Fermi
ResearchAlliance,LLC,underContractNo.DE-AC02-07CH11359withtheUnitedStatesDepart-
mentofEnergy. BrookhavenNationalLaboratoryissupportedbytheUnitedStatesDepartmentof
EnergyundercontractNo.DE-SC0012704.
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