Form factors in the $\mathrm{B}_\mathrm{s} \rightarrow \mathrm{K} \ell \nu$ decays using HQET and the lattice

0.2 MB·

by  Debasish Banerjee

Form factors in the B → K(cid:96)ν decays using HQET
s
and the lattice
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1
0
2 Debasish Banerjee∗

n
JohnvonNeumannInstituteforComputing(NIC),DESY,Platanenallee6,15738Zeuthen,
a
J Germany.

7 fortheALPHAcollaboration.
1 E-mail: [email protected]

]
t
a
We report on a recent computation of the form factors in semi-leptonic decays of the B using
l s
-
p HeavyQuarkEffectiveTheory(HQET)formalismappliedonthelattice. Theconnectionofthe
e
formfactorswiththe2-pointand3-pointcorrelatorsonthelatticeisexplained,andthesubsequent
h
[ non-perturbative renormalization of HQET and it’s matching to Nf =2 QCD is outlined. The

2 resultsofthe(static)leading-ordercalculationinthecontinuumlimitispresented.
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34thannualInternationalSymposiumonLatticeFieldTheory
24-30July2016
UniversityofSouthampton,UK
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
FormfactorsintheB →K(cid:96)ν decaysusingHQETandthelattice DebasishBanerjee
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1. Motivations
TestingtheconsistencyandthecorrectnessoftheStandardModelisacentralgoalofparticle
physics. The decays of the B-mesons and B-baryons can be used to improve the determination of
several poorly known matrix elements of the Cabibo-Kobayashi-Maskawa (CKM) quark mixing
matrix, and in particular |V |. The mean values of this fundamental parameter of the Standard
ub
Model extracted from inclusive decays agree with those extracted from the different exclusive
decays(suchasB→π(cid:96)ν andB→τν)onlywhenthequoteduncertaintiesarestretchedbyafactor
three [1]. It needs to be resolved whether this arises due to systematic uncertainties inherent in
differenttreatments,orduetobeyondStandardModel(BSM)physics.
We report on the determination of the form factors in the B →K(cid:96)ν decay in N =2 QCD,
s f
and the heavy quark effective theory (HQET) to account for the heavy quark on the lattice [2].
The formalism will be outlined here, especially the non-perturbative renormalization procedure
andcontinuumlimitextrapolation. Theinvariantmassoftheleptonsiskeptfixedasthecontinuum
limitistaken. Thedifferentialdecayratesatagivenq2 isrelatedtotherenormalizedformfactors
(at the same q2) and |V |. A precise experimental determination of this differential decay rate
ub
would then allow for a reliable and accurate determination of |V |, when all the errors have been
ub
systematicallyaccountedfor. Inourtheoreticalcalculationweidentifyallthepossiblesystematic
sources of error, providing a robust estimate of the exclusive observable. The complementary
proceeding[3]describesthedetailsinvolvedintheextractionofthebareformfactorsonthelattice.
On the lattice, the most challenging aspect for this reliable computation is the heavy quark
itself [4]. Due it’s mass m ∼ 5GeV being comparable to that of inverse lattice spacing of the
b
finest lattice ensembles in use today, the discretization effects due to the heavy quark are particu-
larlysevere. Atheoreticallycleanwaytoavoidthisproblemistheuseofaneffectivetheoryforthe
heavyquark. However,thestate-of-artcomputations[5,6,7,8]useeitherarelativisticheavyquark
action or a non-relativistic formulation of QCD on the lattice, which are affected by perturbative
renormalization or uncontrolled discretion effects. A fully non-perturbative programme to renor-
malize the currents does not yet exist. Conventional discretization errors are estimated only by
power-countingmethods, whiletheactualextrapolationtothecontinuumlimitwithheavyquarks
may involve a complicated dependence on the lattice spacing. Our computation seeks to address
theseissuesanddemonstrateacleanextrapolationtothecontinuumlimitoftheformfactors.
2. Methodology
Tomakecontactwithphenomenology,welistthefollowing"five-point"program:
(i) reliablecomputationofgroundstatematrixelements(cid:104)K|Vµ(0)|B (cid:105),
s
(ii) renormalizationofthe matrixelements, eitherin fullQCD,or ifaneffectivetheory isbeing
used,thenintheeffectivetheory,whichisthenmatchedtoQCD,
(iii) extrapolationoftherenormalizedquantityatfinitelatticespacingstothecontinuumlimit,
(iv) extrapolationofthe(light)quarkmassestotheirphysicalvalues,
(v) mappingouttheq2 dependenceoftheformfactors,sinceexperimentsmeasurevaluesofthe
differentialcross-section.
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FormfactorsintheB →K(cid:96)ν decaysusingHQETandthelattice DebasishBanerjee
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We employ lattice discretized HQET to compute the form factors on the lattice. While the first
challengeisthesubjectof[3],thisproceedingisconcernedwiththesecondandthirditems. Work
onthefourthpointisinprogress. Atthemoment,ourcomputationfocusesonafixedvalueofq2.
3. HQETonthelatticeandnon-perturbativerenormalization
A B state with a momentum p decays into a Kaon K with momentum p mediated by
s Bs K
thevectorcurrent,Vµ(x)=ψ¯ (x)γ ψ (x). InQCD,weparameterizethematrixelementintotwo
u µ b
equivalentform-factordecompositions:
(cid:32) (cid:33)µ
m2 −m2 m2 −m2
(cid:104)K(p )|Vµ(0)|B (p )(cid:105) = p +p − Bs Kq · f (q2)+ Bs Kqµ· f (q2)
K s Bs Bs K q2 + q2 0
= (cid:112)2m (cid:2)vµ·h (p ·v)+pµ ·h (p ·v)(cid:3) (3.1)
Bs (cid:107) K ⊥ ⊥ K
Thelastlinecanbeseenasadefinitionoftheformfactorsh andh whichwecompute. Theveloc-
(cid:107) ⊥
ity,momentumandsquaredmomentumtransfer(q2)variablesarerelatedas: vµ = pµ /m , pµ =
Bs Bs ⊥
pµ −(v·p )vµ, qµ = pµ −pµ, p ·v= m2Bs+m2K−q2. Usingnon-relativisticstatenormalizationin
K K Bs K K 2mBs
HQET removes the mass dependence present in the relativistic normalization: (cid:104)B (p(cid:48))|B (p)(cid:105)=
s s
2E(p)(2π)3δ(p−p(cid:48)). Consequently, h ,h are independent of the heavy quark mass, modulo a
⊥ (cid:107)
logarithmic dependence coming from the matching function between QCD and HQET. Note that
inthelatter,theeffectiveheavyquarkfieldsaremassindependent,andhencenoadditionaldepen-
denceoriginatesfromthevectorcurrent.
ThelatticecomputationisperformedintherestframeoftheB meson: vµ =(1,0,0,0),where
s
theQCDmatrixelementsaresimplyrelatedtotheformfactorsvia:
(2m )−1/2(cid:104)K(p )|V0(0)|B (p )(cid:105) = h (E ) (3.2)
Bs K s Bs (cid:107) K
(2m )−1/2(cid:104)K(p )|Vk(0)|B (p )(cid:105) = pkh (E ) (3.3)
Bs K s Bs K ⊥ K
TheonlykinematicvariableisE = p ·v,theenergyofthefinalstateKaon. Neglectingtermsof
K K
O(m2/m2 ,m2/q2),thedifferentialdecayratecanbedirectlyrelatedtotheformfactor:
(cid:96) Bs (cid:96)
dΓ(B →K(cid:96)ν) G2
s = F |V |2|p |3[f (q2)]2 (3.4)
dq2 24π3 ub K +
Withtheexperimentalmeasurementofthedifferentialdecayrateandareliableestimateof f (q2),
+
|V | can be accurately measured. We now discuss the renormalized form factors hstat,RGI, related
ub (cid:107),⊥
totheQCDformfactorsh viathematchingfunction(s):
(cid:107),⊥
h (E ) = C (M /Λ )hstat,RGI(E )·[1+O(1/m )] (3.5)
(cid:107),⊥ K V0,Vk b MS (cid:107),⊥ K b
Our computations use the N =2 QCD ensembles. A parallel project is non-perturbatively
f
matching the currents V between QCD and HQET, and will allow for the 1/m corrections to
0,k
be computed [12]. For now, we remain with the static order, where most of the non-perturbative
resultsareavailable. ThebarecurrentsVstat are:
0,k
←−S ←−S
Vstat=ψ¯ γ ψ +ac (g )ψ¯ ∑∇ γ ψ ; Vstat=ψ¯ γ ψ −ac (g )ψ¯ ∑∇ γ γ ψ . (3.6)
0 u 0 h V0 0 u l h k u k h Vk 0 u l k h
l l
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FormfactorsintheB →K(cid:96)ν decaysusingHQETandthelattice DebasishBanerjee
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The leading term is the same as in QCD, except that the heavy quark fields are denoted by ψ¯ ,
h
and the HYP1 and HYP2 heavy quark actions are considered [9], which have an exponentially
bettersignal-to-noiseratioascomparedtotheclassicEichten-Hillactionfortheheavyquarks. In
addition, there is the O(a) improvement term, whose coefficients are currently known to 1-loop
order: c =c(1)g2+O(g4). They are relatively small [10]: for HYP1, c =0.0223(6), c =
x x 0 0 V0 Vk
0.0029(2)whileforHYP2,c =0.0518(2), c =0.0380(6).
V0 Vk
InHQET,thecurrentsgetrenormalizedmultiplicatively. Atthestaticorder,thespinsymmetry
ofHQETresultsinthesamerenormalizationofthevectorcurrentastheaxialcurrent: Zstat,RGI =
Vk
Zstat,RGI,butanextrafactorZstat (g )forV ,asWilsonfermionsbreakchiralsymmetry:
A0 V/A 0 0
Vstat,RGI=Zstat,RGI(g )Zstat (g )Vstat; Vstat,RGI=Zstat,RGI(g )Vstat. (3.7)
0 A0 0 V/A 0 0 k A0 0 k
Thecrucialaspectofournon-perturbativerenormalizationsetupproceedsviathecalculationofthe
RGI(renormalizationgroupinvariant)quantitiesΦRGI,whicharebothschemeandscaleindepen-
dent. For a renormalized static heavy-light current, the the differential equation for the physical
quantityatdifferentscalesµ,canbeintegratedinperturbationtheory:
(ARGI) = lim[2b g¯2(µ)]−γ0/2b0(Astat) (µ) (3.8)
0 0 R 0
µ→∞
The integration constant in the left hand side is completely independent of scale of the running
coupling g¯2(µ), the renormalized current and the scheme used to compute them. γ and b are
0 0
the universal 1-loop coefficients of the running mass and the couplings. The goal therefore is to
determinetheintegrationconstant. Thenon-perturbativeanalogisexpressedas:
ZAst,aRtGI(g0)= ΦΦ(RµG)I ×ZAstat(g0,aµ)(cid:12)(cid:12)µ=2Lm1ax . (3.9)
The first universal factor relates the renormalization of Astat at a scale µ =1/L calculated in
0 0 max
a scheme to the RGI operator, while the second factor knows about the lattice discretization. The
computationofbothfactorswasdonein[11], wheretheSchrödingerfunctionalschemewasused
to compute the RGI quantity, and yielded ΦRGI/Φ(µ)=0.880(7). Finally, for our computation,
we used a a generous range [Zstat (g )]−1 = 0.97(3), motivated by comparing the value for the
V/A 0
quenchedapproximation,notingtheabsenceofN dependenceatthe1-looporder. Thisuncertainly
f
onlyaffectsthe1/m terms,andwillbeeliminatedinthenon-perturbativematchingprogram[12].
h
4. MatchingtoQCD
TherenormalizedformfactorsneedtobematchedtothatofQCDviatheconversionfunctions
C . Thesecanbeobtainedbycalculatingamatrixelementuptoagivenorderinboththeoriesand
x
then matching the expressions. As a pedagogical example, consider the matrix element M of the
renormalizedaxialcurrentinboth1-loopQCDandHQETintheleadingorder(stat). Tomakethe
theoriesagree,wematchtheexpressions[4]obtainedforQCD,withthatofHQET:
1
M (L,m ) = [1+g2(−γ ln(m L)+B )]M(0)+O(g4)+O( ), (4.1)
QCD h 0 h QCD
m L
h
M (µL) = [1+g2(−γ ln(µL)+B )]M(0)+O(g4). (4.2)
stat 0 lat
Expressedintermsoftherunningcouplingg2,andrenormalizedmass,equatingtheleadingorder
givesthemultiplicativematchingfunctionbetweenthetwotheories:C =1+g2γ ln(µ/m )+
match 0 h
(B −B )+O(g4),motivatingthethelogarithmicdependenceontheheavyquarkmass.
QCD lat
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FormfactorsintheB →K(cid:96)ν decaysusingHQETandthelattice DebasishBanerjee
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A non-perturbative matching of the currents is underway by the ALPHA collaboration [12].
Meanwhile,forourpurposes,weusethebestavailableperturbativeresults. Amajoradvantagein
ourprocedureofusingtheRGIoperatorsisthatwecandirectlyusetheresultsfromrenormalized
continuum perturbation theory [13], as opposed to other results in the literature which use bare
perturbation theory. The limitation here is on the knowledge of the running quark mass, which
haveuncertaintiesofO(α3),andtranslatestothesamelevelofuncertaintiesonourestimateofthe
s
totalZ =C ×Zstat,RGI[4]. Thisisalreadybetterbyonemoreorderinα thantheotherapproaches
x x x s
usedintheliteraturewhichhaveuncertaintiesofatleastO(α2).
s
5. Thecontinuumlimit
Wenowoutlineourresultsforthecontinuumlimitsoftheformfactors. Computationsofthe
bareformfactorsarerepeatedfordifferentlatticeensembleswithdecreasinglatticespacings. For
this, the N =2 ensembles by the Coordinated Lattice Simulations (CLS) effort [14] were used.
f
Thechosenensembles(withlabelsA5,F6andN6)allhaveroughlythesamepionmass(330,310,
340 MeV respectively), and the lattice spacing decreases like 0.0749(8), 0.0652(6) and 0.0483(4)
fmrespectively. Thespatialdimensionofthelatticessatisfym L>4andanaspectratioof2. The
π
ensembleshavedegeneratelightquarks. ThestrangequarkmassisfixedbyfixingtheKaonmass
inunitsoftheKaondecayconstanttoit’sphysicalvalueatourlightquarkmasses.
0.33
HYP1 HYP1
HYP2 3.6 HYP2
0.32
3.4
RGI√h/2 M||Bs 00..3301 RGI√h 2 M⊥ Bs 3 .32
0.29
2.8
0.28
0 0.001 0.002 0.003 0.004 0.005 0.006 0 0.001 0.002 0.003 0.004 0.005 0.006
a2 [fm2] a2 [fm2]
Figure 1: The linear extrapolation in a2 to the continuum limit of the 1-loop O(a) improved RGI form
factorshstat,RGI(left)andhstat,RGI(right).ThedifferentdiscretizationsHYP1/2areshiftedtoensurevisibility.
(cid:107) ⊥
Since the computations keep the physical momentum p =0.535 GeV fixed, flavor twisted
K
boundaryconditionsarenecessarytoimpartthesame(three-)momentumtotheKaonasthelattice
spacing varies: ψ (x+L1ˆ)=eiθψ (x). The aforementioned Kaon momentum is obtained with a
s s
lattice momentum of (1,0,0) on the finest lattice (N6). On the F6 and A5, therefore, we choose
flavor twisted boundary conditions, such that, p = (1,0,0)(2π+θ)/L. The F6 gets a larger
K
(θ/(2π)=0.350)twistascomparedtoA5(θ/(2π)=0.034)duetotherespectivelatticesize. The
B mesonisalwayskeptatrest. Forourvalueofp ,themomentumtransferredisq2=21.22GeV2
s K
onalllattices,withanerrorof0.03−0.05GeV2 duetothelatticespacing.
Whiletheextractionofthebareformfactorsisexplainedin[3],weshowhowthecontinuum
limit can be extracted with the RGI form factors. In the error analysis, the statistical correlations
4
FormfactorsintheB →K(cid:96)ν decaysusingHQETandthelattice DebasishBanerjee
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andautocorrelationsarealltakenintoaccount[15,16]. Theresultsofthecontinuumextrapolation
isshowninFig1. Alistofthevaluescanbefoundin[2]. Thecontinuumlimitisobtainedwithan
extrapolationlinearina2,withthec =c(1)g2. Whiletheh givesacompatibleestimatebyfitting
x x 0 (cid:107)
ittoaconstant(andnaturallywithsmallererrorbars),wekeepthea2 extrapolationsbecausethere
is no reason why these should disappear. Also, the O(a) corrections from perturbation theory do
not matter at the precision of our data. This gives us: hstat,RGI =0.976(41)GeV1/2 and hstat,RGI =
(cid:107) ⊥
0.876(43)GeV−1/2.
Wecannowestimatetheformfactor f usingdifferentsystematicsfor1/m suppressedterms.
+ h
Two such estimates are f and f . In the first one, all the known terms and kinematic factors
+,1 +,2
are controlled, expect in hstat,RGI (which have suppressed uncontrolled 1/m contributions), while
⊥,(cid:107) h
inthesecondone f allthe1/m termsaresystematicallydropped:
+,2 h
(cid:114) (cid:18) (cid:19)
m E 1
f = Bs (1− K )C hstat,RGI(E )+ C hstat,RGI(E ) (5.1)
+,1 2 m Vk ⊥ K m V0 (cid:107) K
Bs Bs
(cid:114)
m
f = BsC hstat,RGI(E ). (5.2)
+,2 2 Vk ⊥ K
Numerically, the values are: f = 1.77(7)(7) and f = 1.63(8)(6). Comparing these esti-
+,1 +,2
mates, wesee that theO(1/m ) termscontribute like(1− EK ), andwe have asystematic∼15%
h m
Bs
ambiguity/uncertainty due to the dropping of these terms. Thus we have a preliminary estimate:
f (21.22GeV2)= f ±0.15f =1.63(8)(6)±0.24. TheseconderrorbarsaretheO(α3)per-
+ +,2 +,2 s
turbativeuncertaintiesinthematchingfunctions. Wenotethatthissystematicerrorisexpectedto
dropto∼1−2%whenallthe1/m willbesystematicallyincluded.
h
Thequantity f isalsooftenquotedintheliterature,whichisdefinedas:
0
(cid:112)2/m (cid:18) E p2 (cid:19)
f = Bs (1− K )C hstat,RGI(E )+ K C hstat,RGI(E ) (5.3)
0 1−m2/m2 m V0 (cid:107) K m Vk ⊥ K
K Bs Bs Bs
Forthisnumber,weestimate f =0.663(3)(1). Theresultsfor f agreewellwiththoseexisting
0 +,0
intheliterature: ForFlynnetal. [6],theformfactorsextractedatourvaluesofq2are f (cid:39)1.65(10)
+
and f (cid:39)0.62(5),whileBouchardetal. [7]report f (cid:39)1.80(20)and f (cid:39)0.66(5). Giventhatour
0 + 0
calculation has a very different source of systematic uncertainty, this agreement is very necessary
forphenomenologicalapplications.
6. ConclusionandOutlook
Wereportthefirststudyofthecontinuumlimitofnon-perturbativelyrenormalizedformfac-
tors. Since the stress is on the continuum limit we have kept the momentum transfer fixed at
q2 =21.22GeV2. In the static limit, we have precise non-perturbative determinations of the RGI
formfactors. ThematchingtoQCDisperturbative,andonlyhasuncertaintiesoforderO(α3). The
s
discretizationeffectsarenotverylarge,allowingasmoothlineara2extrapolationtothecontinuum.
Ontheotherhand1/m effectsneedtoincorporated.
h
The agreement of the various methods in the extraction of the form factors increase our con-
fidence in their extraction using lattice techniques. Once the 1/m are included they will be of
h
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FormfactorsintheB →K(cid:96)ν decaysusingHQETandthelattice DebasishBanerjee
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direct phenomenological interest. This is envisaged as a direct follow-up of the project. Prelim-
inary investigations are already in progress. For connection to phenomenology, the extrapolation
to the physical light quarks will also be considered, as well as the computation at another value
of the momentum. The basic results for the continuum extrapolation presented here inspire the
confidencethatpreciselatticeresults, canbeusedtogetherwithexperimentalmeasurementsfora
reliableextractionofV inthenearfuture.
ub
References
[1] TheReviewofParticlePhysics(2016),C.Patrigiannietal.,Chin.Phys.C,40,100001(2016).
[2] F.Bahretal.,ContinuumlimitoftheleadingorderHQETformfactorinB →K(cid:96)ν decays,Phys.
s
Lett.B757(2016),473.
[3] F.Bahretal.,Extractionofthebareformfactorsforthesemi-leptonicB decays,
s
PoS(LATTICE2016)295.
[4] R.Sommer,Non-perturbativeHeavyQuarkEffectiveTheory: IntroductionandStatus,Nucl.Part.
Phys.Proc.261(2015)338.
[5] FermilabLattice,MILCcollaboration,J.A.Baileyetal.,|V |fromB→π(cid:96)ν decaysand(2+1)-flavor
ub
latticeQCD,Phys.Rev.D92(2015)014024.
[6] J.M.Flynnetal.,B→π(cid:96)ν andB →K(cid:96)ν formfactorsand|V |from2+1-flavorlatticeQCDwith
s ub
domain-walllightquarksandrelativisticheavyquarks,Phys.Rev.D91(2015),074510.
[7] C.Bouchard,G.P.Lepage,C.Monahan,H.NaandJ.Shigemitsu,B →K(cid:96)ν formfactorsfrom
s
latticeQCD,Phys.Rev.D90(2014),054506.
[8] B.Colquhoun,R.J.Dowdall,J.Koponen,C.T.H.DaviesandG.P.Lepage,B→π(cid:96)ν atzerorecoil
fromlatticeQCDwithphysicalu/dquarks,Phys.Rev.D93(2016),034502.
[9] M.DellaMorte,A.ShindlerandR.Sommer,Onlatticeactionsforstaticquarks,JHEP0508(2005)
051.
[10] A.Grimbach,D.Guazzini,F.KnechtliandF.Palombi,O(a)improvementoftheHYPstaticaxialand
vectorcurrentsatone-looporderofperturbationtheory,JHEP03(2008)039.
[11] M.DellaMorte,P.FritzschandJ.Heitger,Non-perturbativerenormalizationofthestaticaxialcurrent
intwo-flavourQCD,JHEP0702(2007)079.
[12] M.DellaMorte,J.Heitger,P.Korcyl,H.Simma,C.Wittemeier,Non-perturbativematchingofHQET
heavy-lightaxialandvectorcurrentsinN =2latticeQCD,PoS(LATTICE2016)199.
f
[13] S.Bekavacetal.,MatchingQCDandHQETheavy-lightcurrentsatthreeloops,NPB833(2010)46.
[14] P.Fritzschetal.,ThestrangequarkmassandLambdaparameteroftwoflavorQCD,NPB865(2012)
397.
[15] F.Bernardonietal.,Theb-quarkmassfromnon-perturbativeN =2HeavyQuarkEffectiveTheory
f
atO(1/m ),Phys.Lett.B730(2014)171.
h
[16] ALPHAcollaboration,U.Wolff,MonteCarloerrorswithlesserrors,Comput.Phys.Commun.156
(2004)143;
ALPHAcollaboration,S.Schaefer,R.SommerandF.Virotta,Criticalslowingdownanderror
analysisinlatticeQCDsimulations,NPB845(2011)93.
6