Towards precision heavy flavour physics from lattice QCD

0.38 MB·English

by  Jochen Heitger

MS-TP-11-01
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Towards precision heavy flavour physics from lattice QCD
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Jochen Heitger

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t Westf¨alische Wilhelms-Universit¨at Mu¨nster, Institut fu¨r Theoretische Physik
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l Wilhelm-Klemm-Straße 9, D-48149 Mu¨nster, Germany
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h LPHA
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Collaboration

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Abstract
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1 I convey an idea of the significant recent progress, which opens up good perspectives for high-precision ab-initio
0 computationsinheavyflavourphysicsbasedonlatticeQCD.Thisreportfocusesonthestrategyandthechallenges
1
of fully non-perturbative investigations in the B-meson sector, where the b-quark is treated within an effective
1
theory, as followed by the ALPHA Collaboration. As an application, I outline its use to determine the b-quark
:
v mass and summarize the status of our ongoing project in the two dynamical flavour theory.
i
X
r
a
Invited talk at the
Third Workshop on Theory, Phenomenology and Experiments in Heavy Flavour Physics
July 5 – 7, 2010, in Anacapri, Capri, Italy
To appear in the Proceedings (Nucl. Phys. B Proc. Suppl.)
November 2010

Towards precision heavy flavour physics from lattice QCD
Jochen Heitgera (ALPHA Collaboration)
aWestf¨alische Wilhelms-Universit¨at Mu¨nster, Institut fu¨r Theoretische Physik,
Wilhelm-Klemm-Straße 9, D-48149 Mu¨nster, Germany
I convey an idea of the significant recent progress, which opens up good perspectives for high-precision ab-
initio computations in heavy flavour physics based on lattice QCD. This report focuses on the strategy and the
challenges of fully non-perturbative investigations in the B-meson sector, where the b-quark is treated within an
effective theory, as followed by the ALPHA Collaboration. As an application, I outline its use to determine the
b-quark mass and summarize the status of our ongoing project in the two dynamical flavour theory.
1. B-physics and lattice QCD thekeyfeaturesofthelatticeapproachisthatall
approximations can be systematically improved.
For the plenty of beautiful results from recent
For an overview of the different formulations of
andcurrentB-physicsexperiments[1,2]—aswell
heavy quarks on the lattice that have been pro-
as from what is to be expected from LHC —, to
posed in the literature and are being used to-
lead to feasible precision tests of the Standard
day, and of results from the field of heavy flavour
ModelandtrialsofseveralNewPhysicsscenarios,
physics,whichreflectsomeoftheseimprovements
requires the knowledge of QCD matrix elements
by the small error bars quoted for many quanti-
for their interpretation in terms of parameters of
ties, I refer to the reviews of past Lattice Confer-
the Standard Model and its possible extensions.
ences [3,4,5,6,7] and references therein.
Unfortunately, the uncertainty on the theoreti-
cal side in this interplay of experiment and the- 1.1. Challenges
ory in flavour physics predominantly originates Among the various considerable challenges one
from hardly computable long-distance effects of faces in an actual lattice QCD calculation on
the strong interaction that confines quarks and the theoretical and technical levels, let us only
gluonswithinhadrons. Thispotentiallylimitsthe highlight the multi-scale problem, which is also
impact of future experimental measurements on particularly relevant in view of B-physics appli-
New Physics models and motivates calculations cations. This is illustrated in Figure 1. There
in lattice QCD, which is a powerful approach
µ−1/fm
to reach a few-% theoretical error on those non- 100 10 1 0.1 0.01 0.001
perturbative hadronic contributions. ΛIR ΛMS ΛUV
u d s c b t
Still, some care is needed to obtain reliable re-
sultsforb-quarkphysicsfromaMonteCarloeval- 0.001 0.01 0.1 1 10 100
µ/GeV
uation of the discretized Euclidean path integral.
Figure 1. Large range of energy (µ) scales in lat-
Onehastokeepundercontrolsimultaneouslythe
ticeQCD,whereshadedareasrefertoquarkmass
finite-sizeeffectsand, particularly, thediscretiza-
values(intheMSscheme)quotedbytheParticle
tioneffects,sincethelatticespacingshouldnotbe
DataGroup[8]. Redmarksindicatethepion,the
larger than the Compton length of the b-quark.
D- and the B-meson mass.
In practice, it is not possible to control both ef-
fects by brute force numerical simulations such
are many disparate physical scales to be cov-
that dedicated methods have to be devised.
ered simultaneously, ranging from the lightest
While the numerical computations in lattice
hadronmassofm 140MeV overm 2GeV
QCD necessarily involve approximations, one of π ≈ D ≈
to m 5GeV, plus the ultraviolet cutoff of
B
≈
1
2
Λ = a−1 of the lattice discretization that has sition (DD) applied to QCD [18,19,20], just to
UV
tobelargecomparedtoallphysicalenergyscales name a few. In addition, low-mode deflation [21]
forthediscretizedtheorytobeanapproximation (together with chronological inverters [22]) has
tothecontinuumone. Moreover,thefinitenessof ledtoasubstantialreductionofthecriticalslow-
the linear extent of space-time, L, in a numerical ing down with the quark mass in the DD-HMC.
treatmententailsaninfraredcutoffΛ =L−1 so Finally, in parallel to the continuous increase
IR
that the following scale hierarchy is met: of computer speed (at an exponential rate) over
the last 25 years and the recent investments
Λ =L−1 m ,...,m ,m a−1 =Λ .
IR π D B UV into high performance computing at many places
(cid:28) (cid:28)
This implies L (cid:38) 4/m 6fm to suppress of the world, the Coordinated Lattice Simula-
π
≈ tions [23] (CLS) initiative is a community effort
finite-size effects in the light quark sector and
a (cid:46) 1/(2m ) 0.05fm to still properly resolve to bring together the human and computer re-
D
≈ sources of several teams in Europe interested in
the propagation of a c-quark in the heavy sector.
Lattices with L/a (cid:38) 120 sites in each direction lattice QCD. The present goal are large-volume
simulations with N = 2 dynamical quarks, us-
wouldthusbeneededtosatisfytheseconstraints, f
ing the rather simple O(a) improved Wilson ac-
and since the scale of hadrons with b-quarks was
tion to profit from the above algorithmic devel-
notevenincludedtoarriveatthisfigure,itisob-
opments such as DD-HMC, and lattice spacings
vious that the b-quark mass scale has to be sep-
a=(0.08 0.05)fm,sizesL=(2 4)fmandpion
arated from the others in a theoretically sound − −
massesdowntom =250MeV,whichaltogether
way before simulating the theory. In Section 2 I π
helptodiminishsystematicandstatisticalerrors.
briefly describe, how this is achieved by recours-
Amongst others, the B-physics programme out-
ing to an effective theory for the b-quark.
lined here is investigated on these lattices.
Anothernon-trivialtaskistherenormalization
of QCD operators composed of quark and gluon
2. Non-perturbative HQET
fields, which appear in the effective weak Hamil-
tonian,validatenergiesfarbelowtheelectroweak HeavyQuarkEffectiveTheory(HQET)atzero
scale. Besides perturbation theory (see, e.g., [9]), velocity on the lattice [24] offers a reliable solu-
powerful non-perturbative approaches have been tion to the problem of dealing with the two dis-
developed (and reviewed, e.g., in [10]), and I will parate intrinsic scales encountered in heavy-light
comebacktothenon-perturbativesubtractionof systems involving the b-quark, i.e., the lattice
power-lawdivergencesinthecontextoftheeffec- spacing a, which has to be much smaller than
tive theory for the b-quark later. 1/m to allow for a fine enough resolution of the
b
states in question, and the linear extent L of the
1.2. Perspectives
lattice volume, which has to be large enough for
As for the challenges with light quarks, it
shouldonlybenotedthattheconditionL(cid:38)6fm finite-size effects to be under control (Figure 1).
Sincetheheavyquarkmass(m )ismuchlarger
may be relaxed by simulating at unphysically b
than the other scales such as its 3–momentum or
large pion masses, combined with a subsequent
Λ 500MeV, HQET relies upon a system-
extrapolation guided by chiral perturbation the- QCD ∼
aticexpansionoftheQCDactionandcorrelation
ory [11] and its lattice-specific refinements.
functions in inverse powers of the heavy quark
RegardingthealgorithmicsideofalatticeQCD
mass around the static limit (m ). The
simulation, the Hybrid Monte Carlo [12] (HMC) b → ∞
lattice HQET action S at O(1/m ) reads:
as the first exact and still state-of-the-art algo- HQET b
rithmhasreceivedconsiderableimprovementsby a4(cid:80) ψ (cid:8)D +δm ω D2 ω σB(cid:9)ψ ,
multiple time-scale integration schemes [13,14], x h 0 − kin − spin h
theHasenbuschtrickofmass-preconditioning[15, with ψ satisfying P ψ = ψ , P = 1+γ0,
h + h h + 2
16], supplemented by a sensible tuning of the al- andtheparametersω andω beingformally
kin spin
gorithm’s parameters[17], and domaindecompo- O(1/m ). At leading order (static limit), where
b
3
theheavyquarkactsonlyasastaticcoloursource finitevolume. Applicationsofthisstrategytothe
andthelightquarksareindependentoftheheavy determination of the b-quark mass and (a subset
quark’sflavourandspin,thetheoryisexpectedto of all) HQET parameters at O(1/m ) [30,31], to
b
have 10%precision,whilethisreducesto 1% a study of the B -meson spectrum [32] and to a
s
∼ ∼
at O(1/m ) representing the interactions due to computationof theB -meson decayconstant[33]
b s
the motion and the spin of the heavy quark. As were realized in the quenched approximation by
crucial advantage (e.g., over NRQCD), HQET our collaboration and have been extended to the
treats the 1/m –corrections to the static theory more realistic N =2 situation [34,35,36,37].
b f
asspace-timeinsertionsincorrelationsfunctions.
Forcorrelationfunctionsofsomemulti-localfields 3. The b-quark mass via HQET at O(1/m )
b
andupto1/m –correctionstotheoperatorit-
b
O We first note [38] that in order not to spoil the
self (irrelevant when spectral quantities are con-
asymptotic convergence of the series, the match-
sidered), this means
ing must be done non-perturbatively — at least
= +a4(cid:88)(cid:110)ω (x) for the leading, static piece — as soon as the
stat kin kin stat
(cid:104)O(cid:105) (cid:104)O(cid:105) (cid:104)OO (cid:105) 1/m –corrections are included, since as m
x b b →∞
(cid:111) theperturbative truncationerrorfromthematch-
+ω (x) ,
spin(cid:104)OOspin (cid:105)stat ing coefficient of the static term becomes much
larger than the power corrections Λ /m of
where (cid:104)O(cid:105)stat denotes the expectation value in the HQET expansion. ∼ QCD b
the static approximation and and are
kin spin
O O
given by ψ D2ψ and ψ σBψ . In this way,
h h h h
HQETatagivenorderis(power-counting)renor-
malizable and its continuum limit well defined,
once the mass counterterm δm and the coeffi-
cientsω andω arefixednon-perturbatively
kin spin
by a matching to QCD.
Still,forlatticeHQETanditsnumericalappli-
cations to lead to precise results with controlled
systematic errors in practice, two shortcomings
had to be left behind first.
1.) The exponential growth of the noise-to-
signal ratio in static-light correlators, which is
overcomebyaclevermodificationoftheEichten- Figure 2. Idea of lattice HQET computations
Hill discretization of the static action [25]. via a non-perturbative determination of HQET
2.) As in HQET mixings among operators of parameters from small-volume QCD simulations.
different dimensions occur, the power-divergent For each fixed L , the steps are repeated at
i
additivemassrenormalizationδm g2/aalready smaller a to reach the continuum limit.
∼ 0
affects its leading order. Unless HQET is renor-
malized non-perturbatively [26], this divergence In the framework introduced in [29], match-
— and those g2/a2 arising at O(1/m ) — im- ingandrenormalizationareperformedsimultane-
∼ 0 b
plythatthecontinuumlimitdoesnotexistowing ouslyand non-perturbatively. Letushereexplain
to a remainder, which, at any finite perturbative the general strategy, illustrated in Figure 2, for
order [27,28], diverges as a 0. A general so- thesampleapplicationofcalculatingtheb-quark
→
lution to this theoretically serious problem was mass. S : Starting from a finite volume with
1
workedoutandimplementedforadetermination L 0.5fm, one chooses lattice spacings a suffi-
1
≈
of the b-quark’s mass in the static and quenched ciently smaller than 1/m such that the b-quark
b
approximationsasatestcase[29]. Itisbasedona propagatescorrectlyuptocontrollablediscretiza-
non-perturbative matching of HQET and QCD in tion errors of order a2. The relation between the
4
renormalization group invariant (RGI) and the We introduce observables Φ casted into
k=1,...,5
bare mass in QCD being known, suitable finite- a vector Φ ΦQCD, where in the continuum and
≡
volumeobservablesΦ (L ,M )canbecalculated largevolumelimits,thefirsttwoareproportional
k 1 h
as a function of the RGI heavy quark mass, M , to the meson mass and to the logarithm of the
h
and extrapolated to the continuum limit. S : decay constant, respectively, while Φ is used to
2 3
Next, the power-divergent subtractions are per- fix the counterterm of the axial current and Φ
4,5
formed non-perturbatively by a set of matching forthedeterminationofthekineticandmagnetic
conditions, in which the results obtained for Φ terms in S . The continuum extrapolations
k HQET
are equated to their representation in HQET. of Φ in the small QCD volume (L 0.5fm,
1,2 1
≈
At the same physical value of L but for reso- S in Figure 2), for nine values M M of the
1 1 h
≡
lutions L /a = O(10), the previously computed RGIheavyquarkmassfromthecharmtobeyond
1
heavy-quark mass dependence of Φ (L ,M ) in the bottom region [36], are shown in Figure 3.
k 1 h
finite-volumeQCDmaybeexploitedtodetermine
the bare parameters of HQET for a (0.025 14 0.5
0.05)fm. S3: To evolve the HQET o≈bservable−s D 12 D 0.45
C 10 C
to large volumes, where contact with some phys- QΦ18 QΦ20.4
ical input from experiment can be made, one
6 0.35
also computes them at these lattice spacings in a 4 0.3
0 5 10 15 0 5 10 15
larger volume, L2 = 2L1. The resulting relation (a/L1)2 x 10−4 (a/L1)2 x 10−4
betweenΦ (L )andΦ (L )isencodedinassoci-
k 1 k 2
Figure 3. Continuum extrapolation of the finite-
atedstepscalingfunctionsdenotedasσ . S ,S :
k 4 5
volume observables Φ and Φ , where for Φ we
By using the knowledge of Φ (L ,M ) one fixes 1 2 1
k 2 h
have included the error (cross on the left) stem-
the bare parameters of the effective theory for
mingfromtherenormalizationofthequarkmass.
a (0.05 0.1)fmsothataconnectiontolattice
≈ −
spacings is established, where large-volume ob-
servables,suchastheB-mesonmassordecaycon- When the effective theory is simulated in the
stant, can be calculated. This sequence of steps same physical volume (S2 in Figure 2), a set of
yieldsanexpressionofm ,thephysicalinput,as matching conditions for lim ΦQCD(L ,M,a),
B a→0 i 1
a function of M via the quark mass dependence
h ΦQCD(L ,M)=η(L ,a)+φ(L ,a)ω˜(M,a),
ofΦ (L ,M ),whicheventuallyisinvertedtoar- 1 1 1
k 1 h
riveatthedesiredvalueoftheRGIb-masswithin
is imposed; the r.h.s. represents the heavy quark
HQET. The whole construction is such that the
mass expansion of the ΦQCD at O(1/m ). Hav-
b
continuum limit can be taken for all pieces.
ing computed η and φ from these simulations for
different values of a, the matching equations de-
3.1. Computation of HQET parameters
terminethesetofparametersω˜(M,a);e.g.,inthe
Following the strategy sketched above and ap-
simple case of the static meson mass, and up to
pliedtothequenchedcasein[31],thedetermina-
a kinematic constant, η is the static energy, φ a
tion of the parameters of the HQET Lagrangian
constantandω thebarestaticquarkmass. After
and of the time component of the isovector axial
step scaling to L = 2L , the observables in this
2 1
currentisperformedwithintheSchr¨odingerfunc-
volume are now obtained, thanks to the parame-
tional, i.e., QCD with Dirichlet boundary condi-
ters ω(M,a) fixed by the previous step, as
tions in time and periodic ones in space, where
suitable matching observables Φk, such as finite- Φ(L2,M,0)= lim[η(L2,a)+φ(L2,a)ω˜(M,a)] ,
a→0
volume meson energies and matrix elements, can
be readily defined. Relativistic quarks are sim- and the continuum limit can be taken, since the
ulated as clover-improved Wilson fermions with powerdivergencesinHQETcancelouthere. Fig-
N =2dynamicalquarks; forthestaticquarkwe ure 4 depicts examples of corresponding contin-
f
use the so-called HYP1/2 actions [25]. uum extrapolations in the static approximation,
5
and the results for observables sensitive to the Upon chiral extrapolation in the light quark
1/m –corrections are of similar quality [37]. mass and including a conservative uncertainty in
b
thelatticescale(r =0.475(25)fm[40]),wequote
0
30 0.75
as our preliminary result for the b-quark’s mass
0.7
25 in HQET at O(1/m ) for the N =2 theory:
0.65 b f
statΦ11250 statΦ200.5.65 mbMS(mb)=4.276(25)r0(50)stat+renorm(?)aGeV.
0.5
10
0.45 The first error states the scale uncertainty, while
0 2 4 6 0 2 4 6
(a/L2)2 x 10−3 (a/L2)2 x 10−3 the second covers the statistical errors of HQET
energies,thechiralextrapolationuncertaintyand
Figure 4. Continuum extrapolation of the static
the error on the quark mass renormalization en-
approximation of Φ and Φ in the volume of ex-
1 2
tering the small-volume QCD part of the compu-
tent L . Red (blue) symbols refer to the HYP1
2
tation (S ). More details are found in [37].
(HYP2) discretization of the static propagator. 1
For comparison, we cite the previous N = 0
f
HQET result mMS(m ) = 4.320(40) (48)GeV
b b r0
Finally, the HQET parameters to be employed by our collaboration [30] and the recent sum-rule
in the large volume, L∞, are estimated from S3: determination, mbMS(mb)=4.163(16)GeV [41].
ω(M,a)=φ−1(L2,a)[Φ(L2,M,0) η(L2,a)] . 4. Outlook
−
3.2. Preliminary large-volume results The non-perturbative treatment of HQET in-
Toapplythenon-perturbativematchingresults cluding 1/mb–terms can lead to results with un-
tocalculatetheb-quarkmass, wewritedownthe precedentedprecisionforB-physicsonthelattice.
HQET expansion (to first order in 1/m ) of m It also greatly improves our confidence in the use
b B
in terms of HQET parameters and energies as of the effective theory. Our project to extract
from N = 2 lattice simulations relevant quanti-
f
m =m +Estat+ω Ekin+ω Espin. (1) ties for B-phenomenology within HQET is well
B bare kin spin
advanced. While the non-perturbative matching
HQET energies and matrix elements have been of HQET with QCD through small-volume sim-
extracted from measurements on a subset of con- ulations is almost done, the evaluation of HQET
figuration ensembles produced within CLS [23] energiesandmatrixelementshasstartedrecently
solving the Generalized Eigenvalue Problem [39], on the CLS ensembles, but still awaits a bet-
which allows for a clean quantification of sys- ter control of the cutoff effects. Our first results
tematicerrorsfromexcitedstatecontaminations. for the b-quark mass in HQET at O(1/mb) are
So far, only a single lattice spacing a 0.07fm promising, and further applications of the once
≈
(β = 5.3) has been analyzed so that the size of determined HQET parameters to calculate the
discretizationeffectscannotbeassessedyet. Fig- B-meson decay constant, the spectrum of heavy-
ure 5 shows elements of the computations in L . light mesons and the form factors of the B π
∞ →
semi-leptonic decay are expected in the future.
1,08
17
1,04 16 Acknowledgments
statE [GeV]0,961 statrm0B111345 IamindebtedtomycolleaguesinCLSandAL-
0,92 NNNNNfffff ===== 22220,,,,, mmmmmππππq ===== m54325005st0000ra nMMMMgeeeeeVVVV 1112 13 14 15 16 17 18 PBH. BAlofossriear,frJu.itBfuullacvoal,laMbo.rDateilolna,Minorptaer,tMicu.lDarotno-
0,4 0,6 0,t8 [fm] 1 1,2 z=L1M nellan, P. Fritzsch, N. Garron, G. von Hippel,
Figure 5. Left: Comparison of plateaux of static B. Leder, N. Tantalo, H. Simma and R. Som-
energies at β = 5.3 to earlier quenched results. mer in the context of our common project on B-
Right: Graphical solution of (1) in static approx-
imation; M M is the RGI heavy quark mass.
h
≡
6
physicsphenomenologyfromN =2latticesimu- 18. M. Lu¨scher, J. High Energy Phys. 05 (2003)
f
lations. WeacknowledgesupportbytheDeutsche 052, hep-lat/0304007.
Forschungsgemeinschaft in the SFB/TR 09-03, 19. M. Lu¨scher, Comput. Phys. Commun. 156
“Computational Particle Physics”, and under (2004) 209, hep-lat/0310048.
grant HE 4517/2-1, as well as by the European 20. M. Lu¨scher, Comput. Phys. Commun. 165
Community through EU Contract No. MRTN- (2005) 199, hep-lat/0409106.
CT-2006-035482, “FLAVIAnet”. We thank CLS 21. M. Lu¨scher, J. High Energy Phys. 12 (2007)
for the joint production and use of gauge con- 011, 0710.5417.
figurations [23]. Our simulations are performed 22. R.C. Brower et al., Nucl. Phys. B484 (1997)
on BlueGene, PC clusters and apeNEXT of the 353, hep-lat/9509012.
John von Neumann Institute for Computing at 23. twiki.cern.ch/twiki/bin/view/CLS/WebHome.
FZJu¨lich,atHLRN,Berlin,DESY,Zeuthen,and 24. E. Eichten and B. Hill, Phys. Lett. B234
INFN, University of Rome “Tor Vergata”. We (1990) 511.
thankfully acknowledge the computer resources 25. ALPHA, M. Della Morte, A. Shindler and R.
and support provided by these institutions. Sommer, J.HighEnergyPhys.08(2005)051,
hep-lat/0506008.
26. L. Maiani, G. Martinelli and C.T. Sachrajda,
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