Nonperturbative HQET at Order $1/m$

0.24 MB·

by  Benoit Blossier

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International JournalofModernPhysicsD
(cid:13)c WorldScientificPublishingCompany
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2 Nonperturbative HQET at Order 1/m

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J BenoˆıtBlossier

0 LaboratoiredePhysiqueTh´eorique,Bˆatiment210,Universit´eParisXI,
3 F-91405OrsayCedex,France
GeorgvonHippel
] Institutfu¨rKernphysik,UniversityofMainz,D-55099Mainz,Germany
t
a
NicolasGarron
l
- SchoolofPhysicsandAstronomy,UniversityofEdinburgh,EdinburghEH93JZ,UK
p TerezaMendes∗
e
IFSC,UniversityofS˜aoPaulo,C.P.369,CEP13560-970, S˜aoCarlosSP,Brazil
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v CommunicatedbyManagingEditor
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We summarize first results for masses and decay constants of bottom-strange (pseudo-
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scalar and vector) mesons from nonperturbatively renormalized heavy-quark effective
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theory(HQET),usinglattice-QCDsimulationsinthequenched approximation.
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0 Keywords: Bphysics;latticeQCD;heavy-quarkeffective theory.
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: 1. Introduction
v
i ThestudyofBphysicsisessentialtodeterminetheflavorstructureoftheStandard
X
Model, through knowledge of the Cabibbo-Kobayashi-Maskawa(CKM) matrix de-
r
a scribing quark mixing and CP violation, which may be associated with the lack
of symmetry between matter and anti-matter in the Universe. In fact, since the
amount of baryons in the Universe predicted using the CKM mechanism is sev-
eral orders of magnitude smaller than what is observed by astronomers,extensions
of the Standard Model propose additional sources of CP violation, which must be
tested against Standard-Model predictions. B mesons provide the ideal environ-
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ment for such tests. In particular,high-precisiontheoretical inputs are needed for
hadronic matrix elements, which may be computed starting from the gauge theory
itself using numericalsimulations of lattice QCD. At present,however,it is not yet
feasible to perform simulations on lattices that can simultaneously represent the
tworelevantscalesofBphysics:thelowenergyscaleΛ ,requiringlargephysical
QCD
lattice size, and the high energy scale of the b-quark mass m , requiringvery small
b
lattice spacing a. An approximate framework is therefore needed, but one should
∗Speaker.
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2 B. Blossier etal. [Alpha Collaboration]
strive to achieve sufficiently precise results, otherwise the task of overconstraining
the parameters of the Standard Model is compromised.
Apromisingsuchframeworkistoconsider(lattice)heavy-quarkeffectivetheory
(HQET), which allows for an elegant theoretical treatment, with the possibility
of fully nonperturbative renormalization.2,3 The approach is briefly described as
follows. HQET provides a valid low-momentum description for systems with one
heavy quark, with manifest heavy-quark symmetry in the limit m . The
b
→ ∞
heavy-quarkflavorandspinsymmetriesarebrokenatfinitevaluesofm respectively
b
by kinetic and spin terms, with first-order corrections to the static Lagrangian
parametrized by ω and ω
kin spin
HQET = ψ (x)D ψ (x) ω ω , (1)
L h 0 h − kinOkin − spinOspin
where
= ψ (x)D2ψ (x), = ψ (x)σ Bψ (x). (2)
Okin h h Ospin h · h
TheseO(1/m )correctionsareincorporatedbyanexpansionofthestatisticalweight
b
in 1/m such that , are treated as insertions into static correlation func-
b kin spin
O O
tions. This guarantees the existence of a continuum limit, with results that are
independent of the regularization, provided that the renormalization be done non-
perturbatively.
As a consequence, expansions for masses and decay constants are given respec-
tively by
m = m + Estat + ω Ekin + ω Espin (3)
B bare kin spin
and
m
f B = ZHQETpstat(1 + cHQETpδA + ω pkin + ω pspin), (4)
Br 2 A A kin spin
where the parameters m and ZHQET are written as sums of a static and an
bare A
O(1/m ) term (denoted respectively with the superscripts “stat” and “1/m ” be-
b b
low), and cHQET is of order 1/m . Bare energies (Estat, etc.) and matrix elements
A b
(pstat, etc.) are computed in the numerical simulation.
The divergences (with inverse powers of a) in the above parameters are can-
celled through the nonperturbative renormalization, which is based on a matching
of HQET parameters to QCD on lattices of small physical volume — where fine
lattice spacings can be considered — and extrapolation to a large volume by the
step-scalingmethod.Suchananalysishasbeenrecentlycompletedforthequenched
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case. In particular, there are nonperturbative (quenched) determinations of the
static coefficients mstat and Zstat for HYP1 and HYP2 static-quark actions5 at
bare A
the physical b-quark mass, and similarly for the O(1/m ) parameters ω , ω ,
b kin spin
m1/mb, Z1/mb and cHQET.
bare A A
The newly determined HQET parameters are very precise (with errors of a
couple of a percent in the static case) and show the expected behavior with a.
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Nonperturbative HQETat Order 1/m 3
They are used in our calculations reported here, to perform the nonperturbative
renormalization of the (bare) observables computed in the simulation. Of course,
in order to keep a high precision, also these bare quantities have to be accurately
determined. This is accomplished by an efficient use of the generalized eigenvalue
problem(GEVP)forextractingenergylevelsE andmatrixelements,asdescribed
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below.
Asignificantsourceofsystematic errorsinthe determinationofenergylevels in
lattice simulations is the contamination from excited states in the time correlators
∞
C(t) = O(t)O(0) = n Oˆ 0 2 e−Ent (5)
h i |h | | i|
nX=1
of fields O(t) with the quantum numbers of a given bound state.
Insteadofstarting fromsimple localfields O and getting the (ground-state)en-
ergyfromaneffective-massplateauinC(t)asdefinedabove,itisthenadvantageous
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to consider all-to-all propagators and to solve, instead, the GEVP
C(t)v (t,t ) = λ (t,t )C(t )v (t,t ), (6)
n 0 n 0 0 n 0
where t>t and C(t) is now a matrix of correlators,given by
0
∞
C (t) = O (t)O (0) = e−EntΨ Ψ , i,j =1,...,N. (7)
ij i j ni nj
h i
nX=1
The chosen interpolators O are taken (hopefully) linearly independent, e.g. they
i
may be built from the smeared quark fields using N different smearing levels. The
matrix elements Ψ are defined by
ni
Ψ (Ψ ) = nOˆ 0 , mn = δ . (8)
ni n i i mn
≡ h | | i h | i
One thus computes C for the interpolator basis O from the numerical simu-
ij i
lation, then gets effective energy levels Eeff and estimates for the matrix elements
n
Ψ from the solution λ (t,t ) of the GEVP at large t. For the energies
ni n 0
1 λ (t,t )
Eeff(t,t ) log n 0 (9)
n 0 ≡ a λ (t+a,t )
n 0
it is shown7 that Eeff(t,t ) convergesexponentially as t (and fixed t ) to the
n 0 →∞ 0
true energy E . However, since the exponential falloff of higher contributions may
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be slow, it is also essential to study the convergence as a function of t in order
0
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to achieve the requiredefficiency for the method. This has been recently done, by
explicit application of (ordinary) perturbation theory to a hypothetical truncated
problemwhereonlyN levelscontribute.Thesolutioninthiscaseisexactlygivenby
thetrueenergies,andcorrectionsduetothehigherstatesaretreatedperturbatively.
We get
Eeff(t,t ) = E + ε (t,t ) (10)
n 0 n n 0
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4 B. Blossier etal. [Alpha Collaboration]
for the energies and
∞
e−Hˆt(Qˆenff(t,t0))†|0i = |ni + πnn′(t,t0)|n′i (11)
nX′=1
for the eigenstates of the Hamiltonian, which may be estimated through
ˆeff(t,t )= R (Oˆ, v (t,t )), (12)
Qn 0 n n 0
t/2
λ (t +a,t )
R = (v (t,t ), C(t)v (t,t ))−1/2 n 0 0 . (13)
n n 0 n 0
(cid:20)λ (t +2a,t )(cid:21)
n 0 0
In our analysis we see that, due to cancellations of t-independent terms in the
effective energy, the first-order corrections in ε (t,t ) are independent of t and
n 0 0
very strongly suppressed at large t. We identify two regimes: 1) for t < t/2, the
0
2nd-order corrections dominate and their exponential suppression is given by the
smallestenergygap E E ∆E betweenlevelnanditsneighboringlevels
m n m,n
| − |≡
m; and 2) for t t/2, the 1st-order corrections dominate and the suppression
0
≥
is given by the large gap ∆EN+1,n. Amplitudes πnn′(t,t0) get main contributions
from the first-order corrections. For fixed t t these are also suppressed with
0
−
∆E . Clearly, the appearance of large energy gaps in the second regime im-
N+1,n
proves convergence significantly. We therefore work with t, t combinations in this
0
regime.
A very important step of our approach is to realize that the same perturba-
tive analysis may be applied to get the 1/m corrections in the HQET correlation
b
functions mentioned previously
C (t) = Cstat(t) + ωC1/mb(t) + O(ω2), (14)
ij ij ij
where the combined O(1/m ) corrections are symbolized by the expansion param-
b
eter ω.Following the same procedure as above,we get similar exponentialsuppres-
sions (with the static energy gaps) for static and O(1/m ) terms in the effective
b
theory. We arrive at
Eneff(t,t0) = Eneff,stat(t,t0)+ωEneff,1/mb(t,t0)+O(ω2) (15)
with
Eneff,stat(t,t0)= Enstat + βnstate−∆ENsta+t1,nt+... , (16)
Eneff,1/mb(t,t0)= En1/mb + [βn1/mb − βnstatt∆EN1/+m1b,n]e−∆ENsta+t1,nt+... . (17)
and similarly for matrix elements. Preliminary results of our application of the
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methods described in this section were presented recently and are summarized in
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thenextsection.Amoredetailedversionofthisstudywillbepresentedelsewhere.
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Nonperturbative HQETat Order 1/m 5
2. Results
We carried out a study of static-light B -mesons in quenched HQET with the non-
s
perturbative parameters described in the previous section, employing the HYP1
and HYP2 lattice actions for the static quark and anO(a)-improvedWilson action
for the strange quark in the simulations. The lattices considered were of the form
L3 2Lwithperiodicboundaryconditions.WetookL 1.5fmandlatticespacings
× ≈
0.1 fm, 0.07 fm and 0.05 fm, corresponding respectively to β =6.0219, 6.2885 and
6.4956.We usedall-to-allstrange-quarkpropagatorsconstructedfromapproximate
lowmodes,with100configurations.Gaugelinksininterpolatingfieldsweresmeared
with 3 iterations of (spatial) APE smearing, whereas Gaussian smearing (8 levels)
was used for the strange-quark field. A simple γ γ structure in Dirac space was
0 5
taken for all 8 interpolating fields. Also, the local field (no smearing) was included
in order to compute the decay constant.
Theresulting(8 8)correlationmatrixmaybeconvenientlytruncatedtoanN
× ×
N oneandtheGEVPsolvedforeachN,sothatresultscanbestudiedasafunction
ofN.Wethenpickabasisfromunprojectedinterpolators,bysamplingthedifferent
smearing levels (from 1 to 7) as 1,7 , 1,4,7 ,etc. We perform fits of the various
{ }{ }
energy levels and values of N to the behavior in Eq. (16) and extract our results
from the predicted plateaus. Next, we take the continuum limit, extrapolating our
resultstoa 0.Weseethatthecorrectiontotheground-stateenergyduetoterms
→
of order 1/m , which is positive for finite a, is quite small (consistent with zero)
b
in the continuum limit. Our results for the pseudoscalar meson decay constant,
both in the static limit and including O(1/m ) corrections, are shown in terms
b
of the combination ΦHQET FPS√mPS/CPS, where CPS(M/ΛQCD) is a known
≡
matching function and ΦRGI denotes the renormalization-group-invariant matrix
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elementofthestaticaxialcurrent. Thesetwocontinuumextrapolationsareshown
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in comparison with fully relativistic heavy-light (around charm-strange) data in
Fig. 1 below. Note that, up to perturbative corrections of order α3 in C , HQET
PS
predicts a behaviorconst.+O(1/r m ) inthis graph.Surprisinglyno 1/(r m )2
0 PS 0 PS
terms are visible, even with our rather small errors.
3. Conclusions
The combined use of nonperturbatively determined HQET parameters (in action
and currents) and efficient GEVP allows us to reach precisions of a few percent
in matrix elements and of a few MeV in energy levels, even with only a moder-
ate number of configurations. The method is robust with respect to the choice of
interpolator basis. All parameters have been determined nonperturbatively and in
particular power divergences are completely subtracted. We see that HQET plus
O(1/m )correctionsatthe b-quarkmassagreeswellwithaninterpolationbetween
b
thestaticpointandthecharmregion,indicatingthatlinearityin1/mextendseven
to the charm point. A corresponding study for N =2 is in progress.
f
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6 B. Blossier etal. [Alpha Collaboration]
Fig.1. ComparisonofthecontinuumvaluesforthepseudoscalarmesondecayconstantfromFig.
4tofullyrelativisticdatainthecharmregion.Thesolidlineisalinearinterpolationbetweenthe
staticlimitandthepointsaroundthecharm-quarkmass,whichcorrespondsto 1/r0mPS≈0.2.
Acknowledgements.Thisworkis supportedby theDFG inthe SFB/TR09,and
by the EU Contract No. MRTN-CT-2006-035482,“FLAVIAnet”. T.M. thanks the
A.vonHumboldtFoundation;N.G.thankstheMICINNgrantFPA2006-05807,the
Comunidad Aut´onoma de Madrid programme HEPHACOS P-ESP-00346 and the
Consolider-Ingenio 2010 CPAN (CSD2007-00042).
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