Page 1

Spectroscopy and Decay Constants from

Nonperturbative HQET at Order 1/m

LPHA

A

Collaboration

IFT-UAM/CSIC-09-49

DESY 09-173

SFB/CPP-09-97

MKPH-T-09-26

Benoît Blossier

Laboratoire de Physique Théorique, Bâtiment 210, Université Paris XI,

F-91405 Orsay Cedex, France

E-mail: benoit.blossier@desy.de

Michele Della Morte

Institut für Kernphysik, University of Mainz, D-55099 Mainz, Germany

E-mail: morte@kph.uni-mainz.de

Nicolas Garron∗

Dpto. de Física Teórica and Instituto de Física Teórica UAM/CSIC,

Universidad Autónoma de Madrid, Cantoblanco E-28049 Madrid, Spain

E-mail: nicolas.garron@desy.de

Georg von Hippel, Tereza Mendes†‡, Hubert Simma, Rainer Sommer

NIC, DESY, Platanenallee 6, 15738 Zeuthen, Germany

E-mail: Georg.von.Hippel@desy.de, mendes@ifsc.usp.br,

hubert.simma@desy.de, Rainer.Sommer@desy.de

We carry out a thorough analysis with the GEVP method to obtain ground-state and first-excited-

state masses and decay constants of bottom-strange (pseudo-scalar and vector) mesons. This

computation is done for quenched, nonperturbatively renormalized HQET, including order 1/mb

terms. The continuum limit is obtained using three lattice spacings and two static actions.

The XXVII International Symposium on Lattice Field Theory - LAT2009

July 26-31 2009

Peking University, Beijing, China

∗Current address: School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, UK.

†Speaker.

‡Permanent address: IFSC, University of São Paulo, C.P. 369, CEP 13560-970, São Carlos SP, Brazil.

c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.

http://pos.sissa.it/

arXiv:0911.1568v2 [hep-lat] 9 Nov 2009

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Nonperturbative HQET at Order 1/mTereza Mendes

1. Introduction

It is not yet feasible to perform simulations on lattices that can simultaneously represent the

two relevant scales of B-physics: the low energy scale ΛQCD, requiring large physical lattice size,

and the high energy scale of the b-quark mass mb, requiring very small lattice spacing a.

A promising alternative is to consider (lattice) heavy-quark effective theory (HQET), which

allows for an elegant theoretical treatment, with the possibility of fully nonperturbative renormal-

ization [1] (see [2] for a review). 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 mb→ ∞. The heavy-quark flavor and spin symmetries are broken at fi-

nite values of mbrespectively by kinetic and spin terms, with first-order corrections to the static

Lagrangian parametrized by ωkinand ωspin

LHQET= ψh(x)D0ψh(x) − ωkinOkin− ωspinOspin,

(1.1)

where

Okin = ψh(x)D2ψh(x),

Ospin = ψh(x)σ ·Bψh(x).

(1.2)

These O(1/mb) corrections are incorporated by an expansion of the statistical weight in 1/mb

such that Okin, Ospinare treated as insertions into static correlation functions. This guarantees the

existence of a continuum limit, with results that are independent of the regularization, provided that

the renormalization be done nonperturbatively.

As a consequence, expansions for masses and decay constants are given respectively by

mB = mbare+ Estat+ ωkinEkin+ ωspinEspin

(1.3)

and

fB

?mB

2

= ZHQET

A

pstat(1 + cHQET

A

pδA+ ωkinpkin+ ωspinpspin),

(1.4)

where the parameters mbareand ZHQET

respectivelywiththesuperscripts“stat”and“1/mb”below), andcHQET

for unexplained notation). Bare energies (Estat, etc.) and matrix elements (pstat, etc.) are computed

in the numerical simulation.

The divergences (with inverse powers of a) in the above parameters are cancelled 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 extrapo-

lation to a large volume by the step-scaling method. Such an analysis has been recently completed

for the quenched case [4]. In particular, there are nonperturbative (quenched) determinations of

the static coefficients mstat

A

for HYP1 and HYP2 static-quark actions [5] at the physical

b-quark mass, and similarly for the O(1/mb) parameters ωkin, ωspin, m1/mb

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. 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

A

are written as sums of a static and an O(1/mb) term (denoted

A

isoforder1/mb(seee.g.[3]

bareand Zstat

bare, Z1/mb

A

and cHQET

A

.

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Nonperturbative HQET at Order 1/mTereza Mendes

accurately determined. This is accomplished by an efficient use of the generalized eigenvalue

problem (GEVP) for extracting energy levels Enand matrix elements, as described below.

A significant source of systematic errors in the determination of energy levels in lattice simu-

lations is the contamination from excited states in the time correlators

C(t) = ?O(t)O(0)? =

∞

∑

n=1

|?n|ˆO|0?|2e−Ent

(1.5)

of fields O(t) with the quantum numbers of a given bound state.

Instead of starting from simple local fields O and getting the (ground-state) energy from an

effective-mass plateau in C(t) as defined above, it is then advantageous to consider all-to-all prop-

agators [6] and to solve, instead, the GEVP

C(t)vn(t,t0) = λn(t,t0)C(t0)vn(t,t0),

(1.6)

where t >t0andC(t) is now a matrix of correlators, given by

Cij(t) = ?Oi(t)Oj(0)? =

∞

∑

n=1

e−EntΨniΨnj,

i, j = 1,...,N.

(1.7)

The chosen interpolators Oiare taken (hopefully) linearly independent, e.g. they may be built from

the smeared quark fields using N different smearing levels. The matrix elements Ψniare defined by

Ψni ≡ (Ψn)i = ?n|ˆOi|0? ,

One thus computes Cijfor the interpolator basis Oifrom the numerical simulation, then gets

effective energy levels Eeff

the GEVP at large t. For the energies

?m|n? = δmn.

(1.8)

n and estimates for the matrix elements Ψnifrom the solution λn(t,t0) of

Eeff

n(t,t0) ≡1

alog

λn(t,t0)

λn(t +a,t0)

(1.9)

it is shown [7] that Eeff

However, since the exponential falloff of higher contributions may be slow, it is also essential to

study the convergence as a function of t0in order to achieve the required efficiency for the method.

This has been done in [3], by explicit application of (ordinary) perturbation theory to a hypothetical

truncated problem where only N levels contribute. The solution in this case is exactly given by the

true energies, and corrections due to the higher states are treated perturbatively. We get

n(t,t0) converges exponentially as t → ∞ (and fixed t0) to the true energy En.

Eeff

n(t,t0) = En+ εn(t,t0)

(1.10)

for the energies and

e−ˆ Ht(ˆ

Qeff

n(t,t0))†|0? = |n? +

∞

∑

n?=1

πnn?(t,t0)|n??

(1.11)

for the eigenstates of the Hamiltonian, which may be estimated through1

ˆ

Qeff

n(t,t0) = Rn(ˆO, vn(t,t0)),

(1.12)

Rn= (vn(t,t0),C(t)vn(t,t0))−1/2

?λn(t0+a,t0)

λn(t0+2a,t0)

?t/2

.

(1.13)

1The choice of Rnwe make here leads to smaller statistical errors than the one in [3], while the form of the correc-

tions π remains unchanged.

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Nonperturbative HQET at Order 1/mTereza Mendes

In our analysis we see that, due to cancellations of t-independent terms in the effective energy,

the first-order corrections in εn(t,t0) are independent of t0and very strongly suppressed at large

t. We identify two regimes: 1) for t0< t/2, the 2nd-order corrections dominate and their expo-

nential suppression is given by the smallest energy gap |Em−En| ≡ ∆Em,nbetween level n and its

neighboring levels m; and 2) fort0≥ t/2, the 1st-order corrections dominate and the suppression is

given by the large gap ∆EN+1,n. Amplitudes πnn?(t,t0) get main contributions from the first-order

corrections. For fixed t −t0these are also suppressed with ∆EN+1,n. Clearly, the appearance of

large energy gaps in the second regime improves convergence significantly. We therefore work

with t, t0combinations in this regime.

A very important step of our approach is to realize that the same perturbative analysis may be

applied to get the 1/mbcorrections in the HQET correlation functions mentioned previously

Cij(t) = Cstat

ij(t) + ωC1/mb

ij

(t) + O(ω2),

(1.14)

wherethecombinedO(1/mb)correctionsaresymbolizedbytheexpansionparameterω. Following

the same procedure as above, we get similar exponential suppressions (with the static energy gaps)

for static and O(1/mb) terms in the effective theory. We arrive at

Eeff

n(t,t0) = Eeff,stat

n

(t,t0)+ωEeff,1/mb

n

(t,t0)+O(ω2)

(1.15)

with

Eeff,stat

n

(t,t0) = Estat

n

+ βstat

n

e−∆Estat

N+1,nt+... ,

(1.16)

Eeff,1/mb

n

(t,t0) = E1/mb

n

+ [β1/mb

n

− βstat

n

t ∆E1/mb

N+1,n]e−∆Estat

N+1,nt+... .

(1.17)

and similarly for matrix elements.

Preliminary results of our application of the methods described in this section are presented

next. A more detailed version of this study will be presented elsewhere [8].

2. Results

We carried out a study of static-light Bs-mesons in quenched HQET with the nonperturbative

parameters described in the previous section, employing the HYP1 and HYP2 lattice actions for

the static quark and an O(a)-improved Wilson action for the strange quark in the simulations. The

lattices considered were of the form L3×2L with periodic boundary conditions. We took L ≈ 1.5

fm and lattice spacings 0.1 fm, 0.07 fm and 0.05 fm, corresponding respectively to β = 6.0219,

6.2885and6.4956. Weusedall-to-allstrange-quarkpropagatorsconstructedfromapproximatelow

modes, with 100 configurations. Gauge links in interpolating fields were smeared with 3 iterations

of (spatial) APE smearing, whereas Gaussian smearing (8 levels) was used for the strange-quark

field. A simple γ0γ5structure in Dirac space was taken for all 8 interpolating fields. Also, the local

field (no smearing) was included in order to compute the decay constant.

The resulting (8×8) correlation matrix may be conveniently truncated to an N ×N one and

the GEVP solved for each N, so that results can be studied as a function of N. We have considered

4

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Nonperturbative HQET at Order 1/mTereza Mendes

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

0.2 0.4 0.6 0.8 1 1.2

a E1

stat(t,t0)

t (fm)

E1

stat(t,4a), from 2x2

stat(t,5a), from 2x2 (shifted)

E1

E1

E1

plateau at 0.303

E1

stat(t,4a), from 3x3

stat(t,4a), from 4x4

stat(t,4a), from 5x5

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.2 0.3 0.4 0.5 0.6

t (fm)

0.7 0.8 0.9 1

a E2

stat(t,t0)

E2

stat(t,4a), from 2x2

stat(t,5a), from 2x2 (shifted)

E2

E2

E2

plateau at 0.485

E2

stat(t,4a), from 3x3

stat(t,4a), from 4x4

stat(t,4a), from 5x5

Figure 1: Static energy spectrum obtained from the first basis of interpolators for the case of HYP2 action

and β ≈ 6.3: ground state (left) and first excited state (right).

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

0.2 0.4 0.6 0.8 1 1.2

a E1

stat(t,t0)

t (fm)

E1

stat(t,4a), from 2x2

stat(t,5a), from 2x2 (shifted)

E1

E1

E1

plateau at 0.303

E1

stat(t,4a), from 3x3

stat(t,4a), from 4x4

stat(t,4a), from 5x5

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.2 0.3 0.4 0.5 0.6

t (fm)

0.7 0.8 0.9 1

a E2

stat(t,t0)

E2

stat(t,4a), from 2x2

stat(t,5a), from 2x2 (shifted)

E2

E2

E2

plateau at 0.480

E2

stat(t,4a), from 3x3

stat(t,4a), from 4x4

stat(t,4a), from 5x5

Figure 2: Static energy spectrum obtained from the second basis of interpolators for the case of HYP2

action and β ≈ 6.3: ground state (left) and first excited state (right).

two bases of interpolators. The first basis was obtained by projecting (at ti≈ 0.2 fm) with the N

eigenvectors of C(ti) with the largest eigenvalues

C(ti)bn= λnbn

⇒

C(N×N)

nm

(t) = b†

nC(t)bm,

n,m ≤ N.

(2.1)

For N not too large, this avoids numerical instabilities and large statistical errors in the GEVP. This

basis was used in [3], where also the normalization of the different smeared fields is specified.

Note that the (relative) normalization does matter in Eq. (2.1). The second basis was picked from

unprojected interpolators, sampling the different smearing levels (from 1 to 7) as {1,7},{1,4,7},

etc. Our results for the effective energies using the two bases are shown in Figs. 1 and 2, where

the solid lines correspond to a simultaneous fit of the various energy levels and values of N to the

behavior2in Eq. (1.16). In both cases, we observe the predicted t0independence in the GEVP

solution for the energies. We see a much stronger dependence on N for the ground state in the first

case. Nevertheless, our final results remain unchanged within their errors, which are determined as

described in [3].

We then take the continuum limit, as shown in Figs. 3 and 4. We see that the correction to

the ground-state energy due to terms of order 1/mb, which is positive for finite a, is quite small

2For N = 2 care must be taken, since the subleading corrections may not be negligible in the available t interval.

This happens for our first basis above. In any case, we do not include data from N = 2 in our analysis.

5

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Nonperturbative HQET at Order 1/mTereza Mendes

0.0000.0020.0040.0060.008

a2[fm2]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

r0∆En,1

HYP1

HYP2

Figure 3: Continuum extrapolation (joint limit for HYP1, HYP2) of the energy splittings (in MeV). Shown

are ∆E21(lower two curves, respectively the static and the full values) and ∆E31(static, uppermost curve).

0.0000.0020.004

a2[fm2]

0.0060.008

1.5

1.6

1.7

1.8

1.9

2.0

2.1

r3/2

0ΦRGI

HYP1

HYP2

0.0000.0020.004

a2[fm2]

0.0060.008

1.5

1.6

1.7

1.8

1.9

2.0

2.1

r3/2

0ΦHQET

HYP1

HYP2

Figure 4: Continuum extrapolation (joint limit for HYP1, HYP2) of the pseudoscalar meson decay constant

in the static limit (left) and to order 1/mb(right).

(consistent with zero) in the continuum limit. Our results for the pseudoscalar meson decay con-

stant, both in the static limit and including O(1/mb) corrections, are shown in terms of the com-

bination ΦHQET≡ FPS√mPS/CPS, where CPS(M/ΛQCD) is a known matching function and ΦRGI

denotes the renormalization-group-invariant matrix element of the static axial current [9]. These

two continuum extrapolations are shown in comparison with fully relativistic heavy-light (around

charm-strange) data from [9] in Fig. 5 below. Note that, up to perturbative corrections of order α3

in CPS, HQET predicts a behavior const.+O(1/r0mPS) in this graph. Surprisingly no 1/(r0mPS)2

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

6

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Nonperturbative HQET at Order 1/mTereza Mendes

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.05 0.1 0.15 0.2 0.25 0.3

1/(r0 mPS)

r0

1/2 / CPS

1/2 / CPS

3/2 φRGI

r0

3/2 FPS mPS

3/2 FBs mBs

r0

Figure 5: Comparison of the continuum values for the pseudoscalar meson decay constant from Fig. 4 to

fully relativistic data in the charm region. The solid line is a linear interpolation between the static limit and

the points around the charm-quark mass, which corresponds to 1/r0mPS≈ 0.2.

few MeV in energy levels, even with only a moderate number of configurations. The method is

robust with respect to the choice of interpolator basis. All parameters have been determined non-

perturbatively and in particular power divergences are completely subtracted. We see that HQET

plus O(1/mb) corrections at the b-quark mass agrees well with an interpolation between the static

point and the charm region, indicating that linearity in 1/m extends even to the charm point. A

corresponding study for Nf= 2 is in progress.

Acknowledgements. This work is supported by the DFG in the SFB/TR 09, and by the EU Con-

tract No. MRTN-CT-2006-035482, “FLAVIAnet”. T.M. thanks the A. von Humboldt Foundation;

N.G. thanks the MICINN grant FPA2006-05807, the Comunidad Autónoma de Madrid programme

HEPHACOS P-ESP-00346 and the Consolider-Ingenio 2010 CPAN (CSD2007-00042).

References

[1] J. Heitger and R. Sommer [ALPHA Collaboration], JHEP 0402, 022 (2004) [arXiv:hep-lat/0310035].

[2] R. Sommer, arXiv:hep-lat/0611020.

[3] B. Blossier et al., JHEP 0904, 094 (2009) [arXiv:0902.1265 [hep-lat]]; PoS LATTICE2008, 135

(2008) [arXiv:0808.1017 [hep-lat]].

[4] B. Blossier, M. Della Morte, N. Garron and R. Sommer, to appear.

[5] M. Della Morte et al., Phys. Lett. B 581, 93 (2004) [Erratum-ibid. B 612, 313 (2005)]

[arXiv:hep-lat/0307021].

[6] J. Foley et al., Comput. Phys. Commun. 172, 145 (2005) [arXiv:hep-lat/0505023].

[7] M. Lüscher and U. Wolff, Nucl. Phys. B 339, 222 (1990).

[8] B. Blossier et al., in preparation.

[9] M. Della Morte et al., JHEP 0802, 078 (2008) [arXiv:0710.2201 [hep-lat]].

7