We present a precise lattice QCD determination of the b-quark mass, of the B
and Bs decay constants and first preliminary results for the B-mesons bag
parameter. Simulations are performed with Nf = 2 Wilson twisted mass fermions
at four values of the lattice spacing and the results are extrapolated to the
continuum limit. Our calculation benefits from the use of improved
interpolating operators for the B-mesons and employs the so-called ratio
method. The latter allows a controlled interpolation at the b-quark mass
between the relativistic data around and above the charm quark mass and the
exactly known static limit.Comment: 7 pages, 4 figures, 1 table. Proceedings of the 30th International
  Symposium on Lattice Field Theory - Lattice 2012; June 24-29, 2012; Cairns,
  Australi

Carrasco, N

Dimopoulos, P

Frezzotti, R

Gimenez, V

Herdoiza, G

Lubicz, V

Martinelli, G

Michael, C

Palao, D

Rossi, G

Sanfilippo, F

Shindler, A

Simula, S

Tarantino, C

English

arXiv

Cineca Institutional Research Information System

B-physics from the ratio method with Wilson twisted mass fermions

We present a precise lattice QCD determination of the b-quark mass, of the B and Bs decay constants and first preliminary results for the B-mesons bag parameter. Simulations are performed with Nf = 2 Wilson twisted mass fermions at four values of the lattice spacing and the results are extrapolated to the continuum limit. Our calculation benefits from the use of improved interpolating operators for the B-mesons and employs the so-called ratio method. The latter allows a controlled interpolation at the b-quark mass between the relativistic data around and above the charm quark mass and the exactly known static limit

FREZZOTTI, ROBERTO

ROSSI, GIANCARLO

Tarantino, C.

Archivio della Ricerca - Università di Roma Tor vergata

P
o
S(Lattice 2012)104
B-physics from the ratio method with Wilson twisted
mass fermions
N. Carrasco(a), P. Dimopoulos(b,c), R. Frezzotti(b,c), V. Giménez(a), G. Herdoiza(d),
V. Lubicz(e, f ), G. Martinelli(g,h), C. Michael(i), D. Palao(c), G.C. Rossi(b,c),
F. Sanfilippo( j), A. Shindler∗† (k), S. Simula( f ), C. Tarantino(e, f )
(a)Departament de Física Teòrica and IFIC, Univ. de València-CSIC
(b)Dipartimento di Fisica, Università di Roma “Tor Vergata”
(c)INFN, Sezione di “Tor Vergata” c/o Dipartimento di Fisica, Universita‘ di Roma “Tor
Vergata”
(d)Departamento de Física Teòrica and Instituto de Física Teòrica UAM/CSIC,
(e)Dipartimento di Fisica, Università Roma Tre
( f )INFN, Sezione di Roma Tre c/o Dipartimento di Fisica, Università Roma Tre
(g)SISSA
(h)INFN, Sezione di Roma,
(i)Theoretical Physics Division, Dept. of Mathematical Sciences, University of Liverpool
( j)Laboratoire de Physique Théorique (Bat. 210), Université Paris Sud,
(k)Institut für Elementarteilchenphysik, Fachbereich Physik, Humbolt Universität zu Berlin,
We present a precise lattice QCD determination of the b quark mass, of the B and Bs decay con-
stants and first preliminary results for the B-mesons bag parameter. Simulations are performed
with N f = 2 Wilson twisted mass fermions at four values of the lattice spacing and the results
are extrapolated to the continuum limit. Our calculation benefits from the use of improved inter-
polating operators for the B-mesons and employs the so-called ratio method. The latter allows
a controlled interpolation at the b-quark mass between the relativistic data around and above the
charm quark mass and the exactly known static limit.
The 30 International Symposium on Lattice Field Theory - Lattice 2012,
June 24-29, 2012
Cairns, Australia
∗Speaker.
†Heisenberg Fellow
c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/
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B-physics from the ratio method A. Shindler
1. Introduction
Stringent tests of the Standard Model and the search for new physics pass through a detailed
study of physical processes involving the b quark. It is clear that, despite the present (B-factories,
LHCb) and future (SuperB factories) experimental programs, lattice QCD results of hadronic pa-
rameters need to have reduced uncertainties at the level of∼ 1% (see refs [1, 2] for recent reviews).
In this proceeding we report on the ongoing project, within the European Twisted Mass Collabora-
tion (ETMC), to compute B-physics hadronic parameters with Wilson twisted mass fermions. We
extend and improve our previous analysis [3, 4] in two ways:
• we optimize the interpolating operators for heavy-light systems to better project onto the
fundamental state
• we extend the range of heavy masses, µh, considered reaching values of approximately µh ∼
2.5µc, where µc is the charm quark mass.
2. Improved interpolating operators
We use, for this analysis, the N f = 2 dynamical gauge configurations with up and down mass
degenerate quarks, µu/d = µl , generated by ETMC [5]. The lattice action is the tree-level improved
Symanzik gauge action [6] and the twisted mass quark action [7] at maximal twist [8]. The strange,
µs, and the charm (heavy), µc (µh), quarks are quenched in this work.
We have used four lattice spacings a= {0.098(3),0.085(2),0.067(2),0.054(1)} fm [9] correspond-
ing to β = {3.80,3.90,4.05,4.20}. We also use the values of the renormalized quark masses
µu/d = 3.6(2) MeV and µs = 95(6) MeV [9] and the pseudoscalar density renormalization con-
stants ZMSP (2GeV) = {0.411(12),0.437(7),0.477(6),0.501(20)} [10] at the four beta values. We
denote by a “bar” the quark masses renormalized in the MS scheme at a renormalization scale of 2
GeV. The values of the bare valence quark masses used in this calculation are given in tab. 1.
To keep the noise-to-signal ratio under control we extract meson masses at relatively small
temporal separations. To improve the projection onto the fundamental state, we need trial states
with large overlap with the lowest lying energy eigenstate. We have constructed a smeared inter-
polating operator using Gaussian smearing [11], i.e. we have inverted on a source ΦS
ΦS ∝
(
1+κGa2∇2APE
)NG ΦL , κG = 4 NG = 30 , (2.1)
whereΦL is the standard local source and∇APE is the lattice covariant derivative with APE smeared
gauge links (αAPE = 0.5, NAPE = 20) [12]. We have used both smeared sources and sinks finding
that the best correlator in respect of signal to noise ratio is the smeared-local (SL) correlator where
the source is smeared and the sink is local.
To further improve the overlap with the ground state of the B-system we have constructed the
source
Φ(ω) ∝ ωΦS+(1−ω)ΦL , (2.2)
dependent on a tunable parameter ω . We have computed correlators where the source is Φ(ω) and
the sink is smeared (ωS) for several values of ω . We then optimize the value of ω to achieve a
projection onto the ground state at earlier Euclidean times than with SL correlators 1. In the left plot
1Details on the optimization procedure will be given in a forthcoming publication.
2
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B-physics from the ratio method A. Shindler
β aµ` aµs aµh
3.80 0.0080, 0.0110 0.0175, 0.0194 0.1982, 0.2331, 0.2742, 0.3225, 0.3793 0.4461
0.0213 0.5246
3.90 0.0030, 0.0040 0.0159, 0.0177 0.1828, 0.2150, 0.2529, 0.2974, 0.3498
0.0064, 0.0085, 0.0100 0.0195 0.4114, 0.4839
4.05 0.0030, 0.0060 0.0139, 0.0154 0.1572, 0.1849, 0.2175, 0.2558, 0.3008
0.0080 0.0169 0.3538, 0.4162
4.20 0.0020, 0.0065 0.0116, 0.0129 0.13315, 0.1566, ,0.1842, 0.2166, 0.2548
0.0142 0.2997, 0.3525
Table 1: Bare parameters used in this computation. The volumes L3 × T simulated are L/a =
{24,24,32,48}, with T/L = 2, respectively for β = {3.80,3.90,4.05,4.20}. Only for the ensemble with
aµ` = 0.003 and β = 3.9 we have L/a= 32.
of fig. 1 we show the Euclidean time dependence of the effective masses for a particular simulation
point obtained from local-local (LL), SL and ωS correlation functions. The plot describes well
the general properties for heavy-light effective masses: the SL correlators are already substantially
improved for a safe extraction of the ground state mass compared with the LL correlators. The ωS
correlation functions perform even better. This can be fully appreciated in the right plot of fig. 1
where we compare for different plateau regions the effective masses extracted from the SL and the
ωS correlation functions. It is clear that with ωS correlation functions we can extract the mass of
the ground state using results at Euclidean times smaller than with the SL ones. In the same plot
we also compare these two determinations with the one obtained with the GEVP [13], obtaining
perfectly consistent results within statistical uncertainties. We prefer not to use the GEVP in our
final analysis because we have found that it gives noisier estimates for the pseudoscalar decay
constant. All the results presented in the following sections take advantage of these improvements.
3. The b-quark mass and decay constants fB and fBs
To determine the b-quark mass, fB, fBs and the bag parameters, we implement the ratio
method [4]. We refer to this ref. for the details of the method. Here we briefly recall the basic
steps for the determination of the b-quark mass. For the other quantities discussed in these pro-
ceedings the same strategy applies. The method is suggested by the HQET asymptotic behavior of
the heavy-light meson mass Mhl in the pole heavy quark mass µ
pole
h(
Mhl
µpoleh
)
= 1+O
 1(
µpoleh
)
 . (3.1)
We consider a sequence of heavy-quark masses µ(1)h < µ
(2)
h < · · · < µ(N)h with fixed ratio λ , i.e.
µ(n)h = λµ
(n−1)
h . The key quantities of the method are ratios of heavy-light meson masses at subse-
quent values of the heavy-quark mass, y(µ(n)h ,λ ;µ l,a), properly normalized [4].
From eq. (3.1) and QCD asymptotic freedom it follows that the ratios y(µ(n)h ,λ ;µ l,a) have an
exact static limit:
lim
µh→∞
lim
a→0
y(µ(n)h ,λ ;µ l,a) = 1 . (3.2)
3
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B-physics from the ratio method A. Shindler
0 5 10 15 20 25 301.1
1.2
1.3
1.4
1.5
1.6
1.7
Local Local
Smeared Local
ω-method SL-SS
β = 3.80   aµ = 0.080  aµ
 h = 0.5246 ~ 2.5 m c
a m
 eff
x
 0/a
1.34
1.35
1.36
1.37 ω-S  ω=0.55GEVP   x
 0/a = [2:5]
Smeared Local
β = 3.80  aµ = 0.0080  aµ
 h = 0.5246 ~ 2.5 m c
[8:13]
[6:13]
[6:16]
[5:8] [6:9] [7:10] [8:11] [9:12]
a mhl
Figure 1: Left plot: Euclidean time dependence of the effective masses at β = 3.8, aµl = 0.08, aµh= 0.5246
obtained from LL, SL andωS correlation functions. Right plot: effective masses at the same simulation point
as for the left plot extracted from the SL and the ωS correlation for different plateau ranges. The ranges, in
units of the lattice spacing a, are indicated in square brackets. We also plot for comparison the result of a
GEVP analysis.
The value of λ is chosen in such a way that after a finite number of steps the heavy-light meson
mass assumes the experimental value MB = 5.279 GeV. In order to implement this condition, the
lattice data at the four lattice spacings are interpolated at specific values of the heavy quark mass.
In the left plot of fig. 2 we show the chiral and continuum extrapolation of the ratio of the heavy-
light meson masses evaluated at the heaviest quark mass. A phenomenological fit linear in the light
quark mass and in a2 described the lattice data rather well. We observe that discretization errors are
well under control. The values of the ratios y(µh,λ ) extrapolated to the chiral and continuum limit
have a non-perturbative heavy-quark mass dependence that is well described by the HQET-inspired
function
y(µh,λ ) = 1+
η1(λ )
µh
+
η2(λ )
µ2h
, (3.3)
as it can be seen from the right plot in fig. 2. From this plot it is clear that the improvements we
have implemented in this analysis allow to have an accurate description of the non-perturbative
dependence on µh of the ratio y. In particular there is little doubt that the numerical data are
well described by the fit ansatz in eq. (3.3). The formula (3.3), with the fit parameters η1 and η2,
provides the non-perturbative description of the heavy-quark mass dependence of the ratio y(µh,λ )
over the whole range of heavy-masses. We can thus compute the value of the b-quark mass simply
using the knowledge of y(µh,λ ) to iterate the value of the triggering heavy-light meson mass
Mhl(µ
(1)
h ) from the charm region to the experimental value of the B meson mass MB. This is
always possible with a slight tuning of λ and µ(1)h . One finds λ = 1.1784, µ
(1)
h = 1.14 GeV⇒
µb = λKµ
(1)
h (K = 9) , that leads after a 4-loop evolution to
µMSb (µb)|N f=2 = 4.35(12) GeV . (3.4)
This result is consistent with our previous determination [3] with slightly reduced error, and a fully
equivalent one is obtained if we use as input the heavy-strange meson mass Mhs. The 2.7% relative
error is dominated by the ones associated to the scale and ZP determinations. Note that a b-quark
mass value smaller by∼ 120 MeV would be found by setting N f = 4 (rather than 2) in the evolution
from 2 GeV to the b-scale. The statistical and systematic uncertainties coming from the application
4
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B-physics from the ratio method A. Shindler
0 0.01 0.02 0.03 0.04 0.05 0.061.08
1.1
1.12
1.14
β = 3.80
β = 3.90
β = 4.05
β = 4.20
Cont. Lim.
µ
 l [GeV]
M
 hl(µ h
(8))/M
 hl(µ h
(7))
0 0.2 0.4 0.6 0.80.96
0.97
0.98
0.99
1
1.01
NLL ETMC 2011
NLL Preliminary 2012
1/ µ
 h [GeV
 -1]
y(µ
 h, λ)
1/ µ
 b
Figure 2: Left plot: chiral-continuum extrapolation of the ratio of heavy-light meson masses at the heav-
iest quark masses, µ(7,8)h . Right plot: non-perturbative heavy-quark mass dependence of the ratio function
y(µh,λ ), for λ = 1.1784 in the continuum. For comparison we show our previous results [3] without the
interpolating operators improvement discussed in sect. 2.
of the ratio method turn out to be negligible, after the improvements just discussed. To compute
heavy-light and heavy-strange decay constants a completely analogous strategy can be adopted.
We find advantageous to define ratios with exactly known static limit for the heavy-strange decay
constant, zs(µ
(n)
h ,λ ;µ l,µs,a), and for the ratio fBs/ fB, ζ (µ
(n)
h ,λ ;µ l,µs,a) [3]. As for the b-quark
mass, fhs at the triggering point and the ratio zs have a smooth chiral-continuum extrapolation with
cutoff effects always well under control. The heavy-quark mass dependence in the continuum of zs
can again be described by a formula as the one in eq. (3.3), as shown in the left plot of fig. 3. The
application of the ratio method using as input the value we determined for µb leads us to
fBs = 234(6) MeV . (3.5)
The chiral-continuum extrapolation for the double ratio ζ (µ(n)h ,λ ;µ l,µs,a) does not pose any
problem and its heavy-quark mass dependence is very weak easing the interpolation at the b-quark
mass. At the triggering mass, µ¯(1)h , the data for fhs/ fhl as a function of µ¯l suggest that the h-quark
still behaves as a relativistic one. Thus for the chiral-continuum extrapolation of fhs/ fhl(µ¯
(1)
h ) we
find convenient to study the double ratio [ fhs/ fhl] ·
[
fsl/ f
exp
K
]
, for which a smooth dependence on
the light quark mass is expected [14, 15]. We find that a linear fit in the light quark mass describes
well the data as it can be seen from the right plot in fig. 3. This leads to the values
fBs
fB
= 1.19(5), fB = 197(10) MeV , (3.6)
where we add in quadrature the statistical and systemtic uncertainties.
4. Bag parameters
The B-parameter of the renormalized operator in QCD is defined as
〈Bq|O∆B=2q |Bq〉MS ≡
8
3
f 2BqB
2
Bq(µ)M
2
Bq (4.1)
where O∆B=2q ≡
[
bγµ(1− γ5)q
][
bγµ(1− γ5)q
]
and µ is the renormalization scale. With Wtm
fermions the renormalization of this operator is multiplicative [16]. The ratio of renormalized
5
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B-physics from the ratio method A. Shindler
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.98
1
1.02
1.04
1.06
1.08
1.1
NLL ETMC 2011
NLL Prelimin. 2012
1/ µ
 h [GeV
 -1]
z
s
(µ
 h,λ)
1/ µ
 b
0 0.02 0.04 0.06 0.08
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
β = 3.80
β = 3.90
β = 4.05
β = 4.20
Cont. Lim.
β = 3.80
µ
 l [GeV]
[f
 hs(µ l
(1))/f
 hl(µ l
(1))] [f
sl/fK
exp]
Figure 3: Left plot: same as the right plot in fig. 2 fot the ratio function zs(µh,λ ). Right plot: chiral
extrapolation for the double ratio [ fhs/ fhl ] ·
[
fsl/ f
exp
K
]
at the triggering point, µ(1)h .
B-parameters evaluated in QCD is expected to approach unity as 1/µ¯h → 0, This follows from
standard HQET arguments, which also predict the leading deviations for asymptotically small
1/µ¯h-values to be of the order 1/log(µ¯h/ΛQCD). Such corrections to the power scaling in 1/µ¯h
are expected to be tiny in the µ¯h-range of our data and can be estimated in PT by matching HQET
to QCD. A possible way of doing so is to remove them from the data by considering the modified
ratio
ωq(µh,λ ;µ l,a) =
BBq(µh,µ l,a;µ)
BBq(µh/λ ,µ l,a;µ)
· C(µh;µ
∗,µ)
C(µh/λ ;µ∗,µ)
, (4.2)
where the C-factors ratio contains the info on the 1/log(µ¯h)-corrections at a fixed order in RG-
improved PT. We consider here HQET-to-QCD matching only at tree level and LL order in PT
(thereby avoiding the complications of operator mixing in HQET [17]), and confirm the impact
of 1/log(µ¯h)-corrections on the final results to be at the level of one standard deviation [18]. In
the left plot of fig. 4 we show the chiral-continuum limit for the ratio ωs at the heaviest quark
mass. Both the chiral and continuum extrapolation is well under control. See also ref. [18] for
additional details. In the right plot we show the heavy-quark mass dependence and the formula
ωs(µh) = 1+ c1(λ )/µh+ c2(λ )/µ
2
h describes the numerical results very well. Applying the ratio
method one finally gets
BMSBs (µb) = 0.90(5) , (4.3)
where the error is the sum in quadrature of statistical and systematic uncertainties. To compute
BBd , as we have done for the decay constants, we define a ratio for BBs/BBd . Also for this ratio
the chiral-continuum extrapolation and the heavy-quark mass dependence is very smooth and well
under control, leading to the preliminary estimate of
BBs
BBd
= 1.03(2) , BMSBd (µb) = 0.87(5) . (4.4)
Given this result we can also give a preliminary estimate for the parameter
ξ =
fBs
√
BBs
fB
√
BB
= 1.21(6) , (4.5)
where the error is the sum in quadrature of all uncertainties.
6
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B-physics from the ratio method A. Shindler
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080.96
0.98
1
1.02
1.04
1.06
1.08
β=3.80
β=3.90
β=4.05
Cont. Lim.
Cont. Lim.
ωBs(µ h
(4)) @ LO
µ
 l [GeV]
ω
s
(µ
 h
(7))
0 0.2 0.4 0.6 0.8 10.98
1
1.02
1.04
1.06
1.08
1/µ
 h [GeV
-1]
1/µ
 bωs(µ h)
Figure 4: Left plot: chiral-continuum extrapolation of the ratio of the bag parameters, ωs, for the heaviest
quark mass analyzed, µ(7)h . Right plot: non-perturbative heavy-quark mass dependence of the ratio function
ωs(µh,λ ), for λ = 1.1784 in the continuum.
Acknowledgements
The computer time was made available to us by the Italian SuperComputing Resource Alloca-
tion (ISCRA) under the class A project HP10A7IBG7 "A New Approach to B-Physics on Current
Lattices" and the class C project HP10CJTSNF "Lattice QCD Study of B-Physics" at the CINECA
supercomputing service. We also acknowledge computer time made available to us by HLRN in
Berlin.
References
[1] C. Tarantino, (2012), 1210.0474.
[2] J. Zanotti, PoS ICHEP 2012 (2012) 036.
[3] ETM Collaboration, P. Dimopoulos et al., JHEP 1201 (2012) 046, 1107.1441.
[4] ETM Collaboration, B. Blossier et al., JHEP 1004 (2010) 049, 0909.3187.
[5] ETM Collaboration, R. Baron et al., JHEP 1008 (2010) 097, 0911.5061.
[6] P. Weisz, Nucl.Phys. B212 (1983) 1.
[7] Alpha collaboration, R. Frezzotti et al., JHEP 0108 (2001) 058, hep-lat/0101001.
[8] R. Frezzotti and G. Rossi, JHEP 0408 (2004) 007, hep-lat/0306014.
[9] ETM Collaboration, B. Blossier et al., Phys.Rev. D82 (2010) 114513, 1010.3659.
[10] ETM Collaboration, M. Constantinou et al., JHEP 1008 (2010) 068, 1004.1115.
[11] S. Gusken, Nucl.Phys.Proc.Suppl. 17 (1990) 361.
[12] ETM Collaboration, K. Jansen et al., JHEP 0812 (2008) 058, 0810.1843.
[13] B. Blossier et al., JHEP 0904 (2009) 094, 0902.1265.
[14] A. Roessl, Nucl.Phys. B555 (1999) 507, hep-ph/9904230.
[15] RBC-UKQCD Collaboration, C. Allton et al., Phys.Rev. D78 (2008) 114509, 0804.0473.
[16] R. Frezzotti and G. Rossi, JHEP 0410 (2004) 070, hep-lat/0407002.
[17] D. Becirevic et al., JHEP 0408 (2004) 022, hep-lat/0401033.
[18] N. Carrasco et al., PoS Lattice 2012 (2012) 105.
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B-physics from the ratio method with Wilson twisted mass fermions

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