Physics Letters B 757 (2016) 473–479
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Continuum limit of the leading-order HQET form factor in B
s
→Kν
decays
ALPHA Collaboration
Felix Bahr
a
, Debasish Banerjee
a
, Fabio Bernardoni
a,b
, Anosh Joseph
c
, Mateusz Koren
a,∗
,
Hubert Simma
a
, Rainer Sommer
a
a
John von Neumann Institute for Computing (NIC), DESY, Platanenallee 6, D-15738 Zeuthen, Germany
b
Medizinische Fakultät, Carl Gustav Carus, TU Dresden, Fetscherstraße 74, D-01307 Dresden, Germany
c
Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge, Cambridge, CB3 0WA, UK
a r t i c l e i n f o a b s t r a c t
Article history:
Received
29 January 2016
Accepted
31 March 2016
Available
online 11 April 2016
Editor: A.
Ringwald
Keywords:
Lattice
QCD
Heavy
quark effective theory
Semileptonic
decays of bottom mesons
We discuss the computation of form factors for semi-leptonic decays of B-, B
s
-mesons in lattice QCD.
Considering in particular the example of the static B
s
form factors we demonstrate that after non-
perturbative
renormalization the continuum limit can be taken with confidence. The resulting precision
is of interest for extractions of V
ub
. The size of the corrections of order 1/m
b
is just estimated at present
but it is expected that their inclusion does not pose significant difficulties.
© 2016 CERN for the benefit of the ALPHA Collaboration. Published by Elsevier B.V. This is an open
access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Weak decays of B-mesons are a very important piece in the
puzzle of understanding about how well the Standard Model of
particle physics describes Nature. One relevant question concerns
the determination of the Cabibbo–Kobayashi–Maskawa matrix el-
ement
V
ub
from different decays. This fundamental parameter of
the Standard Model is not known very precisely yet. Testing for
consistent values provides a check of the Standard Model. In fact
results extracted from inclusive decays agree with those from dif-
ferent
exclusive decays, like B →πν or B →τν [1–3], only after
stretching the presently estimated uncertainties by around a fac-
tor
three. We avoid calling this a three-sigma tension since the
uncertainties are largely systematic, coming from the theoretical
computation of form factors in lattice QCD on one side and the
perturbative treatment of inclusive decays on the other side. But
also experimental uncertainties contribute.
In
this letter we consider the determinations of semi-leptonic
form factors for B
s
-mesons from lattice QCD. A review with some
discussion of the challenges involved is found in [4]. It appears that
the most relevant challenge is the presence of a (large) mass scale
*
Corresponding author.
E-mail
address: mateusz.koren@desy.de (M. Koren).
m
b
∼ 5GeV. Together with inverse lattice spacings below 4GeV
this
distorts the continuum physics considerably in a straight ap-
plication
of lattice QCD. We do not want to review here this issue
in detail, but just mention that this leads one to consider effective
field theories for the b-quark or extrapolations in its mass, again
guided by effective field theory considerations. The most advanced
computations [5–8], use either a relativistic heavy quark action
or employ non-relativistic QCD on the lattice. There the challenge
is twofold. First, a fully non-perturbative renormalization program
for the heavy-light currents does not (yet) exist. It is replaced by
“mostly non-perturbative” renormalization [9,10], where the fac-
tor
Z
hl
/
√
Z
hh
Z
ll
is taken from 1-loop perturbation theory and this
approximation is expected to be a good one [9,10]; alternatively
straight 1-loop perturbation theory is used. Second, discretization
errors
are estimated only by power-counting arguments because
continuum limit extrapolations may involve a complicated func-
tional
dependence on the lattice spacing. As a consequence we are
not aware of the computation of a non-perturbatively renormalized
heavy-light form factor extrapolated to the continuum.
In order to place the present work into context, let us briefly
list the steps which are necessary to come to a trustworthy result
of interest to phenomenology:
a) obtain
the ground state matrix elements K|V
μ
(0)|B
s
that me-
diate
the transition,
http://dx.doi.org/10.1016/j.physletb.2016.03.088
0370-2693/
© 2016 CERN for the benefit of the ALPHA Collaboration. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.