Flavour physics and Lattice QCD: averages of lattice inputs for the Unitarity Triangle Analysis

Abstract

We review recent results of Lattice QCD calculations relevant for flavour physics. We discuss in particular the hadronic parameters entering the amplitudes of K0-K0bar, D0-D0bar and B0-B0bar mixing, the B- and D-meson decay constants and the form factors controlling B-meson semileptonic decays. On the basis of these lattice results, which are extensively collected in the paper, we also derive our averages of the relevant hadronic parameters.

Full text

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arXiv:0807.4605v1 [hep-lat] 29 Jul 2008
Flavour physics and Lattice QCD: averages of lattice inputs for
the Unitarity Triangle Analysis
V. Lubicz(1)(∗) and C. Tarantino(1)
(1) Dip. di Fisica, Universit` a di Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy.
Summary. — We review recent results of Lattice QCD calculations relevant for
flavour physics. We discuss in particular the hadronic parameters entering the am-
plitudes of K0−¯ K0, D0−¯D0and B0−¯B0mixing, the B- and D-meson decay
constants and the form factors controlling B-meson semileptonic decays. On the
basis of these lattice results, which are extensively collected in the paper, we also
derive our averages of the relevant hadronic parameters.
PACS 12.38.Gc – Lattice QCD calculations.
The aim of this talk is to illustrate the status of the art of lattice QCD calculations
relevant for flavour physics and to provide averages of the hadronic quantities which are
useful for phenomenological analysis of flavour physics, within or beyond the Standard
Model. These averages will be used in particular by the UTfit collaboration in the
unitarity triangle analysis (UTA) [1]-[4]. We believe, however, that the whole discussion
given in the present contribution can be of more general interest. Even though the
process of averaging lattice results unavoidably involves some degree of subjectivity, for
all the quantities discussed in this work we also provide extensive collections of lattice
results, as well as the references to the original papers. In addition, we try to enrich the
discussion with some technical considerations about the accuracy of the various lattice
calculations, which can be useful for the non-expert of the field in order to have a deeper
view and to draw their conclusions.
The status of the art of Lattice QCD calculations is represented by unquenched si-
mulations performed with either Nf= 2 or Nf= 2 + 1 dynamical quarks and values of
the light (i.e. up/down) quark masses well below ms/2, where msis the physical strange
quark mass. Both these features are particularly important to keep the lattice systematic
uncertainties under control. On the one hand, it is well known that the quenched appro-
ximation introduces a systematic error which, besides depending on the specific quantity
under investigation, is also difficult to evaluate (if not by performing the corresponding
unquenched calculation). On the other hand, lattice simulations performed with suffi-
ciently light values of the up and down quark masses (mud< ms/2) allow to rely on the
(∗) Speaker
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V. LUBICZ and C. TARANTINO
predictions of chiral perturbation theory when performing the chiral extrapolations of
the lattice results to the physical masses.
If the possibility of performing unquenched calculations at light quark masses repre-
sents a major advance of Lattice QCD in the last few years, it has also to be noted that
these calculations have not always reached yet the same accuracy in controlling other
sources of systematic effects as done in the more recent (and less expensive) quenched
calculations. For instance, some of the unquenched results have not yet been extrapo-
lated to the continuum limit, and non-perturbative renormalization techniques have not
always been implemented. Moreover, while most of the quenched results have been de-
rived by many lattice collaborations, often using different approaches, in several cases
only few (if any) unquenched results are already available for a given quantity. For in-
stance, the unquenched results for the kaon B-parameter have been only obtained so far
at fixed (and rather large) values of the lattice spacing, whereas the matrix elements
of the full basis of four-fermion operator relevant for K0−¯K0mixing in generic New
Physics models have been only obtained so far within the quenched approximation.
In what follows we illustrate in some detail the results for various lattice parameters
and present our averages. We will not be able to discuss in the present contribution the
whole set of hadronic parameters available from the lattice and which are relevant for
flavour physics. In particular, we are not going to review the accurate lattice results for
the quark masses, for the ratio of decay constants fK/fπand for the vector form factor
controlling the semileptonic Kℓ3 decays. For the latter two quantities, which allow the
important determination of Vus(the Cabibbo angle), we refer to the review by A. Juttner
at the Lattice’07 conference [5].
1. – The kaon B-parameter BK
A collection of quenched and unquenched lattice results for BMS
shown in fig.1 as a function of the square of the lattice spacing a. Among the quenched
results we have selected all determinations which are O(a)-improved, i.e. that are affected
by discretization errors of O(a2) or smaller, and have been obtained at several values
of the lattice spacing, thus allowing an extrapolation to the physical continuum limit.
An additional feature of all results shown in fig.1 is that they have been obtained with
lattice discretizations of the fermionic action such that the ∆S = 2 four-fermion operator
defining BKis subject to multiplicative renormalization only. This allows to significantly
increase the accuracy of the determination of the kaon B-parameter.
As illustrated in fig.1, the unquenched results for BK, represented by filled points in
the plot, have been all obtained so far at a fixed value of the lattice spacing. Thus, at
variance with the quenched determinations, the unquenched results cannot be extrapo-
lated to the continuum limit, and the size of discretization errors affecting them cannot
be easily estimated. On general grounds, we note that there is no reason to expect dis-
cretization errors in the unquenched case to have smaller size than those affecting the
quenched results.
The comparison between quenched and unquenched determinations of BK obtained
with the same lattice action and at similar values of the lattice spacing does not show
any evidence of the quenching effect within the present uncertainties. Moreover, the
comparison between Nf = 0 and Nf = 2 results obtained with the same lattice action
but at different values of the lattice spacing, see for instance the case of the DBW2-DWF
action (empty and filled circles in the plot), suggests that quenched and unquenched
results could be affected by similar discretization errors. This observation has motivated
K(2 GeV) [6]-[12] is

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FLAVOUR PHYSICS AND LATTICE QCD: AVERAGES OF LATTICE INPUTS FOR THE UTA
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0,0000
0,00250,0050 0,0075
0,0100
0,0125 0,01500,0175
a2 (fm2)
0,40
0,45
0,50
0,55
0,60
0,65
0,70
0,75
0,80
BKMS(2 GeV)
Nf=0, KS, PT [JLQCD’97]
Nf=0, DBW2-DWF, RI [RBC’05]
Nf=0, Iwasaki-DWF, SF [CP-PACS’08]
Nf=0, KS-AsqTad, PT [HPQCD/UKQCD’06]
Nf=2, DBW2-DWF, RI [RBC’04]
Nf=2, Overlap, RI [JLQCD’08]
Nf=2+1, KS-AsqTad, PT [HPQCD/UKCD’06]
Nf=2+1, Iwasaki-DWF, RI [RBC/UKQCD’07]
Fig. 1. – Lattice QCD results for BMS
square. All results selected in the plot are O(a)-improved and required only multiplicative
renormalization of the relevant ∆S = 2 four-fermion operator. Filled points are unquenched
results obtained with either Nf = 2 or Nf = 2+1 dynamical quarks. Empty points are quenched
results. Our average is shown by the horizontal band.
K(2 GeV) [6]-[12] plotted as a function of the lattice spacing
the RBC collaboration to consider their Nf = 0 and Nf = 2 results together in order
to perform a combined extrapolation to the continuum limit [7]. By taking into account
the scale dependence suggested by the quenched results, we quote as final average for
BKthe value
BMS
K(2 GeV) = 0.55 ± 0.05 ,(1)
which corresponds to the renormalization group invariant parameter
?BK= 0.75 ± 0.07 . (2)
Clearly, a better estimate of discretization errors in the unquenched case will be obtained
once the results of unquenched simulations performed at different values of the lattice
spacing, which are currently in progress, will be available.
Our average in eq. (1) is shown by the horizontal band in fig.1. It can be compared
with the average BMS
K(2 GeV) = 0.58±0.03±0.06 quoted by Dawson at Lattice’05 [13],
where the central value and the first error corresponded to the average of the quenched
results in the continuum limit, while the second error was an estimate of the quenching
effect. The recent unquenched results for BKallow to remove the latter uncertainty and
slightly decrease the previous central value.
2. – The B-mesons decay constants fBsand fB
The unquenched lattice QCD results for the B-mesons decay constants fBsand fB
and for their ratio fBs/fB [14]-[20] are shown in fig.2. They have been obtained by
treating the b quark on the lattice with two different approaches, either FNAL [21] or
NRQCD.

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V. LUBICZ and C. TARANTINO
160
180200220 240
260
280 300320340
360
fBs (MeV)
CP-PACS 00
MILC 02
FNAL,MILC 07 *
HPQCD 03
CP-PACS 01
Nf=2
Nf=2+1
JLQCD 03
140
160
180200220 240
260
280300320 340
fB (MeV)
CP-PACS 00
MILC 02
FNAL,MILC 07 *
HPQCD 05
CP-PACS 01
Nf=2
Nf=2+1
JLQCD 03
1,00
1,05
1,10
1,15
1,20
1,25
fBs/fB
1,30
1,35
1,40
1,451,50
CP-PACS 00
MILC 02
FNAL,MILC 07 *
HPQCD 05
CP-PACS 01
Nf=2
Nf=2+1
JLQCD 03
Fig. 2. – Lattice QCD results for the B-mesons decay constants fBs(top left), fB (top right) and
the ratio fBs/fB (bottom) obtained with Nf = 2 and Nf = 2 + 1 unquenched simulations [14]-
[20]. A star in the legends labels preliminary results. Our averages are shown by the vertical
bands.
Our average for the Bsdecay constant,
fBs= 245 ± 25 MeV , (3)
is based on the results shown in the upper plot of fig.2. The lattice determination of
fB is more delicate, because its value is enhanced by chiral logs effects relevant at low
quark masses. In order to properly account for these effects, simulations at light values
of the quark mass (typically mud < ms/2) are required. For this reason, we derive
our average for fB and for the ratio fBs/fB by taking into account only the results
HPQCD’05 [19] and FNAL/MILC’07 [20] (see fig.2), which use the MILC gauge field
configurations generated at light quark masses as low as ms/8. In this way we obtain
fBs/fB= 1.21 ± 0.04 .(4)
Finally, we derive directly from the previous averages of fBsand fBs/fBthe estimate of

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FLAVOUR PHYSICS AND LATTICE QCD: AVERAGES OF LATTICE INPUTS FOR THE UTA
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180 200220 240
260
280 300320 340
360
380
fDs (MeV)
CP-PACS 00
MILC 02
ETMC 08*
FNAL,MILC 07*
HPQCD 07
EXP.
Nf=2
Nf=2+1
140
160
180 200220240
260
280300320340
fD (MeV)
CP-PACS 00
MILC 02
ETMC 08*
FNAL,MILC 07*
HPQCD 07
EXP.
Nf=2
Nf=2+1
1,05
1,10
1,15
1,20
1,25
1,30
1,35
1,40
1,451,501,55
fDs/fD
CP-PACS 00
MILC 02
ETMC 08*
FNAL,MILC 07*
HPQCD 07
EXP.
Nf=2
Nf=2+1
Fig. 3. – Lattice QCD results for the D-mesons decay constants fDs(top), fD (center) and for
the ratio fDs/fD (bottom) obtained with Nf = 2 and Nf = 2 + 1 unquenched simulations [14,
16, 20, 23, 24]. A star in the legends labels preliminary results. Our averages are shown by the
vertical bands. The corresponding experimental averages [22] are also shown for comparison.
the B-meson decay constant
fB= 200 ± 20 MeV .(5)
3. – The D-mesons decay constants fDsand fD
The CKM matrix elements Vcsand Vcd, which control the rate of leptonic D+
and D+→ ℓ+νℓdecays, are well constrained by the unitarity of the CKM matrix, which
predicts |Vcs| ≃ 1 − λ2/2 and |Vcd| ≃ λ (up to O(λ4) corrections), where λ is the
Wolfenstein parameter. By using these constraints, the measurements of the leptonic
decay rates then allow a determination of the D-mesons decay constants, fDsand fD.
The current experimental averages, as evaluated in ref. [22], are
s→ ℓ+νℓ
fEXP.
Ds
= 273 ± 10 MeV
(fDs/fD)EXP.= 1.33 ± 0.07 .
,fEXP.
D
= 205.8 ± 8.9 MeV,(6)
The unquenched lattice QCD results for these constants [14, 16, 20, 23, 24] are shown
in fig.3, where they are also compared with the experimental averages given in eq. (6).
Among the various lattice determinations, which show overall a very good consistency
among each other, it deserves to be noted the result of the HPQCD collaboration [24],
which quotes fDsand fDwith the impressive accuracy of about 1% and 2% respectively.
Remarkably, while the result for fDof ref. [24] is in good agreement with the experimental
average, the prediction for fDsdiffers from the experimental value by approximately 3
standard deviations. This discrepancy has been interpreted in [25] as an evidence for
New Physics effects in leptonic Dsdecay.
A critical review of the lattice study of ref. [24] is not possible yet, since important
details on the analysis are not given in the paper, being postponed to a longer publi-