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arXiv:1009.2074v1 [hep-lat] 10 Sep 2010
SFB/CPP-10-80, DESY 10-127, HU-EP-10/48
Kaon and D meson masses with Nf= 2+1+1
twisted mass lattice QCD
ETM Collaboration: Rémi Barona, Philippe Boucaudb, Jaume Carbonellc,
Vincent Drachc, Federico Farchionid, Gregorio Herdoizae, Karl Jansene,
Chris Michaelf, István Montvayg, Elisabetta Pallanteh, Siebren Rekerh,
Carsten Urbachi, Marc Wagner∗j, Urs Wengerk
aCEA, Centre de Saclay, IRFU/Service de Physique Nucléaire, F-91191 Gif-sur-Yvette, France
bLaboratoire de Physique Théorique (Bât. 210), CNRS et Université Paris-Sud XI, Centre
d’Orsay, 91405 Orsay-Cedex, France
cLaboratoire de Physique Subatomique et Cosmologie, 53 avenue des Martyrs, 38026 Grenoble,
France
dUniversität Münster, Institut für Theoretische Physik, Wilhelm-Klemm-Straße 9,
D-48149 Münster, Germany
eNIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germany
fDivision of Theoretical Physics, University of Liverpool, L69 3BX Liverpool, United Kingdom
gDeutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22603 Hamburg, Germany
hCentre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen,
Netherlands
iHelmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for
Theoretical Physics, Universität Bonn, 53115 Bonn, Germany
jHumboldt-Universität zu Berlin, Institut für Physik, Newtonstraße 15, D-12489 Berlin, Germany
kAlbert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of
Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland
We discuss the computation of the kaon and D meson masses in the Nf= 2+1+1 twisted mass
lattice QCD setup, where explicit heavy flavor and parity breaking occurs at finite lattice spacing.
We present three methods suitable in this context and verify their consistency.
The XXVIII International Symposium on Lattice Field Theory, Lattice2010
June 14-19, 2010
Villasimius, Italy
∗Speaker.
c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence.
http://pos.sissa.it/
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Kaon and D meson masses with Nf= 2+1+1twisted mass lattice QCD
Marc Wagner
1. Introduction
The European Twisted Mass Collaboration (ETMC) is currently performing large scale sim-
ulations with Nf= 2+1+1 flavors of dynamical quarks using Wilson twisted mass lattice QCD
[1, 2, 3].
Particular problems are caused by the non-degenerate strange/charm quark doublet, since be-
sides the usual twisted mass parity breaking the strange and charm quark numbers are not con-
served. The latter amounts to contamination of correlation functions by intermediate states with
wrong flavor quantum numbers. In particular, correlation functions of charmed mesons and baryons
obtain small contributions from similar strange systems, which are significantly lighter and, there-
fore, inevitably dominate at large temporal separations. In this paper we discuss these problems
in the context of the kaon and D meson masses, which are important quantities already at the
stage of generating gauge field configurations, when tuning the strange and charm quark masses to
their physical values. We present three methods to overcome these problems and demonstrate their
consistency. This work is a summary of a more detailed recent paper [4].
2. Simulation setup
The gauge action is the Iwasaki action [5]. The fermion action is Wilson twisted mass,
SF,light[χ(l), ¯ χ(l),U] = a4∑
x
¯ χ(l)(x)
?
DW(m0)+iµγ5τ3
?
χ(l)(x)
(2.1)
SF,heavy[χ(h), ¯ χ(h),U] = a4∑
x
¯ χ(h)(x)
?
DW(m0)+iµσγ5τ1+τ3µδ
?
χ(h)(x)
(2.2)
for the light degenerate up/down doublet [6] and the heavy non-degenerate strange/charm doublet
[7] respectively, where DWdenotes the standard Wilson Dirac operator. κ = 1/(2m0+8) is tuned
to maximal twist by requiring mPCAC
χ(l)
physical quantities, e.g. the here considered kaon and D meson masses. We also refer to [8], where
this Nf= 2+1+1 twisted mass setup has been pioneered.
All results presented in the following correspond to computations on 1042 gauge field con-
figurations from ensemble B35.32 [3] characterized by gauge coupling β = 1.95, lattice exten-
sion L3×T = 323×64 and bare untwisted and twisted quark masses κ = 0.161240, µ = 0.0035,
µσ= 0.135 and µδ= 0.170. The corresponding lattice spacing is a ≈ 0.078fm, the pion mass
mPS≈ 318MeV.
= 0, which guarantees automatic O(a) improvement for
3. Quantum numbers, physical and twisted basis meson creation operators
In Wilson twisted mass lattice QCD parity is not a symmetry and the heavy flavors cannot be
diagonalized – both symmetries are broken at O(a). Consequently, instead of the four QCD heavy-
light meson sectors labeled by heavy flavor and parity, (s,−), (s,+), (c,−) and (c,+), there is only
a single combined heavy-light meson sector (s/c,−/+) in twisted mass lattice QCD. In contrast to
QCD, where the D meson is the lightest state in the (c,−) sector, it is a highly excited state in the
combined (s/c,−/+) sector of twisted mass lattice QCD. This in turn causes severe problems for
2
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Kaon and D meson masses with Nf= 2+1+1twisted mass lattice QCD
Marc Wagner
the computation of mD, since the determination of excited states is inherently difficult, when using
lattice methods.
To create kaons and D mesons a suitable set of operators is
Oj ∈
?
+i¯ χ(d)γ5χ(s), −i¯ χ(d)γ5χ(c), +¯ χ(d)χ(s), −¯ χ(d)χ(c)?
.
(3.1)
In the continuum the twisted basis quark fields χ and their physical basis counterparts ψ are related
by the twist rotation
?
ψ(u)
ψ(d)
?
= exp
?
iγ5τ3ωl/2
??
χ(u)
χ(d)
?
,
?
ψ(s)
ψ(c)
?
= exp
?
iγ5τ1ωh/2
??
χ(s)
χ(c)
?
. (3.2)
ωland ωhdenote the light and heavy twist angles, which are π/2 at maximal twist. At finite lattice
spacing the procedure is more complicated, since renormalization factors have to be included. The
heavy-light meson creation operators (3.1) and the corresponding renormalized operators in the
physical basis transform into each other via
+i ¯ ψ(d)γ5ψ(s)
−i ¯ ψ(d)γ5ψ(c)
+ ¯ ψ(d)ψ(s)
− ¯ ψ(d)ψ(c)
R
=
?
+clch −slsh −slch −clsh
−slsh +clch−clsh −slch
+slch +clsh+clch −slsh
+clsh +slch −slsh +clch
??
?
=M(ωl,ωh)
+iZP¯ χ(d)γ5χ(s)
−iZP¯ χ(d)γ5χ(c)
+ZS¯ χ(d)χ(s)
−ZS¯ χ(d)χ(c)
,
(3.3)
where cx= cos(ωx/2), sx= sin(ωx/2) and ZPand ZSare operator dependent renormalization con-
stants.
The starting point for the three analysis methods presented in the following sections 4 to 6 are
the 4×4 correlation matrices
Cjk(t) = ?Ω|Oj(t)
?
Ok(0)
?†|Ω?
(3.4)
of spatially extended, i.e. APE and Gaussian smeared versions of twisted basis heavy-light meson
creation operators (3.1). The smearing parameters have been optimized by minimizing effective
masses at small temporal separations (cf. [4] for details).
4. Method 1: solving a generalized eigenvalue problem
One possibility to determine mKand mDis to solve the generalized eigenvalue problem
Cjk(t)v(n)
j(t,t0) = Cjk(t0)λ(n)(t,t0)v(n)
j(t,t0),
(4.1)
cf. e.g. [9] and references therein. From the eigenvalues λ(n)one then computes four effective
masses m(n)
effectiveby solving
λ(n)(t,t0)
λ(n)(t +1,t0)
=
exp(−m(n)
effective(t,t0)(t +1))+exp(−m(n)
effective(t,t0)t)+exp(−m(n)
effective(t,t0)(T −t))
effective(t,t0)(T −(t +1)))
exp(−m(n)
,
(4.2)
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Kaon and D meson masses with Nf= 2+1+1twisted mass lattice QCD
Marc Wagner
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30
meffective
t
effective masses of the K (green) and D (magenta)
m = 0.8286 ± 0.0084 (χ2/dof = 0.40)
m = 0.2184 ± 0.0003 (χ2/dof = 0.83)
0.214
0.216
0.218
0.22
0.222
0.224
0 5 10 15 20 25 30
meffective
t
zoomed effective mass of the K
Figure 1: left: effective masses obtained from the 4×4 correlation matrix (3.4) by solving the generalized
eigenvalueproblem(4.1)and(4.2)andcorrespondingplateaufits; right: zoomedeffectivemass ofthelowest
state, which has been identified as the kaon.
where T is the temporal extension of the lattice and the exponentials e(−m(n)
e(−m(n)
Heavy-light meson masses are finally determined by fitting constants to effective mass plateaus at
temporal separations t ≫ 1. Results are shown in Figure 1.
It is well known that for sufficiently large temporal separations t the generalized eigenvalue
problem (4.1) and (4.2) yields the lowest states in the sector considered, i.e. in our case the four
lowest states in the combined (s/c,−/+) sector. The D meson (m(D) ≈ 1868MeV), however, is
not among them. Lighter states include
effective(t,t0)(T −t)) and
effective(t,t0)(T −(t +1))) take care of effects due to the temporal periodicity of the lattice.
• the kaon and its radial excitations,
m(K) ≈ 496MeV, m(K(1460)) = 1400MeV−1460MeV, ...
• parity partners of the kaon
m(K∗
0(800)) = 672(40)MeV, m(K∗
0(1430)) = 1425(50)MeV, ...
• multi particle states
m(K +π), m(K+2×π), ...
At first glance it seems that the 4×4 correlation matrix (3.4) is not sufficient to determine mD, but
that one needs a significantly larger correlation matrix, which is able to resolve all states below the
D meson. Note, however, that in the continuum an exact diagonalization ofCjkis possible yielding
one correlator for each of the four sectors (s,−), (s,+), (c,−), (c,+). Hence the generalized
eigenvalue problem would not yield the four lowest masses ofthe (s/c,−/+) sector, but mK, m(s,+),
mDand m(c,+). At finite lattice spacings such a diagonalization is, of course, only approximately
possible. However, discretization artefacts, which are responsible for that, only appear at O(a)
and are thus expected to be small. Therefore, at not too large temporal separations one of the four
effective masses should be dominated by the D meson and, consequently, provide an estimate for
mD.
One can check that this is indeed the case by twist rotating the eigenvectors v(n)to the pseudo
physical basis, which is defined by (3.3) with ZP= ZS= 1. Then one can read off the approximate
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Kaon and D meson masses with Nf= 2+1+1twisted mass lattice QCD
Marc Wagner
flavor and parity content of each of the four states corresponding to the four effective mass plateaus.
As shown in Figure 2, the lowest state is dominated by the physical basis operator ¯ ψ(d)γ5ψ(s)and,
therefore, interpreted as the kaon, while the second excited state is dominated by ¯ ψ(d)γ5ψ(c), i.e.
corresponds to the D meson.
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30
|vj
(0)|2
t
eigenvector components of the K
γ5 strange
γ5 charm
1 strange
1 charm
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30
|vj
(2)|2
T
eigenvector components of the D
γ5 strange
γ5 charm
1 strange
1 charm
Figure 2: left: squared pseudo physical basis eigenvector components of the lowest state |v(0)
interpreted as the kaon; right: squared pseudo physical basis eigenvector components of the second excited
state |v(2)
j|2, which is
j|2, which is interpreted as the D meson.
5. Method 2: fitting exponentials
An alternative approach to determine mKand mDis to perform a χ2minimizing fit of
N
∑
n=1
(a(n)
j)†a(n)
k
?
exp
?
−mnt
?
+exp
?
−mn(T −t)
??
,
(5.1)
i.e. of N exponentials to the computed correlation matrix Cjkin a suitably chosen window of tem-
poral separations t. Notice that T is the temporal extension of the lattice and the second exponential
exp(−mn(T −t)) takes care of effects due to the lattice temporal periodicity. The masses mncan
be interpreted by analyzing the prefactors a(n)
in more detail in the previous section for the eigenvector components provided by the generalized
eigenvalue problem.
As before we find that the lowest state is a kaon, while the second excited state is dominated
by the physical basis operator ¯ ψ(d)γ5ψ(c)and, hence, should correspond to the D meson.
j, which is very similar to what has been explained
6. Method 3: heavy flavor/parity restoration
Our third approach is based on the twist rotation of heavy-light meson creation operators (3.3).
In a first step we express the correlation matrixCjkin the physical basis in terms of the twist angles
ωland ωhand the ratio of renormalization factors ZPand ZS:
Cphysical,R(t;ωl,ωh,ZP/ZS) =
= M(ωl,ωh)diag(ZP,ZP,ZS,ZS)C(t)diag(ZP,ZP,ZS,ZS)M†(ωl,ωh),
(6.1)
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Kaon and D meson masses with Nf= 2+1+1twisted mass lattice QCD
Marc Wagner
where the matrix M has been defined in (3.3). We then determine ωl, ωhand ZP/ZSby requiring
Cphysical,R
jk
(t;ωl,ωh,ZP/ZS)
???j?=k
= 0.
(6.2)
At finite lattice spacing and small temporal separations t this cannot be achieved exactly, because
of O(a) heavy flavor and parity breaking effects and the presence of excited states. However, at
sufficiently large t, where only the kaon survives, the condition (6.2) can be realized. It amounts to
removing any kaon contribution from the diagonal correlators Cphysical,R
Finally we analyze the diagonal correlators Cphysical,R
each of the four sectors (s,−), (s,+), (c,−) and (c,+). The effective mass plateaus corresponding
to the (s,−) and the (c,−) diagonal correlator yield the heavy-light meson masses mKand mD,
respectively.
jj
, j ?= (s,−).
jj
individually. There is one correlator for
7. Conclusions and outlook
Results for mKand mDobtained with our three methods agree within statistical and system-
atic errors, see Table 1 and Figure 3. For a detailed discussion, of how statistical and systematic
errors have been determined, we refer to [4]. We are able to determine mKin a rigorous way
with rather high statistical precision (statistical error<
require assumptions, when computing mD, which amount to a systematical error being involved.
Nevertheless, the combined statistical and systematical error for mDis<
∼0.4%). On the other hand all three methods
∼2.5%.
method 1method 2 method 3
mK
mD
0.2184(3)
0.829(8)
0.2177(8)
0.835(20)
0.2184(3)
0.823(15)
Table 1: comparison of kaon and D meson masses determined with the three methods presented.
0.217
0.2175
0.218
0.2185
0.219
0.2195
1 2 3
mK
method
mass of the K
method 1 (generalized eigenvalue problem)
method 2 (fitting exponentials)
method 3 (heavy flavor/parity restoration)
0.81
0.82
0.83
0.84
0.85
0.86
0.87
1 2 3
mD
method
mass of the D
method 1 (generalized eigenvalue problem)
method 2 (fitting exponentials)
method 3 (heavy flavor/parity restoration)
Figure 3: comparison of kaon and D meson masses determined with the three methods presented.
Being able to determine mKand mDis very important for tuning the strange and charm quark
masses to their physical values. For precision charm physics, however, we intend to use a mixed
6
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Kaon and D meson masses with Nf= 2+1+1twisted mass lattice QCD
Marc Wagner
action Osterwalder-Seiler setup [10]. First steps in this direction are currently under way and have
been reported during this conference [11].
Acknowledgments
Thecomputer timeforthis project wasmade available to usbythe John von Neumann-Institute
for Computing (NIC) on the JUMP, Juropa and Jugene systems in Jülich and apeNEXT system in
Zeuthen, BG/P and BG/L in Groningen, by BSC on Mare-Nostrum in Barcelona (www.bsc.es)
and by the computer resources made available by CNRS on the BlueGene system at GENCI-IDRIS
Grant 2009-052271 and CCIN2P3 in Lyon. We thank these computer centers and their staff for all
technical advice and help.
This work has been supported in part by the DFG Sonderforschungsbereich TR9 Comput-
ergestützte Theoretische Teilchenphysik and the EU Integrated Infrastructure Initiative Hadron
Physics (I3HP) under contract RII3-CT-2004-506078. We also thank the DEISA Consortium (co-
funded by the EU, FP6 project 508830) for support within the DEISA Extreme Computing Initia-
tive (www.deisa.org).
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