Kaon and D meson masses with Nf = 2 + 1 + 1 twisted mass lattice QCD
Remi Barona, Philippe Boucaudb, Jaume Carbonellc, Vincent Drachc, Federico Farchionid, Gregorio Herdoizae, Karl Jansene, Chris Michaelf , Istvan Montvayg , Elisabetta Pallanteh, Olivier Pèneb, Siebren Rekerh,
Carsten Urbachi, Marc Wagnerj , Urs Wengerk
a
f
b
CEA, Centre de Saclay, IRFU/Service de Physique Nucléaire, F-91191 Gif-sur-Yvette, France;
Laboratoire de Physique Théorique (Bât. 210), CNRS et Université Paris-Sud XI, Centre d’Orsay, 91405 Orsay-Cedex, France;
c
d
e
Laboratoire de Physique Subatomique et Cosmologie, 53 avenue des Martyrs, 38026 Grenoble, France;
Universität Münster, Institut für Theoretische Physik, Wilhelm-Klemm-Straße 9, D-48149 Münster, Germany;
NIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germany;
g
h
Division of Theoretical Physics, University of Liverpool, L69 3BX Liverpool, United Kingdom;
Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22603 Hamburg, Germany;
Centre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen, the Netherlands;
i
j
Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics, Universität Bonn, 53115 Bonn, Germany;
Humboldt-Universität zu Berlin, Institut für Physik, Newtonstraße 15, D-12489 Berlin, Germany;
k
Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland
Abstract
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.
Method 1: generalized eigenvalue problem
Method 3: heavy flavor/parity restoration
• Generalized eigenvalue problem (GEP), effective meson masses:
• Express the correlation matrix Cjk (t) in the physical basis in terms of ωl and ωh and ZP /ZS :
(n)
(n)
Cjk (t)vj (t, t0) = Cjk (t0)vj (t, t0)λ(n)(t, t0)
Nf = 2 + 1 + 1 ETMC simulation setup
,
(n)
meffective(t, t0) = ln
Ã
λ(n)(t, t0)
λ(n)(t + a, t0)
!
.
C physical,R(t; ωl , ωh, ZP /ZS ) = M(ωl , ωh)diag(ZP , ZP , ZS , ZS )C(t)diag(ZP , ZP , ZS , ZS )M†(ωl , ωh).
• Determine ωl , ωh and ZP /ZS by requiring
• Fitting constants to effective mass plateaus at t ≫ a yields meson masses.
effective masses of the K (green) and D (magenta)
• Nf = 2 + 1 + 1 flavors of dynamical Wilson twisted mass quarks:
´
³
X
SF,light[χ(l), χ̄(l), U ] = a4
χ̄(l)(x) DW(m0) + iµγ5τ3 χ(l)(x)
x
´
³
X
SF,heavy[χ(h), χ̄(h), U ] = a4
χ̄(h)(x) DW(m0) + iµσ γ5τ1 + τ3µδ χ(h)(x)
m = 0.8286 ± 0.0084 (χ /dof = 0.40)
m = 0.2184 ± 0.0003 (χ2/dof = 0.83)
meffective
meffective
0.8
0.6
0.22
physical,R
0.216
0.2
0.214
• κ = 1/(2m0 + 8) is tuned to maximal twist by requiring mPCAC
=0
χ(l)
→ automatic O(a) improvement for physical quantities.
physical,R
– Amounts to removing any K contribution from the diagonal correlators Cjj
0.218
0.4
• Analyze the diagonal correlators Cjj
sectors (s, −), (s, +), (c, −), (c, +).
0
0
5
10
15
20
25
30
0
5
15
20
25
30
t
• After rotating the eigenvectors v(n) to the pseudo physical basis (physical basis with ZP = ZS = 1)
one can read off the quantum numbers heavy flavor and parity, i.e. (s, −), (s, +), (c, −) or (c, +).
eigenvector components of the K
1
0.8
0.8
γ5 strange
γ5 charm
1 strange
1 charm
(2)
(0)
Conclusions
• Results obtained with our three methods agree within statistical and systematic errors.
method 1 method 2
γ5 strange
γ5 charm
1 strange
1 charm
0.6
0.4
0.2195
0.4
• Goal: compute the mass of the Kaon and the D meson.
• Twist rotation in the continuum:
!
!
Ã
Ã
µ
¶
χ(u)
ψ (u)
= exp iγ5τ3ωl /2
(d)
χ(d)
ψ
,
Ã
ψ (s)
ψ (c)
!
µ
¶ Ã (s) !
χ
= exp iγ5τ1ωh/2
χ(c)
• We use spatially extended versions of the twisted basis meson creation operators
½
¾
Oj ∈ + iχ̄(d)γ5χ(s) , −iχ̄(d)γ5χ(c) , +χ̄(d)χ(s) , −χ̄(d)χ(c) ,
to access the J = 0, (s/c, −/+) sector (in particular the Kaon and the D).
• Twist rotation of local meson creation operators at finite lattice spacing:
R
+iψ̄ (d)γ5ψ (s)


(d)
(c)
 −iψ̄ γ5ψ 


(d)
(s)

 +ψ̄ ψ
−ψ̄ (d)ψ (c)

+cl ch
 −sl sh
= 
 +sl ch
+cl sh
|
0
0
5
10
15
20
25
30
−sl sh −sl ch
+cl ch −cl sh
+cl sh +cl ch
+sl ch −sl sh
{z
=M(ωl ,ωh)


+iZP χ̄(d)γ5χ(s)
−cl sh


(d)
(c)

−sl ch   −iZP χ̄ γ5χ 


−sl sh   +ZS χ̄(d)χ(s) 
(d)χ(c)
+cl ch
−Z
χ̄
S
}
(cx = cos(ωx/2), sx = sin(ωx/2) and ZP and ZS are operator dependent renormalization constants).
• Starting point for all our analysis methods: 4 × 4 correlation matrices
³
´†
Cjk (t) = hΩ|Oj (t) Ok (0) |Ωi.
mass of the D
method 1 (generalized eigenvalue problem)
method 2 (fitting exponentials)
method 3 (heavy flavor/parity restoration)
0.87
0.86
0
5
10
t
15
20
25
0.218
30
T
method 1 (generalized eigenvalue problem)
method 2 (fitting exponentials)
method 3 (heavy flavor/parity restoration)
0.85
0.2185
mK
0.2
0.84
0.83
0.2175
0.82
Problem:
(ψ denotes quark fields in the physical basis, χ in the twisted basis).

0.2
0
method 3
mass of the K
0.219
• In twisted mass lattice QCD at finite lattice spacing parity is not a symmetry and the heavy flavors
cannot be diagonalized
→ instead of the four sectors (s, −), (s, +), (c, −), (c, +) there is only a single combined sector
(s/c, −/+) in twisted mass lattice QCD.
separately; there is one correlator for each of the four
mK 0.2184(3) 0.21768(84) 0.21842(33)
mD 0.829(8) 0.835(20) 0.823(15)
|vj |2
0.6
, j 6= (s, −).
eigenvector components of the D
1
|vj |2
Quantum numbers, physical and twisted basis
10
t
which amounts to a ≈ 0.078 fm and mPS ≈ 318 MeV.
• Cf. talk by Siebren Reker “Light hadrons from Nf = 2 + 1 + 1 dynamical twisted mass fermions”.
j6=k
= 0:
– At finite lattice spacing and small t this cannot be achieved exactly (excited states, O(a) effects).
– It can be realized at large t (when only the K survives).
0.222
1
(DW is the standard Wilson Dirac operator).
β = 1.95 , L3 × T = 323 × 64 , µ = 0.0035 , κ = 0.161240 , µσ = 0.135 , µδ = 0.170,
0.224
2
1.2
x
• All results shown in the following are for the ensemble with
zoomed effective mass of the K
mD
• Iwasaki gauge action.
¯
¯
physical,R
Cjk
(t; ωl , ωh, ZP /ZS )¯
0.217
• For t ≫ a GEP yields the lowest four states in the combined (s/c, −/+) sector; the D is not among
them:
– m(K) ≈ 496 MeV,
m(K(1460)) = 1400 MeV − 1460 MeV,
... (J P = 0−).
– m(K0∗(800)) = 672(40) MeV,
m(K0∗(1430)) = 1425(50) MeV,
... (J P = 0+).
– m(K + π),
m(K + 2 × π),
...
– m(D) ≈ 1868 MeV (J P = 0−).
Why can we still expect to get an estimate for m(D) from GEP?
• In the continuum an exact diagonalization of Cjk is possible yielding one correlator for each of the
four sectors (s, −), (s, +), (c, −), (c, +)
→ GEP would not yield the four lowest masses but mK , m(s,+), mD and m(c,+).
• At finite lattice spacing corrections are O(a); at not too large temporal separations one of the four
effective masses should be dominated by the D.
Method 2: fitting exponentials
• Perform a χ2 minimizing fit of
N ³
³
´
´†
X
(n)
(n)
aj
C̃jk (t) =
ak exp − mnt ,
n=1
i.e. of N exponentials to the computed correlation matrix Cjk (t), tmin ≤ t ≤ tmax.
0.81
1
2
method
3
1
2
3
method
< 0.4%.
• Precise results for mK ; statistical error ∼
• All three methods require assumptions for mD , i.e. there is a systematical error involved; combined
< 2.5%.
statistical and systematical error ∼
• For precision charm physics we intend to use a mixed action Osterwalder-Seiler setup (cf. talk by
Carsten Urbach, “Pseudoscalar decay constants from Nf = 2 + 1 + 1 twisted mass lattice QCD”);
mK and mD are needed/helpful for matching bare quark masses.
Acknowledgments
The computer time for this project was made available to us by the 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/Transregio SFB/TR9-03
and the EU Integrated Infrastructure Initiative Hadron Physics (I3HP) under contract RII3-CT-2004506078. We also thank the DEISA Consortium (co-funded by the EU, FP6 project 508830), for support
within the DEISA Extreme Computing Initiative (www.deisa.org).