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

Baron, Rémi

Boucaud, Philippe

Carbonell, Jaume

Drach, Vincent

Farchioni, Federico

Herdoiza, Gregorio

Jansen, Karl

Michael, Chris

Montvay, István

Pallante, Elisabetta

Reker, Siebren

Urbach, Carsten

Wagner, Marc

Wenger, Urs

English

Hochschulschriftenserver - Universität Frankfurt am Main

PoS(Lattice 2010)119Kaon and D meson masses with Nf = 2+1+1twisted mass lattice QCDETM 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 Wengerka CEA, Centre de Saclay, IRFU/Service de Physique Nucléaire, F-91191 Gif-sur-Yvette, Franceb Laboratoire de Physique Théorique (Bât. 210), CNRS et Université Paris-Sud XI, Centred’Orsay, 91405 Orsay-Cedex, Francec Laboratoire de Physique Subatomique et Cosmologie, 53 avenue des Martyrs, 38026 Grenoble,Franced Universität Münster, Institut für Theoretische Physik, Wilhelm-Klemm-Straße 9,D-48149 Münster, Germanye NIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germanyf Division of Theoretical Physics, University of Liverpool, L69 3BX Liverpool, United Kingdomg Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22603 Hamburg, Germanyh Centre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen,Netherlandsi Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center forTheoretical Physics, Universität Bonn, 53115 Bonn, Germanyj Humboldt-Universität zu Berlin, Institut für Physik, Newtonstraße 15, D-12489 Berlin, Germanyk Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University ofBern, Sidlerstr. 5, CH-3012 Bern, SwitzerlandWe discuss the computation of the kaon and D meson masses in the N f = 2+1+1 twisted masslattice 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, Lattice2010June 14-19, 2010Villasimius, Italy∗Speaker.†Email: Marc.Wagner@physik.hu-berlin.dec  Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/PoS(Lattice 2010)119Kaon and D meson masses with Nf = 2+1+1twisted mass lattice QCD Marc Wagner1. IntroductionThe 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 withwrong ﬂavor quantum numbers. In particular, correlation functions of charmed mesons and baryonsobtain 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 problemsin the context of the kaon and D meson masses, which are important quantities already at thestage of generating gauge ﬁeld conﬁgurations, when tuning the strange and charm quark masses totheir physical values. We present three methods to overcome these problems and demonstrate theirconsistency. This work is a summary of a more detailed recent paper [4].2. Simulation setupThe 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 DW denotes the standard Wilson Dirac operator. κ = 1/(2m0 +8) is tunedto maximal twist by requiring mPCACχ(l) = 0, which guarantees automatic O(a) improvement forphysical quantities, e.g. the here considered kaon and D meson masses. We also refer to [8], wherethis 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 massmPS ≈ 318MeV.3. Quantum numbers, physical and twisted basis meson creation operatorsIn Wilson twisted mass lattice QCD parity is not a symmetry and the heavy ﬂavors cannot bediagonalized – 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 onlya single combined heavy-light meson sector (s/c,−/+) in twisted mass lattice QCD. In contrast toQCD, where the D meson is the lightest state in the (c,−) sector, it is a highly excited state in thecombined (s/c,−/+) sector of twisted mass lattice QCD. This in turn causes severe problems for2PoS(Lattice 2010)119Kaon and D meson masses with Nf = 2+1+1twisted mass lattice QCD Marc Wagnerthe computation of mD, since the determination of excited states is inherently difﬁcult, when usinglattice methods.To create kaons and D mesons a suitable set of operators isOj ∈ +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 relatedby 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)ωl and ωh denote the light and heavy twist angles, which are π/2 at maximal twist. At ﬁnite latticespacing the procedure is more complicated, since renormalization factors have to be included. Theheavy-light meson creation operators (3.1) and the corresponding renormalized operators in thephysical 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 ZP and ZS are operator dependent renormalization con-stants.The starting point for the three analysis methods presented in the following sections 4 to 6 arethe 4×4 correlation matricesCjk(t)= Ω|Oj(t) Ok(0) †|Ω  (3.4)of spatially extended, i.e. APE and Gaussian smeared versions of twisted basis heavy-light mesoncreation operators (3.1). The smearing parameters have been optimized by minimizing effectivemasses at small temporal separations (cf. [4] for details).4. Method 1: solving a generalized eigenvalue problemOne possibility to determine mK and mD is to solve the generalized eigenvalue problemCjk(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 effectivemasses m(n)effective by solvingλ(n)(t,t0)λ(n)(t +1,t0)=exp(−m(n)effective(t,t0)t)+exp(−m(n)effective(t,t0)(T −t))exp(−m(n)effective(t,t0)(t +1))+exp(−m(n)effective(t,t0)(T −(t +1))), (4.2)3PoS(Lattice 2010)119Kaon 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  30meffectiveteffective 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  30meffectivetzoomed effective mass of the KFigure 1: left: effective masses obtained from the 4×4 correlation matrix (3.4) by solving the generalizedeigenvalueproblem(4.1)and(4.2)andcorrespondingplateauﬁts; right: zoomedeffectivemass oftheloweststate, which has been identiﬁed as the kaon.where T is the temporal extension of the lattice and the exponentials e(−m(n)effective(t,t0)(T −t)) ande(−m(n)effective(t,t0)(T −(t +1))) take care of effects due to the temporal periodicity of the lattice.Heavy-light meson masses are ﬁnally determined by ﬁtting constants to effective mass plateaus attemporal separations t   1. Results are shown in Figure 1.It is well known that for sufﬁciently large temporal separations t the generalized eigenvalueproblem (4.1) and (4.2) yields the lowest states in the sector considered, i.e. in our case the fourlowest states in the combined (s/c,−/+) sector. The D meson (m(D) ≈ 1868MeV), however, isnot among them. Lighter states include• the kaon and its radial excitations,m(K) ≈ 496MeV, m(K(1460)) = 1400MeV−1460MeV, ...• parity partners of the kaonm(K∗0(800)) = 672(40)MeV, m(K∗0(1430)) = 1425(50)MeV, ...• multi particle statesm(K+π), m(K+2×π), ...At ﬁrst glance it seems that the 4×4 correlation matrix (3.4) is not sufﬁcient to determine m D,b u tthat one needs a signiﬁcantly larger correlation matrix, which is able to resolve all states below theD meson. Note, however, that in the continuum an exact diagonalization ofCjk is possible yieldingone correlator for each of the four sectors (s,−), (s,+), (c,−), (c,+). Hence the generalizedeigenvalue problem would not yield the four lowest masses ofthe (s/c,−/+) sector, but m K,m(s,+),mD and m(c,+).A tﬁnite lattice spacings such a diagonalization is, of course, only approximatelypossible. 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 foureffective masses should be dominated by the D meson and, consequently, provide an estimate formD.One can check that this is indeed the case by twist rotating the eigenvectors v (n) to the pseudophysical basis, which is deﬁned by (3.3) with ZP = ZS = 1. Then one can read off the approximate4PoS(Lattice 2010)119Kaon and D meson masses with Nf = 2+1+1twisted mass lattice QCD Marc Wagnerﬂavor 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)|2teigenvector components of the Kγ5 strangeγ5 charm1 strange1 charm 0 0.2 0.4 0.6 0.8 1 0  5  10  15  20  25  30|vj(2)|2Teigenvector components of the Dγ5 strangeγ5 charm1 strange1 charmFigure 2: left: squared pseudo physical basis eigenvector components of the lowest state |v(0)j |2, which isinterpreted as the kaon; right: squared pseudo physical basis eigenvector components of the second excitedstate |v(2)j |2, which is interpreted as the D meson.5. Method 2: ﬁtting exponentialsAn alternative approach to determine mK and mD is to perform a χ2 minimizing ﬁto fN∑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 Cjk in a suitably chosen window of tem-poral separations t. Notice that T is the temporal extension of the lattice and the second exponentialexp(−mn(T −t)) takes care of effects due to the lattice temporal periodicity. The masses m n canbe interpreted by analyzing the prefactors a(n)j , which is very similar to what has been explainedin more detail in the previous section for the eigenvector components provided by the generalizedeigenvalue problem.As before we ﬁnd that the lowest state is a kaon, while the second excited state is dominatedby the physical basis operator ¯ ψ(d)γ5ψ(c) and, hence, should correspond to the D meson.6. Method 3: heavy ﬂavor/parity restorationOur third approach is based on the twist rotation of heavy-light meson creation operators (3.3).In a ﬁrst step we express the correlation matrixCjk in the physical basis in terms of the twist anglesωl and ωh and the ratio of renormalization factors ZP and 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)5PoS(Lattice 2010)119Kaon and D meson masses with Nf = 2+1+1twisted mass lattice QCD Marc Wagnerwhere the matrix M has been deﬁned in (3.3). We then determine ω l, ωh and ZP/ZS by requiringCphysical,Rjk (t;ωl,ωh,ZP/ZS)   j =k= 0. (6.2)At ﬁnite lattice spacing and small temporal separations t this cannot be achieved exactly, becauseof O(a) heavy ﬂavor and parity breaking effects and the presence of excited states. However, atsufﬁciently large t, where only the kaon survives, the condition (6.2) can be realized. It amounts toremoving any kaon contribution from the diagonal correlators Cphysical,Rjj , j  =( s,−).Finally we analyze the diagonal correlators Cphysical,Rjj individually. There is one correlator foreach of the four sectors (s,−), (s,+), (c,−) and (c,+). The effective mass plateaus correspondingto the (s,−) and the (c,−) diagonal correlator yield the heavy-light meson masses m K and mD,respectively.7. Conclusions and outlookResults for mK and mD obtained 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 systematicerrors have been determined, we refer to [4]. We are able to determine m K in a rigorous waywith rather high statistical precision (statistical error <∼0.4%). On the other hand all three methodsrequire assumptions, when computing mD, which amount to a systematical error being involved.Nevertheless, the combined statistical and systematical error for m D is <∼2.5%.method 1 method 2 method 3mK 0.2184(3) 0.2177(8) 0.2184(3)mD 0.829(8) 0.835(20) 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  3mKmethodmass of the Kmethod 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  3mDmethodmass of the Dmethod 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 mK and mD is very important for tuning the strange and charm quarkmasses to their physical values. For precision charm physics, however, we intend to use a mixed6PoS(Lattice 2010)119Kaon and D meson masses with Nf = 2+1+1twisted mass lattice QCD Marc Wagneraction Osterwalder-Seiler setup [10]. First steps in this direction are currently under way and havebeen reported during this conference [11].AcknowledgmentsThecomputer timeforthis project wasmade available to usbythe John von Neumann-Institutefor Computing (NIC) on the JUMP, Juropa and Jugene systems in Jülich and apeNEXT system inZeuthen, 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 CNRSon the BlueGene system at GENCI-IDRISGrant 2009-052271 and CCIN2P3 in Lyon. We thank these computer centers and their staff for alltechnical 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 HadronPhysics (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).References[1] R. Baron et al. [ETM Collaboration], PoS LATTICE2008, 094 (2008) [arXiv:0810.3807[hep-lat]].[2] R. Baron et al., PoS LATTICE2009, 104 (2009) [arXiv:0911.5244[hep-lat]].[3] R. Baron et al. [ETM Collaboration], JHEP 1006, 111 (2010) [arXiv:1004.5284[hep-lat]].[4] R. Baron et al. [ETM Collaboration], arXiv:1005.2042[hep-lat].[5] Y. Iwasaki, Nucl. Phys. B 258, 141 (1985).[6] ALPHA Collaboration, R. Frezzotti, P. A. Grassi, S. Sint and P. Weisz, JHEP 0108, 058 (2001)[arXiv:hep-lat/0101001].[7] R. Frezzotti and G. C. 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[ETM Collaboration], talk by C. Urbach at “XXVIIIth International Symposium on Lattice Field Theory”, Villasimius,

Kaon and D meson masses with Nf = 2+1+1 twisted mass lattice QCD

Kaon and D meson masses with Nf = 2+1+1 twisted mass lattice QCD

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