QCD results are presented for a 2+1 flavour fermion clover action (which we
call the SLiNC action). A method of tuning the quark masses to their physical
values is discussed. In this method the singlet quark mass is kept fixed, which
solves the problem of different renormalisations (for singlet and non-singlet
quark masses) occuring for non-chirally invariant lattice fermions. This
procedure enables a wide range of quark masses to be probed, including the case
with a heavy up-down quark mass and light strange quark mass. Preliminary
results show the correct splittings for the baryon (octet and) decuplet
spectrum.Comment: 7 pages; talk given at the XXVII International Symposium on Lattice
  Field Theory, July 26-31 2009, Peking University, Beijing, Chin

Bietenholz, W.

Bornyakov, V.

Cundy, N.

Göckeler, M.

Horsley, R.

Kennedy, A. D.

Nakamura, Y.

Perlt, H.

Pleiter, D.

Rakow, P. E. L.

Schierholz, G.

Schiller, A.

Schäfer, A.

Stüben, H.

Zanotti, J. M.

English

arXiv

Conference details: The XXVII International Symposium on Lattice Field Theory, held July 26-31, 2009, Peking University, Beijing, China.QCD results are presented for a 2+1 flavour fermion clover action (which we call the SLiNC action). A method of tuning the quark masses to their physical values is discussed. In this method the singlet quark mass is kept fixed, which solves the problem of different renormalisations (for singlet and non-singlet quark masses) occuring for non-chirally invariant lattice fermions. This procedure enables a wide range of quark masses to be probed, including the case with a heavy up-down quark mass and light strange quark mass. Preliminary results show the correct splittings for the baryon (octet and) decuplet spectrum.W. Bietenholz, V. Bornyakov, N. Cundy, M. Göckeler, R. Horsley, A. D. Kennedy, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, A. Schäfer, G. Schierholz, A. Schiller, H. Stüben and J. M. Zanottihttp://pos.sissa.it/cgi-bin/reader/conf.cgi?confid=9

Gockeler, M.

Kennedy, A.

Rakow, P.

Schafer, A.

Stuben, H.

Zanotti, J.

Adelaide Research &amp; Scholarship

PUBLISHED VERSION  http://hdl.handle.net/2440/80258   W. Bietenholz, V. Bornyakov, N. Cundy, M. Göckeler, R. Horsley, A. D. Kennedy, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, A. Schäfer, G. Schierholz, A. Schiller, H. Stüben and J. M. Zanotti Results from 2+1 flavours of SLiNC fermions PoS - Proceedings of Science (LATTICE 2009), 2009:102 © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-Non Commercial-Share Alike Licence.                    PERMISSIONS   http://creativecommons.org/licenses/by-nc-nd/4.0/      PoS(LAT2009)102Results from 2+1 flavours of SLiNC fermionsW. Bietenholza, V. Bornyakovb, N. Cundyc, M. Göckelerc, R. Horsley∗d , A. D.Kennedyd, Y. Nakamurac, H. Perlte, D. Pleiter f , P. E. L. Rakowg, A. Schäferc,G. Schierholzch, A. Schillere, H. Stübeni and J. M. Zanottida Instituto de Ciencias Nucleares, Universidad Autónoma de México, A.P. 70-543, C.P. 04510Distrito Federal, Mexicob Institute for High Energy Physics, 142281 Protovino, Russia andInstitute of Theoretical and Experimental Physics, 117259 Moscow, Russiac Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germanyd School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, UKe Institut für Theoretische Physik, Universität Leipzig, 04109 Leipzig, Germanyf John von Neumann Institute NIC / DESY Zeuthen, 15738 Zeuthen, Germanyg Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool,Liverpool L69 3BX, UKh Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germanyi Konrad-Zuse-Zentrum für Informationstechnik Berlin, 14195 Berlin, GermanyE-mail: rhorsley@ph.ed.ac.ukQCDSF–UKQCD CollaborationsQCD results are presented for a 2+1 flavour fermion clover action (which we call the SLiNCaction). A method of tuning the quark masses to their physical values is discussed. In this methodthe singlet quark mass is kept fixed, which solves the problem of different renormalisations (forsinglet and non-singlet quark masses) occuring for non-chirally invariant lattice fermions. Thisprocedure enables a wide range of quark masses to be probed, including the case with a heavyup-down quark mass and light strange quark mass. Preliminary results show the correct splittingsfor the baryon (octet and) decuplet spectrum.The XXVII International Symposium on Lattice Field TheoryJuly 26-31, 2009Peking University, Beijing, China∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(LAT2009)102Results from 2+1 flavours of SLiNC fermions R. Horsley1. IntroductionThere has been a steady progression of lattice results from a quenched sea to two-flavourand more recently 2 + 1 flavour sea in an attempt to provide a more complete and quantitativedescription of hadronic phenomena. (By 2+1 flavours we mean here 2 mass degenerate up-down,ml , quarks and one strange, ms, quark.) In this talk we shall consider a 2 + 1 flavour clover actionand discuss some ways of approaching in the ml–ms plane the physical point (m∗l ,m∗s ), where thenatural starting point for these paths is an SUF(3) flavour symmetric point ml = ms = m(0)sym. Forclover (i.e. non-chiral) fermions a problem arises because the measured pseudoscalar masses areproportional to the renormalised quark masses. The singlet, S, and nonsinglet, NS quark massesrenormalise differently which means that the relation to the bare quark masses and hence κ , whichis the adjustable simulation parameter, is more complicated [1]. Choosing the path such that thesinglet quark mass is kept fixed provides an elegant solution to this problem. This procedurealso has the advantage that it enables a wide range of quark masses to be probed (including themass of the strange quark) and is thus particularly useful for strange quark physics and SUF(3)–chiral perturbation theory, as the kaon mass is never larger than its physical value. (Note thatthis path choice although favourable for clover–type fermions is not restricted to them.) Indeedthis includes the case with a heavy light quark mass and light strange quark mass. As a ‘proofof concept’ preliminary results given here show the correct splittings for the baryon (octet and)decuplet spectrum.The particular clover action used here has a single iterated mild stout smearing for the hoppingterms together with thin links for the clover term (this is in an attempt to ensure that the fermionmatrix does not become too extended). Together with the (tree level) Symanzik improved gluonaction this constitutes the Stout Link Non-perturbative Clover or SLiNC action for which the clovercoefficient, csw, has recently been non-perturbatively (NP) determined, [2], using the SchrödingerFunctional, or SF, formalism. Further details about the action may be found in this reference.Simulations have been performed using HMC with mass preconditioning for 2 mass-degenerateflavours and the rational approximation for the 1-flavour. Two programmes were used, a Fortranprogramme, [3], and also the Chroma programme, [4]. Quark mass degenerate runs (denoted by thesubscript ‘sym’) on 163×32 lattices followed by 243×48 lattices have located a suitable β -range,see Fig. 1. We see that there is good agreement between SF and pseudoscalar mass determinationsof the critical value of κsym, κsym;c. The simulations reported here have been performed at β = 5.50which gives a ∼ 0.08fm (taking the scale r0 = 0.5fm and a linear extrapolation of r0/a to thechiral limit). We have furthermore checked that there is a distinct gap between the distribution of(the modulus of) the lowest eigenvalue and 0 indicating that the simulations are stable on presentvolumes. Further details of the results in this write-up will be given in [5].2. Non-degenerate quark massesAs mentioned before the problem (at least for clover-like fermions, with no chiral symmetry)is that singlet and non-singlet quark mass can renormalise differently,mRq = ZNSm (mq−m)+ZSmm2PoS(LAT2009)102Results from 2+1 flavours of SLiNC fermions R. Horsley8.20 8.22 8.24 8.26 8.28 8.30 8.32 8.34 8.36 8.38 8.401/κsym0.000.050.100.150.200.250.300.35(ampi)2β=5.40, 163x32β=5.50, 163x32+243x48β=5.60, 163x32Figure 1: Results for the pseudoscalar mass against 1/κsym for β = 5.40, 5.50, 5.60. For β = 5.50 a linearfit (line) using the lightest three masses is made. For comparison a quadratic fit (dashed line) is also made toall the masses. The stars represent the SF determination of κsym;c; the open circle is the result from the linearfit.= ZNSm (mq +αZm) with αZ =ZSm−ZNSmZNSm, (2.1)with q ∈ {sym, l,s,v} (also including possible different valence quarks to sea quarks) andm =13(2ml +ms) . (2.2)(We are assuming here that the quadratic improvement terms, [6], are small, [5].) The bare quarkmass in eq. (2.1) is defined byamq =12(1κq−1κsym;c). (2.3)Now if mq1q2ps (l,s) is the measured pseudoscalar mass (with quarks q1, q2) then we expect that(amK)2 ≡ (amq1q2ps )2∝ amRq1 +amRq2 ∝ amq1 +amq2 +2αZam . (2.4)In particular(ampi)2 ≡ (amqqps)2∝ 2(amq +αZam) , (2.5)which is 6∝ amq unless q = l = s = sym (i.e. mq = m). That αZ is non-zero may be easily seenby considering partially quenched results (i.e. results for the pseudoscalar mass where the valencequark masses may differ from the sea quark masses). In Fig. 2 we show some partially quenchedresults and compare the extrapolation with the SUF(3)-symmetric results. The lines clearly havedifferent gradients. αZ can be estimated as (ampi)2 vanishes at κ pqc say (pq for ‘partially quenched’),givingαZ =−amq|κ=κ pqcam=(1κsym;c− 1κ pqc)(1κsym− 1κsym;c) , (2.6)3PoS(LAT2009)102Results from 2+1 flavours of SLiNC fermions R. Horsley−0.01 0.00 0.01 0.02amq0.000.020.040.060.080.100.12(ampi)2β=5.50, q=symβ=5.50, q=v−0.01 0.00 0.01 0.02amq0.000.020.040.060.080.100.12(ampi)2mq=0.005mq=0.010symmq=0.005mq=0.010Figure 2: The left plot shows the SUF(3) symmetric pseudoscalar masses versus amq (in red) together withthe partially quenched results (in black) using the mass degenerate sea quark κsym = 0.12090. (This is thekappa value at the crossing point.) The right plot shows equivalent domain wall results from [7] table V.(setting l = s = sym in m). This gives here αZ ∼ 1.2 (but the determination is quite sensitive tosmall changes in κsym;c and κ pqc ). This is to be compared with domain-wall fermions, also shownin Fig. 2, where the results line up.We wish to approach the physical point along some path in the mRl –mRs plane (mRq is consideredas it is related to the measurable pseudoscalar mass) from an SUF(3) symmetric point (mRl = mRs ≡mR(0)sym ). This is depicted in Fig. 3. Two possibilities are mRs = const. = mR∗s (i.e. strange quark mass0mlR0msRmsR=msR*(mlR*,msR*)msR=mlRm−R*=constmsR=msR*(ml*,ms*)ms=mlm−=constmlmsFigure 3: The left sketch shows the mRl –mRs plane. The physical quark masses are denoted by (mR∗l ,mR∗s ).The dashed diagonal line is the SUF(3)-symmetric line. Two possible paths from this line are shown, mRs =mR∗s and mR = const. . The right sketch shows the equivalent results in the ml–ms plane.being held constant) or mR = const. (i.e. singlet quark mass being held constant). Note that theregion covered is mRl ≥ 0, mRs ≥ 0. For simulations we need to translate this to unrenormalisedquantities also shown in Fig. 3. Note that the physical domain is now ml ≥−( 13 αZ/(1+ 23 αZ))ms,ms ≥−(23 αZ/(1+ 13 αZ))ml . While mRs = const. = mR∗s translated to bare quark masses now dependson the difficult-to-determine αZ , the singlet quark mass mR ∝ m or ms = (2m∗l + m∗s )− 2ml isindependent of the value of αZ . This motivates the choice of this path. As ml ց m∗l then ms ր m∗si.e. the ms–ml splitting or mK increases to its physical value. Other potential advantages include:the singlet quark mass is correct from the very beginning; flavour singlet quantities are flat at the4PoS(LAT2009)102Results from 2+1 flavours of SLiNC fermions R. Horsleysymmetric point – allowing simpler extrapolations; numerically the HMC cost change should bemoderate along this path. (These points will be further discussed in [5].)Of course practically we must now determine the initial κ(0)sym to complete the relation betweenκs and κl ,κs =13κ(0)sym− 2κl. (2.7)For our path choice, κ(0)sym can be implicitly found by using eqs. (2.4), (2.5) together with a singletscale X (so X∗ = X (0)sym here) to relate the known physical point to the initial symmetric point,1cX≡13(2m2K +m2pi)X2∣∣∣∣∣∗∝ mR∗ = mR(0)sym ∝(am(0)pi )2(aX (0)sym)2. (2.8)For X we have several choices. For example• The centre of mass2 of the octet baryons: X2N ≡ 13(m2N +m2Σ +m2Ξ) (stable under strong interac-tion) giving 13(2m2K +m2pi)/ 13(m2N +m2Σ +m2Ξ)|∗ = 0.169/1.34 = 1/7.93 = (am(0)pi )2/(am(0)N )2,• Centre of mass2 of the decuplet baryons: X2∆ ≡ 13(2m2∆ +m2Ω) (which decay under strong inter-actions) giving 13(2m2K +m2pi)/ 13(2m2∆ +m2Ω)|∗ = 0.169/1.94 = 1/11.5 = (am(0)pi )2/(am(0)∆ )2,• A gluonic quantity X2r ≡ 1/r20 so r20(2m2K +m2pi)|∗ = 0.169/0.3952 = 1.083 = (r(0)0 /a)2(am(0)pi )2(r0 = 0.5fm).Thus where the numerically determined SUF(3)–symmetric line crosses with the line (aXsym)2 =cX(ampi)2 gives our initial point. The results are shown in Fig. 4. We see that they are all consistentaround the lightest pseudoscalar mass, namely for κ(0)sym = 0.12090 which we shall take as our start-ing value. From eq. (2.7) we now have a relation between κs and κl . After some experimentationwe chose the κl , κs values given in Table 1. Note that is possible to choose κl , κs values (hereκl κs0.12083 0.12104 ml > ms0.12090 0.12090 ml = ms0.12095 0.12080 ml < ms0.12100 0.12070 ml < ms0.12104 0.12062 ml < msTable 1: Present (κl ,κs) values (simulated on 243×48 lattices).(0.12083,0.12104)) such that ml > ms. In this strange world we would expect to see an inversionof the particle spectrum, with, for example, the nucleon being the heaviest octet particle.3. Hadron spectrumAs an example we now give some results in Fig. 5 for the the baryon decuplet: Ω, Ξ∗, Σ∗, ∆.These are at present low statistics results and are meant to illustrate the ‘proof of concept’ only.Noteworthy is that the correct ordering of the particle spectrum has been achieved.5PoS(LAT2009)102Results from 2+1 flavours of SLiNC fermions R. Horsley0.00 0.05 0.10 0.15(ampi)20.000.250.500.75(amN)20.00 0.05 0.10 0.15(ampi)20.00.20.40.60.81.0(am∆)20.00 0.05 0.10 0.15(ampi)24.05.06.07.0r 0/aFigure 4: (amN)2 against (ampi)2 (upper left picture) together with the line (amN)2 = 7.93(ampi)2; (am∆)2against (ampi)2 (upper right picture) together with the line (am∆)2 = 11.5(ampi)2; r0/a against (ampi)2 (lowerpicture) together with the line (r0/a)2 = 1.083/(ampi)2. All data comes from the symmetric points.4. ConclusionsFor the NP O(a)-improved 2+1 flavour clover action discussed here we have first found thatthere is consistency for the critical κsym between the SF approach and the determination from thevanishing of the hadron pseudoscalar mass. The quark mass renormalisation suggests that thesimplest way of approaching the physical point is to hold the singlet quark mass fixed. Exploratoryresults for the hadron mass spectrum give splittings in the correct order (including inversion whenml > ms i.e. we can simulate a strange world where, for example, the nucleon can decay). Furtherresults will be published elsewhere, [5].AcknowledgementsThe numerical calculations have been performed on the IBM BlueGeneL at EPCC (Edin-burgh, UK), the BlueGeneL and P at NIC (Jülich, Germany), the SGI ICE 8200 at HLRN (Berlin-Hannover, Germany), the SGI Altix 4700 at LRZ (Munich, Germany) and JSCC (Moscow, Rus-sia). We thank all institutions. The BlueGene codes were optimised using Bagel, [8]. This workhas been supported in part by the EU Integrated Infrastructure Initiative Hadron Physics (I3HP)6PoS(LAT2009)102Results from 2+1 flavours of SLiNC fermions R. Horsley0.0 0.5 1.0 1.5 2.0(r0mpi)28101214161820(r 0m∆)2∆(lll)0.0 0.5 1.0 1.5 2.0(r0mpi)28101214161820(r 0mΣ*)2Σ*(lls)0.0 0.5 1.0 1.5 2.0(r0mpi)28101214161820(r 0mΞ*)2Ξ*(lss)0.0 0.5 1.0 1.5 2.0(r0mpi)28101214161820(r 0mΩ)2Ω(sss)Figure 5: The decuplet baryons. The upper left picture shows the ∆, upper right Σ∗, lower left Ξ∗ and lowerright the Ω. The common SUF(3) symmetric quark mass values are shown in black. The experimental valuesare shown with stars.under contract RII3-CT-2004-506078 and by the DFG under contracts FOR 465 (ForschergruppeGitter-Hadronen-Phänomenologie) and SFB/TR 55 (Hadron Physics from Lattice QCD).References[1] M. Göckeler et al., Phys. Lett. B639, 307 (2006) [arXiv:hep-ph/0409312]; P. E. L. Rakow,Nucl. Phys. Proc. Suppl. 140, 34 (2005) [arXiv:hep-lat/0411036].[2] N. Cundy et al., Phys. Rev. D79, 094507 (2009) [arXiv:0901.3302[hep-lat]].[3] M. Göckeler et al., PoS 041, (2007) [arXiv:0712.3525 [hep-lat]].[4] R. Edwards and B. Joó, Nucl. Phys. Proc. Suppl. 140, 832 (2005) [arXiv:hep-lat/0409003].[5] W. Bietenholz et al., QCDSF–UKQCD Collaborations, in preparation.[6] T. Bhattacharya et al., Phys. Rev. D73, 034504 (2006) [arXiv:hep-lat/0511014].[7] C. Allton et al., Phys. Rev. D78, 114509 (2008) [arXiv:0804.0473[hep-lat]].[8] P. A. Boyle, http://www.ph.ed.ac.uk/∼paboyle/Bagel.html (2005).7

Results from 2+1 flavours of SLiNC fermions

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Results from 2+1 flavours of SLiNC fermions

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Results from 2+1 flavours of SLiNC fermions

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Adelaide Research & Scholarship