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UKQCD Collaboration Flavor Blin PUBLISHED VERSION Bietenholz, W.;...; Zanotti, James Michael; ... et al.; QCDSF Collaboration; UKQCD Collaboration Flavor blindness and patterns of flavor symmetry breaking in lattice simulations of up, down and strange quarks Physical Review D, 2011; 84(5):054509 ©2011 American Physical Society http://prd.aps.org/abstract/PRD/v84/i5/e054509 PERMISSIONS http://publish.aps.org/authors/transfer-of-copyright-agreement http://link.aps.org/doi/10.1103/PhysRevD.62.093023 “The author(s), and in the case of a Work Made For Hire, as defined in the U.S. Copyright Act, 17 U.S.C. §101, the employer named [below], shall have the following rights (the “Author Rights”): [...] 3. The right to use all or part of the Article, including the APS-prepared version without revision or modification, on the author(s)’ web home page or employer’s website and to make copies of all or part of the Article, including the APS-prepared version without revision or modification, for the author(s)’ and/or the employer’s use for educational or research purposes.” 26th April 2013 http://hdl.handle.net/2440/76522 PHYSICAL REVIEW D 84, 054509 (2011) Flavor blindness and patterns of flavor symmetry breaking in lattice simulations of up, down, and strange quarks W. Bietenholz,1 V. Bornyakov,2 M. Go¨ckeler,3 R. Horsley,4 W. G. Lockhart,5 Y. Nakamura,6 H. Perlt,7 D. Pleiter,8 P.E. L. Rakow,5 G. Schierholz,3,9 A. Schiller,7 T. Streuer,3 H. Stu¨ben,10 F. Winter,4 and J. M. Zanotti4 (QCDSF-UKQCD Collaboration) 1Instituto de Ciencias Nucleares, Universidad Auto´noma de Me´xico, A.P. 70-543, C.P. 04510 Distrito Federal, Mexico 2Institute for High Energy Physics, 142281 Protovino, Russia and Institute of Theoretical and Experimental Physics, 117259 Moscow, Russia 3Institut fu¨r Theoretische Physik, Universita¨t Regensburg, 93040 Regensburg, Germany 4School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, UK 5Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK 6RIKEN Advanced Institute for Computational Science, Kobe, Hyogo 650-0047, Japan 7Institut fu¨r Theoretische Physik, Universita¨t Leipzig, 04109 Leipzig, Germany 8Deutsches Elektronen-Synchrotron DESY, 15738 Zeuthen, Germany 9Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germany 10Konrad-Zuse-Zentrum fu¨r Informationstechnik Berlin, 14195 Berlin, Germany (Received 9 March 2011; published 26 September 2011) QCD lattice simulations with 2 þ 1 flavors (when two quark flavors are mass degenerate) typically start at rather large up-down and strange quark masses and extrapolate first the strange quark mass and then the up-down quark mass to its respective physical value. Here we discuss an alternative method of tuning the quark masses, in which the singlet quark mass is kept fixed. Using group theory the possible quark mass polynomials for a Taylor expansion about the flavor symmetric line are found, first for the general 1 þ 1 þ 1 flavor case and then for the 2 þ 1 flavor case. This ensures that the kaon always has mass less than the physical kaon mass. This method of tuning quark masses then enables highly constrained polynomial fits to be used in the extrapolation of hadron masses to their physical values. Numerical results for the 2 þ 1 flavor case confirm the usefulness of this expansion and an extrapolation to the physical pion mass gives hadron mass values to within a few percent of their experimental values. Singlet quantities remain constant which allows the lattice spacing to be determined from hadron masses (without necessarily being at the physical point). Furthermore an extension of this program to include partially quenched results is given. DOI: 10.1103/PhysRevD.84.054509 PACS numbers: 12.38.Gc I. INTRODUCTION and find that flavor blindness is particularly helpful if we approach the physical point, denoted by ðmRÃ;mRÃÞ, along The QCD interaction is flavor blind. Neglecting electro- l s a path in the mR À mR plane starting at a point on the magnetic and weak interactions, the only difference be- l s SUð Þ mR ¼ mR ¼ mR tween quark flavors comes from the quark mass matrix, 3 flavor symmetric line ( l s 0 ) and holding the sum of the quark masses m R ¼ 1 ðmR þ mR þ mRÞ which originates from the coupling to the Higgs field. We 3 u d s 1 R R R investigate here how flavor blindness constrains hadron 3 ð2ml þ ms Þ constant [1], at the value m0 as sketched in masses after flavor SUð3Þ symmetry is broken by the Fig. 1. The usual procedure (path) is to estimate the physical mass difference between the strange and light quarks. strange quark mass and then try to keep it fixed, i.e. R The flavor structure illuminates the pattern of symmetry ms ¼ constant, as the light quark mass is reduced towards breaking in the hadron spectrum and helps us extrapolate its physical value. However on that path the problem is that 1 2 þ 1 flavor lattice data to the physical point. (By 2 þ 1 we the kaon mass is always larger than its physical value. mean that the u and d quarks are mass degenerate.) Choosing instead a path such that the singlet quark mass We have our best theoretical understanding when all 3 is kept fixed has the advantage that we can vary both quark flavors have the same masses (because we can use quark masses over a wide range, with the kaon mass always the full power of flavor SUð3Þ symmetry); nature presents being lighter than its physical value along the entire trajec- R R tory. Starting from the symmetric point when masses are us with just one instance of the theory, with ms =ml 25 (where the superscript R denotes the renormalized mass). We are interested in interpolating between these two cases. 1In this article quark masses will be denoted by m, and hadron We consider possible behaviors near the symmetric point, masses by M. 1550-7998=2011=84(5)=054509(43) 054509-1 Ó 2011 American Physical Society W. BIETENHOLZ et al. PHYSICAL REVIEW D 84, 054509 (2011) background for this article, discuss the action used here and give tables of the hadron masses found. m R=m R s l Mostly we restrict ourselves to the constant surface. R* R* (ml ,ms ) However, in a few sections, we also consider variations in m R (for example in the derivation of the quark mass R s expansion polynomials, Sec. II C, the discussion of OðaÞ m ⎯mR=const improvement in Sec. II D, and in Sec. IV D where we generalize a constant m R formula). R R (m0 ,m0 ) II. THEORY FOR 1 þ 1 þ 1 FLAVORS 0 Our strategy is to start from a point with all three sea 0 R ml quark masses equal, R R R R FIG. 1 (color online). Sketch of the path (red, solid line) in the mu ¼ md ¼ ms m0 ; (1) mR À mR plane to the physical point denoted by ðmRÃ;mRÃÞ. The l s l s RÃ RÃ RÃ dashed diagonal line is the SUð3Þ-symmetric line. and extrapolate towards the physical point, ðmu ;md ;ms Þ, keeping the average sea quark mass degenerate is particularly useful for strange quark physics R 1 R R R m ¼ 3ðmu þ md þ ms Þ; (2) as we can track the development of the strange quark mass. R Also if we extend our measurements beyond the symmetric constant at the value m0 . For this trajectory to reach point we can investigate a world with heavy up-down the physical point we have to start at a point where R 1 RÃ quarks and a lighter strange quark. m0 3 ms . As we approach the physical point, the u and The plan of this article is as follows. Before considering d quarks become lighter, but the s quark becomes heavier. the 2 þ 1 quark flavor case, we consider the more general Pions are decreasing in mass, but K and increase in mass 1 þ 1 þ 1 case in Sec. II. This also includes a discussion of as we approach the physical point. the renormalization of quark masses for nonchiral fermi- ons. Keeping the singlet quark mass constant constrains the A. Singlet and nonsinglet renormalization extrapolation and, in particular, it is shown in this section Before developing the theory, we first briefly comment that flavor singlet quantities remain constant to leading on the renormalization of the quark mass. While for chiral order when extrapolating from a flavor symmetric fermions the renormalized quark mass is directly propor- point. This motivates investigating possible quark mass R tional to the bare quark mass, mq ¼ Zmmq, the problem, polynomials—we are able to classify them here to third at least for Wilson-like fermions which have no chiral order in the quark masses under the SUð3Þ and S3 (flavor) symmetry, is that singlet and nonsinglet quark mass can groups. In Sec. III we specialize to 2 þ 1 flavors and give renormalize differently [2,3]2 quark mass expansions to second order for the pseudosca- R NS S lar and vector meson octets and baryon octet and decuplet. mq ¼ Zm ðmq À m ÞþZmm; q ¼ u; d; s; (3) (The relation of this expansion to chiral perturbation theory is discussed later in Sec. V.) In Sec. IV we extend the where mq are the bare quark masses, formalism to the partially quenched case (when the valence m ¼ 1ðm þ m þ m Þ; quarks of a hadron do not have to have the same mass as 3 u d s (4) the sea quarks).
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