Journal ArticleThe fourth root approximation in LQCD simulations with dynamical staggered fermions requires justification. We test its validity numerically in the interacting theory in a renormalization group framework

Bernard, C.

DeTar, C.

English

The University of Utah: J. Willard Marriott Digital Library

arXiv:hep-lat/0509176 v1   29 Sep 2005The locality of the fourth root of staggered fermiondeterminant in the interacting case.C. BernardDepartment of Physics, Washington University, St. Louis, MO 63130, USAE-mail: cb@lump.wustl.eduC. DeTar and F. Maresca∗Department of Physics, University of Utah, Salt Lake City, UT 84112, USAE-mail: detar@physics.utah.edu, maresca@physics.utah.eduSteven Gottlieb and L. LevkovaDepartment of Physics, Indiana University, Bloomington, IN 47405, USAE-mail: sg@physics.indiana.edu, llevkova@indiana.eduU.M. HellerAmerican Physical Society, One Research Road, Box 9000, Ridge, NY 11961, USAE-mail: heller@aps.orgJ.E. HetrickPhysics Department, University of the Pacific, Stockton, CA 95211, USAE-mail: jhetrick@pacific.eduR. SugarDepartment of Physics, University of California, Santa Barbara, CA 93106, USAE-mail: sugar@vulcan2.physics.ucsb.eduD. Renner and D. ToussaintDepartment of Physics, University of Arizona, Tucson, AZ 85721, USAE-mail: doug@physics.arizona.edu, dru@physics.arizona.eduThe fourth root approximation in LQCD simulations with dynamical staggered fermions requiresjustification. We test its validity numerically in the interacting theory in a renormalization groupframework.XXIIIrd International Symposium on Lattice Field Theory25-30 July 2005Trinity College Dublin, Ireland∗Speaker.PoS(LAT2005)114c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/The fourth root approximation: interacting theory. F. Maresca1. IntroductionNumerical simulations of dynamical fermions within the framework of the staggered formal-ism are both computationally cost effective and phenomenologically successful. Exploiting theadvantageous properties of an improved staggered fermion formulation, various collaborations areperforming high-precision lattice QCD calculations that are in excellent agreement with experi-mentally known measurements [1]. However, this success is clouded by the long-standing prob-lematic issue of the validity of the fourth root approximation. The staggered fermion describes, infact, four tastes on the lattice, so in order to study QCD with N f = 2 or N f = 2 + 1, a Boltzmannweighting is used that contains a fractional power of the fermion determinant. Because of taste-breaking at nonzero lattice spacing, taking the fractional power of the staggered determinant beforerestoring taste symmetry is conceptually nontrivial. The issue whether the fourth root prescriptiongives a lattice theory in the right universality class to reproduce QCD is yet unresolved and thephenomenological success of the staggered formulation has heated up the discussion. In the pastyears various numerical investigations [2] have addressed this concern. Recently a direct approachwas adopted towards establishing whether the universality class is the right one: if a single-tastelocal fermion action can be found, whose determinant is equal to the fourth root of the staggeredfermion determinant up to cutoff effectslima→0( det Dst)1/4 = det D det H (1.1)(where D is a local Dirac operator and H is local and contains only cutoff effects) then the fourthroot prescription can be consistently fitted into the framework of a local field theory. In the freetheory numerical [3] and analytical studies [4, 5] showed that such a local operator exists. Inparticular, in [5] Shamir applied renormalization group blocking to the free staggered operator D0in the spin ⊗ taste representation. After n blocking steps Qn, the fermionic degrees of freedom liveon the coarse lattice with lattice spacing ac = 2na and the determinant of the staggered operatordecomposes as det Dst(a = 2−nac) = det Dn det G−1n . More details can be found in [5], whereit was proved analytically that the blocked propagator D−1n = α−1 + QnD−10 Q†n factorizes, in thelimit n → ∞, as D−1∞= Drg⊗ I and that G−1n = D0 + α Q†nQn is a local operator that contains onlycutoff effects. This completes the proof that in the continuum limit, i.e. a(= 2−nac) → 0, thedecompositions of Eq. (1.1) holds in the free case.2. Renormalization group transformation in the interacting caseIn this section we briefly discuss the renormalization group (RG) program adopted to testthe interacting case. The new complication is that a mapping from the one-component staggeredfermion basis to the spin ⊗ taste basis is not unique. We define the interacting theory in theone-component formalism. The first fermion RG blocking transformation is used to perform thetransition to the spin ⊗ taste representation:(Q(1)ψ)aα(2x) =12 ∑rµ=0,1(γr11 γr22 γr33 γr44 )(aα) Wr(2x,2x+ r)ψ(2x+ r), (2.1)PoS(LAT2005)114114 / 2The fourth root approximation: interacting theory. F. Marescawhere a and α are Dirac spin and taste indices respectively. We require the parallel transportersWr(2x,2x+ r) to be a sum over all the shortest paths 1 between the origin 2x and the other sixteensites 2x + r of the hypercube. The sum over different paths is thought to reduce the breakingof hypercubic symmetry; however, the choice to parallel transport all the points within a givenhypercube to the corner r = 0 unavoidably introduces some asymmetries. The subsequent RGsteps are a covariant generalization of the arithmetic mean on a 24 hypercube used in [5] :(Q(n)ψ)(2x) = 2−4 ∑rµ=0,1Wr(2x,2x+ r)ψ(2x+ r) n > 1. (2.2)The shortest paths that compose the parallel transporters Wr(2x,2x+ r) in Eq. (2.2) are constructedfrom links of the blocked lattice (with lattice spacing 2n a). These blocked links are built followinga program suggested in [7]. In detail, the links that live on the fine lattice (with lattice spacing2n−1 a) are twice APE smeared and projected back to SU(3). The blocked links are then builtmultiplying two of these fine smeared links in line. Once the blocking kernels are defined weproject the nondiagonal piece of the blocked propagator D−1n onto the spin ⊗ taste Clifford space:(QnD−10 Q†n)(x,y = 0) = ∑S,TΓS ⊗Γ†T MST (x,y = 0). (2.3)Here x and y = 0 2 live on the coarsest lattice and Qn can be regarded as a ’big’ blocking stepQn = Q(n)Q(n−1) · · ·Q(1) that transforms the original lattice with lattice spacing a into the blockedlattice with lattice spacing ac = 2na. The coefficients MST (x,y = 0) of the projection in Eq. (2.3) areevaluated numerically and averaged over two different lattice ensembles. We fixed Lorentz gaugebefore doing any RG blocking. Parameters of the simulations are shown in Table 1. Applying adifferent number of RG steps, n, on each of these two ensembles, the resulting blocked latticeshave the same coarse-lattice spacing of ac = 0.72 fm. This allowed us to see how the coefficientsMST scale with one additional RG blocking step.unblocked lattice 163 ×48 403 ×96a 0.18 fm 0.09 fmms a 0.125 0.05number of RGT’s 2 3blocked lattice 43×12 53×12ac 0.72 fm 0.72 fmnumber of cfgs 148 56Table 1: Simulation parametersAs a consistency check on the blocking procedure described above, we evaluated the massof the (ss) meson. In fact, the spectrum on the blocked lattice is expected to be identical to thespectrum in the original theory. The meson mass can be evaluated directly from the coefficientsMST (x,y) of Eq. (2.3):1These are built using APE smeared links of the original lattice to be consistent with the RG program adopted forthe subsequent RG steps.2The fermion source is defined at the origin.PoS(LAT2005)114114 / 3The fourth root approximation: interacting theory. F. MarescaCTss(t) = ∑S,T ′,~x< Tr(τ5ΓT ΓT ′ΓT τ5Γ†T ′) |MST ′(t,~x)|2 >, (2.4)where the ΓS’s and the ΓT,T ′’s are the 16 Dirac matrices in the normal and adjoint (τµ = γ⋆µ ) rep-resentation respectively. In Figure 1 we present the taste splittings for the meson masses evaluatedon the two blocked lattices. At this level of statistics, the decrease in splitting from n = 2 to n = 3is consistent with the expected O(a2α) or O(a2α2), but the preliminary RG blocked splittings aresystematically higher than values obtained in direct measurements at much lighter valence quarkmasses. Further study is needed.000.250.50.7511.251.51.752( MT2- M52) r12τ05τi5τijτi0τiτ01n = 2 n = 3Figure 1: Pseudoscalar taste splittings after 2 and 3 RG blockings in units of r1, the distance at which thestatic quark potential satisfies r2dV/dr = 1.3. Preliminary resultsIn this preliminary study we have analyzed for various displacements |x− y| the dominantterms in Eq. (2.3) that are also present in the free theory:∑µ(γµ ⊗ I)A(n)µ (x,y)+(γ5 ⊗ τ5τµ)B(n)µ (x,y)+ ∑ν 6=µ(γ5⊗ τ5τν)B′ (n)ν (x,y)+(I ⊗ I) C(n)(x,y)+ · · · · · ·(3.1)Comparing Figures 2 and 3, it is remarkable to see that increasing by only one unit the numberof RG blocking transformations the mass term (I⊗ I) (  ) becomes larger than the taste breakingterm γ5⊗ τ5τi ( ⋄ ) for any direction i for displacement |x− y|= 1. This is in agreement with whatwe expect from the free theory: in the limit n → ∞ the only terms in MST that should survive arediagonal in taste space.Other terms in Eq. (3.1) are seen in our simulations. Their contribution is 10− 1000 timessmaller than the leading kinetic term A(n)µ in Eq. (3.1) and this make it difficult to identify theterms that are statistically relevant. We need to increase the number of configurations to have abetter understanding of the statistics. Besides this issue, it is crucial to know how these termsscale with n to determine whether the blocked propagator becomes diagonal in taste space in theinteracting theory as it does in the free theory. However an element that can play an importantrole in this scaling is the breaking of the hypercubic symmetry, caused by the introduction of theparallel transporters, as it has been briefly discussed in the previous section. In fact, Figures 2PoS(LAT2005)114114 / 4The fourth root approximation: interacting theory. F. Maresca00.00050.0010.00150.0020.00250.003| MST |γi ⊗ Ιγ5  ⊗ τ5τi Ι ⊗ Ιγ5  ⊗ τ5τk  (k ≠ i) Displacement of 1 163x48 latticen = 2;(+)cfgs = 148(-)(i)+x +y +z +t -x -y -z -tFigure 2: MST after n = 2 RG steps.05e-050.00010.000150.00020.00025| MST |γi ⊗ ΙΙ ⊗ Ιγ5  ⊗ τ5τi γ5  ⊗ τ5τk  (k ≠ i) Displacement of 1 403x 96 latticen = 3;(+)cfgs = 56(-)(i)+x +y +z +t -x -y -z -tFigure 3: MST after n = 3 RG steps.and 3 suggest that the blocking procedure is introducing some hypercubic asymmetries, since thecoefficients MST (x,y) are not perfectly degenerate under reflections about the principal axes. Inorder to investigate further these terms, present only in the interacting theory, it is necessary eitherto quantify the hypercubic symmetry breaking or to redefine the RG blocking transformations inorder to guarantee hypercubic invariance (one suggestion is presented at this conference [6]).4. Interacting theory vs free theoryIn [5] the scaling properties of the taste breaking terms B(n)µ (p) were evaluated analytically. Itwas shown in momentum space that they scale like 2−n, uniformly in p, so the blocked propagatorbecomes diagonal in taste space, when n → ∞. Postponing the issue of the breaking of hypercubicsymmetry to a later study, we consider here a linear combination of the amplitudes |MST (x,y)|under reflections and rotations about the principal axes. In Figures 4 and 5 we show how the ratiosbetween the taste violating terms and the leading kinetic term A(n) scale with n.n00.250.50.751B / AB(n=2) / A(n=2)B(n=3) / A(n=3)B(n=2) / A(n=2) (free)B(n=3) / A(n=3) (free)2 3 2 3Displacement 1+1Displacement 1Figure 4: Taste breaking term B(n) over the ki-netic term.n00.20.40.60.8B’ / AB’(n=2) / A(n=2)B’(n=3) / A(n=3)B’(n=2) / A(n=2) (free)B’(n=3) / A(n=3) (free)2 3 2 3Displacement 1+1Displacement 1Figure 5: Taste breaking term B′(n) over the ki-netic term.Our data show that they diminish when n is increased and that their magnitudes are also re-markably in agreement with the free theory represented by solid lines. In Figures 6 and 7 we showPoS(LAT2005)114114 / 5The fourth root approximation: interacting theory. F. Marescathe scaling properties of the taste breaking term divided by the mass term. The interacting theoryagrees with the free theory for displacements |x− y| = 1,2, however a discrepancy is seen at zerodisplacement where the interacting theory does not scale as expected. The zero displacement is notinteresting when discussing locality thus this behavior, even if not well understood, does not spoilthe good scaling properties we are seeing in the interacting theory.n00.511.52B / CB(n=2) / C(n=2)B(n=3) / C (n=3)2 3 2 3Displ. 1+1Displ. 1Displ. 02 3Figure 6: Taste breaking term divided by the massterm: interacting theoryn00.511.522.53B / CB(n=2) / C(n=2) (free)B(n=3) / C (n=3) (free)2 3 2 3Displ. 1+1Displ. 1Displ. 02 3Figure 7: Taste breaking term divided by the massterm: free theory.The conclusion that we can draw from this preliminary study is that the blocked propagator inthe interacting case is dominated by the same terms present in the free theory and that these termsscale as expected to make the decomposition of Eq. (1.1) possible. The statistical relevance and thescaling properties of other terms seen in the interacting theory is still under investigation.We thank Yigal Shamir and Maarten Goltermann for helpful discussions during the confer-ence. Computing support from the Utah Center for High Performance Computing is gratefullyacknowledged. This work was supported by the US NSF and DOE.References[1] C.T.H. Davies et al.[HPQCD, UKQCD and MILC Collaborations] High-precision lattice QCDconfronts experiment, Phys.Rev.Lett.92:022001,2004, [hep-lat/0304004][2] S. Dürr and C. Hoelbling, Scaling tests with dynamical overlap and rooted staggered fermions,Phys.Rev.D71:054501,2005, [hep-lat/0411022]; E. Follana et al., The index theorem and universalityproperties of the low-lying eigenvalues of improved staggered quarks, Phys.Rev.Lett.93:241601,2004,[hep-lat/0406010][3] F. Maresca and M. Peardon, A path-integral representation of the free one-flavor staggered-fermiondeterminant, [hep-lat/0411029][4] D. Adams, On the fourth root prescription for dynamical staggered fermions, [hep-lat/0411030][5] Y. Shamir, Locality of the fourth root of the staggered fermion determinant: renormalization-groupapproach, Phys.Rev.D71:034509,2005, [hep-lat/0412014][6] Y. Shamir, These proceedings.[7] T. DeGrand et al. The classically perfect fixed point action for SU(3) gauge theories,Nucl.Phys.B454:587-614,1995, [hep-lat/9506030]PoS(LAT2005)114114 / 6

Locality of the fourth root of staggard fermion determinant in the interacting case

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Locality of the fourth root of staggard fermion determinant in the interacting case

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