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arXiv:hep-lat/0209066v1 4 Sep 2002 1 Chiral logs with staggered fermions∗ C. Aubin,a C. Bernard,a C. DeTar,b Steven Gottlieb,c Urs M. Heller,d K. Orginos,e R. Sugarf and D. Toussaintg a Department of Physics, Washington University, St. Louis, MO 63130, USA bPhysics Department, University of Utah, Salt Lake City, UT 84112, USA cDepartment of Physics, Indiana University, Bloomington, IN 47405, USA dCSIT, Florida State University, Tallahassee, FL 32306-4120, USA eRIKEN BNL Research Center, Upton, New York 11973, USA fDepartment of Physics, University of California, Santa Barbara, CA 93106, USA gDepartment of Physics, University of Arizona, Tucson, AZ 85721, USA We compute chiral logarithms in the presence of “taste” symmetry breaking of staggered fermions. The lagrangian of Lee and Sharpe is generalized and then used to calculate the logs in π and K masses. We correct an error in Ref. [1]; the issue turns out to have implications for the comparison with simulations, even at tree level. MILC data with three light dynamical flavors can be well fit by our formulas. However, two new chiral parameters, which describe O(a2) hairpin diagrams for taste-nonsinglet mesons, enter in the fits. To obtain precise results for the physical O(p4) coefficients, these new parameters will need to be bounded, at least roughly. It has become clear that simulation at rather light quark mass is crucial for accurate determi- nation of physical parameters, e.g., heavy-light decay constants [2]. Since staggered fermions provide the fastest known method of simulating low-mass quarks, the systematic effects associ- ated with this fermion choice should be studied. We adopt the following nomenclature: A single staggered field describes 4 equivalent “tastes” of quarks in the continuum limit; taste symmetry is broken at O(a2). We use the word “flavor” for different staggered fields; lattice flavor symmetry is exact for equal masses. MILC’s improved stag- gered (“a2-tad”) simulations [3] use three fields (“u, d, s”) and reduce the tastes to one per fla- vor by taking 4√ Det. We call these “2 + 1” flavor simulations since we take mu = md ≡mℓ. MILC data for m2 π vs. quark mass show clear deviations from linearity, as expected from chi- ral logarithms. However, the detailed behavior at a ≈0.13 fm does not agree with continuum chi- ∗talk presented by C. Bernard at Lattice 2002; to be pub- lished in the proceedings ral perturbation theory (χPT) [4]. Figure 1 shows a fit to the continuum forms for m2 X/(m1 + m2), where X = π or K, and m1, m2 are quark masses. (We define “pions” to have m1 = m2 and “kaons” to have m2 ≈mphys s and m1 ̸= m2.) The fit is very poor, with a confidence level of 5 × 10−5. To get good fits, one needs to include the O(a2) taste-breaking effects into the chiral calculations, i.e., we need “staggered chiral perturbation the- ory” (SχPT). Such calculations start with the chiral lagrangian of Lee and Sharpe [5] for 1 stag- gered flavor (4 tastes). One defines Σ(x) ≡exp(iφ/f) ; φ = φ5ξ5 + φµξµ + · · · , (1) where φ is a 4 × 4 matrix of pseudoscalar mesons of various tastes, and ξ5, ξµ, … are taste matrices. The Lee-Sharpe lagrangian is then L = f 2 8 tr(∂µΣ∂µΣ†)−µmf 2 4 tr(Σ+Σ†)+a2V (2) The taste-breaking potential V has various op- erators with explicit taste matrices: −V
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2 Figure 1. m2 π,K/(m1 + m2) vs. m1 + m2, in units of the potential scale r1, for 2 + 1 flavor lattices at a ≈0.13 fm. The fit is to the continuum chiral log forms. The upper branch for (m1 + m2)r1 ∼0.2 uses the kaon form; the rest of the curve is the pion form. P6 i=1 CiOi. For example, O2
1 2[tr(ΣΣ) −tr(ξ5Σξ5Σ) + h.c.] (3) O5
1 2[tr(ξνΣξνΣ†) −tr(ξνξ5Σξ5ξνΣ†)] . (4) To get NLO chiral results relevant to 2+1 sim- ulations, we follow a three step procedure:
- Generalize the Lee-Sharpe lagrangian to the 3- flavor case, i.e., include three lattice staggered fields (u, d, s), each with four tastes, and masses mu = md ≡mℓ̸= ms. This is an “8 + 4” theory.
- Compute the desired quantities (here, m2 π and m2 K) at one loop in the 8 + 4 case.
- Adjust the 8+4 answer “by hand” to correspond to the 2 + 1 case of interest by identifying the chi- ral contributions that correspond to n virtual quark loops and dividing them by 4n. Step 1 turns out to be non-trivial: Ref. [1] em- ployed Fierz transformations to simplify the fla- vor structure of the generalized potential V. How- ever, Ref. [5] already used the Fierz freedom in deriving V. The net result is that the generalized operators O2 and O5 are incorrect in Ref. [1]. There are actually two operators that corre- spond to each of O2 and O5: O21 = 1 4[Tr(ξµΣ)Tr(ξµΣ) + h.c.] O22 = 1 4[Tr(ξµξ5Σ)Tr(ξ5ξµΣ) + h.c.] O51 = 1 2Tr(ξµΣ)Tr(ξµΣ†) O52 = 1 2Tr(ξµξ5Σ)Tr(ξ5ξµΣ†) . (5) In the absence of flavor indices, the combinations O21 + O22 and −O51 + O52 can be Fierzed into Lee-Sharpe O2 and O5, respectively; the other linear combinations turn into other Oi. The new operators have a surprising effect: they generate hairpin (disconnected) diagrams for taste-nonsinglet (but flavor neutral) mesons. For taste vector and taste axial vector mesons, this occurs at chiral tree level. The magnitudes of the hairpins are δ′ V ≡16a2 f 2 (C21 −C51); δ′ A ≡16a2 f 2 (C22 −C52) (6) for vector and axial taste, respectively. The disconnected hairpin diagrams have not been included in any simulations to date of taste- nonsinglet mesons. Thus the Lee-Sharpe la- grangian does not apply to such simulations: The chiral lagrangian requires more than one flavor to describe particles, e.g., π+ = u ¯d, that have no disconnected contributions. At 1-loop, we need to iterate three kinds of hairpins on flavor-neutral internal lines: the stan- dard anomaly hairpin (m2
- for taste singlets, and the new hairpins for taste vector and axial vector. Unlike m2 0, δ′ V,A cannot be taken to infinity. We therefore have to rediagonalize the mass matrix in the taste V, A channels. We call, for example, the mass eigenstates in the flavor-neutral, taste vec- tor channel π0 V , ηV and η′ V . Techniques in Ref. [6] are very useful for reexpressing the iterated prop- agator as a sum of simple poles. Adjustment to go from the 8 + 4 to the 2 + 1 theory is easy. Every hairpin interaction on a meson line after the first introduces an additional virtual quark loop. Therefore, if F(∆) is the sum of hairpin diagrams, just take F(∆) →4F( ∆ 4 ) for ∆= δ′ V , δ′ A or δ (where δ ≡m2 0/(24π2f 2)). Additional diagrams that were identified as va- lence diagrams in Ref. [1] turn out to be discon- nected and contribute only to flavor-neutral cor- relators. We define βK+ 5 to be the chiral log term for the Goldstone K+ mass (m1−loop K+ 5 )2/(µ(mℓ+ ms)) =
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3 1 + βK+ 5 /(16π2f 2) + · · ·, and similarly for βπ+ 5 . Our results are then: βπ+ 5 = 2δ′ V m2 η′ V −m2 ηV ” m2 η′ V −m2 SV m2 η′ V −m2 πV m2 η′ V ln m2 η′ V − m2 ηV −m2 SV m2 ηV −m2 πV m2 ηV ln m2 ηV
−4m2 πV ln m2 πV +(V →A) + m2 πI ln m2 πI −1 3m2 ηI ln m2 ηI βK+ 5 = 2δ′ V ” m2 η′ V ln m2 η′ V −m2 ηV ln m2 ηV m2 η′ V −m2ηV
+(V →A) + 2 3m2 ηI ln m2 ηI . (7) Here S is the s¯s meson. The continuum result is in the taste-singlet channel (I); the remainder vanishes in the limit δ′ V,A →0. With δ′ V,A as free parameters, these results give excellent fits to the MILC data. However, the parameters are poorly constrained: δ′ V and δ′ A wander offin opposite directions to large values (an order of magnitude larger than known taste- violating terms in the π+ sector). On the other hand, if we fix δ′ V or δ′ A to be of reasonable magni- tude, the fits find reasonable values for the other parameter, too, and the confidence levels are still excellent. Figure 2 shows the fit with δ′ Ar2 1 fixed to −0.1; δ′ V r2 1 is then found to be 0.25(11). Based on the measured taste violations, the natural size for these parameters is δ′r2 1 ∼0.2. We conclude that SχPT is necessary to fit cur- rent staggered lattice data. The taste-violating hairpins introduce two new free parameters (δ′ V,A) into the chiral theory; other taste violations, which split flavor-nonsinglet mesons, are not free parameters in the fits because they are measured directly in simulations. Unfortunately, the hair- pin parameters are not well constrained, at least at present. It does however appear to us that even a rough constraint on δ′ V,A (e.g., demanding that they be no more than 3 times the known taste- violating terms) will be sufficient to determine p4 analytic coefficients (Gasser-Leutwyler “Li”) with good precision. We may be able to constrain the hairpin param- eters in a variety of ways. Additional SχPT cal- culations are underway [7] that generalize these results to the quenched and partially quenched Figure 2. Same as Fig. 1, but using the SχPT forms, eq. (7). The parameter δ′ A is held fixed to a reason- ably small value in the fit. cases, and extend them to pseudoscalar decay constants (of both light-light and heavy-light mesons). With these calculations in hand, the direct constraints from all the fits may prove suf- ficient. Another approach is to compute the four- quark, taste-violating operators perturbatively [8]; the hairpin parameters may then be estimated by vacuum saturation or lattice calculation of the matrix elements. Finally, a direct lattice evalua- tion of the disconnected hairpin graphs may be possible. We are grateful to M. Golterman, G. P. Lepage, and S. Sharpe for very useful discussions. Com- putations were performed at LANL, NERSC, NCSA, ORNL, PSC and SDSC. This work was supported by the U.S. DOE and NSF. REFERENCES 1. C. Bernard, Phys. Rev. D65 (2002) 054031. 2. A. Kronfeld and S. Ryan, hep-ph/0206058; N. Yamada, review talk, these proceedings. 3. See, e.g., C. Bernard et al., Phys. Rev. D 64, 054506 (2001). 4. J. Gasser and H. Leutwyler, Nucl. Phys. B250, 465 (1985). 5. W. Lee and S.R. Sharpe, Phys. Rev. D60, 114503 (1999). 6. S. Sharpe and N. Shoresh, Phys. Rev. D62 (2000) 094503.
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4 7. C. Aubin and C. Bernard, work in progress. 8. G. P. Lepage, et al., work in progress.
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