REVIEW
Pion and Kaon masses in Staggered Chiral Perturbation Theory
High Energy Physics - Lattice · Submitted 21 Apr 2003 · 20 references
Pith's verdict: A genuine correction to the multi-flavor Lee-Sharpe potential, with honest new parameters; the operator-basis completeness argument is the softest part, not the acknowledged root-trick step.
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Referee report
The paper
The paper claims
A corrected multi-flavor chiral Lagrangian lets lattice simulations of pions and kaons include taste-breaking lattice artifacts, at the price of two new unknown parameters.
The reading
What carries the argument
The load-bearing device is the distinction between single-trace and two-trace operators in the taste-breaking potential, combined with a resummation of the two-trace ('hairpin') vertices into exact flavor-neutral propagators. Because the single-trace part $U$ is flavor blind, it shifts all tastes of a given meson equally; because the two-trace part $U'$ couples flavor-neutral mesons of taste vector, axial vector, and singlet, it must be summed to all orders. The resummed propagator is written as $G_V = G_{0,V} + D_V$ with $D^V_{MN} = -a^2delta'_V prod_L(q^2+m^2_{LV}) / [(q^2+m^2_{MV})(q^2+m^2_{NV}) prod_F(q^2+m^2_{FV})]$, where $L$ runs over unmixed sea-quark states and $F$ over the true mass eigenstates. The identity $sum_M D^V_{XM} = -1/(q^2+m^2_{XV}) - (q^2+m^2_{XV}) D^V_{XX}/(a^2delta'_V)$ is what makes the many diagrams of Fig. 3 combine into the single proportional-to-mass term of eq. (48). The step from four tastes per flavor to one taste per flavor is then implemented by the rule that each sea-quark loop costs a factor of 4, i.e. $delta' to delta'/4$ before diagonalizing the mass matrix.
Extended reading notes
Core claim
The central claim is that the taste-breaking potential of the multi-flavor Lee–Sharpe Lagrangian, when derived by promoting the taste matrices to block-diagonal $4n times 4n$ spurions before any Fierz transformation, is not the naive promotion of the six single-trace operators. Operators $O_2$ and $O_5$ of the earlier treatment are not invariant under an axial rotation of a single flavor, and the correct potential instead contains four two-trace operators, eq. (14), of which only the combinations $C_{2V}-C_{5V}$ and $C_{2A}-C_{5A}$ enter at one loop. These produce tree-level mixing among flavor-neutral taste-vector and taste-axial mesons, resummed into 'disconnected' propagators $D_V$, $D_A$, $D_I$ given explicitly in eqs. (27), (30) and (31). The one-loop self-energy of a flavor-nonsinglet Goldstone meson then collapses to the compact form $sigma_{rm disc} = m^2_{P^+_5}/2 int (4D^V_{XY} + 4D^A_{XY} - D^I_{XY})$, eq. (48), which is proportional to $m^2_{P^+_5}$ as the separate axial symmetries for the valence quarks require. Evaluating the pole integrals with Lagrange's formula gives the explicit chiral logarithms of eqs. (61), (64) and (69), and adding the analytic $O(p^4)$ terms gives the fit-ready expressions of eqs. (71)–(76). A by-product is that the connected valence-only logarithms claimed in the earlier work do not exist for a charged pion; they belong only to flavor-neutral propagators.
Load-bearing premise
The conversion of the four-tastes-per-flavor result into the physical one-taste-per-flavor result assumes that taking the fourth root of the quark determinant corresponds, diagram by diagram, to dividing by four for every sea-quark loop; the paper calls this a by-hand adjustment justified by quark-flow reasoning rather than derived.
Editorial extensions
If this is right
- Lattice fits to staggered pion and kaon masses can now be done with the correct $O(a^2)$ chiral logarithms, so the residual lattice-spacing dependence of extrapolated results becomes a controlled systematic rather than an unknown.
- The corrected formulas contain two new low-energy constants, $a^2delta'_V$ and $a^2delta'_A$, which are not fixed by measured taste splittings and must be fit or estimated; a 2+1 fit to eq. (72) carries seven free parameters.
- The one-flavor comparison of tree-level staggered chiral perturbation theory to simulation data in the earlier literature is not justified, because simulations without disconnected propagators describe charged pions built from two flavors, not single-flavor neutral mesons.
- The same machinery extends to decay constants and, with heavy-quark symmetries added, to heavy-light meson quantities.
- If $delta'_A$ or $delta'_V$ is negative and large enough — the paper estimates the critical value in eq. (89) — a taste-axial or taste-vector flavor-neutral mass can vanish before the chiral limit, signalling an unusual phase of the staggered theory.
Reading between the lines
- The whole 1+1+1 and 2+1 program rests on an unproven correspondence: that the fourth-root trick acts on chiral diagrams as a simple factor of 1/4 per sea-quark loop. A direct lattice test — comparing a partially quenched quantity computed with four tastes against the rooted theory at the same parameters — would probe this step independently of the fit quality.
- Because $delta'_V$ and $delta'_A$ enter only through disconnected, flavor-neutral channels, they are exactly the quantities that lattice calculations of disconnected correlators could measure directly; the paper notes this as a possibility, and it would convert two nuisance parameters into predictions.
- The residual discretization error after continuum extrapolation at fixed quark mass is nonanalytic in the lattice spacing, growing like the chiral logs of the non-Goldstone tastes; this suggests that quoting a purely polynomial $a^2$ extrapolation error will underestimate the true systematic until $k a^2 Lambda_{QCD}^2 ll m_pi^2/Lambda^2$ is reached.
- The possible phase transition at negative $delta'$ is interesting because it would show up as a non-Goldstone pion mass vanishing at nonzero quark mass; a scan of the taste-axial flavor-neutral mass in existing ensembles would settle whether the fitted negative $delta'_A$ is physical or an artifact of truncated fitting.
Original abstract
We show how to compute chiral logarithms that take into account both the $\cO(a^2)$ taste-symmetry breaking of staggered fermions and the fourth-root trick that produces one taste per flavor. The calculation starts from the Lee-Sharpe Lagrangian generalized to multiple flavors. An error in a previous treatment by one of us is explained and corrected. The one loop chiral logarithm corrections to the pion and kaon masses in the full (unquenched), partially quenched, and quenched cases are computed as examples.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We show how to compute chiral logarithms that take into account both the $\cO(a^2)$ taste-symmetry breaking of staggered fermions and the fourth-root trick that produces one taste per flavor. The calculation starts from the Lee-Sharpe Lagrangian generalized to multiple flavors. An error in a previous treatment by one of us is explained and corrected. The one loop chiral logarithm corrections to the pion and kaon masses in the full (unquenched), partially quenched, and quenched cases are computed as examples.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We show how to compute chiral logarithms that take into account both the $\cO(a^2)$ taste-symmetry breaking of staggered fermions and the fourth-root trick that produces one taste per flavor. | contribution 0.98 | 0.94 | 0.59 | asserted 1.00 |
| s1 | The calculation starts from the Lee-Sharpe Lagrangian generalized to multiple flavors. | method 0.98 | 0.39 | 0.03 | asserted 1.00 |
| s2 | An error in a previous treatment by one of us is explained and corrected. | contribution 0.82 | 0.66 | 0.04 | asserted 0.99 |
| s3 | The one loop chiral logarithm corrections to the pion and kaon masses in the full (unquenched), partially quenched, and quenched cases are computed as examples. | contribution 0.69 | 0.95 | 0.28 | asserted 1.00 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:hep-lat/0304014 by the authors.
References
How the literature uses it
44 arXiv papers cite this in 120 sentences
Papers citing it · 76 sentences without a confident label
Builds on it 2 papers
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(Ref. [ref] only dealt with the case $n_r=1$, but it follows from the discussion in Ref. [ref] that an analogous form can be written down for integer $n_r1$ as well.) If we now set $n_r=n_s/4$, the underlying lattice theory is just lattice QCD with $n_s$ rooted staggered quarks, and there is only one lattice spacing $a_f$.
Effective field theories for rooted staggered fermions · Generalized theory · 0.98
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The results generalize those of Refs. [ref] to the mixed-action case; the results could be used to improve determinations of LECs poorly determined by existing analyses and to improve determinations of light-quark masses, the Gasser-Leutwyler couplings, and the pion and kaon decay constants.
Masses and decay constants of pions and kaons in mixed-action staggered chiral perturbation theory · Introduction · 1.00
Uses its method or tool 12 papers
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The errors introduced by the chiral and (light-quark) continuum extrapolations are controlled with rooted staggered chiral perturbation theory () [ref] applied to heavy-light mesons.
B- and D-meson decay constants from three-flavor lattice QCD · Chiral Perturbation Theory framework · 0.96
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The extrapolation of our lattice results to the physical quark masses and the continuum is guided by rS$χ$PT [ref] .
Semileptonic form factor ratio B_s->D_s/B->D and its application to BR(B^0_s->\mu^+\mu^-) · Results · 1.00
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To extrapolate our data to the physical light quark masses and to the continuum limit we use the functional form predicted by SChPT [ref] .
Finite volume effects in $B_K$ with improved staggered fermions · Introduction · 0.98
Uses its data or values 1 paper
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Appendix [ref] lists the integrals used in the form factor calculations; while Appendix [ref] collects necessary wavefunction renormalization factors that were calculated in Refs. [ref] .
Heavy-Light Semileptonic Decays in Staggered Chiral Perturbation Theory · Introduction · 0.89
Compares with it 2 papers
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The end result is an contribution from the connected diagram, and a non- contribution from the disconnected diagrams that matches that of Ref. [ref] if $m_P_5^+^2$ in their Eq. (48) is replaced with $B_0 (m_x + m_y)$. where $m_x$ and $m_y$ are the masses of valence quarks $x$ and $y$, respectively.
Electromagnetic Corrections in Staggered Chiral Perturbation Theory · Staggered Chiral Perturbation Theory with Electromagnetic Contributions · 0.85
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The functions that appear in the form factors are[For ease of comparison to Ref. [ref] , we use $I_1(m)$ instead of $ℓ(m^2)$ (as in Refs. [ref] ) for the chiral logarithm.] [equation] The main difference in these formulae with those of [ref] is that they keep the divergence pieces, while we have renormalized as in Refs. [ref] .
Heavy-Light Semileptonic Decays in Staggered Chiral Perturbation Theory · Form factors for 3-flavor partially quenched · 1.00
Cites it as background 12 papers
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The form of the taste breaking has been worked out as corrections to the chiral Lagrangian by Lee and Sharpe [ref] , with extension to multiple flavors by Aubin and Bernard [ref] .
Staggered chiral random matrix theory · Introduction · 0.92
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Staggered ChPT (SChPT) was developed to analyze results of lattice calculations with staggered fermions [ref] ; it has been used extensively to control extrapolations to physical light-quark masses and to remove dominant light-quark and gluon discretization errors [ref] .
Masses and decay constants of pions and kaons in mixed-action staggered chiral perturbation theory · Introduction · 0.86
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In the LCE regime the LO Lagrangian, which is of order $𝒪(p^2, m, a^2)$, reads as [ref]
Exploring the Aoki regime · Introduction · 0.91
Discussion
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