REVIEW
Applications of chiral perturbation theory to lattice QCD
High Energy Physics - Lattice · Submitted 20 Dec 2009 · 121 references
Pith's verdict: A careful, honest set of lecture notes on extending ChPT to the lattice; the value is in the systematic spurion machinery and the explicit flagging of what is assumed rather than proved.
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Referee report
The paper
The paper claims
Chiral perturbation theory can be carried over to lattice QCD's unphysical regimes — nonzero lattice spacing, partially quenched and mixed actions, and a heavy kaon — with a controlled power counting, so lattice data can be fit with the non
The reading
What carries the argument
The load-bearing device is the spurion construction layered on the Symanzik effective theory. One first writes the lattice theory as a continuum theory plus a tower of higher-dimension operators suppressed by powers of $a$; one then makes each symmetry-breaking operator invariant by letting a spurion field transform under the chiral group in the same way the quark mass matrix does; finally one maps those operators onto the chiral lagrangian, where each becomes a new term with a new low-energy constant that must vanish as $a to 0$. This is what turns 'all operators allowed by the surviving lattice symmetries' — an unusable statement — into a finite list at each order in $p$ and $a$. For partial quenching the second device is Morel's ghost trick: adding bosonic quark fields whose determinant cancels the valence determinant gives a genuine euclidean path integral for the valence/sea split, and the graded symmetry group $SU(N_s+N_v|N_v)$ then dictates which terms can appear.
Extended reading notes
Core claim
The central claim is that scaling violations, partial quenching, mixed actions and a heavy kaon can each be absorbed into an effective chiral lagrangian with a well-defined power counting. For Wilson fermions the argument runs through the Symanzik effective theory: the lattice spacing is promoted to a spurion transforming like the quark mass matrix, so that the leading $O(a)$ operator has exactly the chiral structure of a mass term and can be absorbed into a shifted quark mass, $m_i' = m_i + (W_0/B_0)a$. At $O(a^2)$ new operators survive with their own low-energy constants, and the competition between these and the quark mass term produces a genuine phase structure — for two flavors, a first-order transition or, when a particular constant is negative, an Aoki phase of finite width in which isospin and parity break spontaneously and a pion condensate forms. For staggered fermions the same machinery yields taste splittings of order $a^2$ among the would-be Goldstone bosons. For partial quenching, a path integral with ghost quarks of bosonic statistics reproduces the valence/sea split, and the resulting chiral theory predicts flavor-neutral propagators with double poles whose residue is proportional to $m_{sea}^2 - m_{val}^2$, plus infrared divergences when valence and sea masses are taken too far apart. For mixed actions the author shows that, to one loop and for valence-only external legs, continuum partially quenched formulas survive with redefined sea and mixed meson masses.
Load-bearing premise
The whole construction assumes that a local effective field theory — and the Symanzik expansion it is built from — exists nonperturbatively for partially quenched and mixed-action QCD, an assumption the author states outright is unproven.
Editorial extensions
If this is right
- Lattice data at nonzero spacing can be fit with chiral formulas in which $a$-dependent mass shifts sit inside the logarithms, so that the curvature of the chiral logarithm is modified rather than fitted away.
- Wilson-fermion simulations at very small quark mass are predicted to show either a first-order transition or an Aoki phase of width of order $a^3$ in lattice units, depending on the sign of one low-energy constant.
- Partially quenched and mixed-action data carry double-pole and infrared-enhanced contributions whose size is set by the valence-sea mass splitting, giving a quantitative criterion for how far the two masses may be separated before the chiral expansion fails.
- Staggered-fermion meson masses are predicted to split into a small number of taste multiplets with splittings proportional to $a^2$, with the pattern fixed by which low-energy constants dominate.
- Kaon properties can be extrapolated in the light quark masses using a two-flavor theory in which the kaon is an explicit heavy field, without requiring the strange quark mass to be small.
Reading between the lines
- If the mixed-action one-loop reduction holds as claimed, it suggests a practical bookkeeping rule for any new fermion discretization: compute the sea-sector shift of the mass matrix once, and reuse the existing continuum partially quenched code.
- The Aoki-phase analysis implies a testable tension for Wilson simulations that push toward small quark masses: the same low-energy constant that controls whether pions can be massless at fixed $a$ also controls the size of the $a^2$ shift inside the chiral logarithms, so a lattice determination of one constrains the other.
- The double-pole residue proportional to $m_{sea}^2 - m_{val}^2$ is a sharper diagnostic than a generic 'sea and valence masses differ' warning, and could be used to decide empirically how much partial quenching a given ensemble tolerates.
- The heavy-kaon construction could be repurposed for other non-Goldstone hadrons whose mass sits near 1 GeV, provided one restricts to processes where the heavy field goes in and comes out.
Original abstract
These notes contain the written version of lectures given at the 2009 Les Houches Summer School "Modern perspectives in lattice QCD: Quantum field theory and high performance computing." The goal is to provide a pedagogical introduction to the subject, and not a comprehensive review. Topics covered include a general introduction, the inclusion of scaling violations in chiral perturbation theory, partial quenching and mixed actions, chiral perturbation theory with heavy kaons, and the effects of finite volume, both in the p- and epsilon-regimes.
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
These notes contain the written version of lectures given at the 2009 Les Houches Summer School "Modern perspectives in lattice QCD: Quantum field theory and high performance computing." The goal is to provide a pedagogical introduction to the subject, and not a comprehensive review. Topics covered include a general introduction, the inclusion of scaling violations in chiral perturbation theory, partial quenching and mixed actions, chiral perturbation theory with heavy kaons, and the effects of finite volume, both in the p- and epsilon-regimes.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | These notes contain the written version of lectures given at the 2009 Les Houches Summer School "Modern perspectives in lattice QCD: Quantum field theory and high performance computing." The goal is to provide a pedagogical introduction to the subject, and not a comprehensive review. | background 0.48 | 0.54 | 0.03 | asserted 0.96 |
| s1 | Topics covered include a general introduction, the inclusion of scaling violations in chiral perturbation theory, partial quenching and mixed actions, chiral perturbation theory with heavy kaons, and the effects of finite volume, both in the p- and epsilon-regimes. | contribution 0.92 | 0.13 | 0.05 | asserted 0.95 |
What was examined, and what was not
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Figures
Showing 7 of the paper's 9 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:0912.4042 by the authors.
References
How the literature uses it
42 arXiv papers cite this in 47 sentences
Papers citing it · 18 sentences without a confident label
Uses its method or tool 2 papers
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At leading order in CHPT, i.e. $𝒪(p^2)$, with $p$ representing a small momentum or a Goldstone boson mass, the separation of the different Wick contractions can be done by merely extending the normal SU(3) CHPT to SU(4) with one auxiliary quark flavor [At higher orders, one needs to use partially quenched CHPT [ref] (for reviews, see Refs. [ref] ), and examples may be found in Refs. [ref] .], to be denoted by $j$.
Tetraquarks, hadronic molecules, meson-meson scattering and disconnected contributions in lattice QCD · Introduction · 0.80
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The derivation of the propagators for the Lagragian Eq. [ref] can be performed following a similar procedure as in [ref] .
Hadronic light-by-light contribution to $(g-2)_\mu$ from lattice QCD: a complete calculation · U(N|M) · 0.94
Compares with it 1 paper
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This is very much unlike QCD, where, if one plots the differences between the squares of the masses of the different tastes, one would find virtually no dependence on the fermion mass, i.e., horizontal lines (see, for example, Fig. 3 in Ref. [ref] ).
Fits of $SU(3)$ $N_f=8$ data to dilaton-pion effective field theory · Taste breaking · 0.83
Cites it as support 1 paper
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There are analytic studies that show that the reliable low energy portions from such loops are very small [ref] , and lattice studies have definitively shown that the chiral loop effects are not strongly present at such large masses [ref] .
Equivalence Principle Violations and Couplings of a Light Dilaton · Appendix: The strange quark mass · 0.97
Cites it as background 23 papers
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Staggered chiral perturbation theory (SChPT) has been used extensively to control extrapolations from unphysical light-quark simulation masses to physical masses and to remove dominant light quark and gluon discretization effects [ref] .
Flavored pions and kaons at next-to-leading order in mixed-action staggered chiral perturbation theory · Introduction · 0.85
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Staggered ChPT (SChPT) [For a review, see Ref. [ref] .] predicts that taste splittings are of order $α^2_s a^2$, and that, to that order in the lattice spacing, the $SU(4)$ taste symmetry is only partially broken to $SO(4)$, while it breaks down completely to the lattice symmetry group at order $a^4$ and beyond [ref] .
Theory review for hadronic corrections to $g-2$ · Staggered fermions · 0.86
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In staggered QCD, taste splittings can be understood in terms of staggered ChPT [ref] (for reviews, see Refs. [ref] ).
Fits of $SU(3)$ $N_f=8$ data to dilaton-pion effective field theory · Taste breaking · 0.88
Discussion
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