REVIEW
Electromagnetic Splittings and Light Quark Masses in Lattice QCD
High Energy Physics - Lattice · Submitted 6 Feb 1996 · 14 references
Pith's verdict: A genuinely new lattice method with an honest preliminary result; the soft spot is the unquantified truncation of the charge expansion, not circularity.
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Referee report
The paper
The paper claims
By putting a live electromagnetic field on the lattice alongside the gluons, the authors compute how much of the pion and kaon mass splittings comes from photons, and use the kaon splitting to extract a light quark mass ratio of 0.512.
The reading
What carries the argument
The load-bearing device is the separation of the pseudoscalar mass-squared into a charge-symmetric constant plus a term linear in each bare quark mass, with the constant fixed by Dashen's theorem to scale as the square of the total meson charge and the slope expanded as a power series in the quark charges. This turns a set of measured meson masses at unphysical charges into a small set of coefficients, from which the physical-charge splittings follow by setting $e$ to its true value. Two supporting pieces do real work: the noncompact photon action, which keeps the pure-gauge sector free and lets configurations be drawn directly in momentum space, and a resummed tadpole estimate of the electromagnetic self-energy that shifts the critical hopping parameter separately for each quark charge so that the chiral extrapolation is anchored correctly.
Extended reading notes
Core claim
The central claim is that electromagnetic contributions to light hadron masses can be computed nonperturbatively in lattice QCD and are large enough, when the quark charges are artificially enhanced, to be fitted reliably. The authors generate 200 quenched SU(3)$times$U(1) configurations at $beta=5.7$ on a $12^3times24$ lattice, with the photon field given a noncompact Gaussian action in Coulomb gauge and coupled to quarks through the usual compact link phase. Propagators for four charges ($e_q = 0, -0.4, +0.8, -1.2$ in electron units) and three bare masses yield 78 independent meson states, whose squared masses are fitted to $m_P^2 = A(e_q,bar e_q) + m_q B(e_q,bar e_q) + bar m_q B(bar e_q,e_q)$ with $A$ proportional to $(e_q+bar e_q)^2$ and $B$ expanded through order $e^4$. The fit gives $m_{pi^+}-m_{pi^0} = 4.9(3)$ MeV against an experimental $4.6$ MeV, and, using the measured $K^0$-$K^+$ splitting, $m_u/m_d = 0.512(6)$ together with $m_u = 3.86(3)$, $m_d = 7.54(5)$ and $m_s = 147(1)$ MeV. The authors also decompose the $4.0$ MeV kaon splitting into $5.9$ MeV from the up-down mass difference and $-1.9$ MeV from electromagnetism, where the lowest-order treatment gives $5.3$ and $-1.3$ MeV.
Load-bearing premise
The whole result rests on the assumption that the response of meson masses to quark charge, measured at two to six times the physical electric charge, can be extrapolated back to the physical charge using only the first few terms of a power series in $e^2$.
Editorial extensions
If this is right
- The $pi^+$-$pi^0$ splitting comes out at $4.9(3)$ MeV against an experimental $4.6$ MeV, so the method reproduces a purely electromagnetic mass difference from first principles in QCD plus QED.
- The kaon splitting decomposes into $5.9$ MeV from the up-down mass difference and $-1.9$ MeV from electromagnetism, shifting the electromagnetic share away from the value assumed in the standard lowest-order analysis.
- The extracted ratios $m_u/m_d = 0.512(6)$ and $(m_d-m_u)/m_s = 0.0249(3)$ are renormalisation-scheme independent and can be compared directly with chiral perturbation theory and with the $eta to 3pi$ decay rate.
- Because the photon is massless and unconfined, finite-volume corrections fall as inverse powers of the box size rather than exponentially; the authors estimate a $0.3$ MeV upward correction to the splitting, about six percent, on their $12^3times24$ box.
- The same configuration set can be reused for electromagnetic splittings of vector mesons, baryons and heavy-light systems, and for magnetic moments and form factors.
Reading between the lines
- The quoted statistical errors, in the third decimal place, are far smaller than the systematic uncertainties the authors flag, so the practical precision of $m_u/m_d$ will be set by continuum extrapolation, box size and the quenched approximation rather than by statistics.
- If the electromagnetic shift of the kaon splitting is genuinely $-1.9$ MeV rather than $-1.3$ MeV, the same correction should show up in other isospin-breaking quantities such as the $eta to 3pi$ amplitude and the $pi^+$-$pi^0$ mass difference in nuclei; a consistent pattern across those would be a strong independent check.
- The enhancement trick, running at several times the physical coupling and extrapolating back, is a template that could be applied to other small couplings in lattice simulations where the physical effect is buried under statistical noise.
- A massless up quark remains a live possibility in the authors' framing; the same fit, repeated with $m_u to 0$, would test whether the data prefer a nonzero up mass rather than assuming it.
Original abstract
A method for computing electromagnetic properties of hadrons in lattice QCD is described and preliminary numerical results are presented. The electromagnetic field is introduced dynamically, using a noncompact formulation. Employing enhanced electric charges, the dependence of the pseudoscalar meson mass on the (anti)quark charges and masses can be accurately calculated. At $\beta=5.7$ with Wilson action, the $\pi^+-\pi^0$ splitting is found to be $4.9(3)$ MeV. Using the measured $K^0-K^+$ splitting, we also find $m_u/m_d = .512(6)$. Systematic errors are discussed.
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
A method for computing electromagnetic properties of hadrons in lattice QCD is described and preliminary numerical results are presented. The electromagnetic field is introduced dynamically, using a noncompact formulation. Employing enhanced electric charges, the dependence of the pseudoscalar meson mass on the (anti)quark charges and masses can be accurately calculated. At $\beta=5.7$ with Wilson action, the $\pi^+-\pi^0$ splitting is found to be $4.9(3)$ MeV. Using the measured $K^0-K^+$ splitting, we also find $m_u/m_d = .512(6)$. Systematic errors are discussed.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A method for computing electromagnetic properties of hadrons in lattice QCD is described and preliminary numerical results are presented. | contribution 1.00 | 0.88 | 0.11 | asserted 0.98 |
| s1 | The electromagnetic field is introduced dynamically, using a noncompact formulation. | method 1.00 | 0.82 | 0.03 | asserted 0.99 |
| s2 | Employing enhanced electric charges, the dependence of the pseudoscalar meson mass on the (anti)quark charges and masses can be accurately calculated. | method 0.99 | 0.76 | 0.14 | asserted 0.78 |
| s3 | At $\beta=5.7$ with Wilson action, the $\pi^+-\pi^0$ splitting is found to be $4.9(3)$ MeV. | result 1.00 | 0.97 | 0.26 | asserted 1.00 |
| s4 | Using the measured $K^0-K^+$ splitting, we also find $m_u/m_d = .512(6)$. | result 1.00 | 0.93 | 0.27 | asserted 1.00 |
| s5 | Systematic errors are discussed. | limitation 0.94 | 0.56 | 0.02 | unclear 0.68 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:hep-lat/9602005 by the authors.
References
How the literature uses it
40 arXiv papers cite this in 60 sentences
Papers citing it · 26 sentences without a confident label
Uses its method or tool 12 papers
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We use the electro-quenched approximation [ref] to partially include the dynamics of QED.
Calculating the QED correction to the hadronic vacuum polarisation on the lattice · Simulation Details · 0.99
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The $U(1)$ photon fields are then multiplied with the $SU(3)$ gauge links [ref] [equation] In the following we will refer to this method as the stochastic method.
Electromagnetic Corrections to Meson Masses and the HVP · Introduction · 0.90
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The methods initially introduced in [ref] make it possible to include QED effects on the lattice explicitly and thereby enable us to differentiate between isospin breaking through charge and different up and down quark masses.
Electromagnetic corrections to pseudoscalar decay constants · Introduction · 0.86
Compares with it 1 paper
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In the pioneering work by Duncan, Eichten, and Thacker [ref] which used the quenched approximation for both QCD and QED and employed Wilson fermions at a single coarse lattice spacing, the pion mass splitting was found to be 4.9(3) MeV.
Determination of light quark masses from the electromagnetic splitting of pseudoscalar meson masses computed with two flavors of domain wall fermions · summary · 0.85
Cites it as background 16 papers
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In order to circumvent this problem, many regularizations of the lattice QED action have been proposed, that either put contraints on the Fourier modes of the photon field, like $$ [ref] and its infrared improvements [ref] , $$ [ref] and $$ [ref] , or provide a small mass to the photon like $$ [ref] , or adopt instead different boundary conditions like $$ [ref] .
Isospin-breaking corrections to weak decays: the current status and a new infrared improvement · Introduction · 0.95
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Examples include $K_L$-$K_S$ mixing [ref] and $ϵ_K$ [ref] , rare kaon decays [ref] , neutrinoless double-beta decays [ref] , electroweak box contributions to semileptonic decays [ref] , inclusive $B$-meson decays [ref] , nucleon Compton amplitudes [ref] , as well as isospin-breaking effects in hadronic spectra [ref] , the hadronic vacuum polarization function [ref] , leptonic decays [ref] and $K→ππ$ decays [ref] .
Finite-volume effects in long-distance processes with massless leptonic propagators · Introduction · 0.93
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The inclusion of quenched QED effects was first performed in [ref] on quenched QCD configurations, in [ref] on $N_f=2$ QCD configurations and in [ref] on $N_f=2+1$ configurations, at a single lattice spacing, with rather large pion masses and in small volumes.
Up and down quark masses and corrections to Dashen's theorem from lattice QCD and quenched QED · Introduction · 0.94
Discussion
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