REVIEW 4 major objections 5 minor 3 cited by
Proton and neutron electromagnetic form factors from lattice QCD in the continuum limit
T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read This paper computes proton and neutron electromagnetic form factors directly at the physical pion mass in the continuum limit of lattice QCD, with disconnected contributions included, yielding radii and magnetic moments for both nucleons.
desk verdict A serious lattice calculation that likely gives the first physical-pion continuum limit for nucleon EM form factors, but the model-averaged systematic error understates the spread between fit families and needs reworking. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing mechanism is a one-step continuum extrapolation: for each ensemble the Sachs form factors are extracted at many values of $Q^2$ by multi-state fits to the two- and three-point nucleon correlation functions, and then all three ensembles are fit together to a $z$-expansion whose coefficients depend linearly on $a^2$, so that radii, moments, and continuum limit come out of one combined fit. Disconnected quark-loop contributions are computed with the local vector current renormalized in the RI'/MOM scheme; the paper confirms that the singlet and nonsinglet renormalization factors coincide, allowing the disconnected and connected pieces to be added safely. Excited-state systematics are handled by allowing different excited-state energies in the two- and three-point functions and by Akaike-weighted model averaging over fit ranges. Dipole and Galster-like parameterizations are used as cross-checks, and the one-step and two-step continuum extrapolations agree.
What would settle it
Compute the same observables on a fourth, finer ensemble, for example at $a\approx0.04$ fm, or refit the present data with an $a^4$ term added to the $z$-expansion coefficients; if $\sqrt{\langle r_E^2\rangle^p}$ shifts by more than the quoted $0.023$ fm systematic error, the linear-in-$a^2$ extrapolation is the fragile step.
Extended reading notes
Core claim
The central claim is that proton and neutron electromagnetic form factors can be obtained in the continuum limit directly at the physical pion mass. With $N_f=2+1+1$ twisted-mass clover-improved fermions, the paper obtains the electric and magnetic Sachs form factors $G_E(Q^2)$ and $G_M(Q^2)$ for both nucleons on three ensembles at $a=0.080$, $0.068$, and $0.057$ fm, including disconnected contributions renormalized with the flavor-singlet vector current. The final model-averaged results are $\sqrt{\langle r_E^2\rangle^p}=0.860(38)(23)$ fm, $\langle r_E^2\rangle^n=-0.147(48)$ fm$^2$, $\sqrt{\langle r_M^2\rangle^p}=0.870(53)(15)$ fm, $\sqrt{\langle r_M^2\rangle^n}=0.913(67)(19)$ fm, $\mu^p=2.849(92)(52)$, and $\mu^n=-1.819(76)(29)$, with the $z$-expansion fits carrying the largest model-averaging weight. The neutron electric form factor, which is hard to measure directly, comes out more precise than the experimental one, and the computed Zemach and Friar radii agree with other determinations.
Load-bearing premise
All lattice artifacts are assumed to be captured by a straight-line dependence on the square of the lattice spacing ($a^2$) across only three lattice spacings, with the lattice spacing values taken from an earlier paper; a bend in that line would shift every quoted radius and moment.
Editorial extensions
If this is right
- Lattice QCD results for nucleon form factors can now be compared with electron-scattering and muonic-hydrogen measurements without any chiral extrapolation.
- The proton electric radius from first principles, $\sqrt{\langle r_E^2\rangle^p}=0.860(38)(23)$ fm, has errors that cover both the electron-scattering and muonic-hydrogen values, so it provides an independent data point without resolving the proton radius puzzle.
- The neutron electric form factor, measured only indirectly in experiment, is obtained more precisely from lattice QCD than from experiment, making lattice QCD the more accurate source for that quantity.
- The same continuum-limit form factors yield Zemach and Friar radii, giving lattice-based input for hydrogen hyperfine-splitting and Lamb-shift determinations of the proton radius.
- Adding a fourth, finer ensemble or larger physical volumes, as the paper suggests, is the specific next step that would shrink the dominant systematic errors in the radii.
Reading between the lines
- If the linear-in-$a^2$ assumption is right, a fourth ensemble near $a\approx0.04$ fm would be the cleanest test; visible curvature there would push every continuum radius outside its quoted error.
- The SVD-based use of nonzero sink momenta for the disconnected diagram adds many $Q^2$ points at no extra inversions, a technique that could be applied to other isoscalar nucleon matrix elements such as the axial or scalar charges.
- Because the three ensembles have similar physical volumes, finite-volume effects are constrained mainly by comparison with other calculations rather than by the data itself; a dedicated larger-volume study at the finest spacing would quantify them.
- Interpreting the systematic error as dominated by model and $Q^2_{\rm cut}$ choice, the quickest precision gain may come from denser low-$Q^2$ coverage, for example from larger volumes, rather than from more statistics at existing momenta.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper presents a lattice QCD calculation of the proton and neutron electromagnetic form factors using three Nf=2+1+1 twisted-mass clover ensembles at the physical pion mass with lattice spacings a=0.080, 0.068, and 0.057 fm. The authors perform a detailed multi-state excited-state analysis of the connected three-point functions, include disconnected isoscalar contributions computed with stochastic techniques, renormalize the local current in RI'/MOM, and extract radii and magnetic moments from dipole, z-expansion, and Galster-like fits to the Q^2 dependence, followed by a linear a^2 continuum extrapolation. The final results, including proton and neutron electric and magnetic radii and magnetic moments, as well as Zemach and Friar radii, agree with experiment within the quoted errors.
Significance. If the quoted systematic errors are robust, this is a valuable step: it is one of the first attempts to take the continuum limit directly at the physical pion mass for nucleon electromagnetic form factors, with a careful treatment of excited states and disconnected contributions. The main strengths are the physical-point ensembles, the extensive sink-source separation analysis, the stochastic disconnected-loop methodology, and the transparent comparisons with the Mainz collaboration and with experimental data. However, the central error budget is not fully established because the final model average and the three-point continuum extrapolation rest on assumptions that are only weakly tested by the data.
major comments (4)
- [Sec. V.A, Table XIV, Eq. (45)] The final quoted systematic error for <r_E^2>^p is dominated by an AIC average in which a single z-expansion(a^2) fit with Q^2_cut=0.85 GeV^2 receives 82% of the weight, while the dipole(a^2) and z-expansion-without-a^2 fits receive essentially zero weight. The continuum values of those down-weighted fits are 0.650(52) fm^2 and 0.636-0.667 fm^2, respectively, compared with 0.747-0.816 fm^2 for the adopted z-exp(a^2) fits. The ~0.1 fm^2 offset is comparable to the statistical error and is several times larger than the quoted 0.039 fm^2 model systematic. These fits are not excluded by their reduced chi^2 values, so the AIC weighting of Eq. (45) does not cover the ansatz dependence. I recommend adding an explicit systematic from the spread over fit families, for example an envelope over all fits with acceptable chi^2 or a prior over model families, and checking the effect on the final errors.
- [Sec. IV.A.1, Eq. (27)] The AIC weights used in Table XIV are computed with log(w_j) = -chi^2_j/2 + N_dof,j, where N_dof = N_data - N_params. Because Q^2_cut changes N_data, this formula adds a term linear in N_data to the log-weight. For the differences in data counts between Q^2_cut=0.4 and 1.0 GeV^2, this term overwhelmingly favors the largest Q^2_cut even when the fit quality per degree of freedom is comparable. The near-100% probabilities for single Q^2_cut values in Table XIV therefore partly reflect dataset-size effects rather than relative support of the fits. A fair model average over Q^2_cut should either use a common Q^2 range or a likelihood normalization that does not reward extra data points exponentially.
- [Sec. V.A, Eqs. (35) and (41)] The continuum extrapolation assumes a linear a^2 dependence and uses only three lattice spacings covering 0.057-0.080 fm. The one-step and two-step comparisons in Sec. V.D test the fitting procedure, not the functional form of the cutoff dependence. A curvature term such as a^4, or a residual O(a) effect, would shift the extrapolated radii and moments by an amount not included in the quoted systematics. I ask the authors to estimate this sensitivity, for example by adding an a^4 term, by dropping the coarsest ensemble, or by quoting a cutoff-scale systematic; without this, the claimed continuum-limit precision is underdetermined.
- [Sec. V.F and Table XIV] The neutron electric radius is extracted from a single Galster-like fit with no a^2 dependence and is quoted as <r_E^2>^n = -0.147(48) fm^2 with no systematic error in the abstract. The statement that the data cannot resolve a lattice-spacing dependence does not imply that the discretization systematic is zero; a cutoff effect at the level of the other observables would directly bias this central quantity. I request a conservative estimate of the continuum systematic, for example from fits to individual ensembles or from a fit with an a^2 slope, and that this be included in the final error.
minor comments (5)
- [Sec. VI.A, Eq. (46)] The proton magnetic moment is quoted in Eq. (46) as mu_p = 2.849(92)(25), while the abstract, Table XIV, and the surrounding text give 2.849(92)(52). This is an inconsistency that must be corrected.
- [Sec. III.B] There is a typo in 'the the heavy quark parameters' in the paragraph after Table I.
- [Table VII] The ensemble label 'cB211.72.64' omits the zero in 'cB211.072.64' used elsewhere; please make the labels consistent.
- [Sec. IV.B and Table VII] The disconnected contributions are extracted from a single fit-range combination, with no model averaging over fit ranges. The text states that varying the fit ranges gives effects suppressed by the larger statistical errors, but the error budget should state explicitly that this contribution to the disconnected systematic is not included in Eq. (45).
- [Fig. 19] The horizontal axis labels for the proton electric and magnetic radii are written as 'r2_E^p [fm]' and 'r2_M^p [fm]' but the plotted quantities are the square roots of the mean-square radii. Please relabel the axes to avoid ambiguity.
Circularity Check
No significant circularity: the central form factors, radii, and moments are extracted directly from lattice correlators computed in this work, with only standard input parameters taken from prior literature.
full rationale
The paper's central claim—continuum-limit proton and neutron electromagnetic form factors, radii, and magnetic moments—rests on lattice QCD two- and three-point correlation functions computed on three physical-pion ensembles analyzed in this work. The radii and moments are fit parameters in dipole, z-expansion, and Galster-like descriptions of the Q²-dependence, combined with a linear-in-a² continuum extrapolation. These are outputs of fits to data produced in this paper, not quantities fed into the analysis and then relabeled as predictions. The lattice spacings and pion masses are taken from Ref. [16], a prior ETMC publication, but those are meson-sector inputs independent of the nucleon form factors being calculated; using them as inputs is standard practice and does not make the form-factor extraction circular. The renormalization factors Z_V are determined in this work via RI'/MOM, and the equality Z_s^V = Z_ns^V is both cited from Ref. [16] and numerically confirmed in Table V, so the isoscalar current renormalization does not reduce to an unverified self-citation. The AIC model averaging in Eq. (45) is a statistical weighting of fits to the same lattice data; the skeptic's concern that the dipole and no-a² fits receive near-zero weight while differing by about 0.1 fm² in the proton radius is a legitimate correctness or systematic-error criticism, but it is not circularity: the model weights are derived from chi² values, not from the desired final observables. No equation in the paper is equivalent by construction to its own input, and no fitted parameter is renamed as a prediction. Therefore the derivation is self-contained with respect to the final form-factor results, and no circular step is exhibited.
Assumptions & free parameters
free parameters (6)
- z-expansion coefficients c_k,0 and c_k,2 =
e.g., c1,0=-0.936(84), c2,0=-1.16(26), c3,0=-0.54(45) for G_E^p at Q2_cut=0.85 GeV2 (Table XX)
- Dipole mass parameter M^2 (or radius) and magnetic moment g =
e.g., for G_E^p: r2_E^p=0.650(52) fm2 (one-step dipole, Table VIII); for G_M^p: mu_p=2.66(12), r2_M^p=0.576(59) fm2…
- Galster-like parameters A and B for G_E^n =
A=2.23(73), B=23(14) (Table XX)
- Linear a^2 slope parameters g_2 and <r^2>_2 in continuum extrapolation =
Not quoted directly; part of one-step fits (Eq. 35)
- Priors width parameter w and expansion order k_max =
w varied from 1 to 5; k_max=3 for magnetic, 4 for electric proton (chosen for stability)
- Q^2_cut (maximum momentum transfer included in fits) =
0.4 to 1.0 GeV2 depending on form factor; final preferred values 0.85 or 1.0 GeV2
assumptions (8)
- domain assumption Lattice QCD with N_f=2+1+1 twisted mass clover-improved fermions provides a valid nonperturbative regularization of QCD, and the continuum limit can be taken after extrapolating linearly in a^2.
- domain assumption The lattice spacings and pion masses from Ref. [16] are correct and apply to these ensembles.
- domain assumption Z_V^s = Z_V^ns (flavor singlet and nonsinglet vector renormalization factors coincide), following the symmetry argument in Ref. [16].
- ad hoc to paper The z-expansion with Gaussian priors and the condition sum(c_k)=0 provides a convergent parameterization of the form factors in the fit range.
- ad hoc to paper The Galster-like parameterization G(Q^2)=Q^2 A/(4m_N^2+Q^2 B) * 1/(1+Q^2/0.71 GeV^2)^2 describes G_E^n in the range Q^2<0.3 GeV^2.
- domain assumption Isospin symmetry limit: QED corrections and u-d quark mass difference are neglected.
- standard math The dispersion relation E_N(q)=sqrt(m_N^2+q^2) is exact for the nucleon ground state energy at non-zero momentum.
- domain assumption Finite volume effects are negligible at m_pi L ~ 3.6 to 3.9.
Cite this review
Pith. "Pith review of Proton and neutron electromagnetic form factors from lattice QCD in the continuum limit." pith.science (2026). https://pith.science/paper/WRA7ROGZ
@misc{pith2026250720910,
author = {Pith},
title = {Pith review of: Proton and neutron electromagnetic form factors from lattice QCD in the continuum limit},
year = {2026},
howpublished = {\url{https://pith.science/paper/WRA7ROGZ}},
note = {Machine review of arXiv:2507.20910}
}
abstract
We compute the electromagnetic form factors of the proton and neutron using lattice QCD. We employ $N_\mathrm{f}$=2+1+1 twisted mass clover-improved fermions with quark masses tuned to their physical values. Three ensembles with lattice spacings of $a$=0.080 fm, 0.068 fm, and 0.057 fm, and approximately the same physical volume allow us to obtain the continuum limit directly at the physical pion mass. For each ensemble, we use several values of the sink-source time separation, ranging from 0.5 fm to 1.5 fm, to allow for a thorough analysis of excited state effects via multi-state fits. The disconnected contributions are also analyzed using high statistics combined with techniques to mitigate stochastic noise in the estimation of the fermion loop. These techniques include low-mode deflation, dilution in the color and spin components, and hierarchical probing. We study the momentum transfer dependence of the form factors using the $z$-expansion and dipole Ans\"atze, thereby enabling the extraction of the electric and magnetic radii and the magnetic moments, as well as the Zemach and Friar radii in the continuum limit. Results for the proton and neutron electric and magnetic mean square radii are $\sqrt{\langle r_E^2\rangle^p} = 0.860(38)(23)$ fm, $\langle r_E^2\rangle^n = -0.147(48)$ fm$^2$, $\sqrt{\langle r_M^2\rangle^p} = 0.870(53)(15)$ fm and $\sqrt{\langle r_M^2\rangle^n} = 0.913(67)(19)$ fm, and for the proton and neutron magnetic moments $\mu^p=2.849(92)(52)$ and $\mu^n=-1.819(76)(29)$, respectively. In all cases, the first error is statistical and the second systematic, where the latter includes an estimate of the error from the fits to the momentum dependence of the form factors and from the continuum extrapolation.
Figures
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Reference graph
Works this paper leans on
-
[1]
Dipole analysis Since we use the lattice conserved current, the form factor is equal to 1 atQ 2 = 0 for any lattice spacing by symmetry, and we can therefore setg= 1 in Eq. (34) andg 0 = 1,g 2=0 in Eq. (35). In Table VIII, we summarize the values of⟨r 2 E⟩p ob- tained when fitting the individual ensembles and at the TABLE VIII. Results on the proton elect...
-
[2]
Energy spectrum and dispersion relation Since in our fits we allow for different excited state energies in the two- and the isoscalar and isovector three- point functions, it is worthwhile checking and comparing 10 the spectrum with the Roper andπ +Nfree energies. In Fig. 5, we collect the excited state energies obtained from the fits to the isovector and...
-
[3]
Dipole results The same analysis is carried out for the proton mag- netic form factor, starting with the dipole fits. In this case, the proton magnetic moment (µ p) is a fit parame- ter, in addition to the magnetic radius⟨r 2 M ⟩p. In Fig. 12, we show the fit results using the same convention as used 16 0.0 0.2 0.4 0.6 0.8 1.0 Q2 [GeV2] 0.0 0.5 1.0 1.5 2....
-
[4]
Dipole results Our analysis of the neutron magnetic form factor fol- lows that of the proton. The dipole fits to the form fac- tor are shown in Fig. 16 and in Fig. 17, we compare the continuum extrapolation between the one- and two-step approaches, showing that also in this case the two ap- proaches yield consistent results. In Table XII, we sum- 17 0.0 0...
-
[5]
J. C. Bernaueret al.(A1), Electric and magnetic form factors of the proton, Phys. Rev. C90, 015206 (2014), arXiv:1307.6227 [nucl-ex]
arXiv 2014
-
[6]
V. Punjabi, C. F. Perdrisat, M. K. Jones, E. J. Brash, and C. E. Carlson, The Structure of the Nucleon: Elastic Electromagnetic Form Factors, Eur. Phys. J. A51, 79 (2015), arXiv:1503.01452 [nucl-ex]
arXiv 2015
-
[7]
Pohlet al., The size of the proton, Nature466, 213 (2010)
R. Pohlet al., The size of the proton, Nature466, 213 (2010)
2010
- [8]
Show all 79 references
-
[9]
Xionget al., A small proton charge radius from an electron–proton scattering experiment, Nature575, 147 (2019)
W. Xionget al., A small proton charge radius from an electron–proton scattering experiment, Nature575, 147 (2019)
2019
-
[10]
Djukanovic, G
D. Djukanovic, G. von Hippel, H. B. Meyer, K. Ottnad, M. Salg, and H. Wittig, Electromagnetic form factors of the nucleon from Nf=2+1 lattice QCD, Phys. Rev. D 109, 094510 (2024), arXiv:2309.06590 [hep-lat]
2024 arXiv
-
[11]
Djukanovic, G
D. Djukanovic, G. von Hippel, H. B. Meyer, K. Ot- tnad, M. Salg, and H. Wittig, Precision Calculation of the Electromagnetic Radii of the Proton and Neutron from Lattice QCD, Phys. Rev. Lett.132, 211901 (2024), arXiv:2309.07491 [hep-lat]
2024 arXiv
-
[13]
Tsuji, Y
R. Tsuji, Y. Aoki, K.-I. Ishikawa, Y. Kuramashi, S. Sasaki, K. Sato, E. Shintani, H. Watanabe, and T. Ya- mazaki (PACS), Nucleon form factors in Nf=2+1 lattice QCD at the physical point: Finite lattice spacing effect on the root-mean-square radii, Phys. Rev. D109, 094505 (2024...
2024 arXiv
-
[14]
Y.-C. Jang, R. Gupta, H.-W. Lin, B. Yoon, and T. Bhat- tacharya, Nucleon electromagnetic form factors in the continuum limit from ( 2+1+1 )-flavor lattice QCD, Phys. Rev. D101, 014507 (2020), arXiv:1906.07217 [hep- lat]
2020 arXiv
-
[15]
Frezzotti and G
R. Frezzotti and G. C. Rossi, Chirally improving Wil- son fermions. 1. O(a) improvement, JHEP08, 007, arXiv:hep-lat/0306014
-
[16]
Frezzotti, P
R. Frezzotti, P. A. Grassi, S. Sint, and P. Weisz (Alpha), Lattice QCD with a chirally twisted mass term, JHEP 08, 058, arXiv:hep-lat/0101001
-
[17]
M. Constantinouet al.(ETM), Non-perturbative renor- malization of quark bilinear operators withN f = 2 (tmQCD) Wilson fermions and the tree-level improved gauge action, JHEP08, 068, arXiv:1004.1115 [hep-lat]
-
[18]
Alexandrouet al., Simulating twisted mass fermions at physical light, strange and charm quark masses, Phys
C. Alexandrouet al., Simulating twisted mass fermions at physical light, strange and charm quark masses, Phys. Rev. D98, 054518 (2018), arXiv:1807.00495 [hep-lat]
2018 arXiv
-
[19]
Finkenrathet al., Twisted mass gauge ensem- bles at physical values of the light, strange and charm quark masses, PoSLA TTICE2021, 284 (2022), arXiv:2201.02551 [hep-lat]
J. Finkenrathet al., Twisted mass gauge ensem- bles at physical values of the light, strange and charm quark masses, PoSLA TTICE2021, 284 (2022), arXiv:2201.02551 [hep-lat]
2022 arXiv
-
[20]
C. Alexandrouet al.(Extended Twisted Mass), Lattice calculation of the short and intermediate time-distance hadronic vacuum polarization contributions to the muon magnetic moment using twisted-mass fermions, Phys. Rev. D107, 074506 (2023), arXiv:2206.15084 [hep-lat]
2023 arXiv
-
[21]
Alexandrouet al.(Extended Twisted Mass), Quark masses using twisted-mass fermion gauge ensembles, Phys
C. Alexandrouet al.(Extended Twisted Mass), Quark masses using twisted-mass fermion gauge ensembles, Phys. Rev. D104, 074515 (2021), arXiv:2104.13408 [hep- lat]
2021 arXiv
-
[22]
Gusken, A Study of smearing techniques for hadron correlation functions, Nucl
S. Gusken, A Study of smearing techniques for hadron correlation functions, Nucl. Phys. B Proc. Suppl.17, 361 (1990)
1990
-
[23]
Alexandrou, S
C. Alexandrou, S. Gusken, F. Jegerlehner, K. Schilling, and R. Sommer, The Static approximation of heavy - light quark systems: A Systematic lattice study, Nucl. Phys. B414, 815 (1994), arXiv:hep-lat/9211042
1994 arXiv
-
[24]
Alexandrouet al., Moments of nucleon generalized parton distributions from lattice QCD simulations at physical pion mass, Phys
C. Alexandrouet al., Moments of nucleon generalized parton distributions from lattice QCD simulations at physical pion mass, Phys. Rev. D101, 034519 (2020), arXiv:1908.10706 [hep-lat]
2020 arXiv
-
[25]
Albaneseet al.(APE), Glueball Masses and String Tension in Lattice QCD, Phys
M. Albaneseet al.(APE), Glueball Masses and String Tension in Lattice QCD, Phys. Lett. B192, 163 (1987)
1987
-
[26]
Alexandrouet al., Moments of the nucleon transverse quark spin densities using lattice QCD, Phys
C. Alexandrouet al., Moments of the nucleon transverse quark spin densities using lattice QCD, Phys. Rev. D 107, 054504 (2023), arXiv:2202.09871 [hep-lat]
2023 arXiv
-
[27]
McNeile and C
C. McNeile and C. Michael (UKQCD), Decay width of light quark hybrid meson from the lattice, Phys. Rev. D 73, 074506 (2006), arXiv:hep-lat/0603007
2006 arXiv
-
[28]
Alexandrou, M
C. Alexandrou, M. Constantinou, V. Drach, K. Had- jiyiannakou, K. Jansen, G. Koutsou, A. Strelchenko, and A. Vaquero, Evaluation of disconnected quark loops for hadron structure using GPUs, Comput. Phys. Commun. 185, 1370 (2014), arXiv:1309.2256 [hep-lat]
2014 arXiv
-
[29]
Alexandrou, M
C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, and A. Vaquero Aviles-Casco, Nucleon axial form factors usingN f = 2 twisted mass fermions with a physical value of the pion mass, Phys. Rev. D96, 054507 (2017), arXiv:1705.03399 [hep-lat]
2017 arXiv
-
[30]
Alexandrouet al., Nucleon scalar and tensor charges using lattice QCD simulations at the physical value of the pion mass, Phys
C. Alexandrouet al., Nucleon scalar and tensor charges using lattice QCD simulations at the physical value of the pion mass, Phys. Rev. D95, 114514 (2017), [Erratum: Phys.Rev.D 96, 099906 (2017)], arXiv:1703.08788 [hep- lat]
2017 arXiv
-
[31]
Alexandrou, M
C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, A. Vaquero Avil´ es- Casco, and C. Wiese, Nucleon Spin and Momentum De- composition Using Lattice QCD Simulations, Phys. Rev. Lett.119, 142002 (2017), arXiv:1706.02973 [hep-lat]
2017 arXiv
-
[32]
Stathopoulos, J
A. Stathopoulos, J. Laeuchli, and K. Orginos, Hierar- chical Probing for Estimating the Trace of the Matrix Inverse on Toroidal Lattices, SIAM J. Sci. Comput.35, S299 (2013), arXiv:1302.4018 [hep-lat]
2013 arXiv
-
[33]
A. S. Gambhir, A. Stathopoulos, and K. Orginos, Defla- tion as a Method of Variance Reduction for Estimating the Trace of a Matrix Inverse, SIAM J. Sci. Comput.39, A532 (2017), arXiv:1603.05988 [hep-lat]
2017 arXiv
-
[34]
Martinelli, C
G. Martinelli, C. Pittori, C. T. Sachrajda, M. Testa, and A. Vladikas, A General method for nonperturbative renormalization of lattice operators, Nucl. Phys. B445, 81 (1995), arXiv:hep-lat/9411010
1995 arXiv
-
[35]
Extended Twisted Mass Collaboration, Non-perturbative renormalisation of quark bilinear operators withN f = 25 2 + 1 + 1 Wilson-clover twisted mass fermions, In prepa- ration
-
[36]
Gockeler, R
M. Gockeler, R. Horsley, H. Oelrich, H. Perlt, D. Pet- ters, P. E. L. Rakow, A. Schafer, G. Schierholz, and A. Schiller, Nonperturbative renormalization of compos- ite operators in lattice QCD, Nucl. Phys. B544, 699 (1999), arXiv:hep-lat/9807044
1999 arXiv
-
[37]
Alexandrou, S
C. Alexandrou, S. Bacchio, J. Finkenrath, C. Iona, G. Koutsou, Y. Li, and G. Spanoudes, Nucleon charges andσ-terms in lattice QCD, Phys. Rev. D111, 054505 (2025), arXiv:2412.01535 [hep-lat]
2025 arXiv
-
[38]
Alexandrou, M
C. Alexandrou, M. Constantinou, and H. Panagopoulos (ETM), Renormalization functions for Nf=2 and Nf=4 twisted mass fermions, Phys. Rev. D95, 034505 (2017), arXiv:1509.00213 [hep-lat]
2017 arXiv
-
[39]
Bar and H
O. Bar and H. Colic, Nπ-state contamination in lattice calculations of the nucleon electromagnetic form factors, Phys. Rev. D103, 114514 (2021), arXiv:2104.00329 [hep- lat]
2021 arXiv
-
[40]
W. I. Jay and E. T. Neil, Bayesian model averaging for analysis of lattice field theory results, Phys. Rev. D103, 114502 (2021), arXiv:2008.01069 [stat.ME]
2021 arXiv
-
[41]
E. T. Neil and J. W. Sitison, Improved information cri- teria for Bayesian model averaging in lattice field the- ory, Phys. Rev. D109, 014510 (2024), arXiv:2208.14983 [stat.ME]
2024 arXiv
-
[42]
Alexandrou, S
C. Alexandrou, S. Bacchio, M. Constantinou, J. Finken- rath, R. Frezzotti, B. Kostrzewa, G. Koutsou, G. Spanoudes, and C. Urbach (Extended Twisted Mass), Nucleon axial and pseudoscalar form factors using twisted-mass fermion ensembles at the physical point, Phys. Rev. D109, 03...
2024 arXiv
-
[43]
Galster, H
S. Galster, H. Klein, J. Moritz, K. H. Schmidt, D. We- gener, and J. Bleckwenn, Elastic electron-deuteron scat- tering and the electric neutron form factor at four- momentum transfers 5fm −2 < q2 <14fm −2, Nucl. Phys. B32, 221 (1971)
1971
-
[44]
G. Lee, J. R. Arrington, and R. J. Hill, Extraction of the proton radius from electron-proton scattering data, Phys- ical Review D92, 10.1103/physrevd.92.013013 (2015)
2015 doi
-
[45]
A. S. Meyer, M. Betancourt, R. Gran, and R. J. Hill, Deuterium target data for precision neutrino- nucleus cross sections, Phys. Rev. D93, 113015 (2016), arXiv:1603.03048 [hep-ph]
2016 arXiv
-
[46]
R. J. Hill and G. Paz, Model independent extraction of the proton charge radius from electron scattering, Phys. Rev. D82, 113005 (2010), arXiv:1008.4619 [hep-ph]
2010 arXiv
-
[47]
Navaset al.(Particle Data Group), Review of particle physics, Phys
S. Navaset al.(Particle Data Group), Review of particle physics, Phys. Rev. D110, 030001 (2024)
2024
-
[48]
Z. Ye, J. Arrington, R. J. Hill, and G. Lee, Proton and Neutron Electromagnetic Form Factors and Uncertain- ties, Phys. Lett. B777, 8 (2018), arXiv:1707.09063 [nucl- ex]
2018 arXiv
-
[49]
Beckeret al., Determination of the neutron electric form-factor from the reaction He-3(e,e’ n) at medium mo- mentum transfer, Eur
J. Beckeret al., Determination of the neutron electric form-factor from the reaction He-3(e,e’ n) at medium mo- mentum transfer, Eur. Phys. J. A6, 329 (1999)
1999
-
[50]
Edenet al., Electric form factor of the neutron from the 2H(− →e , e′− →n) 1Hreaction atQ 2 = 0.255 (GeV/c) 2, Phys
T. Edenet al., Electric form factor of the neutron from the 2H(− →e , e′− →n) 1Hreaction atQ 2 = 0.255 (GeV/c) 2, Phys. Rev. C50, R1749 (1994)
1994
-
[51]
Meyerhoffet al., First measurement of the electric form-factor of the neutron in the exclusive quasielas- tic scattering of polarized electrons from polarized He-3, Phys
M. Meyerhoffet al., First measurement of the electric form-factor of the neutron in the exclusive quasielas- tic scattering of polarized electrons from polarized He-3, Phys. Lett. B327, 201 (1994)
1994
-
[52]
Passchieret al., The Charge form-factor of the neutron from the reaction polarized H-2(polarized e, e-prime n) p, Phys
I. Passchieret al., The Charge form-factor of the neutron from the reaction polarized H-2(polarized e, e-prime n) p, Phys. Rev. Lett.82, 4988 (1999), arXiv:nucl-ex/9907012
1999 arXiv
-
[53]
Warrenet al.(Jefferson Lab E93-026), Measurement of the electric form-factor of the neutron atQ 2 = 0.5 and 1.0GeV 2/c2, Phys
G. Warrenet al.(Jefferson Lab E93-026), Measurement of the electric form-factor of the neutron atQ 2 = 0.5 and 1.0GeV 2/c2, Phys. Rev. Lett.92, 042301 (2004), arXiv:nucl-ex/0308021
2004 arXiv
-
[54]
Zhuet al.(E93026), A Measurement of the elec- tric form-factor of the neutron through polarized-d (polarized-e, e-prime n)p at Q**2 = 0.5-(GeV/c)**2, Phys
H. Zhuet al.(E93026), A Measurement of the elec- tric form-factor of the neutron through polarized-d (polarized-e, e-prime n)p at Q**2 = 0.5-(GeV/c)**2, Phys. Rev. Lett.87, 081801 (2001), arXiv:nucl- ex/0105001
2001
-
[55]
Madeyet al.(E93-038), Measurements of G(E)n / G(M)n from the H-2(polarized-e,e-prime polarized-n) re- action to Q**2 = 1.45 (GeV/c)**2, Phys
R. Madeyet al.(E93-038), Measurements of G(E)n / G(M)n from the H-2(polarized-e,e-prime polarized-n) re- action to Q**2 = 1.45 (GeV/c)**2, Phys. Rev. Lett.91, 122002 (2003), arXiv:nucl-ex/0308007
2003 arXiv
-
[56]
Roheet al., Measurement of the neutron electric form- factor G(en) at 0.67-(GeV/c)**2 via He-3(pol.)(e(pol.),e’ n), Phys
D. Roheet al., Measurement of the neutron electric form- factor G(en) at 0.67-(GeV/c)**2 via He-3(pol.)(e(pol.),e’ n), Phys. Rev. Lett.83, 4257 (1999)
1999
-
[57]
Bermuthet al., The Neutron charge form-factor and target analyzing powers from polarized-He-3 (polarized- e,e-prime n) scattering, Phys
J. Bermuthet al., The Neutron charge form-factor and target analyzing powers from polarized-He-3 (polarized- e,e-prime n) scattering, Phys. Lett. B564, 199 (2003), arXiv:nucl-ex/0303015
2003 arXiv
-
[58]
D. I. Glazieret al., Measurement of the electric form- factor of the neutron at Q**2 = 0.3-(GeV/c)**2 to 0.8- (GeV/c)**2, Eur. Phys. J. A24, 101 (2005), arXiv:nucl- ex/0410026
2005
-
[59]
Herberget al., Determination of the neutron electric form-factor in the D(e,e’ n)p reaction and the influence of nuclear binding, Eur
C. Herberget al., Determination of the neutron electric form-factor in the D(e,e’ n)p reaction and the influence of nuclear binding, Eur. Phys. J. A5, 131 (1999)
1999
-
[60]
Schiavilla and I
R. Schiavilla and I. Sick, Neutron charge form-factor at large q**2, Phys. Rev. C64, 041002 (2001), arXiv:nucl- ex/0107004
2001
-
[61]
Ostricket al., Measurement of the neutron elec- tric form-factor G(E,n) in the quasifree H-2(e(pol.),e’ n(pol.))p reaction, Phys
M. Ostricket al., Measurement of the neutron elec- tric form-factor G(E,n) in the quasifree H-2(e(pol.),e’ n(pol.))p reaction, Phys. Rev. Lett.83, 276 (1999)
1999
-
[62]
Andersonet al.(Jefferson Lab E95-001), Extraction of the Neutron Magnetic Form Factor from Quasi-elastic 3 ⃗He(⃗ e, e′) at Q 2 = 0.1 - 0.6 (GeV/c) 2, Phys
B. Andersonet al.(Jefferson Lab E95-001), Extraction of the Neutron Magnetic Form Factor from Quasi-elastic 3 ⃗He(⃗ e, e′) at Q 2 = 0.1 - 0.6 (GeV/c) 2, Phys. Rev. C75, 034003 (2007), arXiv:nucl-ex/0605006
2007 arXiv
-
[63]
Gaoet al., Measurement of the neutron magnetic form-factor from inclusive quasielastic scattering of po- larized electrons from polarized He-3, Phys
H. Gaoet al., Measurement of the neutron magnetic form-factor from inclusive quasielastic scattering of po- larized electrons from polarized He-3, Phys. Rev. C50, R546 (1994)
1994
-
[64]
Anklinet al., Precision measurement of the neutron magnetic form-factor, Phys
H. Anklinet al., Precision measurement of the neutron magnetic form-factor, Phys. Lett. B336, 313 (1994)
1994
-
[65]
Anklinet al., Precise measurements of the neutron magnetic form-factor, Phys
H. Anklinet al., Precise measurements of the neutron magnetic form-factor, Phys. Lett. B428, 248 (1998)
1998
-
[66]
Kubonet al., Precise neutron magnetic form-factors, Phys
G. Kubonet al., Precise neutron magnetic form-factors, Phys. Lett. B524, 26 (2002), arXiv:nucl-ex/0107016
2002 arXiv
-
[67]
Alarcon (BLAST), Nucleon form factors and the BLAST experiment, Eur
R. Alarcon (BLAST), Nucleon form factors and the BLAST experiment, Eur. Phys. J. A32, 477 (2007)
2007
-
[68]
Alexandrou, M
C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, and A. Vaquero Aviles-Casco, Nucleon electromagnetic form factors us- ing lattice simulations at the physical point, Phys. Rev. D96, 034503 (2017), arXiv:1706.00469 [hep-lat]
2017 arXiv
-
[69]
Shintani, K.-I
E. Shintani, K.-I. Ishikawa, Y. Kuramashi, S. Sasaki, and T. Yamazaki, Nucleon form factors and root-mean-square radii on a (10.8 fm) 4 lattice at the physical point, Phys. Rev. D99, 014510 (2019), [Erratum: Phys.Rev.D 102, 019902 (2020)], arXiv:1811.07292 [hep-lat]
2019 arXiv
-
[70]
Antogniniet al., Proton Structure from the Measure- ment of 2S−2PTransition Frequencies of Muonic Hy- 26 drogen, Science339, 417 (2013)
A. Antogniniet al., Proton Structure from the Measure- ment of 2S−2PTransition Frequencies of Muonic Hy- 26 drogen, Science339, 417 (2013)
2013
-
[71]
Borah, R
K. Borah, R. J. Hill, G. Lee, and O. Tomalak, Parametrization and applications of the low-Q 2 nucleon vector form factors, Phys. Rev. D102, 074012 (2020)
2020
-
[72]
M. O. Distler, J. C. Bernauer, and T. Walcher, The RMS Charge Radius of the Proton and Zemach Moments, Phys. Lett. B696, 343 (2011), arXiv:1011.1861 [nucl-th]
2011 arXiv
-
[73]
Lin, H.-W
Y.-H. Lin, H.-W. Hammer, and U.-G. Meißner, New in- sights into the nucleon’s electromagnetic structure, Phys. Rev. Lett.128, 052002 (2022)
2022
-
[74]
Djukanovic, G
D. Djukanovic, G. von Hippel, H. B. Meyer, K. Ottnad, M. Salg, and H. Wittig, Zemach and friar radii of the proton and neutron from lattice qcd, Phys. Rev. D110, L011503 (2024)
2024
-
[75]
K. M. Graczyk and C. Juszczak, Zemach moments of the proton from Bayesian inference, Phys. Rev. C91, 045205 (2015)
2015
-
[76]
A. V. Volotka, V. M. Shabaev, G. Plunien, and G. Soff, Zemach and magnetic radius of the proton from the hy- perfine splitting in hydrogen, Eur. Phys. J. D33, 23 (2005), arXiv:physics/0405118
2005 arXiv
-
[77]
P. J. Mohr, D. B. Newell, and B. N. Taylor, CO- DATA Recommended Values of the Fundamental Physi- cal Constants: 2014, Rev. Mod. Phys.88, 035009 (2016), arXiv:1507.07956 [physics.atom-ph]
2016 arXiv
-
[78]
Alexandrouet al., Large-scale simulations of lattice QCD for nucleon structure using Nf=2+1+1 flavors of twisted mass fermions, Procedia Comput
C. Alexandrouet al., Large-scale simulations of lattice QCD for nucleon structure using Nf=2+1+1 flavors of twisted mass fermions, Procedia Comput. Sci.267, 92 (2025)
2025
-
[79]
J¨ ulich Supercomputing Centre, JUWELS: Modular Tier-0/1 Supercomputer at the J¨ ulich Supercomput- ing Centre, Journal of large-scale research facilities5, 10.17815/jlsrf-5-171 (2019)
2019 doi
-
[80]
J¨ ulich Supercomputing Centre, JUWELS Cluster and Booster: Exascale Pathfinder with Modular Supercom- puting Architecture at Juelich Supercomputing Centre, Journal of large-scale research facilities7, 10.17815/jlsrf- 7-183 (2021). Appendix A: Excited state analysis of connect...
2021 arXiv
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