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High Energy Physics - Lattice · arXiv:hep-lat/0403005 v1 · submitted 3 Mar 2004 · 57 references

Electroweak Matrix Elements in the Two-Nucleon Sector from Lattice QCD

William Detmold, Martin J. Savage

PDF · arXiv · The abstract, sentence by sentence · Cite

The paper says

“We demonstrate how to make rigorous predictions for electroweak matrix elements in nuclear systems directly from QCD. More precisely, we show how to determine the short-distance contributions to low-momentum transfer electroweak matrix elements in the two-nucleon sector from lattice QCD.”

more of the abstract

In potential model descriptions of multi-nucleon systems, this is equivalent to uniquely determining the meson-exchange currents, while in the context of nuclear effective field theory, this translates into determining the coefficients of local, gauge-invariant, multi-nucleon-electroweak current operators. The energies of the lowest-lying states of two nucleons on a finite volume lattice with periodic boundary conditions in the presence of a background magnetic field are sufficient to determine the local four-nucleon operators that contribute to the deuteron magnetic moment and to the threshold cross-section of n + p -> d + gamma. Similarly, the energy-levels of two nucleons immersed in a background isovector axial weak field can be used to determine the coefficient of the leading local four-nucleon operator contributing to the neutral- and charged-current break-up of the deuteron. This is required for the extraction of solar neutrino fluxes at SNO and future neutrino experiments.

from the abstract

Pith says

Machine review · deepseek-v4.1-flash · 13 Nov 2026 · what was read was not recorded · not checked by a person

Reviewed in 2026 with later literature in view.

It can be wrong. Challenge this review or reply to it.

Holds up, with conditions. A clean, honest methods paper: the finite-volume quantization conditions with background fields are the real contribution, but the l̃2 signal is small and the deferred higher-order corrections are not shown to be smaller. This line is wrong

3 major objections · 6 minor

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The paper says, at Abstract and §I

The review did not quote this passage.

Pith says: Objection 1 of 3 · major

The abstract opens with 'We demonstrate how to make rigorous predictions for electroweak matrix elements in nuclear systems directly from QCD' (abstract:s0), and §I states that the work 'establishes a framework.' These are not the same claim. No electroweak matrix element is computed in this paper; the coefficients l̃2, l1 and l1,A are left as quantities to be measured, and the figures use experimental values of the nucleon and deuteron magnetic moments as inputs.

more of objection 1

The word 'predictions' in the abstract and the phrase 'rigorous predictions' should be replaced by language that matches the content, for example 'a rigorous framework for determining' or 'a method to extract.' This is a framing issue rather than a technical error, but it is the sentence a reader will carry away, and it overstates what is delivered.

The paper says, at §IV, Eqs. (17), (20), (21)

The review did not quote this passage.

Pith says: Objection 2 of 3 · major

Eq. (17) contains a field-dependent term ∓(eB0/2)(l̃2 − r3κ0), but the effective scattering length defined in Eq. (21) is 1/A3 = 1/a3 ± (eB0/2)l̃2, with no r3κ0 piece. Since Eq. (20) is written entirely in terms of A3, the reader cannot tell whether the r3κ0 term has been absorbed into a field-dependent effective range, dropped as higher order, or omitted.

more of objection 2

Please state explicitly where this term goes and at what order it contributes to the level positions in Eqs. (20) and (22). The same question applies to the analogous structure in Eqs. (26) and (34).

The paper says, at §IV and §VII

The review did not quote this passage.

Pith says: Objection 3 of 3 · major

The practical extraction of l̃2 rests on a small effect: the authors state that l̃2 = −0.0576 fm and that at eB0 = 10^4 MeV^2 the effective scattering length A3 differs from a3 by about 10%. The same levels receive field-dependent corrections from nucleon magnetic polarizabilities (Eq.

more of objection 3

(7)), from higher-derivative two-nucleon magnetic operators, and from Landau-level admixture in the transverse directions. The paper acknowledges in §VII that a higher-order analysis including these effects is still wanted, but gives no estimate of their size relative to the 10% shift. A short order-of-magnitude estimate, or an explicit statement of the field range and lattice volumes over which the l̃2 signal is expected to dominate the truncation error, would materially strengthen the central feasibility claim. The same remark applies to the quark-mass-tuning 'twist' discussed at the end of §IV, which requires a chiral extrapolation whose uncertainty is not estimated.

6 minor objections

The paper says, at §VI, Eq. (34)

Pith says: minor objection 1 of 6

The right-hand side of Eq. (34) reads [gW l1,A/4 − S1 − S2/(2πL)]^2, whereas the corresponding term in Eq. (26) is [eB0 l1/2 + (S1 − S2)/(2πL)]^2. The sign and grouping of the S1, S2 terms appear to differ between the two equations. Please check and make the two expressions structurally parallel, or explain why the axial case requires a different combination.

The paper says, at §VI, Eq. (33)

Pith says: minor objection 2 of 6

Eq. (33) lists two cross sections of the form σ(νx d → νx np) with different numerical coefficients (1.76 + 0.056 l1,A and 1.66 + 0.052 l1,A), and the accompanying text lists 'νd → νnp' twice. Presumably one of each pair is the antineutrino channel. The bar notation should be made explicit throughout Eq. (33) and the surrounding sentence, since the neutral-current and charged-current neutrino and antineutrino channels are physically distinct and are quoted separately.

show 4 more

The paper says, at Appendix A, Eq. (A9)

Pith says: minor objection 3 of 6

The expansion in Eq. (A9) reads [1 + c′1 tan δ0 + c′2 tan δ0 + ...], with the same power of tan δ0 in both correction terms. By analogy with Eq. (22), where the c′2 term carries (p cot δ̃0)^2, the second term here should presumably be c′2 tan^2 δ0. Please correct or clarify.

The paper says, at Fig. 1 and §IV

Pith says: minor objection 4 of 6

The caption of Fig. 1 states that the background field strength is eB0 = 10^2 MeV^2, while the surrounding discussion of the 10% shift in A3 refers to eB0 = 10^4 MeV^2. The reader is left unsure which field strength the figure illustrates and whether the plotted shifts are representative of the regime in which l̃2 is actually extractable. Please reconcile the caption with the text.

The paper says, at Fig. 4 and Table II

Pith says: minor objection 5 of 6

The 'magic box' at η1 = η2 = 0.268494 where c1 vanishes is an interesting practical observation. It would help to state the value of c2 at that point as well, since a small c1 with a large c2 would limit the usefulness of the convergence improvement claimed in the text.

The paper says, at §VII

Pith says: minor objection 6 of 6

The sentence 'While the result are rigorous where applicable' should read 'results are.' There are also several typographical issues in the displayed equations and in the running text (for example the spacing in 'EFT( π /)' and the broken 'L¨ uscher' in the reference list) that should be cleaned up in production.

What would settle it

Using the physical parameters, compute the lowest continuum level from eq. (20) at eB0 = 10^4 MeV^2 and L = 20 fm with l̃2 = -0.0576 fm. Then recompute including the leading field-dependent corrections the authors defer: the nucleon magnetic polarizability term of eq. (7) (β ~ 10^-4 fm^3) and the leading higher-derivative two-nucleon magnetic operator. If the l̃2-induced shift in E is smaller than or comparable to the deferred-correction shift, the claimed separation is not demonstrated at that field; if it dominates by a comfortable margin, the concern does not land.

from the skeptic's reading

How the review went

4 readings by deepseek-v4.1-flash · 13 Nov 2026

  1. Reader holds up, with conditions confidence high
  2. Skeptic partly agrees 3 min 14 s · 22,278 tokens read, 11,109 written
  3. Editor wrote the verdict above 21 s · 23,236 tokens read, 1,034 written
  4. Referee minor revision: 3 major objections, 6 minor 1 min 28 s · 22,601 tokens read, 5,363 written

Not run on this paper: the simulated authors' defence.

Pith did not check this review's objections against the paper.

The review in full

The review in brief, machine-written

The thing worth knowing is that this is a methods paper, and a good one. Detmold and Savage write down finite-volume quantization conditions for two nucleons in a background magnetic field (eq. 17) and in an isovector axial weak field (eq. 34), with the coupled-channel version for the mixed 1S0–3S1 case (eq. 26). The structural point is that the four-nucleon counterterm (l̃2, l1, l1,A) enters these conditions differently from the one-nucleon couplings, so a lattice measurement of the lowest levels separates short-distance from long-distance physics without computing five-point correlators. That is the new content, and it follows correctly from the Lüscher and EFT(π-slash) machinery. Appendix A is real work: the asymmetric-lattice sums are derived explicitly and tabulated (Tables I and II), including the 'magic box' where c1 vanishes. That is checkable and useful independent of the rest of the paper. The paper is also honest where it counts — it says outright that the electromagnetic half will not tell us anything new experimentally, and the conclusions flag the need for a higher-order analysis including Landau levels and polarizabilities. I take that at face value rather than as a buried admission. The soft spot is quantitative, and the stress-test note lands on it. l̃2 is unnaturally small (−0.0576 fm), so at eB0 = 104 MeV2 the effective scattering length A3 shifts from a3 by only about 10%. Whether that shift is resolvable above the field-dependent corrections the authors defer — polarizabilities, higher-derivative two-nucleon magnetic operators, Landau-level admixture — is not demonstrated. The 'twist' of tuning mq to make a3 infinite would sharpen the fractional l̃2 dependence, but it needs a quark-mass extrapolation whose uncertainty is not estimated. None of this is an internal inconsistency; it is a precision question the paper raises and does not close. The abstract's 'rigorous predictions' is a framing overclaim — nothing is computed — but the body does not oversell. This is for people working on lattice nuclear physics, finite-volume methods, or EFT(π-slash) power counting; a general lattice reader will get less from it. It deserves a serious referee, and I would cite it as the reference for background-field two-nucleon quantization.

The strongest objection

The structural claim — that eqs. (17), (26) and (34) encode the four-nucleon coefficients l̃2, l1, l1,A in a way separable from the one-nucleon couplings — is internally consistent and follows the Lüscher/EFT(π-slash) machinery. The soft spot is quantitative. The authors state l̃2 = -0.0576 fm (Section IV) and that at eB0 = 10^4 MeV^2 the effective scattering length A3 differs from a3 by only ~10%. The same finite-volume levels receive field-dependent corrections from nucleon magnetic polarizabilities (eq. 7), higher-derivative two-nucleon magnetic operators, and Landau-level admixtures. These are not quantified. The authors' own conclusions say: 'One would like to see a higher-order analysis of these finite volume observables in EFT(π-slash) including the effects of the Landau-levels, one- and two-nucleon polarizabilities and other relevant contributions.' So the truncation error relative to the l̃2 signal is an acknowledged, unresolved gap. If the deferred corrections shift the levels by an amount comparable to the ~10% A3 shift, then l̃2 is not cleanly determined at fields where EFT(π-slash) is valid. The 'twist' of tuning mq to make a3 infinite enhances the fractional l̃2 dependence, but it requires a quark-mass extrapolation whose uncertainty is not estimated. This is a precision/control concern, not an internal inconsistency.

The paper is a well-constructed methods proposal. The central structural claim — that finite-volume two-nucleon levels in background fields determine the four-nucleon current coefficients — is supported by internally consistent derivations, and the authors are candid about limitations (the smallness of l̃2, the exceptional-plaquette issue, the fact that the electromagnetic case would not reveal anything new). I agree with the reader that the verdict should remain CONDITIONAL, but I would sharpen the weakest-assumption framing: the binding issue is not only whether the ~10% A3 shift is 'measurable' but whether it is distinguishable from the field-dependent EFT truncation error that the authors themselves defer to future work. The reader's concern and mine overlap but are not identical, hence partial agreement. The paper does not commit circularity and does not overclaim a numerical result; the abstract's 'rigorous predictions' is a framing issue rather than a correctness defect. No change to the verdict is warranted.

The referee's report

If the framework is correct, it converts an intractable five-point-function calculation into a set of two-nucleon energy-level measurements in background fields, which is a genuinely useful reduction in cost and a clean conceptual separation of long- and short-distance electroweak physics. The strengths are real: the quantization conditions are written in closed form, the asymmetric-lattice sums are worked out explicitly with numerical tables (Table I, Table II), the large-volume expansions are given to the order needed for extraction, and the connection to the SNO neutrino-flux program gives the work a concrete phenomenological target. The observation of 'magic' lattice asymmetries where c1 vanishes (Fig. 4) is a nice practical by-product. The paper also makes falsifiable statements: the level positions in Figs. 1-3 and the shifts in the effective scattering length A3 of Eq. (21) are predictions that a lattice calculation can confirm or refute. The main caveat is that the paper delivers a method, not predictions for the coefficients themselves, and the abstract's language should reflect that.

Pith's reading of the paper (a machine's summary, not the authors')

Nuclear theory has long treated the interaction of a photon or weak boson with a nucleus as mostly a one-nucleon affair, with a poorly known two-body correction — the 'meson-exchange current' — added on top. This paper argues that the coefficient of that two-body correction can be extracted directly from lattice QCD without ever computing the expensive five-point correlation function that a brute-force matrix element would need. The proposal is to put two nucleons in a finite periodic box, switch on a weak background magnetic field (or, for the weak case, an isovector axial field), and measure how the lowest energy levels move. Because the one-nucleon magnetic moments and the two-nucleon current enter the finite-volume formula in structurally different ways, the two contributions can be separated from the same set of level measurements. If the scheme works, the lattice data that already fixes nucleon-nucleon scattering would also fix the coefficients needed for the deuteron magnetic moment, for $np$ capture, and for the neutrino breakup of the deuteron used to sort solar neutrino flavours.

What carries the argument. The load-bearing device is L眉scher's finite-volume quantization condition, $p cot delta - frac{1}{pi L} S(eta_1,eta_2; ilde p^2) = 0$, generalized to asymmetric boxes and to background gauge fields. The function $S$ is a regulated sum over the allowed momentum modes of the box, and its argument carries the one-nucleon coupling to the background field while the two-nucleon current appears as a separate additive or off-diagonal term. That structural separation is what makes the extraction possible: the long-distance piece is computable, the short-distance piece is the unknown, and the two do not degenerate in the level spectrum. Supporting machinery includes the dibaryon formulation of the pionless effective field theory, in which the $^3S_1$ and $^1S_0$ channels are represented by auxiliary fields with couplings fixed by the scattering lengths and effective ranges, and the explicit large-volume expansions of the ground and first continuum levels, whose coefficients are tabulated for several lattice asymmetries.

Core claim. The paper's central claim is that the short-distance, four-nucleon electroweak current operators of the pionless effective field theory can be determined from the energies of the lowest-lying two-nucleon states on a finite-volume lattice in the presence of background electroweak fields. The authors derive the quantization conditions that replace L眉scher's formula when a background field is present. For the $^3S_1$ channel in a magnetic field $B_0$ aligned along $z$, the $m = 1$ levels are the solutions of $p cot delta_3 - frac{1}{pi L} S(eta_1,eta_2; ilde p^2 pm ilde u_0^2) mp frac{eB_0}{2}(ilde l_2 - r_3 kappa_0) = 0$, where the one-nucleon magnetic moment $kappa_0$ enters through a shift of the argument of the finite-volume function and through a term linear in $B_0$, while the two-nucleon coefficient $ilde l_2$ enters only through the second. Expanding about the infinite-volume limit, the continuum levels depend on an effective scattering length $1/A_3 = 1/a_3 pm (eB_0/2) ilde l_2$, so measuring a family of levels across volumes and field strengths separates $kappa_0$ from $ilde l_2$. The same construction, with the magnetic field replaced by a fictitious isovector axial background field, gives a coupled-channel condition for the $^1S_0(I_z=0)$–$^3S_1(m=0)$ system in which the weak coefficient $l_{1,A}$ appears as the off-diagonal mixing term; that coefficient controls the neutral- and charged-current breakup of the deuteron. The authors also argue that the analogous magnetic mixing between $^1S_0$ and $^3S_1$ isolates the coefficient $l_1$ that enters the threshold $np to d gamma$ cross-section.

The premise it rests on. The scheme needs the background field to be weak enough that the pionless effective theory still applies, yet strong enough that the resulting energy shifts are large compared with lattice resolution, and the paper's own numbers show the two-nucleon magnetic coefficient is unnaturally small enough that at $eB_0 = 10^4$ MeV$^2$ the effective scattering length moves by only about ten percent.

If the paper is right (Pith's inferences, not the authors')

A lattice measurement of the $m = pm 1$ two-nucleon level splittings at several volumes and magnetic field strengths would yield $ilde l_2$, the short-distance contribution to the deuteron magnetic moment, separately from the proton and neutron magnetic moments.

The same class of measurements in a background magnetic field, once the $^1S_0$–$^3S_1$ mixing is resolved, would fix $l_1$ and hence remove the dominant theoretical uncertainty in the threshold $np to d gamma$ cross-section.

A lattice simulation in a background isovector axial field would determine $l_{1,A}$, which controls the neutral- and charged-current breakup of the deuteron; the paper quotes that a 10 percent determination of $l_{1,A}$ translates into a few-percent uncertainty on the $nu d$ cross-sections at 10 MeV.

Because the method needs only two-nucleon four-point correlators rather than five-point functions with an inserted current, the computational cost is close to that of existing nucleon-nucleon scattering calculations.

The spectra shown for asymmetric volumes with $eta_{1,2} = 0.1$ indicate that for suitable field strengths one or two levels are strongly sensitive to the short-distance coefficient while the continuum levels are not, giving a clean experimental handle.

The proposal is a template rather than a one-off: any short-distance operator that couples differently from its one-body counterpart in a finite-volume quantization condition could in principle be extracted the same way, which suggests the technique generalizes to three-nucleon and hyperon-nucleon channels.

The paper's own numbers make the practical case delicate: $ilde l_2$ is unnaturally small, so at $eB_0 = 10^4$ MeV$^2$ the effective scattering length differs from $a_3$ by only about ten percent, and whether that shift is resolvable on lattices small enough for the pionless theory to hold is an empirical question the paper does not settle.

The suggestion that light-quark masses could be tuned to a point where the scattering lengths diverge, making the two-nucleon coefficient the sole determinant of the effective scattering length, is a concrete and testable refinement of the method that would sidestep the small-$ilde l_2$ problem.

If the same background-field configurations can serve both the electromagnetic and weak extractions, the marginal cost of the weak measurement is small, which would make the neutrino-deuteron channel an attractive early target for lattice groups already generating two-nucleon spectroscopy.

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The paper

Abstract

We demonstrate how to make rigorous predictions for electroweak matrix elements in nuclear systems directly from QCD. More precisely, we show how to determine the short-distance contributions to low-momentum transfer electroweak matrix elements in the two-nucleon sector from lattice QCD. In potential model descriptions of multi-nucleon systems, this is equivalent to uniquely determining the meson-exchange currents, while in the context of nuclear effective field theory, this translates into determining the coefficients of local, gauge-invariant, multi-nucleon-electroweak current operators. The energies of the lowest-lying states of two nucleons on a finite volume lattice with periodic boundary conditions in the presence of a background magnetic field are sufficient to determine the local four-nucleon operators that contribute to the deuteron magnetic moment and to the threshold cross-section of n + p -> d + gamma. Similarly, the energy-levels of two nucleons immersed in a background isovector axial weak field can be used to determine the coefficient of the leading local four-nucleon operator contributing to the neutral- and charged-current break-up of the deuteron. This is required for the extraction of solar neutrino fluxes at SNO and future neutrino experiments.

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. No sentence in this record was written by a model.

abstract compiled · 2026-09-23

The authors' main claim, as the classifier picked it out 0.81

“We demonstrate how to make rigorous predictions for electroweak matrix elements in nuclear systems directly from QCD.”

How the main claim is stated: asserted 1.00

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.25 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.16 · mentions code or data availability 0.02

The abstract, sentence by sentence

This table labels the abstract only (arXiv listing abstract v1). The number is the probability the classifier gave the label.

SentenceRoleThe authors' own workStates the main conclusionStrength
We demonstrate how to make rigorous predictions for electroweak matrix elements in nuclear systems directly from QCD. contribution 1.00 0.92 0.68 asserted 1.00
More precisely, we show how to determine the short-distance contributions to low-momentum transfer electroweak matrix elements in the two-nucleon sector from lattice QCD. contribution 0.97 0.91 0.38 asserted 1.00
In potential model descriptions of multi-nucleon systems, this is equivalent to uniquely determining the meson-exchange currents, while in the context of nuclear effective field theory, this translates into determining the coefficients of local, gauge-invariant, multi-nucleon-electroweak current operators. interpretation 0.94 0.28 0.13 asserted 1.00
The energies of the lowest-lying states of two nucleons on a finite volume lattice with periodic boundary conditions in the presence of a background magnetic field are sufficient to determine the local four-nucleon operators that contribute to the deuteron magnetic moment and to the threshold cross-section of n + p -> d + gamma. method 0.45 0.63 0.24 asserted 0.99
Similarly, the energy-levels of two nucleons immersed in a background isovector axial weak field can be used to determine the coefficient of the leading local four-nucleon operator contributing to the neutral- and charged-current break-up of the deuteron. contribution 0.37 0.64 0.10 asserted 0.80
This is required for the extraction of solar neutrino fluxes at SNO and future neutrino experiments. interpretation 0.65 0.08 0.05 asserted 0.98

What kind of paper the classifier reports this is: method or tool 0.88 · evidence from proof or derivation 0.67

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How the labels were made

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Cited by 24 Pith papers

Reviewed papers on Pith that cite this one. The full citation record

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  2. Proton-proton fusion and tritium $\beta$-decay from lattice quantum chromodynamics hep-lat · Oct 2016
  3. The Magnetic Structure of Light Nuclei from Lattice QCD hep-lat · Jun 2015
  4. Ab initio calculation of the $np \to d \gamma$ radiative capture process hep-lat · May 2015
  5. Two-nucleon systems at $m_{\pi}\approx292$ MeV from lattice QCD hep-lat · May 2026
  6. Preparations for Quantum Simulations of Quantum Chromodynamics in 1+1 Dimensions: (II) Single-Baryon $\beta$-Decay in Real Time quant-ph · Sep 2022
  7. Relativistic, model-independent, multichannel $2\to2$ transition amplitudes in a finite volume hep-lat · Sep 2015
  8. Multichannel 0-to-2 and 1-to-2 transition amplitudes for arbitrary spin particles in a finite volume hep-lat · Feb 2015
  9. Multichannel one-to-two transition amplitudes in a finite volume hep-lat · Jun 2014
  10. Two-particle multichannel systems in a finite volume with arbitrary spin hep-lat · Jan 2014
  11. Two-Baryon Systems with Twisted Boundary Conditions hep-lat · Nov 2013
  12. Two-Nucleon Systems in a Finite Volume: (I) Quantization Conditions hep-lat · May 2013
  13. Matrix elements of unstable states hep-lat · May 2012
  14. Moving Multi-Channel Systems in a Finite Volume with Application to Proton-Proton Fusion hep-lat · Apr 2012
  15. n-Boson Energies at Finite Volume and Three-Boson Interactions hep-lat · Jul 2007
  16. Electromagnetic and spin polarisabilities in lattice QCD hep-lat · Mar 2006
  17. Form factors of two-hadron states from a covariant finite-volume formalism hep-lat · Dec 2018
  18. Two-Nucleon Systems in a Finite Volume: (II) 3S1-3D1 Coupled Channels and the Deuteron hep-lat · Sep 2013
  19. Extracting Scattering Phase-Shifts in Higher Partial-Waves from Lattice QCD Calculations hep-lat · Jan 2011
  20. QCD with background electromagnetic fields on the lattice: a review hep-lat · Jun 2024
  21. Neutrinoless Double-Beta Decay: A Roadmap for Matching Theory to Experiment hep-ph · Mar 2022
  22. Lattice QCD and Neutrino-Nucleus Scattering hep-lat · Apr 2019
  23. The Role of Lattice QCD in Searches for Violations of Fundamental Symmetries and Signals for New Physics hep-lat · Apr 2019
  24. Hadrons and Nuclei hep-lat · Apr 2019
How the literature uses it

42 arXiv papers cite this in 67 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 46 sentences without a confident label

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  • The approach discussed here is based on prior formalism developed by two of us in Ref. [ref] .[This, in turn, was inspired and guided by the work of Refs. [ref] .] In Sec. [ref] we present a slightly modified version of this formalism in which all infinite-volume quantities are Lorentz covariant and the single-particle matrix elements that enter, abbreviated $1 + 𝒥 →1$, are expressed in terms of standard Lorentz-invariant form factors.

    Form factors of two-hadron states from a covariant finite-volume formalism · Introduction · 0.83

  • The matching framework of this work builds upon major developments in recent years in accessing local and non-local transition amplitudes in hadronic physics from the corresponding finite-volume matrix elements in Euclidean spacetime obtained with lattice QCD [ref] , and in particular, a recent work on developing a similar formalism for the two-neutrino process $nn →pp (eeν̅_eν̅_e)$ [ref] .

    The path from lattice QCD to the short-distance contribution to $0\nu\beta\beta$ decay with a light Majorana neutrino · Introduction · 0.88

  • Extending the formalism developed for coupled-channel systems [ref] , the FV formalism describing NN systems with arbitrary CM momenta, spin, angular momentum and isospin has been developed recently, providing expressions for the energy eigenvalues in irreducible representations (irreps) of the FV symmetry groups [ref] .

    Two-Nucleon Systems in a Finite Volume: (II) 3S1-3D1 Coupled Channels and the Deuteron · Introduction · 0.81

Uses its method or tool 3 papers
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