Low-$Q^2$ empirical parametrizations of the $N^\ast$ helicity amplitudes

G. Ramalho

REVIEW 4 major objections 5 minor 61 references

Low-$Q^2$ empirical parametrizations of the $N^\ast$ helicity amplitudes

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that empirical fits to nucleon-resonance helicity amplitudes that ignore the pseudothreshold kinematics are inconsistent at low $Q^2$, and presents an analytic matching method that enforces the constraints while…

desk verdict A useful methods paper that enforces pseudothreshold constraints on empirical N* helicity amplitudes via a matching algorithm, but the Delta(1232) headline claim is stronger than the evidence supports. read the letter →

arxiv 1909.00013 v2 pith:GL33CDGV submitted 2019-08-30 hep-ph hep-exnucl-exnucl-th

G. Ramalho This is my paper · ORCID

classification hep-phhep-exnucl-exnucl-th

keywords helicityamplitudespseudothresholdnucleonresonanceselectromagnetictransitionformfactorsanalyticcontinuationSiegerttheoremDelta(1232)lowmomentumtransfer

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

At the pseudothreshold, where the photon three-momentum vanishes, the $\gamma^\ast N \to N^\ast$ helicity amplitudes are forced by current conservation and kinematics to have specific powers of $|\mathbf q|$ and to satisfy correlations between transverse and longitudinal amplitudes. The paper argues that common empirical parametrizations ignore these constraints and therefore give wrong low-$Q^2$ behavior, especially for resonances whose mass is close to the nucleon mass. It proposes a method that replaces a given parametrization below a matching point $Q_P^2$ with an expansion in powers of $|\mathbf q|$ that respects the pseudothreshold constraints and joins smoothly, with amplitude plus first, second, and sometimes third derivatives continuous. Applying this to nine resonances, the paper finds that the constraints have a definite impact for the $\Delta(1232)$ below $0.3$ GeV$^2$ and for the $N(1520)$ below about $0.1$ GeV$^2$, while other cases remain undecided by current data. If correct, the method supplies a consistency requirement that any future low-$Q^2$ fit or theory comparison should meet.

What carries the argument

The central object is the pseudothreshold expansion $A=\tilde q^n(\alpha_0+\alpha_1\tilde q^2+\alpha_2\tilde q^4+\alpha_3\tilde q^6)$, where $\tilde q=|\mathbf q|/M_R$ is the photon three-momentum in the resonance rest frame normalized by the resonance mass and $n=0,1,2$ encodes the required $|\mathbf q|$ power behavior near the pseudothreshold. The coefficients are fixed by matching the amplitude, its first derivative, and its second (and sometimes third) derivative at the matching point $Q_P^2$, while the leading coefficient is set by pseudothreshold correlations such as the long-wavelength (Siegert) relation between $A_{1/2}$ and $S_{1/2}$. This machinery turns the kinematic constraints into a smooth boundary condition that any data parametrization can be forced to satisfy without refitting the higher-$Q^2$ data.

What would settle it

Precise $\Delta(1232)$ helicity amplitudes at $Q^2=0.05$, $0.1$, and $0.2$ GeV$^2$ from electron-scattering data of the kind used in the paper would settle the issue: the $Q_P^2=0.3$ GeV$^2$ constrained extension and the original parametrization differ visibly in that interval, and data that follow the unconstrained curve would refute the claim that ignoring the pseudothreshold constraints is erroneous below $0.3$ GeV$^2$.

Watch

Extended reading notes

Core claim

The central claim is that the pseudothreshold point $Q^2=-(M_R-M_N)^2$ is not a remote technicality: for resonances close to the nucleon it sits near $Q^2=0$, and the fixed $|\mathbf q|$ power laws, such as $A_{1/2}\propto |\mathbf q|$ or constant and $S_{1/2}\propto |\mathbf q|^2$ or $|\mathbf q|$, together with correlations like $S_{1/2}\propto A_{1/2}|\mathbf q|$, control the amplitudes throughout the low-$Q^2$ region. The paper demonstrates this by constructing, for each of nine resonances, an analytic extension $A=\tilde q^n(\alpha_0+\alpha_1\tilde q^2+\alpha_2\tilde q^4+\alpha_3\tilde q^6)$ on the interval from the pseudothreshold up to a matching point $Q_P^2$, with coefficients fixed by continuity of the amplitude and its derivatives at $Q_P^2$ and by the pseudothreshold correlations. Scanning $Q_P^2=0.1$, $0.3$, and $0.5$ GeV$^2$, the paper shows that the $\Delta(1232)$ data select the $Q_P^2=0.3$ GeV$^2$ extension and that ignoring the constraints below $0.3$ GeV$^2$ gives erroneous amplitudes, while the $N(1520)$ case selects $Q_P^2=0.1$ GeV$^2$ through the $A_{3/2}(0)$ constraint. For the remaining resonances, the available data cannot distinguish the different extensions.

Load-bearing premise

The paper assumes that the simple four-term power series in the photon momentum describes the amplitudes exactly across the whole interval from the pseudothreshold to $Q_P^2$, not just right at the pseudothreshold; if extra terms are needed there, the matched curves and the conclusions about which matching point fits the data would change.

Editorial extensions

If this is right

Any empirical fit aiming to describe nucleon-resonance amplitudes below about $0.3$ GeV$^2$ should build in the $|\mathbf q|$ power laws and amplitude correlations from the pseudothreshold, or it will misrepresent the region between $Q^2=0$ and the first data.
The constrained $Q_P^2=0.3$ GeV$^2$ extension for the $\Delta(1232)$ provides a testable prediction for the amplitudes in the low-$Q^2$ gap.
For the $N(1520)$, only extensions matched at $Q_P^2=0.1$ GeV$^2$ reproduce the measured $A_{3/2}(0)$, so fits matched at higher momentum transfer should be avoided for that resonance.
For the other resonances, the method identifies which amplitudes are most sensitive to the matching point and therefore which new measurements below $Q^2=0.3$ GeV$^2$ would be most informative.
The matching procedure applies to any parametrization whose amplitudes and first derivatives are continuous, so other empirical fits can be made pseudothreshold-consistent without changing their high-$Q^2$ behavior.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The method turns the pseudothreshold constraint into a model-independent boundary condition, so the same matching could be applied to any theoretical transition current to test whether theory curves respect the same low-$Q^2$ behavior.
The $\Delta(1232)$ case suggests a useful rule of thumb: the constraints matter up to roughly a few times $(M_R-M_N)^2$, so resonances with smaller mass splittings should show the largest pseudothreshold effects near the photon point.
The absence of odd powers of $|\mathbf q|$ in the ansatz is a testable assumption; extending the expansion with an additional odd term and refitting would quantify the truncation error and reveal whether the data tolerate such a term.
Accurate longitudinal $S_{1/2}$ data below $Q^2=0.3$ GeV$^2$ would be the sharpest discriminator, because the paper shows that $S_{1/2}$ extensions differ most strongly across the chosen matching-point values.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The genuinely new thing here is the gluing algorithm: take any smooth empirical parametrization of N* helicity amplitudes valid at moderate Q^2, and replace it below a chosen matching point QP^2 by a short power series in |q|^2 that builds in the pseudothreshold factors and amplitude correlations, matching value and derivatives at QP^2. That is a practical tool, and the paper gives enough detail to re-implement it: derivative conversion formulas in the appendices, and coefficient tables for nine states at three QP^2 values. Credit where it is due: the algebra is transparent, the appendices are useful, and the author is candid that most of the nine states are inconclusive.

The soft spots are real but not fatal. First, Eq. (4.2) truncates the |q|^2 series at |q|^6 with no sensitivity test. The even-power-only structure follows from regularity in Q^2, but the number of terms is set by how many derivatives you want to match, not by any physical scale. A different polynomial order or a different variable would give different interpolants, and the visual ranking of QP^2 = 0.3 for the Delta could shift. So "conclusively" is too strong. Second, the Delta comparison replaces older MAMI/MIT-Bates points below 0.15 GeV^2 with JLab/Hall A data, partly on the strength of the author's own reanalysis. That may be right, but it deserves a sensitivity check rather than a footnote. Third, the JLab input parametrizations are declared valid for Q^2 = 0.5–5 GeV^2, while two of the three matching points (0.1 and 0.3 GeV^2) are below that range, so the derivatives at those points are extrapolations of the fit, not data. Finally, agreement with data is judged by eye, with no chi-square or error propagation.

None of this kills the method. As an interpolation tool that enforces kinematic constraints, it works, and the coefficient tables are useful. The paper should be read as a methods proposal with illustrative applications, not as a determination that pseudothreshold constraints cut in at 0.3 GeV^2 for the Delta. The author should supply the input derivatives, add a higher-order sensitivity test, and quantify the Delta comparison.

This deserves peer review. Send it to a serious referee and ask for those revisions; with them, it would be a solid reference for anyone doing low-Q^2 N* fits.

Referee Report

4 major / 5 minor

Summary. The paper addresses the low-Q^2 behavior of empirical parametrizations of the helicity amplitudes for γ*N → N* transitions. It argues that pseudothreshold constraints, which dictate specific powers of the photon three-momentum |q| and correlations among helicity amplitudes, are commonly ignored in data parametrizations and can lead to erroneous low-Q^2 descriptions. The paper proposes a method to modify an existing analytic parametrization below a chosen transition point Q_P^2 by matching the amplitude and its first (and sometimes second/third) derivatives at Q_P^2 to a polynomial expansion in |q|^2 that respects the pseudothreshold behavior. The method is applied to nine resonances using the Jefferson Lab parametrizations of Ref. [33], with Q_P^2 = 0.1, 0.3, and 0.5 GeV^2, and the resulting extensions are compared visually with data. The main quantitative claim is that for the Δ(1232) the pseudothreshold constraints cannot be ignored below 0.3 GeV^2, while for several other resonances the data are insufficient to discriminate among the choices of Q_P^2. The paper includes explicit formulas for the matching coefficients, derivative conversion tables, and full coefficient tables in appendices.

Significance. If the proposed method is sound, it provides a practical and general prescription for incorporating current-algebra constraints into empirical parametrizations of helicity amplitudes, which would be useful for the interpretation of low-Q^2 electroproduction data and for model comparisons. The paper is candid about the inconclusive cases, and the algebraic derivation of the matching conditions is internally consistent; the explicit coefficient tables and derivative relations (Appendices A–C) are a useful resource. The manuscript’s strengths include the generality of the formalism, the explicit treatment of all J^P = 1/2^±, 3/2^± cases, and the transparent discussion of limitations for resonances with scarce data. However, the headline Δ(1232) conclusion rests on an untested truncation of the low-Q^2 ansatz, on matching points that may lie outside the stated validity range of the input parametrization, and on a visual (not quantitative) comparison with data, in a region where the author’s own prior analysis supplies several of the data points. The result is therefore promising but not yet established to the level claimed.

major comments (4)

[Eq. (4.2) and Sec. VI] The central claim in Sec. VI that for the Δ(1232) the pseudothreshold constraints ‘cannot be ignored below 0.3 GeV²’ rests on the ansatz A = ũ^n(α_0 + α_1 ũ² + α_2 ũ⁴ + α_3 ũ⁶) being valid over the whole interval from the pseudothreshold to Q_P². Footnote 2 only shows that odd powers vanish as |q|→0; it does not establish that the bracket is a quartic polynomial in ũ² throughout the interval, and no convergence test or comparison with alternative bases (including odd powers or higher-order terms) is given. Because the coefficients are fixed by matching at Q_P², the curves in Fig. 3 are not fits to the low-Q² data, so the visual preference for Q_P² = 0.3 is specific to this basis and cannot, as it stands, support the ‘conclusive’ wording in Sec. VI.
[Sec. IV and Sec. V vs. Sec. I/VI] The method assumes (Sec. IV) that the original parametrization ‘describe[s] well the data above Q_P²’, but the Jefferson Lab parametrizations are stated in Sec. I to be valid for Q² = 0.5–5 GeV², so Q_P² = 0.1 and 0.3 GeV² lie outside the region where the input is established; Sec. VI later says the same parametrizations ‘cover the region Q² = 0–5 GeV²’, which is internally inconsistent. The derivatives at Q_P² used to fix the extension coefficients are thus not reliable for the two smaller Q_P² values, and the conclusions drawn from comparing Q_P² = 0.1, 0.3, and 0.5 are correspondingly weakened.
[Sec. V.C (Delta(1232))] In Sec. V.C the low-Q² Δ(1232) data (Q² < 0.15 GeV²) are not the original measurements from MAMI and MIT-Bates but are replaced by results from Refs. [41,42] (the author’s own work), converted to helicity amplitudes using MAID 2007. Since the central conclusion concerns which Q_P² best reproduces the data in this region, the use of author-derived pseudo-data introduces a circularity that the text acknowledges but does not quantify; at minimum a comparison with the original MAMI and MIT-Bates data should be shown to demonstrate that the Q_P² preference is not an artifact of the replacement.
[Figs. 1–4 and Sec. V.C] The comparison in Sec. V and Figs. 1–4 is purely visual; no χ² or other quantitative measure is given. Statements such as ‘the parametrization characterized by Q_P² = 0.3 GeV² is the one that better describes the data’ (Sec. V.C) and the corresponding conclusion in Sec. VI require a quantitative comparison that accounts for the data uncertainties, especially because the data are sparse and the curves differ mainly below Q² = 0.3 GeV².

minor comments (5)

[Table II] In the 1/2⁻ row, the expression for S_{1/2} lists c_3 ũ⁵ twice; it should presumably read c_3 ũ⁷ to follow the stated pattern, and the coefficient c_0 should be bold in the table as indicated by the text.
[Sec. IV.D] The phrase ‘The states 3/2⁻ are the exception to this role’ should read ‘exception to this rule’.
[Fig. 2 caption] The caption refers to ‘tick lines’; this should be ‘thick lines’.
[Abstract] The abstract states ‘the invariant four-momentum square became q²’; the verb should be ‘becomes’ or ‘is’.
[Reference [33]] Reference [33] is a URL; it should be replaced by a published source or by a more complete description of the parametrizations, since several conclusions depend on its validity range and functional forms.

Circularity Check

1 steps flagged · score 3.0 of 10

Core method is a genuine matched-extension test, but the Delta(1232) benchmark data come from the author's own prior reanalyses, making the flagship conclusion partly self-referential.

self citation load bearing [Sec. V.C, Delta(1232) data paragraph (Fig. 3); cf. Sec. VI conclusion that pseudothreshold constraints 'cannot be ignored below 0.3 GeV^2']
"As for the Q2 < 0.15 GeV2 data, we replace the results from MAMI and MIT-Bates ... by the recent results from JLab/Hall A ... This procedure is motivated by the conclusion that there are errors in the previous analysis which lead to an overestimation of the results for GE and GC , as discussed in Refs. [41, 49]. The data for A1/2, A3/2 and S1/2 presented here are converted from the results for the form factors presented in Refs. [41, 42]."

The paper's strongest conclusion is that the Delta(1232) pseudothreshold constraints cannot be ignored below 0.3 GeV^2, and that the QP^2 = 0.3 extension best describes the low-Q^2 data. But the low-Q^2 Delta(1232) points used for that comparison are not raw independent data: they are converted from the author's own previous papers [41,42], and the decision to discard the older MAMI/MIT-Bates points is justified by the author's own Ref. [41]. Thus the empirical confirmation is partly a comparison of the present extension against the author's earlier reanalysis, not against an external benchmark. The extension coefficients are not refitted to these points, so the circularity is partial, but the 'conclusive' support for the central claim is self-referential.

full rationale

The core method is not circular by construction. The coefficients in Eq. (4.4) are fixed by matching the amplitude and its derivatives at QP^2 to the JLab parametrizations and by imposing pseudothreshold correlations, and the resulting extensions are then compared with data to see which QP^2 is consistent. That is a genuine interpolation-and-comparison procedure, not a fit of the conclusion, and the pseudothreshold constraints are standard results (Siegert theorem, Jones-Scadron relations) supported by independent references [23-25,30]. The untested truncation of the polynomial ansatz (Eq. 4.2: only even powers beyond qtilde^n, four terms) is a real robustness/accuracy risk, but it is not a circularity, because it is a model assumption rather than an input that already contains the output. The main circularity burden is the Delta(1232) benchmark: the low-Q^2 'data' used to select QP^2 = 0.3 and to draw the flagship conclusion are taken from the author's own Refs. [41,42], with the replacement of older data justified by the author's own Ref. [41]. This makes the headline result partly self-referential, but not fully reducible to its inputs, so a moderate score is appropriate.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper adds no new entities. Its central construction depends on two hand-chosen quantities (Lambda and QP^2) and on four premises: the exactness of the pseudothreshold constraints, the validity of the input parametrizations and their moments at QP^2, the adequacy of the truncated even-power polynomial in |q|^2 over the whole low-Q^2 interval, and the sufficiency of derivative matching. The alpha coefficients are fixed by the matching, not fitted to the target data.

free parameters (2)

Lambda = Lambda^3 = 0.1 GeV^3
Hand-chosen scale to regularize the divergent 1/Q^3 JLab ansatz for Delta(1620) S_1/2; the extension depends on this choice (Sec. V.E).
Q_P^2 = 0.1, 0.3, and 0.5 GeV^2
Transition point chosen by hand; results for several states depend strongly on it, and the paper selects preferred values by visual comparison with data.

assumptions (4)

domain assumption The pseudothreshold constraints (Table I), including Siegert's theorem, are exact consequences of the gauge-invariant structure of the gamma* N to N* current.
These relations from Refs. [23-25,30-32] are imported as prior results; the whole method enforces them, so if they were only approximate or model-dependent, the central claim would weaken.
domain assumption The original empirical parametrizations are regular, continuous, and have continuous derivatives for Q^2 >= QP^2, and remain valid down to QP^2 in the region where they are used to define the matching.
The method requires the moments A(k) at QP^2 to exist and to represent the physical amplitudes; no independent test of this is given (Sec. III).
ad hoc to paper The truncated expansion A = qtilde^n(alpha0 + alpha1 qtilde^2 + alpha2 qtilde^4 + alpha3 qtilde^6) with only even powers in the bracket represents the amplitudes over the whole interval from pseudothreshold to QP^2.
The form is chosen to match the leading pseudothreshold behavior and to allow the matching equations to close; no physical derivation or convergence test establishes it away from the pseudothreshold (Eq. (4.2), Table II).
domain assumption Demanding continuity of the amplitude and its first two (or three) derivatives at QP^2 yields a physically acceptable continuation.
The smoothness conditions are imposed by fiat; the paper does not show that an amplitude satisfying them is unique or physical (Sec. IV).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Low-$Q^2$ empirical parametrizations of the $N^\ast$ helicity amplitudes." pith.science (2026). https://pith.science/paper/GL33CDGV

@misc{pith2026190900013,
  author       = {Pith},
  title        = {Pith review of: Low-$Q^2$ empirical parametrizations of the $N^\ast$ helicity amplitudes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GL33CDGV}},
  note         = {Machine review of arXiv:1909.00013}
}

abstract

The data associated with the electromagnetic excitations of the nucleon ($\gamma^\ast N \to N^\ast$) are usually parametrized by helicity amplitudes at the resonance $N^\ast$ rest frame. The properties of the $\gamma^\ast N \to N^\ast$ transition current at low $Q^2$ can be, however, better understood when expressed in terms of structure form factors, particularly near the pseudothreshold, when the magnitude of the photon three-momentum vanishes ($|{\bf q}| =0$). At the pseudothreshold the invariant four-momentum square became $q^2= (M_R-M_N)^2$, well in the timelike region $Q^2 =- q^2 < 0$ [$M_N$ and $M_R$ are the mass of the nucleon and of the resonance, respectively]. In the helicity amplitude representation, the amplitudes have well-defined dependences on $|{\bf q}|$, near the pseudothreshold, and there are correlations between different amplitudes. Those constraints are often ignored in the empirical parametrizations of the helicity amplitudes. In the present work, we show that the structure of the transition current near the pseudothreshold has an impact on the parametrizations of the data. We present a method which modifies analytic parametrizations of the data at low $Q^2$, in order to take into account the constraints of the transition amplitudes near the pseudothreshold. The model dependence of the parametrizations on the low-$Q^2$ data is studied in detail.

Figures

Figures reproduced from arXiv: 1909.00013 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

61 extracted references · 44 canonical work pages

[33]

A. N. Hiller Blin et al. , Phys. Rev. C 100, 035201 (2019) [arXiv:1904.08016 [hep-ph]]

work page Pith review arXiv 2019
[1]

The amplitude is independent of the pseudothresh- old conditions

work page
[2]

In the ﬁrst case, we can determine all coeﬃcients us- ing the continuity of A and the ﬁrst three derivatives at the point Q2 = Q2 P to ﬁx all the coeﬃcients

The amplitude is correlated with another ampli- tude. In the ﬁrst case, we can determine all coeﬃcients us- ing the continuity of A and the ﬁrst three derivatives at the point Q2 = Q2 P to ﬁx all the coeﬃcients. The last coeﬃcient α3 is in this case determined by A(3) (third derivative of A). In the second case we use the correlation condition to ﬁx the ﬁ...

work page 2007
[3]

The coeﬃcient α3 is determined using the continu- ity of the third derivative: 6α3 = ˜A(3); (A8)

work page
[4]

The coeﬃcient α3 can then be de- termined by Eq

the coeﬃcient α0 is determined by a pseudothresh- old condition. The coeﬃcient α3 can then be de- termined by Eq. (A7) using α3 = ( ˜Ap − α0) 1 ˜q6p , (A9) where ˜Ap ≡ ˜A(0) − ˜A(1) ˜q2 p + 1 2 ˜A(2) ˜q4 p. According to the discussion from Sec. IV, the ﬁrst con- dition is used in case 1, and the second condition is used in case 2. In general Eq. (A8) is u...

work page
[5]

I. G. Aznauryan et al. , Int. J. Mod. Phys. E 22, 1330015 (2013) [arXiv:1212.4891 [nucl-th]]

arXiv 2013
[6]

A. J. R. Puckett et al. , Phys. Rev. C 96, 055203 (2017) Erratum: [Phys. Rev. C 98, 019907 (2018)] [arXiv:1707.08587 [nucl-ex]]

arXiv 2017
[7]

I. G. Aznauryan and V. D. Burkert, Prog. Part. Nucl. Phys. 67, 1 (2012) [arXiv:1109.1720 [hep-ph]]

arXiv 2012

Show all 61 references

[8]

Drechsel, S

D. Drechsel, S. S. Kamalov and L. Tiator, Eur. Phys. J. A 34, 69 (2007) [arXiv:0710.0306 [nucl-th]]

2007 arXiv
[9]

A. Lenz, M. Gockeler, T. Kaltenbrunner and N. Warkentin, Phys. Rev. D 79, 093007 (2009) [arXiv:0903.1723 [hep-ph]]

2009 arXiv
[10]

Wang and F

L. Wang and F. X. Lee, Phys. Rev. D 80, 034003 (2009) [arXiv:0905.1944 [hep-ph]]

2009 arXiv
[11]

I. V. Anikin, V. M. Braun and N. Oﬀen, Phys. Rev. D 92, 014018 (2015) [arXiv:1505.05759 [hep-ph]]

2015 arXiv
[12]

Eichmann, H

G. Eichmann, H. Sanchis-Alepuz, R. Williams, R. Alkofer and C. S. Fischer, Prog. Part. Nucl. Phys. 91, 1 (2016) [arXiv:1606.09602 [hep-ph]]

2016 arXiv
[13]

Segovia and C

J. Segovia and C. D. Roberts, Phys. Rev. C 94, 042201 (2016) [arXiv:1607.04405 [nucl-th]]

2016 arXiv
[14]

G. F. de Teramond and S. J. Brodsky, AIP Conf. Proc. 1432, 168 (2012) [arXiv:1108.0965 [hep-ph]]

2012 arXiv
[15]

S. J. Brodsky, G. F. de Teramond, H. G. Dosch and J. Erlich, Phys. Rept. 584, 1 (2015) [arXiv:1407.8131 [hep-ph]]

2015 arXiv
[16]

Ballon-Bayona, H

A. Ballon-Bayona, H. Boschi-Filho, N. R. F. Braga, M. Ihl and M. A. C. Torres, Phys. Rev. D 86, 126002 (2012) [arXiv:1209.6020 [hep-ph]]

2012 arXiv
[17]

Ramalho and D

G. Ramalho and D. Melnikov, Phys. Rev. D 97, 034037 (2018) [arXiv:1703.03819 [hep-ph]]; G. Ramalho, Phys. Rev. D 96, 054021 (2017) [arXiv:1706.05707 [hep-ph]]

2018 arXiv
[18]

Gutsche, V

T. Gutsche, V. E. Lyubovitskij and I. Schmidt, Phys. Rev. D 97, 054011 (2018) [arXiv:1712.08410 [hep- ph]]; T. Gutsche, V. E. Lyubovitskij and I. Schmidt, arXiv:1911.00076 [hep-ph]

2018 arXiv
[19]

I. T. Obukhovsky, A. Faessler, T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D 89, 014032 (2014) [arXiv:1306.3864 [hep-ph]]; I. T. Obukhovsky, A. Faessler, D. K. Fedorov, T. Gutsche and V. E. Lyubovitskij, arXiv:1909.13787 [hep-ph]

2014 arXiv
[20]

I. G. Aznauryan and V. D. Burkert, Phys. Rev. C 85, 055202 (2012) [arXiv:1201.5759 [hep-ph]]; I. G. Aznau- ryan and V. Burkert, Phys. Rev. C 95, 065207 (2017) [arXiv:1703.01751 [nucl-th]]

2012 arXiv
[21]

Santopinto and M

E. Santopinto and M. M. Giannini, Phys. Rev. C 86, 065202 (2012) [arXiv:1506.01207 [nucl-th]]; M. M. Gian- nini and E. Santopinto, Chin. J. Phys. 53, 020301 (2015) [arXiv:1501.03722 [nucl-th]]

2012 arXiv
[22]

Ramalho and K

G. Ramalho and K. Tsushima, Phys. Rev. D 84, 051301 (2011) [arXiv:1105.2484 [hep-ph]]; G. Ramalho and M. T. Pe˜ na, Phys. Rev. D 89, 094016 (2014) [arXiv:1309.0730 [hep-ph]]; G. Ramalho, Phys. Rev. D 95, 054008 (2017) [arXiv:1612.09555 [hep-ph]]

2011 arXiv
[23]

Ramalho, Phys

G. Ramalho, Phys. Rev. D 90, 033010 (2014) [arXiv:1407.0649 [hep-ph]]

2014 arXiv
[24]

S. S. Kamalov and S. N. Yang, Phys. Rev. Lett. 83, 4494 (1999) [nucl-th/9904072]

1999 arXiv
[25]

V. D. Burkert and T. S. H. Lee, Int. J. Mod. Phys. E 13, 1035 (2004) [nucl-ex/0407020]

2004 arXiv
[26]

Julia-Diaz, T.-S

B. Julia-Diaz, T.-S. H. Lee, T. Sato and L. C. Smith, Phys. Rev. C 75, 015205 (2007) [nucl-th/0611033]

2007 arXiv
[27]

J. D. Bjorken and J. D. Walecka, Annals Phys. 38, 35 (1966)

1966
[28]

H. F. Jones and M. D. Scadron, Annals Phys. 81, 1 (1973)

1973
[29]

R. C. E. Devenish, T. S. Eisenschitz and J. G. Korner, Phys. Rev. D 14, 3063 (1976)

1976
[30]

Eichmann and G

G. Eichmann and G. Ramalho, Phys. Rev. D 98, 093007 (2018) [arXiv:1806.04579 [hep-ph]]

2018 arXiv
[31]

Ramalho, Phys

G. Ramalho, Phys. Lett. B 759, 126 (2016) [arXiv:1602.03444 [hep-ph]]

2016 arXiv
[32]

Ramalho, Phys

G. Ramalho, Phys. Rev. D 93, 113012 (2016) [arXiv:1602.03832 [hep-ph]]. 17

2016 arXiv
[34]

Drechsel and L

D. Drechsel and L. Tiator, J. Phys. G 18, 449 (1992)

1992
[35]

Amaldi, S

E. Amaldi, S. Fubini, and G. Furlan, Pion- Electroproduction Electroproduction at Low Energy and Hadron Form Factor , Springer Berlin Heidelberg (1979)

1979
[36]

Tiator, Few Body Syst

L. Tiator, Few Body Syst. 57, 1087 (2016)

2016
[37]

https://userweb.jlab.org/~isupov/couplings/
[38]

V. I. Mokeev, https://userweb.jlab.org/~mokeev/ resonance_electrocouplings/
[39]

I. G. Aznauryan et al. [CLAS Collaboration], Phys. Rev. C 80, 055203 (2009) [arXiv:0909.2349 [nucl-ex]]

2009 arXiv
[40]

V. I. Mokeev et al. [CLAS Collaboration], Phys. Rev. C 86, 035203 (2012) [arXiv:1205.3948 [nucl-ex]]

2012 arXiv
[41]

V. I. Mokeev et al. , Phys. Rev. C 93, 025206 (2016) [arXiv:1509.05460 [nucl-ex]]

2016 arXiv
[42]

A. J. Buchmann, E. Hernandez, U. Meyer and A. Faessler, Phys. Rev. C 58, 2478 (1998)

1998
[43]

Ramalho and K

G. Ramalho and K. Tsushima, Phys. Rev. D 81, 074020 (2010) [arXiv:1002.3386 [hep-ph]]; G. Ramalho and K. Tsushima, Phys. Rev. D 89, 073010 (2014) [arXiv:1402.3234 [hep-ph]]

2010 arXiv
[44]

Ramalho, Phys

G. Ramalho, Phys. Rev. D 94, 114001 (2016) [arXiv:1606.03042 [hep-ph]]

2016 arXiv
[45]

Ramalho, Eur

G. Ramalho, Eur. Phys. J. A 54, 75 (2018) [arXiv:1709.07412 [hep-ph]]

2018 arXiv
[46]

Ramalho, Eur

G. Ramalho, Eur. Phys. J. A 55, 32 (2019) [arXiv:1710.10527 [hep-ph]]

2019 arXiv
[47]

A. J. Buchmann, E. Hernandez and A. Faessler, Phys. Rev. C 55, 448 (1997) [nucl-th/9610040]

1997 arXiv
[48]

A. J. Buchmann, Phys. Rev. Lett. 93, 212301 (2004) [hep-ph/0412421]

2004 arXiv
[49]

K. A. Olive et al. [Particle Data Group Collaboration], Chin. Phys. C 38, 090001 (2014)

2014
[50]

Stajner et al

S. Stajner et al. , Phys. Rev. Lett. 119, 022001 (2017)

2017
[51]

Park et al

K. Park et al. [CLAS Collaboration], Phys. Rev. C 91, 045203 (2015) [arXiv:1412.0274 [nucl-ex]]

2015 arXiv
[52]

V. I. Mokeev and I. G. Aznauryan, Int. J. Mod. Phys. Conf. Ser. 26, 1460080 (2014) [arXiv:1310.1101 [nucl-ex]]

2014 arXiv
[53]

Blomberg et al

A. Blomberg et al. , Phys. Lett. B 760, 267 (2016)

2016
[54]

Stave et al

S. Stave et al. [A1 Collaboration], Phys. Rev. C 78, 025209 (2008)

2008
[55]

Sparveris et al

N. Sparveris et al. , Eur. Phys. J. A 49, 136 (2013)

2013
[56]

V. I. Mokeev, I. Aznauryan, V. Burkert and R. Gothe, EPJ Web Conf. 113, 01013 (2016) [arXiv:1508.04088 [nucl-ex]]

2016 arXiv
[57]

Dugger et al

M. Dugger et al. [CLAS Collaboration], Phys. Rev. C 79, 065206 (2009) [arXiv:0903.1110 [hep-ex]]

2009 arXiv
[58]

Ramalho, M

G. Ramalho, M. T. Pe˜ na and F. Gross, Eur. Phys. J. A 36, 329 (2008) [arXiv:0803.3034 [hep-ph]]; G. Ra- malho and K. Tsushima, Phys. Rev. D 82, 073007 (2010) [arXiv:1008.3822 [hep-ph]]

2008 arXiv
[59]

Ramalho and M

G. Ramalho and M. T. Pe˜ na, Phys. Rev. D 80, 013008 (2009) [arXiv:0901.4310 [hep-ph]]

2009 arXiv
[60]

N. F. Sparveris et al. [OOPS Collaboration], Phys. Rev. Lett. 94, 022003 (2005)

2005
[61]

Pascalutsa, M

V. Pascalutsa, M. Vanderhaeghen and S. N. Yang, Phys. Rept. 437, 125 (2007) [hep-ph/0609004]

2007 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [33]

A. N. Hiller Blin et al. , Phys. Rev. C 100, 035201 (2019) [arXiv:1904.08016 [hep-ph]]

work page Pith review arXiv 2019

[2] [1]

The amplitude is independent of the pseudothresh- old conditions

work page

[3] [2]

In the ﬁrst case, we can determine all coeﬃcients us- ing the continuity of A and the ﬁrst three derivatives at the point Q2 = Q2 P to ﬁx all the coeﬃcients

The amplitude is correlated with another ampli- tude. In the ﬁrst case, we can determine all coeﬃcients us- ing the continuity of A and the ﬁrst three derivatives at the point Q2 = Q2 P to ﬁx all the coeﬃcients. The last coeﬃcient α3 is in this case determined by A(3) (third derivative of A). In the second case we use the correlation condition to ﬁx the ﬁ...

work page 2007

[4] [3]

The coeﬃcient α3 is determined using the continu- ity of the third derivative: 6α3 = ˜A(3); (A8)

work page

[5] [4]

The coeﬃcient α3 can then be de- termined by Eq

the coeﬃcient α0 is determined by a pseudothresh- old condition. The coeﬃcient α3 can then be de- termined by Eq. (A7) using α3 = ( ˜Ap − α0) 1 ˜q6p , (A9) where ˜Ap ≡ ˜A(0) − ˜A(1) ˜q2 p + 1 2 ˜A(2) ˜q4 p. According to the discussion from Sec. IV, the ﬁrst con- dition is used in case 1, and the second condition is used in case 2. In general Eq. (A8) is u...

work page

[6] [5]

I. G. Aznauryan et al. , Int. J. Mod. Phys. E 22, 1330015 (2013) [arXiv:1212.4891 [nucl-th]]

arXiv 2013

[7] [6]

A. J. R. Puckett et al. , Phys. Rev. C 96, 055203 (2017) Erratum: [Phys. Rev. C 98, 019907 (2018)] [arXiv:1707.08587 [nucl-ex]]

arXiv 2017

[8] [7]

I. G. Aznauryan and V. D. Burkert, Prog. Part. Nucl. Phys. 67, 1 (2012) [arXiv:1109.1720 [hep-ph]]

arXiv 2012