Probing non-unitarity of the PMNS matrix in P2SO and comparison with DUNE

Monojit Ghosh; Rukmani Mohanta; Sambit Kumar Pusty; Samiran Roy

REVIEW 3 major objections 5 minor 71 references

Probing non-unitarity of the PMNS matrix in P2SO and comparison with DUNE

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Two future long-baseline neutrino experiments would split the job of testing whether the neutrino mixing matrix is truly unitary.

desk verdict The paper's headline claim—P2SO's sensitivity to α33 and its improved bound—is likely an artifact of an incorrect matter Hamiltonian in the GLoBES implementation; the rest is a competent but incremental configuration study. read the letter →

arxiv 2603.01031 v2 pith:Z23ZQWGI submitted 2026-03-01 hep-ph

Sambit Kumar Pusty , Samiran Roy , Monojit Ghosh , Rukmani Mohanta This is my paper · ORCID

classification hep-ph PACS 14.60.Pq14.60.St

keywords neutrinooscillationsnon-unitarityPMNSmatrixP2SODUNElong-baselineexperimentsmattereffectsCPviolation

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

This paper asks whether the neutrino mixing matrix is exactly unitary, and how the two next-generation long-baseline experiments—DUNE and P2SO—could find out. Treating non-unitarity (NU) model-independently through a triangular parametrization of the mixing matrix, it finds that DUNE and P2SO probe different NU parameters: DUNE constrains α11 and |α21| more tightly, while P2SO's longer baseline and stronger matter effects give it a superior handle on α22 and especially α33, a parameter that is invisible in vacuum oscillations. The paper also shows that if NU is real, it distorts the apparent sensitivities to mass hierarchy, the θ23 octant, and CP violation, so the flagship measurements of the coming decade could be biased unless NU is included in the fit. A sympathetic reader would care because the two experiments are often treated as interchangeable precision machines; this paper argues they are instead two halves of one measurement.

What carries the argument

The triangular parametrization N = N_NP U, where N_NP is a lower-triangular matrix carrying three real diagonal parameters α11, α22, α33 and complex off-diagonal parameters αij; the paper focuses on α11, α22, α33, and the off-diagonal pair (|α21|, φ21). The analytic oscillation probabilities (Eqs. 7 and 10) show that α11 enters only the νe appearance channel, α22 enters both appearance and disappearance, and α33 enters neither vacuum probability—only matter effects. P2SO's 2595 km baseline and denser detector amplify matter effects, which is the mechanism behind its α33 advantage, while DUNE's larger statistics and beam configuration drive its superior α11 and |α21| bounds.

What would settle it

Measure the actual P2SO detector performance (density, efficiency, background rates) and rerun the α33 sensitivity calculation; alternatively, obtain an independent, matter-free measurement of α33 from a short-baseline disappearance channel or a different experimental setup and compare it with the P2SO-projected bound. If the real detector performs noticeably worse than the 10×-ORCA assumption, or if a matter-free α33 measurement disagrees with the P2SO projection, the paper's central complementarity claim would be falsified.

Watch

Extended reading notes

Core claim

The central claim is that DUNE and P2SO have complementary sensitivity to non-unitarity of the leptonic mixing matrix: DUNE gives stronger bounds on the diagonal parameter α11 and the off-diagonal magnitude |α21|, while P2SO gives stronger bounds on α22 and α33, with the α33 advantage driven by P2SO's longer baseline (2595 km) and stronger matter effects. Within current allowed ranges, the paper finds that DUNE can improve the existing lower bound on α11 and P2SO can improve the existing lower bound on α33. It further shows that the presence of NU alters the experiments' ability to determine the mass hierarchy, the octant of θ23, and the CP-violating phase δCP, sometimes increasing and somet

Load-bearing premise

The entire P2SO advantage, including the claimed α33 improvement, is computed for a specific assumed detector configuration—Super-ORCA ten times denser than ORCA, 450 kW beam, 4×10^20 POT/year, six-year run—with backgrounds and systematics imported from earlier studies by the same authors and never quantified in this paper; if the real detector's density, efficiency, or background rejection differs, the P2SO curves move and the α33 bound could slip below the current limit.

Editorial extensions

If this is right

DUNE should be able to tighten the current bound on α11, while P2SO should tighten the current bound on α33; these are the two concrete improvements the paper claims.
If NU is present at currently allowed levels, mass-hierarchy, octant, and CP-violation sensitivities computed under strict unitarity will be biased—hierarchy sensitivity drops with α11, rises with α22, and octant and CPV sensitivities shift non-monotonically.
The bound on |α21| depends strongly on the phase φ21; marginalizing over φ21 weakens it, and CP-violation measurements must fit δCP and φ21 together rather than fixing the NU phase.
The unusual kink/dip structure in the α33 sensitivity curves is traced to degeneracy with θ23 and to matter effects; removing θ23 or using vacuum would erase those features.
The two experiments are complementary in NU parameter space, so a robust global picture requires combining both rather than relying on either alone.

Reading between the lines

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

A combined fit of DUNE and P2SO with all six NU parameters free would likely sharpen both α11 and α33 bounds beyond either experiment individually, because the two experiments' strongest constraints are nearly orthogonal in parameter space.
Because α33 enters only through matter effects, P2SO's α33 bound carries a direct systematic dependence on the assumed Earth density profile along 2595 km; the paper does not quantify this, so the real-world bound could be looser than quoted.
The 'six dof' scan still omits α31 and α32, so the quoted bounds are not full six-parameter results; reinserting those parameters with current global constraints could shift the allowed regions.
A natural testable extension is to scan over the assumed Super-ORCA detector density (the paper uses 10× ORCA); if the real density or background rejection differs, the relative ordering of DUNE and P2SO on α22 and α33 could change.

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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 first thing you should know: the stress-test note is right, and it is a load-bearing flaw. In the standard non-unitary formalism the paper itself cites (Blennow et al., Ref [25]), α33 does not enter P_μe or P_μμ, not even in matter. With α31=α32=0, the electron row of N is α11 U_e, so the matter Hamiltonian in the mass basis, H_m = M/(2E) + N† V N, becomes M/(2E) + |α11|^2 V_CC U_e† U_e. No α33. The μ and e projections of N also have no α33. So a correct calculation of ν_μ→ν_e and ν_μ→ν_μ gives zero α33 dependence in vacuum and matter. The paper's Sec. VI.A says the opposite ('α33 does not enter vacuum probabilities, but contributes in matter'), which means the simulation is almost certainly evolving with an ad-hoc flavor-basis Hamiltonian like N M N† + V that mixes the τ sector through the kinetic term. That would generate exactly the spurious α33 sensitivity seen in Fig. 1. This is not a minor point: it is the paper's main new result, and it invalidates the complementarity claim.

Credit where due: this is the first NU study for the upgraded P2SO configuration (Super-ORCA, 450 kW beam), and the scans with simultaneous NU parameters plus the |α21|–φ21 and δ_CP–φ21 correlations are new relative to Ref [50]. The internal diagnostics in Figs. 5 and 7 are careful attempts to explain kinks for α11 and α22, and the GLoBES framework with DUNE TDR files and NuFIT 6.0 inputs is standard.

The other problems are real but smaller. The P2SO detector response, systematics, and matter profile are inherited from self-authored refs [51, 58–61] and never quantified, so the P2SO numbers aren't independently reproducible. 'Six dof' is a misnomer—α31 and α32 are dropped. The abstract compares one-dof projections to six-dof current bounds without flagging the mismatch. And no code or parameter files are provided. None of these is fatal if fixed, but combined with the α33 error they describe a paper that overstates its case.

The α11 and |α21| comparisons with DUNE probably survive, since those parameters enter the vacuum probabilities. The α22 advantage is marginal (0.002 difference) and may shift under a corrected matter treatment.

I'd send this to a serious referee: the error is subtle and worth catching in writing, and the paper has salvageable content if the authors redo the calculation with the correct evolution and remove the α33 claims. But I would not cite it now, and the reader's report under-weights the central problem.

Referee Report

3 major / 5 minor

Summary. The manuscript uses GLoBES to simulate the upcoming P2SO and DUNE long-baseline experiments and derives projected constraints on the non-unitarity (NU) parameters α11, α22, α33, and |α21| in the triangular parametrization. It claims that DUNE will give stronger bounds on α11 and |α21|, while P2SO will give stronger bounds on α22 and α33, with DUNE (P2SO) improving the current bound on α11 (α33). It further studies correlations with standard oscillation parameters and quantifies the impact of NU on mass-hierarchy, octant, and CP-violation sensitivities. The methodology is mostly field-standard: Poisson likelihood with pull systematics, NuFIT 6.0 inputs, official DUNE TDR GLoBES files, and published NU probability formulas. The internal diagnostics in Figs. 5 and 7 are careful and valuable. However, the central α33 claim is inconsistent with the standard non-unitary matter evolution, and the P2SO simulation inputs are not quantified. These issues make the main conclusion unsupported as presented.

Significance. If the results were correct, the paper would provide a useful comparison of two complementary long-baseline experiments for NU searches and would quantify how NU distorts standard precision measurements. The authors deserve credit for explicitly investigating the origin of the kinks and dips in their sensitivity curves rather than leaving them unexplained, and for using external global-fit inputs. That said, the headline complementarity claim rests on the α33 sensitivity of P2SO, which appears to be an artifact of an incorrect matter Hamiltonian. Because the abstract and the conclusions rest on this claim, the significance of the paper cannot be assessed as stated; a corrected version could be valuable, but the current central result is not reliable.

major comments (3)

[VI.A, Fig. 1, Table II] The α33 sensitivity is not predicted by the standard non-unitary matter formalism. With the triangular parametrization and α31=α32=0, the matter Hamiltonian in the mass basis is H_m = M/(2E) + N† V_f N = M/(2E) + |α11|² V_CC U_e U_e†, and the amplitudes are (N e^{-iH_m L}N†)_{βα}. For β,α=e,μ, neither N nor H_m contains α33, so P_μe and P_μμ are independent of α33 in vacuum and in matter. The statement that α33 'does not enter the vacuum oscillation probabilities, but contributes in matter' is therefore incorrect for the channels analyzed. No ν_τ appearance sample is described in Secs. IV–V, and the diagnostics in Figs. 5 and 7 use only P_μe/P_μμ. I conclude that the α33 curves in Fig. 1, the α33 rows of Table II, and all α33-dependent sensitivity results are artifacts of the probability engine (apparently replacing U by N in the flavor-basis matter Hamiltonian). The claim that P2SO impr
[IV.A, V] The P2SO projections are not reproducible from the information given. Detector response, energy resolution, efficiencies, background rates, systematic pulls, and matter density profile are not specified; they are only delegated to refs [51,58–61], several of which are co-authored by the present authors. The only quantitative new ingredient is 'Super-ORCA detector 10 times more dense.' Because the P2SO advantage on α22 (and the purported one on α33) depends on these inherited assumptions, the reader cannot judge the robustness of the results. A table listing the simulated channels, signal/background efficiencies, systematic uncertainties, and matter profile should be added.
[Table II, Sec. VI] The column labeled 'six dof' is misleading. In the text, α31 and α32 are fixed to zero and only four NU parameters (plus φ21) are varied. Thus the marginalized bounds are four-degree-of-freedom bounds, not six-parameter marginalized bounds. The comparison with the current limits from [36] is therefore not apples-to-apples. Either include α31 and α32 in the marginalization, or rename the column and explicitly justify fixing them.

minor comments (5)

[Eq. (11)] The symbol 'cos(I_123)' is undefined. It should presumably be cos(δCP + Δm²31 L/(4E)) or an explicitly defined phase.
[Table II] The column structure of Table II is hard to read. Add explicit column headers in the caption so that the one-dof and six-dof entries for DUNE and P2SO are unambiguous.
[Fig. 1] The dashed curves for the α33 panel, corresponding to θ23 fixed at its true value, are not explained in the caption. The caption should identify all curves.
[Sec. VI.B] The phrase 'assuming NU does not exist in Nature' is imprecise. The simulations use benchmark true values with αii=1 and αij=0; this is a choice of true parameters, not an assumption about data.
[Secs. IV–V] The paper should state explicitly whether ν_τ charged-current events are included in the GLoBES simulation. If they are not, α33 cannot be constrained by the channels used; if they are, the channel list and efficiencies should be documented.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the projected NU bounds are forward GLoBES simulations benchmarked against external fits; self-citations supply experimental inputs, not the derived conclusions.

full rationale

The central claims (DUNE constrains α11/α21 better; P2SO constrains α22/α33 better; NU affects hierarchy/octant/CPV sensitivities) are outputs of a forward simulation: GLoBES event generation + Poisson likelihood (Eq. 14) with pull systematics, not results obtained by re-inserting the target quantities. The benchmark for 'improving current bounds' is the external global fit Ref. [36], and the standard oscillation inputs are NuFIT 6.0, so the comparison is externally anchored. The analytical probabilities in Eqs. (7) and (10) are taken from the external formalism of Ref. [27]; the statement that α33 enters only through matter is attributed to Refs. [25,46], with [25] being an independent external paper. Even if the α33 matter implementation in the modified GLoBES engine is wrong (the skeptical concern), that would be a correctness/validity error in the Hamiltonian, not a circular reduction of the prediction to its input. Self-citations are present: S. Roy co-authors Refs. [38,46], and Ghosh/Mohanta co-author the P2SO configuration papers [51,59-61]. However, these are used as experimental inputs and supporting formalism, not as the derivation of the new bounds; the P2SO configuration also rests on the external Ref. [58], and the α31/α32 neglect is consistent with the external current limits of Ref. [36]. Minor caveats (the 'six dof' scan actually varies four NU parameters after dropping α31,α32; the α33 sensitivity is not shown at the probability level) are presentation/completeness issues, not circularity. Therefore no claim in this paper reduces by construction to a fitted parameter, a self-citation chain, or a renamed input.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The analysis imports its formalism (triangular NU parametrization, matter-modified probabilities) from prior literature [25, 27, 46] and its experimental configurations from TDR/design papers; it fits no new free parameters to data and introduces no new physics entities — the heavy neutral leptons are standard seesaw/sterile constructions borrowed from the cited literature. The main trust-bearing inputs are the P2SO design assumptions (self-cited, unquantified here) and the approximations excluding α31/α32. Any re-implementation would need to secure these inputs independently.

free parameters (3)

Standard oscillation parameters (true values) = NuFIT 6.0 best fit (Table I): sin²θ23=0.470, δCP=212°, Δm²31=2.513×10⁻³ eV², etc.
Fixed as true values in the χ² analysis (Sec. V). These are inputs from a prior global fit, not fitted here, but all projected bounds scale with them.
Benchmark NU values for correlation plots = α11=0.95/0.90, α22=0.99/0.98, α33=0.95/0.90, |α21|=0.02/0.04, φ21=0°, ±90°
Hand-chosen benchmark points (Sec. VI.G/H) used to generate θ23–Δm²31 and δCP–φ21 contours; chosen to lie within current allowed ranges.
P2SO detector response, systematics, and matter profile = not stated in the paper
Assumed numbers inherited from refs [51, 58–61] (co-authored by present authors). The P2SO bounds on α22/α33 depend directly on these unquantified inputs.

assumptions (6)

domain assumption The 3×3 mixing matrix is written N = N_NP U (lower-triangular times unitary), with heavy states integrated out (Eq. 3).
Adopted from Escrihuela et al. [27], Sec. II. The entire analysis inherits this parametrization.
domain assumption Pμe = α11²|α21|² + α11²α22² P3×3_μe + α11²α22|α21| P^I_μe (Eq. 7), dropping cubic terms in α21, sinθ13, Δm²21; analogous Pμμ expression (Eqs. 10–13).
Approximation from [27] in Sec. III. It selects α11, α22, α21 as the only NU parameters appearing in vacuum oscillation probabilities.
domain assumption α31 and α32 have negligible effects within current limits [38] and are excluded, including from the 'six dof' scans.
Sec. VI opening. If α31/α32 contribute at P2SO/DUNE sensitivity, the quoted six-dof bounds could shift.
domain assumption Matter evolution is computed in a modified GLoBES engine that includes NU; heavy states do not participate in oscillation (averaged-out regime).
Sec. V; follows [25, 46]. The central claim that P2SO's stronger matter effects drive the α33 sensitivity depends on this treatment.
domain assumption P2SO experimental specifications: 450 kW beam, 4×10^20 POT/yr, six-year run, Super-ORCA 10× ORCA density, 0.2–10 GeV window.
Sec. IV.A, taken from refs [58–61] which include current authors. Assumed inputs not independently validated in this paper.
standard math Poisson log-likelihood with pull systematics; θ23 and Δm²31 marginalized within NuFIT 6.0 3σ ranges; δCP and φ21 fully marginalized.
Secs. V–VI. Standard oscillation-analysis practice.

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Pith. "Pith review of Probing non-unitarity of the PMNS matrix in P2SO and comparison with DUNE." pith.science (2026). https://pith.science/paper/Z23ZQWGI

@misc{pith2026260301031,
  author       = {Pith},
  title        = {Pith review of: Probing non-unitarity of the PMNS matrix in P2SO and comparison with DUNE},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z23ZQWGI}},
  note         = {Machine review of arXiv:2603.01031}
}

abstract

We compare the sensitivity of the upcoming long-baseline neutrino experiments Protvino to Super-ORCA (P2SO) and the Deep Underground Neutrino Experiment (DUNE) to non-unitarity (NU) of the leptonic mixing matrix in a model-independent framework. NU can arise in theories beyond the Standard Model that include heavy neutral leptons. These effects can modify neutrino oscillation probabilities and introduce new sources of CP violation, which may affect precision measurements of neutrino parameters. We find that DUNE provides stronger bounds on $\alpha_{11}$ and $|\alpha_{21}|$, while P2SO shows better sensitivity to $\alpha_{22}$ and $\alpha_{33}$, mainly due to its longer baseline and stronger matter effects. Our results show that DUNE (P2SO) will be able to improve the current bounds of $\alpha_{11}$ ($\alpha_{33}$). We further examine correlations with standard oscillation parameters and quantify the impact of NU on mass hierarchy, octant, and CP-violation sensitivities. Our results show that these sensitivities depend upon NU in a non-trivial way interconnecting the parameter degeneracies and matter effects. Our results demonstrate the complementarity of P2SO and DUNE in probing NU and show that NU can significantly influence next-generation precision oscillation studies.

Figures

Figures reproduced from arXiv: 2603.01031 by the authors.

**Figure 2.** FIG. 2: Allowed parameter space in the [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** FIG. 3: Allowed parameter space between [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4: Mass hierarchy sensitivity in the presence of NU parameters ( [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: FIG. 5: Appearance (left) and disappearance (right) probability difference (NH-IH) as a [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: FIG. 6: Octant sensitivity in the presence of NU parameters ( [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: FIG. 7: Octant sensitivity for P2SO with the NU parameter [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: FIG. 8: CP violation sensitivity in the presence of NU parameters. P2SO (DUNE) and [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: FIG. 9: Allowed regions in the [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

**Figure 10.** Figure 10: FIG. 10: Allowed regions in the [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

**Figure 6.** Figure 6: This difference comes from the strong matter effects in P2SO compared to DUNE. [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]

**Figure 11.** Figure 11: FIG. 11: Allowed parameter space between [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]

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Figure 3 shows how∆m2 31 depends on the NU parameters;α11 (upper-left),α 22 (upper-right),α 33 (lower-left), andα 21 (lower- right)

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It remains unclear if the value is greater (normal hierarchy) or less (inverted hierarchy) than zero. Regarding the CP-violating phase δCP, it is still one of the most unsettled issues in the neutrino field. It is crucial to ascertain the CP violating phase since it offers an indication to answer the matter-antimatter asym- metry of the universe through l...

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