Neutrinoless double beta decay with light sterile neutrinos: the contact terms

Sebasti\'an Urrutia Quiroga; Vincenzo Cirigliano; Wouter Dekens

REVIEW 3 major objections 4 minor 76 references

Neutrinoless double beta decay with light sterile neutrinos: the contact terms

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Sterile neutrino masses below 1 GeV shift the 0νββ short-range coupling by an order of magnitude more than assumed, lengthening predicted half-lives in the 200–800 MeV range.

desk verdict First real calculation of the ms-dependence of the 0νββ short-range couplings—likely right within the model, but the central ms^2 coefficient is a delicate cancellation and deserves a robustness check before it drives phenomenology. read the letter →

arxiv 2412.10497 v2 pith:7B2EJWAG submitted 2024-12-13 hep-ph nucl-th

Vincenzo Cirigliano , Wouter Dekens , Sebastián Urrutia Quiroga This is my paper · ORCID

classification hep-phnucl-th

keywords neutrinolessdoublebetadecaysterileneutrinoschiraleffectivefieldtheoryshort-rangecontacttermsCottinghammatchingleft-rightsymmetricmodelslow-energyconstants

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 establishes how the short-range nuclear contact term that controls neutrinoless double $\beta$ decay (0νββ) depends on the mass $m_s$ of light sterile neutrinos in two motivated extensions of the Standard Model. Generalizing the Cottingham-style matching used for the massless case, the authors derive the polynomial $m_s^2$ and $m_s^4$ coefficients of the $nn\to pp$ coupling $C_1(m_s)$ in the minimal neutrino-extended Standard Model, and the $m_s^2$ coefficient of $C_1+C_2$ in left-right symmetric models. The central quantitative claim is that the $m_s^2$ coefficient is about an order of magnitude larger than the naive dimensional analysis estimate assumed in earlier rate calculations. If this is right, the predicted 0νββ half-life in the 3+1 sterile-neutrino scenario grows by up to a factor of six when $m_s$ lies between 200 and 800 MeV, which would change how experimental limits on sterile-neutrino contributions to 0νββ are interpreted.

What carries the argument

The central object is the matching condition for the dimensionless low-energy constant $\widetilde{C}_1(\mu_\chi, m_s)$, defined from the $nn\to pp$ contact term by Eq. (2.6). The authors build a 'full' amplitude $a_<(|k|, m_s)$ for neutrino virtualities up to an arbitrary scale $\Lambda$ from chiral EFT at low momentum, dipole form factors and a Kaplan–Steele three-Yukawa potential for the two-nucleon half-off-shell amplitude (the momentum dependence of the short-range $^1S_0$ scattering amplitude away from the energy shell) at intermediate momentum, and an operator-product-expansion tail above $\Lambda$; they then equate this to the chiral EFT amplitude, whose singular piece contains the same topology plus the counterterm $\widetilde{C}_1$. The key structural step is subtracting the infrared behavior: the full amplitude contains powers of $m_s \tan^{-1}(\lambda/m_s)$ and logarithms of $m_s^2+|p|_{\rm ext}^2$, while the EFT amplitude reproduces them only after including the NLO term $d_{\rm NLO}\,\pi\, m_s$, so that the difference is the polynomial in $m_s^2$ that defines $C_1(m_s)$. The same machinery, with the $\pi^-\to\pi^+$ amplitude and the vector-like current $J_\mu^V=J_\mu^L+J_\mu^R$, extracts $C_1+C_2$ for left-right models via Eq. (3.28).

What would settle it

A concrete check is to re-run the matching with a different representation of the full amplitude—for instance, including inelastic $NN\pi$ intermediate states or using a different $NN$ potential such as Reid or CD-Bonn—and compare the extracted $m_s^2$ coefficient of $C_1(m_s)$ with the 12.062 GeV$^{-2}$ of Eq. (2.29); a shift larger than the paper's stated 30–50% uncertainty would falsify the prediction. A direct lattice QCD extraction of the $nn\to pp$ short-range amplitude at a few sterile masses between 100 and 500 MeV would settle the same question without the model assumption.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is an explicit, numerically evaluated mass dependence for the leading-order short-range $nn\to pp$ operators of 0νββ. For the minimal $\nu$SM, the dimensionless coupling evaluated at $\mu_\chi=m_\pi$ is $C_1(m_s) \simeq 1.377 + (12.062/\mathrm{GeV}^2)\, m_s^2 - (16.735/\mathrm{GeV}^4)\, m_s^4$, Eq. (2.29); for the left-right scenario, $C_1+C_2 \simeq 2.253 + (7.993/\mathrm{GeV}^2)\, m_s^2$, Eq. (3.34). These polynomials are extracted by matching a modeled full amplitude, split into low-, intermediate-, and high-momentum regions, against the chiral EFT amplitude, after including the NLO linear-in-$m_s$ term that carries the infrared behavior. The paper demonstrates that the extracted coupling is independent of the arbitrary matching scales and external momenta to within about 1%, and that the non-analytic $m_s$ logarithms cancel between the full and EFT amplitudes, so the coupling is the polynomial in $m_s^2$ that EFT principles require. The numerical consequence is that the $m_s^2$ coefficient is roughly an order of magnitude larger than the NDA-based estimate used in Ref. [34], which substantially alters half-life predictions when $200\ \mathrm{MeV} \lesssim m_s \lesssim 800\ \mathrm{MeV}$.

Load-bearing premise

The load-bearing premise is the model chosen for the full hadronic amplitude that is matched onto the EFT — chiral EFT at low momentum, dipole form factors with a three-Yukawa nucleon-nucleon potential in the intermediate region, and an OPE tail at high momentum — so if that interpolation misses an important intermediate state such as inelastic $NN\pi$, or if the assumed cancellation of non-analytic $m_s$ terms in the left-right extraction fails, the quoted polynomial coefficients shift.

Editorial extensions

If this is right

For sterile neutrino masses between 200 and 800 MeV, the half-life of 0νββ in the 3+1 νSM scenario is up to six times longer than earlier NDA-based estimates, so interpreting a null experimental signal as a bound on sterile-neutrino parameters requires the new $C_1(m_s)$.
The $m_s^2$ coefficient of $C_1+C_2$ in left-right symmetric models is also several times larger than NDA would suggest, shifting the predicted 0νββ amplitude in that scenario for $m_s$ up to about 1 GeV.
The extracted low-energy constant remains polynomial in $m_s^2$ up to the breakdown scale $m_s\sim\Lambda_\chi$, confirming that the EFT description of Refs. [32–34] is consistent; only the numerical interpolation of $C_1(m_s)$ needs to be updated.
The new interpolation of $C_1(m_s)$ follows the polynomial expansion up to about $3m_\pi$ before turning over to the $1/m_s^2$ behavior, which changes where the contact term's contribution is maximal relative to the long-range and ultrasoft contributions.
Varying the arbitrary matching scales $\lambda$, $\Lambda$, and external momenta changes the extracted coupling by only about 1%, so the numerical result is stable within the chosen model.

Reading between the lines

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

If the assumed cancellation of non-analytic $m_s$ terms in Section 3.2.2 were found to fail in a complete N2LO calculation, the left-right coefficient $C_1+C_2$ quoted in Eq. (3.34) could shift at the same order as its $m_s^2$ term; a full two-loop EFT calculation would be the direct test.
The same matching approach could be applied to sterile masses above O(1 GeV) with the $1/m_s^2$ dimension-nine operator description restored, in principle connecting the low-mass polynomial and the heavy-mass tail by a single hadronic model.
Because the contact term is isospin-symmetric and nuclear-structure independent, comparing 0νββ rates across different isotopes in the $m_s=200$–$800\ \mathrm{MeV}$ window could separate the $C_1(m_s)$ contribution from the long-range potential, providing an observable cross-check on the nuclear matrix elements used in the interpolation.
If a future lattice QCD calculation confirms the large $m_s^2$ coefficient found here, then previous sterile-neutrino interpretations of 0νββ limits—which used the NDA-sized coefficient—will need to be re-evaluated, and the allowed parameter space will shift toward larger mixings.

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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 paper supplies the first real calculation of the ms^2 (and ms^4) dependence of the leading short-range 0νββ couplings for light sterile neutrinos. That is genuinely new—previous work used NDA or simple interpolations. If you work on 0νββ interpretations, this is the number you need.

What the paper does well: the matching framework is explicit, the scale checks in λ, Λ, and external momentum are reasonable, and the polynomial fits in Eqs. (2.29) and (3.34) are easy to use. The paper also identifies a ~ -0.6 shift in the vector combination relative to Ref. [46], which is the kind of detail that matters. The half-life illustration (factor ~6 for ms in 200-800 MeV) makes clear why the result matters.

Soft spots: the stress-test is right that C~(2)_1 = 12.06 GeV^-2 is a small residual of large opposite-sign terms—the low-momentum analytic piece is around -20 to -40 GeV^-2 and the intermediate-momentum integral around +30 to +50 GeV^-2. The integrand is dominated by |k| just above λ ~ mπ, exactly where the full amplitude is modeled with dipole form factors and the Kaplan-Steel three-Yukawa potential. The ms=0 analysis was cross-checked against Reid and AV18; this paper does not do that for the ms-dependent pieces. The inelastic NNπ contribution is not included, and the cited Ref. [71] addresses only the ms=0 case. So the central coefficient could carry a model error comparable to its value. The left-right part has a second soft spot: the subtraction of non-analytic ms terms in Eq. (3.28) assumes that uncalculated N2LO loops cancel them. The authors say this explicitly, but it is not demonstrated, and the ΛS dependence is a residual.

These are real caveats, not fatal flaws. The paper is honest about the ~30-50% uncertainty, and the central result is likely correct within the adopted model. What is not yet established is whether the ms^2 coefficient is robust against NN-potential and inelastic-channel choices. That should be probed before the factor-of-six half-life impact is used quantitatively.

Recommendation: send to peer review. The referees should push for an alternative-potential check and a bound on the N2LO cancellation, but this is a serious calculation that belongs in the literature. I would cite it if I were writing on sterile-neutrino 0νββ.

Referee Report

3 major / 4 minor

Summary. The paper derives the sterile-neutrino-mass dependence of the leading short-range nn→pp couplings in chiral EFT for 0νββ, for the minimal νSM (LL) scenario and for left-right (LR) scenarios. Using a Cottingham-style matching between a modeled full hadronic amplitude and chiral EFT, it obtains Eq. (2.29) for C1(ms) and Eq. (3.34) for C1+C2(ms), and uses these results to update the prediction of 0νββ half-lives in a simplified 3+1 model, finding a lengthening by up to a factor of roughly six for ms between 200 and 800 MeV.

Significance. If correct, the extracted ms^2 coefficient C~(2)_1 ≈ 12 GeV^-2 is an order of magnitude larger than the NDA estimate used in Ref. [34], and it materially changes the sterile-neutrino interpretation of 0νββ searches in a mass range that is otherwise difficult to probe. The paper's strengths include explicit analytic matching expressions, a transparent polynomial expansion in ms (Eqs. (2.29) and (3.34)), stability checks against the arbitrary scales λ and Λ and against the external momenta, and a falsifiable prediction that can be confronted with future lattice-QCD or alternative NN-potential calculations. The paper also identifies a small (≈ -0.6) shift in the ms=0 LR contact term relative to Ref. [46], which is a useful technical advance. The main limitation is that the full hadronic amplitude is modeled rather than computed from QCD, and the LR extraction relies on an assumed cancellation of non-analytic terms, so the accuracy of the headline coefficients rests on model assumptions that are not yet fully quantified.

major comments (3)

[§2.2, Eqs. (2.27)–(2.29)]
[§3.2.2, Eq. (3.28)]
[§4, Eq. (4.5) and Fig. 6]

minor comments (4)

[§2.2, Eq. (2.29)]
[§3.4, Eq. (3.34)]
[§4, Eq. (4.4)]
[§5, Conclusions]

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; the ms-dependent short-range couplings are extracted from independent hadronic inputs, with reliance on the authors' own matching methodology being methodological rather than input-output circular.

full rationale

The derivation of C1(ms) and C1(ms)+C2(ms) is not circular in the sense required by the analysis. The couplings are obtained by matching chiral EFT amplitudes to a modeled full amplitude built from independent inputs: nucleon form factors, the Kaplan-Steel three-Yukawa NN potential, the OPE tail with gbar_NN^1 treated as an unknown range, and lattice/experimental inputs such as Z(0) and gbar_pi_pi_LR. No low-energy constant is fitted to 0nubb data; the half-life calculation in Sec. 4 is an application of the extracted C1(ms), not an input to it. The paper self-cites Refs. [45,46] for the Cottingham-style matching methodology and states 'we extend the methodology of Refs. [45,46], to which we refer the reader for details'; this is methodological inheritance, and the ms-dependence computed here is new content, so it does not reduce the central claim to the cited work. The largest caveat is explicit in the text: in Sec. 3.2.2 the authors state, 'we simply assume the cancellation takes place and subtract any non-analytic terms in ms from Eq (3.24) in the matching condition.' This is an unproven EFT-consistency assumption that adds model dependence, but it is not circular because the subtracted terms are prescribed by the required polynomial form of the LEC, not by a fitted target observable. Similarly, the conclusion's admission of 'an overall 30-50% fractional error' from missing inelastic NNpi states and NN-potential model dependence is a limitation, not an input-output inversion. Therefore no specific equation reduces by construction to its inputs, and the score reflects only the minor self-citation reliance in the matching framework.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central calculation rests on standard chiral EFT plus a modeled hadronic amplitude. Free parameters are mostly phenomenological inputs from prior fits, such as NN EFT LECs and the NN potential, plus two unknown matrix elements varied over wide ranges. The fitted interpolation coefficients enter only the half-life illustration. The main axiom risk is the model dependence of the full amplitude and the assumed cancellation of non-analytic terms in the left-right matching.

free parameters (5)

dNLO = 8 C2 / (m_N C^2), with C2/C^2 approximately 0.38 = 0.38 (from Ref. [54])
Short-range NN EFT LEC ratio that controls the linear-in-ms term in C1(ms) through Eqs. (2.16) and (A.5). It is fitted to NN scattering data in the cited literature, not derived here.
Kaplan-Steel three-Yukawa potential parameters = not quoted; from Ref. [51]
Used to obtain the half-off-shell NN scattering amplitude fS(q,p), which enters IC and r(|k|) in the intermediate-momentum region. This is a phenomenological model fitted to NN phase shifts in its original work.
gbar_NN^1, nn to pp matrix element of the local operator in the OPE tail = varied over [-10, +10]
Unknown hadronic matrix element in the high-momentum contribution, Eq. (2.10). The paper chooses a wide range because NDA suggests O(1); the numerical effect is small.
gbar_NN^LR, analogous nucleon matrix element for the left-right scenario = varied over [-10, +10]
Same role as gbar_NN^1 in the OPE tail of the vector-like amplitude, Eq. (3.27). Unknown and chosen by hand over a conservative range.
Interpolation coefficients a2, a4, b4, b6 in Eq. (4.5) = a2 = 8.8 GeV^-2, a4 = -1.9 GeV^-4, b4 = 10 GeV^-4, b6 = 11 GeV^-6
Free coefficients in the ansatz for C1(ms) used to estimate the half-life impact. They are fixed by matching A9 at ms = 2 GeV, by the computed series coefficients, and by a chosen maximum at ms approximately 3 m_pi. These are fit parameters, not outputs of the matching calculation.

assumptions (5)

domain assumption Chiral EFT expansion in ms / Lambda_chi is valid for the mass range studied, with the polynomial expansion valid up to approximately m_K or 3 m_pi.
The extraction assumes C1(ms) can be Taylor-expanded in ms^2 and that higher-order ms corrections are suppressed. Invoked in Section 2.1 before Eq. (2.9) and in the uncertainty estimate.
ad hoc to paper The full hadronic amplitude in the Cottingham matching is adequately represented by the low-energy chiral EFT model, the form-factor and NN-potential model, and the OPE tail, with an interpolation between regions.
This is the modeling assumption underlying Eqs. (2.3), (2.10), (2.11), and the numerical evaluation in Section 2.2. It is not derived from QCD and carries an estimated 30 to 50 percent error.
domain assumption The operator product expansion sets in above a scale Lambda of order GeV, and the high-momentum contribution can be computed with the unknown matrix element gbar_NN^1.
Used in Eq. (2.10) and in the alpha>0 coefficients. The scale separation is standard, but the matrix element is only bounded by the chosen range.
ad hoc to paper In the left-right matching, uncalculated N2LO chiral EFT loops exactly cancel the non-analytic ms terms present in the modeled full amplitude.
Stated in Section 3.2.2, third bullet below Eq. (3.26): the authors assume the cancellation takes place and subtract non-analytic terms. This is an admitted gap for the C1+C2 result.
ad hoc to paper The high-mass behavior of C1(ms) is captured by dimension-nine operator matching plus the interpolation ansatz Eq. (4.5), with a maximum near 3 m_pi.
Section 4 uses this to extend C1 beyond the computed ms^4 term. The coefficients beyond the series are fixed by requiring the dimension-nine result at ms = 2 GeV and a chosen maximum, not derived from first principles.

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Cite this review

Pith. "Pith review of Neutrinoless double beta decay with light sterile neutrinos: the contact terms." pith.science (2026). https://pith.science/paper/7B2EJWAG

@misc{pith2026241210497,
  author       = {Pith},
  title        = {Pith review of: Neutrinoless double beta decay with light sterile neutrinos: the contact terms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7B2EJWAG}},
  note         = {Machine review of arXiv:2412.10497}
}

abstract

We study neutrinoless double-beta decay in extensions of the Standard Model that include $n$ right-handed neutrino singlets, with masses $m_s$ below the GeV scale. Generalizing recently developed matching methods, we determine the $m_s$ dependence of the short-range $nn \to pp$ couplings that appear to leading order in the chiral effective field theory description of neutrinoless double beta decay. We focus on two scenarios, corresponding to the minimal $\nu$SM and left-right symmetric models. We illustrate the impact of our new results in the case of the $\nu$SM, showing a significant impact on the neutrinoless double-beta decay half-life when $m_s$ is in the 200-800 MeV range.

Figures

Figures reproduced from arXiv: 2412.10497 by the authors.

**Figure 1.** Topologies relevant for the nn → ppe−e − amplitude to leading order in the chiral expansion. The double solid lines denote nucleons, while the thin solid oriented lines denote leptons (the internal neutrino and the external electrons). The black squares denote the leptonnumber violating vertices: Majorana mass insertions in topologies A, B, and C, and the contact interaction C1(ms) in topology D. The gray circles r… view at source ↗

**Figure 2.** C˜ 1(µχ, ms) as a function of ms (in MeV) for µχ = mπ. The full matching expression in Eq. (2.25) is shown as a solid blue line, whereas the low-mass expansion in Eq. (2.26) is shown as a dashed red line. The naïve uncertainty band given by C˜ 1(ms) 1 ± O(ms/Λχ) is shown as a blue shaded region. For the concreteness of this plot, we set |p| = |p ′ | = 1 MeV, Λ = 2 GeV, λ = 100 MeV, and g¯ NN 1 = 0. For plotting … view at source ↗

**Figure 3.** C˜ 1(µχ, ms) as a function of the scale λ (in MeV) for µχ = mπ and three different values of ms = {1 MeV, 100 MeV, 3mπ}. The solid and dashed lines represent the use of the full matching expression in Eq. (2.25) and its low-mass expansion in Eq. (2.26) to compute C˜ 1, respectively. For the concreteness of this plot, we set |p| = |p ′ | = 1 MeV, Λ = 2 GeV, and g¯ NN 1 = 0 . 4For ms = 100 MeV, the individual variatio… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: C˜ 1+C˜ 2 (µχ, ms) as a function of ms (in MeV) for µχ = mπ. The blue band corresponds to the effects of varying ΛS ∈ [mπ/2, 2mπ]. The full matching expression in Eq. (3.28) is shown as a solid blue line, whereas the low-mass regime expansion in Eq. (3.32) is shown…

**Figure 5.** Figure 5: C˜ 1(ms)/C˜ 1(0) as a function of ms (in MeV), using different approximations and interpolations. The dashed blue line shows the expansion up to m4 s , while the blue solid line depicts the result of Eq. (4.5), both of which use the results derived in this work. Inste…

**Figure 6.** Figure 6: The half-life of 0νββ in the 3 + 1 scenario as a function of the sterile mass m4 = ms, with the left and right panels showing different regions ranges of ms. Here, the blue line employs the interpolation of Eq. (4.5) and the determination of C˜ 1(ms) derived in this wo…

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