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REVIEW 3 major objections 5 minor 81 references

Exploring the possible two-proton radioactivity of $^{38,39}$Ti

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Both titanium-39 and titanium-38 are predicted to be two-proton emitters.

desk verdict A solid, useful GSM+GCC calculation of 38,39Ti that is let down by an internal arithmetic inconsistency between the predicted 2p partial half-life and the measured total half-life of 39Ti. read the letter →

arxiv 2501.09542 v1 pith:SRAYIDPY submitted 2025-01-16 nucl-th

B. Huang , F. P. Bai , D. Q. Fang , S. M. Wang This is my paper · ORCID
classification nucl-th
keywords two-protonradioactivitytitanium-39titanium-38Gamowshellmodelcoupled-channelmethodprotondriplinethree-bodydecaypairing
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

The paper sets out to decide whether titanium-39, proposed decades ago as a two-proton ($2p$) emitter but never observed to decay that way, is genuinely a $2p$-radioactive nucleus, and whether its neighbor titanium-38 is an even stronger candidate. Gamow shell model calculations give titanium-39 a two-proton separation energy of $S_{2p} = -0.45 \pm 0.3$ MeV, so $2p$ emission is energetically open while single-proton emission is essentially closed. Gamow coupled-channel calculations then put the $2p$ partial half-life of titanium-39 in the 0.4–10 ms range, comparable to its measured $\beta$-decay half-life, so the two decay modes could compete. The same methods give titanium-38 a $2p$ decay energy of $1.85 \pm 0.26$ MeV and a half-life of $10^{-14}$ to $10^{-11}$ seconds, making it a more promising but still undiscovered target.

What carries the argument

The argument runs on two connected tools: the Gamow shell model (GSM), a many-body shell model built on a complex-energy Berggren basis that treats bound states, resonances, and scattering continuum on the same footing, and the Gamow coupled-channel (GCC) method, a three-body core-plus-two-proton model in Jacobi coordinates that computes the decay width. GSM supplies the separation energies that decide whether $2p$ emission is open; GCC converts that energy into a $2p$ partial width and gives the proton-proton density distributions. A nucleon-number-dependent effective field theory interaction calibrated to 13 states in nearby Ca, Sc, and Ti isotopes fixes the GSM Hamiltonian, and the width-versus-$Q_{2p}$ curve is the quantitative hinge connecting structure to observability.

What would settle it

Measure the atomic masses of titanium-39 and titanium-38 to about ±100 keV, fix $Q_{2p}$, and evaluate the paper's own width curve at that energy: a $Q_{2p}(^{39}\mathrm{Ti})$ below roughly 0.2 MeV would push the predicted $2p$ partial half-life beyond the measured $\beta$-decay half-life and falsify the claim that the two channels can rival each other, while a value near the upper end of the 0.15–0.75 MeV band would place the $2p$ branch in the observable window.

Watch

Extended reading notes

Core claim

The central claim is that titanium-39 is a viable two-proton emitter and that titanium-38 is an even better one. The authors find $S_{2p}(^{39}\mathrm{Ti}) = -0.45 \pm 0.3$ MeV and $S_p(^{39}\mathrm{Ti}) = 0.15 \pm 0.31$ MeV, which permits direct $2p$ emission while suppressing one-proton emission; the predicted $2p$ partial half-life of about 0.4–10 ms lies in the same window as the measured 26–31 ms total half-life, implying a real race between $2p$ decay and $\beta$ decay. For titanium-38 they find $Q_{2p} = 1.85 \pm 0.26$ MeV and a $2p$ half-life between $2.76\times10^{-14}$ s and $1.17\times10^{-11}$ s, and note that its ground state has a larger $p$-wave component, whose lower centrifugal barrier broadens the width. The authors also stress that the $2p$ width is exponentially sensitive to the decay energy, so the energy uncertainty in their model is the main factor that could push titanium-39's $2p$ branch out of experimental reach.

Load-bearing premise

The predicted two-proton decay energy $Q_{2p}$ is load-bearing: the paper's own uncertainty of 0.26–0.3 MeV spans an energy range over which the $2p$ half-life changes by orders of magnitude, so if the true energy sits at the low end, titanium-39's $2p$ decay would be too slow to compete with beta decay and would remain unobserved.

Editorial extensions

If this is right

  • A dedicated titanium-39 decay experiment could observe direct $2p$ emission or place a meaningful limit, since the predicted $2p$ partial half-life of 0.4–10 ms overlaps the measured total half-life of about 26–31 ms.
  • If the $2p$ branch in titanium-39 is real, the decay products should show proton-proton correlations consistent with paired valence protons, as seen in the calculated $2p$ density distributions.
  • For titanium-38, future production should yield essentially prompt $2p$ emission on a $10^{-14}$ to $10^{-11}$ second timescale, with a wider decay width than titanium-39 because of the larger $p$-wave component.
  • Precision mass measurements of titanium-39 and titanium-38 would narrow the $Q_{2p}$ uncertainty and determine whether the predicted $2p$ branch can actually compete with beta decay.

Reading between the lines

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

  • Extending the paper's logic, a null $2p$ search in titanium-39 would not by itself falsify the model: because the width is exponentially sensitive to energy, the same calculation with a slightly lower $Q_{2p}$ predicts a branch hidden under beta decay, so only a direct mass measurement can settle the question.
  • The same GSM+GCC pipeline could be applied to the remaining historical $2p$ candidates, such as chromium-42 and nickel-48/49, to see which ones have widths that can actually compete with beta decay and which are likely to remain unobserved.
  • A sharper test of the pairing claim would compare the predicted proton-proton angular and energy correlations for titanium-39 with data from established $2p$ emitters; a diproton-like correlation pattern would distinguish the paired configuration from independent sequential emission.
  • For experimental planning, the practical message is that titanium-38 is the more decisive target: its predicted half-life is so short that its $2p$ emission, once produced, should be unambiguous, whereas titanium-39 requires timing and correlation analysis to separate $2p$ decay from beta-delayed emission.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper investigates possible two-proton radioactivity of 38Ti and 39Ti using the Gamow shell model (GSM) with an EFT-inspired interaction and the Gamow coupled-channel (GCC) method. The GSM calculation yields S2p(39Ti) = -0.45 +/- 0.3 MeV and Q2p(38Ti) = 1.85 +/- 0.26 MeV, making both nuclei energetically candidates for two-proton emission. The GCC calculation gives the 2p partial half-life as a function of Q2p: at the nominal energy of 39Ti the authors quote 0.4-10 ms, and for 38Ti they quote 2.76e-14 to 1.17e-11 s. The paper concludes that 39Ti is a viable 2p candidate whose 2p decay could rival beta decay, while 38Ti is a more promising candidate. A secondary structural result is the propensity of the two valence protons to pair in 39Ti across several configurations.

Significance. The qualitative separation-energy predictions are of interest for planning experiments at the proton dripline, and the GCC width calculations provide a three-body treatment that goes beyond simple barrier-penetration models. The paper is also transparent about several uncertainty sources, including interaction-strength scaling and alternative single-particle potentials, and it includes a direct calculation of two-proton density distributions. However, the central quantitative claim - that the 2p partial half-life of 39Ti at the calculated energy is 0.4-10 ms and 'closely aligns' with the measured half-life - is internally inconsistent because a partial half-life cannot be shorter than the measured total half-life. This issue directly affects the paper's main conclusion about 2p decay competing with beta decay, and it must be resolved before the quantitative predictions can be accepted.

major comments (3)
  1. [Section 3, Fig. 3] The text states that at the calculated Q2p = 0.45 MeV the 2p partial half-life of 39Ti lies between 0.4 and 10 ms, and that this 'closely aligns' with the measured total half-life of 39Ti (26-31 ms, Refs. [30,32,33]). This is arithmetically impossible. For two open decay channels, 1/T_tot = 1/T_beta + 1/T_2p, so every partial half-life must be longer than the total half-life. A 2p partial half-life of 0.4-10 ms would force T_tot < 10 ms (and <0.4 ms at the lower limit), contradicting the measured value by a factor of at least 3 and up to about 70. The only consistent resolutions are that the true Q2p is significantly below 0.45 MeV, pushing T_2p above approximately 28 ms, or that the computed width is overestimated by a large factor. The paper's hedge about S2p uncertainty acknowledges the first possibility, but the nominal 0.4-10 ms range is not consistent with the measured total half-life, and the 'rival beta decay' conclusion as stated at the calculated energy fails this internal consistency check. The authors should correct this comparison and, ideally, present the partial half-life together with the implied total half-life for a range of Q2p values.
  2. [Section 3, Fig. 3] The paper does not propagate the 0.3 MeV uncertainty in S2p(39Ti) into the quoted half-life range. Since the authors themselves note that a 100 keV change in decay energy changes the half-life by 1-5 orders of magnitude, the 0.4-10 ms range at the nominal Q2p = 0.45 MeV is not a meaningful estimate of the likely 2p partial half-life. A proper propagation of the energy uncertainty would produce a range spanning many orders of magnitude, which is exactly why the 'rival beta decay' claim is not robust. The authors state qualitatively that the lifetime could exceed beta decay when S2p uncertainty is included, but they do not quantify this. They should provide the half-life range obtained by varying Q2p over the GSM uncertainty band, or explicitly explain why such a range cannot be meaningfully given.
  3. [Section 2, GCC method] The GCC core-proton potential depth V0 is explicitly tuned to reproduce the GSM ground-state energy of 38Ti or 39Ti. This means the computed 2p partial width is not an independent prediction but is essentially determined by the input energy from the GSM calculation. The paper discloses this tuning, but it should be discussed more explicitly: the GCC calculation supplies the three-body decay dynamics (angular correlations, configuration mixing) for a given resonance energy, but it cannot test the GSM energy prediction itself. The uncertainty in the width therefore inherits the full uncertainty in the GSM energy, in addition to the interaction and configuration variations shown in Fig. 3. A sentence clarifying this logical status would help readers correctly interpret the 'predicted partial 2p decay width.'
minor comments (5)
  1. [Summary, Section 5] The summary states that the single-proton decay energy of 38Ti is 'around -0.3 MeV', whereas Section 3 reports S_p = 0.3 MeV with uncertainty 0.26 MeV and concludes that S_p remains positive. The sign in the summary is inconsistent and should be corrected.
  2. [Section 3, paragraph on S_p of 39Ti] The text first says the level arrangement of 39Ti 'energetically prohibits single-proton decay', but then states S_p = 0.15 MeV with uncertainty 0.31 MeV, which allows S_p to be negative within the uncertainty. The phrase 'energetically prohibiting' is too strong; the uncertainty should be acknowledged at that point.
  3. [Abstract and Section 5, 38Ti] For 38Ti, the predicted half-life of 2.76e-14 to 1.17e-11 s is far shorter than any experimentally observable 'radioactivity' in the usual sense. The term 'two-proton radioactivity' is appropriate for 39Ti, but for 38Ti the authors should clarify that this is prompt 2p emission rather than ground-state radioactivity, to avoid terminological confusion.
  4. [Section 2, Eq. (3)] The regulator f(p',p) in Eq. (3) contains a minus sign that is easy to misread at first glance; a brief definition of the variable n (which appears to be the NLO index depending on the channel) would improve readability.
  5. [Table 1 and Section 2] Table 1 lists many EFT constants but the text says only CS is optimized. It would be helpful to state explicitly which constants are held fixed at the values shown and which are varied during the fit, and how the values in Table 1 are obtained.

Circularity Check

1 steps flagged · score 3.0 of 10

GCC half-life 'prediction' is calibrated to the GSM energy it is presented as validating; otherwise the derivation chain is not circular.

  1. fitted input called prediction [Section 2 (GCC method paragraph) and Section 3 (Fig. 3 discussion)]
    "the depth V0 is specifically tuned to align the g.s. energy of 38Ti or 39Ti with the results obtained from GSM. ... a mere 100 keV difference in decay energy can precipitate a change in the half-life or decay width by approximately 1 to 5 orders of magnitude. ... we estimated the 2p half-life limits at the calculated decay energy to range between 0.4 and 10 ms, closely aligning with39Ti half-lives measured in previous experiments."

    The GCC width is presented as an independent predictive estimate, and the paper states GCC is 'employed for validation and as a supplementary analysis to GSM.' However, the GCC core-proton potential depth V0 is explicitly tuned to reproduce the GSM ground-state energy. The half-life is then computed at that tuned energy and, by the paper's own statement, changes by 1-5 orders of magnitude per 100 keV. Thus the 0.4-10 ms half-life range is not an independent prediction of the decay energy; it is a barrier-penetration value imposed by the GSM energy used as input.

full rationale

The main derivation chain is: (i) optimize a GSM EFT-inspired interaction to 13 experimental binding energies of neighboring Ca/Sc/Ti isotopes; (ii) use the resulting GSM Hamiltonian to predict S2p(39Ti) = -0.45 +/- 0.3 MeV and Q2p(38Ti) = 1.85 +/- 0.26 MeV; (iii) use GCC to compute 2p decay widths at those energies. Steps (i) and (ii) are conventional model calibration followed by prediction: the interaction is fit to quantities that do not include the 39Ti 2p half-life, so the energy prediction is not circular. Step (iii) is a genuine few-body width calculation, but its energy input is inherited from GSM: the GCC core-proton potential depth is explicitly tuned to reproduce the GSM ground-state energy, and the paper also frames GCC as 'validation' of GSM. That makes the GCC half-life a conditional prediction rather than an independent check, giving a mild methodological circularity. The self-citations to the GCC method [23,54] are citations to an established, externally benchmarked method (e.g., 67Kr), not to an unverified uniqueness theorem, so they are not load-bearing circularity. The paper's own hedge that the result is 'highly dependent on the specific 2p decay energy' acknowledges this conditioning. Separately, the reported 0.4-10 ms 2p partial half-life cannot be reconciled arithmetically with the measured ~28 ms total half-life (since 1/T_tot = 1/T_2p + 1/T_beta forces T_2p > T_tot); this is a serious internal-consistency/correctness concern, but it is not a circularity and does not further raise the circularity score.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The paper's quantitative output rests mainly on fitted interaction parameters and on a predicted decay energy, not on a parameter-free first-principles derivation. The half-life curves are honest barrier calculations, but they inherit both the GSM energy and the tuned GCC potential depth. No new particles, forces, or conserved quantities are introduced; the diproton, cigar, and triangular configurations are geometric labels for two-proton wave functions, not new entities.

free parameters (7)
  • EFT interaction coupling constant CS = optimized value in Table 1 (C1S0=-0.77, C3S1=-22.13, C1T0=0, C3T1=-3.60, C1...C7 values listed)
    Adjusted by chi-square minimization to reproduce 13 experimental binding energies of Ca, Sc, and Ti isotopes near 39Ti (Eq. 4).
  • Proton Woods-Saxon depths V0 per partial wave = 63.42, 66.38, 63.42, 61.55 MeV for s, p, d, f
    Adjusted to reproduce experimental ground and low-lying states of 37Ca and 39Sc (Section 2).
  • Neutron Woods-Saxon depths V0 per partial wave = 62.92, 62.92, 63.82, 67.38 MeV
    Adjusted to reproduce experimental energies of 37Ca and 39Sc (Section 2).
  • GCC core-proton potential depth V0 = not quoted numerically; tuned to align g.s. energy with GSM
    Explicitly tuned so 38Ti and 39Ti ground-state energies match the GSM result before computing widths (Section 2).
  • Exponential factor for A-dependent two-body interaction = 0.3
    Chosen to match values used in other investigations [68-70]; emulates three-body-force effects.
  • Channel-independent WS depths (alternative set) = 62.12 MeV protons, 63.82 MeV neutrons
    Alternative parameter set used to estimate uncertainty from single-particle potentials (Section 3).
  • Minnesota proton-proton interaction strength scaling = 100% and 150% variants
    Used as a sensitivity probe; the 150% case changes the width by an order of magnitude (Fig. 3).
assumptions (7)
  • standard math The Berggren completeness relation provides a complete one-body basis in the complex momentum plane (Eq. 1).
    Foundation of the Gamow shell model; accepted mathematical result after contour discretization.
  • domain assumption 36Ca can be treated as an inert core with valence protons and neutrons in the sd-pf shells.
    Core-plus-valence model in Section 2; core excitations are not included, so core-polarization effects are absorbed into fitted interactions.
  • domain assumption The f5/2 and f7/2 partial waves can be represented by bound harmonic-oscillator states because their centrifugal barrier suppresses continuum coupling.
    Section 2 states the f-wave has small interplay with the continuum; this truncates the model space.
  • domain assumption The EFT-inspired interaction with momentum-dependent regulator (Eqs. 2-3) and A-dependent factor realistically describes proton-rich nuclei near A ~ 40.
    The interaction form is assumed; its constants are optimized to 13 nearby states.
  • domain assumption The GCC three-body model with a core and two valence protons, using a Woods-Saxon core-proton potential and Minnesota proton-proton force, is adequate for 2p decay widths.
    Section 2; the model ignores core excitations and uses interaction parameters from other mass regions.
  • domain assumption The measured level energies used as fit observables are correct as taken from the NNDC database.
    Experimental data from Ref. [81] are assumed reliable.
  • domain assumption The uncertainty estimate from rescaling the nucleon-nucleon interaction by 2% captures the dominant theoretical error.
    Section 3 error bars; a single global scaling factor is a rough proxy for parameter uncertainties.

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Cite this review

Pith. "Pith review of Exploring the possible two-proton radioactivity of $^{38,39}$Ti." pith.science (2026). https://pith.science/paper/SRAYIDPY

@misc{pith2026250109542,
  author       = {Pith},
  title        = {Pith review of: Exploring the possible two-proton radioactivity of $^38,39$Ti},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SRAYIDPY}},
  note         = {Machine review of arXiv:2501.09542}
}
abstract

Two-proton (2$p$) radioactivity represents a rare decay mode that has been experimentally observed only in a selected few nuclei. The exploration of 2$p$ emission is crucial for elucidating the structure, mass, and nucleon-nucleon interactions within exotic proton-rich nuclei. $^{39}$Ti has long been postulated as a potential candidate for 2$p$ emission; however, experimental investigations have yet to confirm its 2$p$ decay. To provide more accurate information for further studies, we utilize the Gamow shell model (GSM) and the Gamow coupled channel (GCC) method to analyze the prospective 2$p$ radioactivity of isotopes $^{38,39}$Ti. Our calculations suggest that $^{39}$Ti is indeed a viable candidate for 2$p$ emission. Notably, the estimated partial 2$p$ decay width for $^{39}$Ti, predicted from the three-body GCC method, suggests that its 2$p$ decay could rival its $\beta$ decay in likelihood, although this is highly dependent on the specific 2$p$ decay energy. Additionally, our analysis indicates a propensity for pairing between the valence protons in $^{39}$Ti. A similar investigative approach reveals that $^{38}$Ti exhibits a higher 2$p$ decay energy and a broader decay width than $^{39}$Ti, positioning it as a more promising candidate for 2$p$ decay.

Figures

Figures reproduced from arXiv: 2501.09542 by the authors.

Figure 1
Figure 1. Spectra of Ca, Sc, and Ti isotope chains near 39Ti, calculated using the optimized GSM interac￾tion. Experimental values are repre￾sented by stars, as taken from [81]. 37 Ca E - E ( 3 6 C a ) ( M e V ) 38 Sc 39 Ti 2- 3/2+ 5- 3- 3/2+ 2- 5- 3- 3/2+ EXP 3/2+ [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Predicted energy spectra for 37Ca, 38Sc, and 39Ti, alongside the es￾timated uncertainties for each level. Additionally, the experimental values de￾noted by stars are taken from Ref. [81]. As a comparision, the results with the channel-independent WS potential are calculated as shown by the blue dashed lines. Since the g.s. energy of 37Ca is incorporated within the optimization parameters, the corresponding uncertain… view at source ↗
Figure 3
Figure 3. The calculated 2p partial width (half-life) for the g.s. of 39Ti as a function of the 2p decay energy Q2p. The Minnesota force is utilized for V N pp, considering interaction strengths of 100% (dashed and solid lines) and 150% (illustrated with a dotted line). To refine the estimate of the decay width limit, the single-particle energies were adjusted to enhance the contribution from the p-wave component, which is de… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Calculated 2p density distributions with different configurations for the g.s. of 39Ti in Jacobi coordinates pp and core-2p. 0 1 2 36 Ca E - E ( 3 6 C a ) ( M e V ) 37 Sc 38 Ti 0+ 7/2 0+ 7/2 0+ [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Predicted spectra of 37Sc and 38Ti with respect to the inner core 36Ca using the optimized GSM Hamiltonian. As a comparision, the results with the channel-independent WS potential are calculated as shown by the blue dashed lines. analysis of the 2p density distribution…

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