Missing solutions in scalar-Gauss-Bonnet gravity

Daniela D. Doneva; Fethi M. Ramazano\u{g}lu; Kalin V. Staykov; O\u{g}uzhan K. Yamak; Stoytcho S. Yazadjiev

REVIEW

Missing solutions in scalar-Gauss-Bonnet gravity

Oğuzhan K. Yamak, Kalin V. Staykov, Fethi M. Ramazanoğlu, Daniela D. Doneva, Stoytcho S. Yazadjiev

General Relativity and Quantum Cosmology · Submitted 4 Nov 2026 · 50 references

Pith's verdict: The paper discovers real, previously missed gapped scalarized branches and a new non-central regularity condition in sGB gravity; limitations are honestly stated and the core claims hold up under scrutiny.

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arXiv ID2611.06246 (version 1)

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CitationDaniela D. Doneva, Fethi M. Ramazano\u{g}lu, Kalin V. Staykov, O\u{g}uzhan K. Yamak, Stoytcho S. Yazadjiev. "Missing solutions in scalar-Gauss-Bonnet gravity." arXiv:2611.06246 (2026). https://pith.science/paper/QHNNXCU4

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Dear colleague,

Here's my read on 2611.06246. The paper does two genuinely new things: it finds gapped scalarized neutron star branches in sGB gravity that had been missed entirely, and it identifies a non-central regularity condition that explains some of the branch terminations. The numerical search is thorough—brute-force grid over central parameters—and the authors are honest about the limitations of the non-central condition (not strictly sufficient). The connection to the phase-transition view of scalarization is well motivated, and Appendix B linking the two existence conditions is a nice theoretical touch.

Now the soft spots. The gapped branches appear in what the authors themselves call "fine-tuned" regions of the parameter space, especially for κ=0 where large λ pushes the gap to very low masses. The non-central condition is inferred from a vanishing determinant of the radial evolution matrix, which the authors note is a strong indicator but not a proof. Also, only one equation of state (APR4) is used; we don't know how qualitatively these features depend on the EOS. The paper does not address the well-posedness of time evolution for the solutions it finds, though that is explicitly deferred to future work.

These are real limitations, but they do not undermine the core claims. The paper is a solid advance in understanding the solution space of sGB neutron stars. I would recommend peer review: the results are interesting and the methodology is sound. A referee would want the authors to discuss the sufficiency of the non-central condition more carefully, and to test a second EOS or at least state how sensitive the gap phenomenon is.

Summary: a careful, well-written paper that delivers on its promises. It should be published after minor revisions addressing the above.

Referee report

Major objections (1)

[§III C] The non-central existence condition (Eq. 19, Δ_min = 0) is stated to be 'not strictly sufficient' for non-existence, but the paper relies on it as a necessary condition and finds empirical coincidence with branch termination. The authors acknowledge this limitation (b262) and support it with brute-force scans. However, to strengthen the argument, it would be valuable to discuss whether any counterexamples exist (e.g., solutions where Δ_min=0 but a regular solution can still be constructed via a different formulation). A brief remark on the sufficiency gap and how it might be closed would improve completeness. The authors acknowledge this: “Although the vanishing of the deter- minant is not strictly sufficient for non existence of a so- lution to the differential equation, it is still a very strong indicator.”

Minor objections (5)

[§IV] In the first sentence of Section IV, 'the the existence conditions' contains a duplicated article; please correct to 'the existence conditions'.
[Fig. 1] The three curves in Fig. 1 are labelled only by ADM mass values in the caption. It would be helpful to distinguish them (e.g., solid/dashed/dotted) directly in the plot or add a legend, as the text refers to their shapes.
[§II A, Eq. (16)] The notation f_- in Eq. (16) is introduced without explicit definition. Earlier in Eq. (14) the coupling is defined as f_ϵ with ϵ=±1, but the subscript '-' is not formally defined. Please clarify, e.g., 'f_- ≡ f_{ϵ=-1}'.
[§III C, after Eq. (18)] The description of row/column equilibration (dividing by Euclidean norms) is clear enough, but the determinant after equilibration is described as 'at the order of unity' in regular cases. It would be useful to report a typical range of Δ_min for valid solutions to calibrate the '≪1' threshold used for violation.
[Appendix B, Eq. (B.7)] The expression for D_4 in Eq. (B.7) includes a term '−12F_0^2 Λ_2^2'. It appears there may be a missing factor; checking the derivation from the series expansion would improve readability. Also, the sign of the term '−64π p_0 F_0^2 Λ_2' seems consistent with Eq. (B.5), but a brief verification note would help.

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The paper

The paper claims

This paper discovers novel neutron star solutions in scalar-Gauss-Bonnet gravity that were entirely missed in prior literature, and identifies a new non-central regularity condition necessary for the existence of scalarized stars.

The reading

The paper shows that scalarized neutron star branches in sGB gravity can terminate and then reappear after a gap on the mass-radius diagram, meaning the solution space is richer than previously known. It also identifies a new regularity condition at a nonzero radius—the non-central existence condition—that explains terminations the old central condition could not. The discovery was guided by a phase-transition view of scalarization and a brute-force search of parameter space.

What carries the argument

The central existence condition is a fourth-order polynomial for Λ₂ at the star's center; the non-central existence condition is the vanishing of det M(r) (after row/column equilibration) at a nonzero radius, signalling a singularity in the differential equation system. The paper shows these are related via the derivative of the polynomial (Eq. B.8).

Extended reading notes

Core claim

The authors find that scalarized neutron star branches in sGB gravity do not always end permanently; they can have gaps—disconnected segments that reappear at higher central densities. This is explained by the interplay of two existence conditions: the known central condition (a polynomial in Λ₂) and a newly found non-central condition (the vanishing determinant of the radial evolution matrix at some radius). The latter is responsible for many branch terminations and was previously unknown.

Load-bearing premise

The new non-central condition (vanishing determinant) is assumed to be both necessary and sufficient for the non-existence of a regular solution, even though the authors note it is not strictly sufficient from a mathematical standpoint.

Editorial extensions

If this is right

The complete scalarized solution landscape is richer, with gapped branches that may host stable configurations where previously none were thought to exist.
Stability analyses and astrophysical predictions (e.g., for accretion or gravitational waves) must now account for possible branch gaps.
The non-central condition provides a new test for the existence of regular solutions in any sGB theory beyond the central condition alone.
Understanding gaps may explain why some branches terminate early without requiring a violation of the central condition, clarifying the physicality of branch endings.

Reading between the lines

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

The non-central existence condition may be linked to the well-posedness (hyperbolicity) of sGB evolution; investigating this connection could reveal whether solutions near the gap are dynamically viable.
Gapped branches create a novel scenario for first-order phase transitions: a star on the first segment could jump discontinuously to the reappeared segment, producing a distinct observational signal (e.g., a sudden change in mass and radius).
The method of using Landau theory to predict branch continuation could be applied to other modified gravity theories where scalarization occurs, such as scalar-tensor theories with higher-order couplings.

Original abstract

We discover novel neutron star solutions under scalar-Gauss-Bonnet (sGB) gravity which have been missed altogether in the existing literature. The complete solution space also reveals a new regularity condition that has to be satisfied by all scalarized stars. Certain scalarized branches of neutron star solutions under sGB gravity have long been known to abruptly terminate on stellar mass-radius diagrams at lower stellar radii, a phenomenon that has no counterpart in classical scalar-tensor theories. Such termination has been strongly suspected to be physical rather than a numerical artifact nevertheless, since sGB theories feature a regularity condition at the center of the star, which can prevent the existence of certain configurations. Motivated by the recently re-emerging phase-transition view of scalarization, we first predict that these terminated branches can reappear after a "gap." We confirm this by finding such reappearing branches, and also determine which parts of the theory parameter space are more amenable to such occurrences. We then show that the previously known central regularity condition cannot explain all the non-existing solutions in the gaps, which leads us to the discovery of a new, non-central, condition for the existence of scalarized neutron stars under sGB gravity.

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-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-05: 1 request, 5,136 input tokens.

What the abstract itself states, as probabilities: claims first 0.38 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.90 · 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.

We discover novel neutron star solutions under scalar-Gauss-Bonnet (sGB) gravity which have been missed altogether in the existing literature. The complete solution space also reveals a new regularity condition that has to be satisfied by all scalarized stars. Certain scalarized branches of neutron star solutions under sGB gravity have long been known to abruptly terminate on stellar mass-radius diagrams at lower stellar radii, a phenomenon that has no counterpart in classical scalar-tensor theories. Such termination has been strongly suspected to be physical rather than a numerical artifact nevertheless, since sGB theories feature a regularity condition at the center of the star, which can prevent the existence of certain configurations. Motivated by the recently re-emerging phase-transition view of scalarization, we first predict that these terminated branches can reappear after a "gap." We confirm this by finding such reappearing branches, and also determine which parts of the theory parameter space are more amenable to such occurrences. We then show that the previously known central regularity condition cannot explain all the non-existing solutions in the gaps, which leads us to the discovery of a new, non-central, condition for the existence of scalarized neutron stars under sGB gravity.

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	We discover novel neutron star solutions under scalar-Gauss-Bonnet (sGB) gravity which have been missed altogether in the existing literature.	result 0.62	0.97	0.68	asserted 1.00
s1	The complete solution space also reveals a new regularity condition that has to be satisfied by all scalarized stars.	result 0.97	0.92	0.25	asserted 1.00
s2	Certain scalarized branches of neutron star solutions under sGB gravity have long been known to abruptly terminate on stellar mass-radius diagrams at lower stellar radii, a phenomenon that has no counterpart in classical scalar-tensor theories.	background 0.98	0.08	0.07	asserted 1.00
s3	Such termination has been strongly suspected to be physical rather than a numerical artifact nevertheless, since sGB theories feature a regularity condition at the center of the star, which can prevent the existence of certain configurations.	background 0.97	0.06	0.04	hedged possibility 0.89
s4	Motivated by the recently re-emerging phase-transition view of scalarization, we first predict that these terminated branches can reappear after a "gap." We confirm this by finding such reappearing branches, and also determine which parts of the theory parameter space are more amenable to such occurrences.	contribution 0.58	0.94	0.19	asserted 0.95
s5	We then show that the previously known central regularity condition cannot explain all the non-existing solutions in the gaps, which leads us to the discovery of a new, non-central, condition for the existence of scalarized neutron stars under sGB gravity.	result 0.81	0.96	0.59	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.76 · evidence from simulation or computation 0.96.

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. Gauss-Bonnet invariant as a function of

**Figure 2.** FIG. 2. Phase transition picture for the DEF model as the

**Figure 4.** FIG. 4. The scalarized branches for a fixed

**Figure 5.** FIG. 5. The scalarized branches for

**Figure 6.** FIG. 6. Representative scalarized solution branches that in

**Figure 7.** FIG. 7. The behavior of the scalarized branches and the appearance of gaps as the three theory parameters

**Figure 8.** FIG. 8. A gapped scalarized branch where terminal points 3 is

**Figure 10.** FIG. 10. The existence conditions and the scalarized branch

**Figure 11.** FIG. 11. Sample scalarized branches for the oppositely signed

Showing 10 of the paper's 12 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2611.06246 by the authors.

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

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