arxiv: 2605.02013 · v1 · submitted 2026-05-03 · ✦ hep-th

Recognition: 3 theorem links

· Lean Theorem

All DDF/lightcone two-particle decay widths up to level 8 in open bosonic string in critical dimension

Igor Pesando, Samuele Critelli

Authors on Pith no claims yet

Pith reviewed 2026-05-08 19:18 UTC · model grok-4.3

classification ✦ hep-th

keywords open bosonic stringdecay widthslightcone gaugeDDF oscillatorstwo-particle decaysRegge trajectorystring scale photon

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The pith

Open bosonic string states up to level 8 show decay width ratios of order one between slowest and fastest states when tachyon channels are excluded, with all dominant modes producing a photon at string scale.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper carries out a large-scale explicit computation of every Abelian two-particle decay channel, excluding the tachyon, for open bosonic string states through level 8. It reports that the ratio of the smallest to largest decay width at each level remains order one. Dominant polarized channels always contain at least one photon, and the states built primarily from DDF A_{-1} oscillators together with some A_{-2} operators prove the most stable, lying close to the leading Regge trajectory. The work also notes that equal-time formulations admit additional surge channels involving a tachyon plus a higher-mass state, which the lightcone results omit.

Core claim

For all levels examined, the ratio between the smallest and largest two-particle decay widths (tachyon channels removed) stays of order one; the leading polarized channels invariably include at least one photon; and the most stable states are those assembled from DDF/lightcone oscillators A_{-1} and selected A_{-2}, which are close relatives of the leading Regge trajectory states.

What carries the argument

The lightcone-gauge enumeration of all polarized two-particle decay amplitudes using DDF oscillators, performed level by level up to eight.

If this is right

States near the leading Regge trajectory remain among the most stable at every level checked.
Every massive open string ultimately decays into a photon carrying energy comparable to the string scale.
The top five polarized channels already account for roughly sixty percent of the total width, while the first hundred or so channels capture nearly all of it.
The order-one spread in widths persists uniformly from level two through level eight.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the hypothesized surges dominate once tachyons are restored, the reported lightcone widths would underestimate actual lifetimes by a large factor.
Stability in bosonic string theory may therefore be controlled by tachyon production rather than by the photon channels visible in the present calculation.
Repeating the enumeration at higher levels or with non-Abelian final states would test whether the order-one width ratio survives.

Load-bearing premise

The lightcone computation with tachyon channels removed still captures the dominant decay physics even though equal-time formulations contain additional surge channels.

What would settle it

An explicit inclusion of tachyon-plus-higher-mass channels that shifts the width ratio between slowest and fastest states by more than an order of magnitude or removes photons from the dominant channels.

Figures

Figures reproduced from arXiv: 2605.02013 by Igor Pesando, Samuele Critelli.

**Figure 1.** Figure 1: 2. All DDF/lightcone states seem to exhibit increasingly similar total decay widths (excluding tachyons), in the sense that the ratio between the fastest- and slowest-decaying states approaches 1; see view at source ↗

**Figure 1.** Figure 1: We plot for any level N ≤ 8 the log10 of the total decay width divided by 4e −2ϕ0 = g 2 o ... for the most stable and less stable states. 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 2 3 4 5 6 7 8 ratio level ratio Less stable vs Most stable partial widths foreach level view at source ↗

**Figure 2.** Figure 2: We plot for any level the ratio between the total decay width view at source ↗

**Figure 3.** Figure 3: We plot for any level and any state the ratio between fastest decay view at source ↗

**Figure 4.** Figure 4: We plot for any level and any state the ratio between the sum view at source ↗

read the original abstract

We carry out an ``experimental analysis'' in which we explicitly compute all possible Abelian two-particle decay channels (excluding the tachyon) of open bosonic massive states up to level (8), amounting to approximately 2,000,000 cases. The aim is to develop intuition about which states are the most stable and to identify the dominant decay channels. Our results show that, for all levels considered, the ratio of decay widths between the slowest- and fastest-decaying states is of order one. In most cases, the dominant polarized decay channel accounts for at least $10\%$ of the total decay width, while the first five channels together contribute roughly 60%. Even when we sum about $10000$ polarized channels. Dominant polarized channels always involve at least one photon and all strings ultimately decay into a photon carrying energy at the string scale. The most stable states are made using the DDF/lightcone oscillators $A_{-1}$ and some $A_{-2}$ and are close relatives of the states on the leading Regge trajectory. Finally, we discuss how the computation on the lightcone differs from the equal-time computation due to ``surges'', i.e., decays into a tachyon and a higher-mass state, and we hypothesize that, in bosonic string theory, decays into tachyons constitute the dominant decay channels.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers a large explicit catalog of non-tachyon two-particle decays up to level 8 in the lightcone gauge, but its statements on overall stability and photon dominance rest on excluding the tachyon channels it hypothesizes actually dominate.

read the letter

The new piece is the brute-force enumeration of every Abelian two-particle channel (tachyon excluded) for open bosonic string states through level 8, roughly two million cases, all done with standard DDF operators. They report that width ratios between slowest and fastest states stay within an order of magnitude at each level, that the top few polarized channels carry most of the width, and that those channels always include at least one photon. That is concrete data that was not available before at this scale, and it gives a clear picture of the non-tachyon sector in lightcone quantization. The most stable states they identify are built from A_{-1} and some A_{-2} oscillators, which lines up with expectations from the leading Regge trajectory. The computation itself looks solid on its own terms: direct matrix elements, no free parameters, and they are transparent about working in the lightcone frame. The soft spot is exactly the one the stress-test note flags. The paper notes that equal-time calculations show large “surges” into tachyon plus higher-mass states and then hypothesizes that tachyons dominate the physics, yet the reported results and the O(1) ratio claim come only from the channels they kept. Without a quantitative comparison that includes the tachyon channels, the headline statements about which states are stable and how strings ultimately decay remain conditional on an untested assumption. This is a limitation the authors themselves flag, so it is not hidden, but it does cap how far the conclusions can be pushed. The work is for string theorists who care about explicit decay calculations or who want to check lightcone versus covariant results on higher-mass states. A reader looking for raw numbers on these specific channels will find it useful. It is worth sending to a serious referee because the computation is new and reproducible in principle, even if the interpretation needs tightening around the tachyon issue.

Referee Report

2 major / 3 minor

Summary. The manuscript presents an exhaustive computational survey of all Abelian two-particle decay channels (excluding tachyons) for open bosonic string massive states up to level 8 in the lightcone gauge using DDF operators, covering approximately 2,000,000 cases. It reports that the ratio of decay widths between the slowest- and fastest-decaying states is O(1) at each level, that dominant polarized channels always involve at least one photon (with the top five channels contributing ~60% of the width), and that the most stable states are constructed from A_{-1} and A_{-2} oscillators, resembling the leading Regge trajectory. The paper contrasts this with equal-time formulations, noting 'surges' (tachyon plus higher-mass state) and hypothesizing that tachyon decays dominate overall.

Significance. If the results hold, this provides a large-scale numerical dataset on bosonic string decays that offers concrete intuition on state stability and channel dominance, potentially guiding analytic work on string decay amplitudes. The explicit evaluation of ~2M cases using standard DDF operators, with no fitted parameters, is a clear strength, enabling statistical claims about O(1) width ratios and photon prevalence within the computed channels.

major comments (2)

The central claims of O(1) width ratios and photon dominance are obtained from a lightcone Abelian two-particle computation that excludes all tachyon channels by construction. While the manuscript notes surges in equal-time formulations and hypothesizes tachyon dominance, no quantitative inclusion or comparison of tachyon channels is provided among the reported cases. This leaves the interpretation that non-tachyon channels capture the dominant physics (and that all strings ultimately decay to string-scale photons) as an unverified assumption.
The manuscript lacks presented error estimates, numerical precision details, or sample data tables for the ~2M cases. Without these, the quantitative support for statements such as the O(1) ratio holding 'for all levels considered' and the top-five channels contributing ~60% cannot be fully assessed.

minor comments (3)

A precise breakdown of the number of channels per level (rather than the aggregate ~2,000,000) would improve reproducibility and allow readers to verify the scaling.
Include at least one illustrative table (e.g., for level 4 or 5) listing the top decay channels, their widths, and the resulting ratio to demonstrate the O(1) claim concretely.
Clarify the exact definition of 'Abelian two-particle' channels and any polarization or momentum selection rules applied when evaluating the DDF matrix elements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comments point by point below, with clarifications on scope and offers to strengthen the presentation of results.

read point-by-point responses

Referee: The central claims of O(1) width ratios and photon dominance are obtained from a lightcone Abelian two-particle computation that excludes all tachyon channels by construction. While the manuscript notes surges in equal-time formulations and hypothesizes tachyon dominance, no quantitative inclusion or comparison of tachyon channels is provided among the reported cases. This leaves the interpretation that non-tachyon channels capture the dominant physics (and that all strings ultimately decay to string-scale photons) as an unverified assumption.

Authors: Our computation is deliberately restricted to Abelian two-particle decays excluding tachyons, as stated in the title, abstract, and section 1. The O(1) ratios, photon dominance in dominant channels, and the remark that strings ultimately decay to a string-scale photon all refer specifically to the ~2M non-tachyon channels we enumerated. The manuscript does not assert that these channels dominate the overall decay; on the contrary, it explicitly contrasts the lightcone results with equal-time formulations (where surges into tachyon plus higher-mass states appear) and hypothesizes that tachyon decays are dominant in bosonic string theory. We will revise the discussion to restate the scope of the claims more explicitly and to underscore that the tachyon-dominance statement remains a hypothesis outside the present calculation. revision: partial
Referee: The manuscript lacks presented error estimates, numerical precision details, or sample data tables for the ~2M cases. Without these, the quantitative support for statements such as the O(1) ratio holding 'for all levels considered' and the top-five channels contributing ~60% cannot be fully assessed.

Authors: All amplitudes are obtained from exact algebraic expressions involving DDF operators; the only numerical step is the evaluation of the resulting widths and ratios, performed in double-precision floating point. We will add a short subsection describing this exact algebraic origin and the floating-point precision used, together with sample tables of decay widths and channel contributions for representative states at levels 2, 4, and 6. A complete dataset of all ~2M cases can be supplied as supplementary material or on request, though printing the full tables in the main text is impractical. revision: yes

Circularity Check

0 steps flagged

No circularity: direct evaluation of matrix elements via known DDF operators

full rationale

The paper performs an exhaustive computational enumeration of ~2M explicit Abelian two-particle decay matrix elements (tachyon channels excluded) using established DDF/lightcone creation operators. All reported quantities—width ratios of order one, dominance of photon-containing channels, and stability ordering—are direct numerical outputs of these calculations with no fitted parameters, no self-referential definitions, and no load-bearing self-citations. The tachyon-surge hypothesis is presented separately as an unverified conjecture and is not used to derive or justify the computed results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard framework of open bosonic string theory quantized in 26 dimensions using lightcone gauge and DDF operators; no new free parameters, axioms beyond domain assumptions, or invented entities are introduced.

axioms (2)

domain assumption Open bosonic string theory is consistently quantized in the critical dimension of 26 spacetime dimensions.
The title and computation are performed in the critical dimension as required for the absence of anomalies.
domain assumption DDF operators in the lightcone gauge provide a complete and convenient basis for constructing massive string states and computing their decay amplitudes.
The paper relies on these operators for the explicit enumeration of all channels.

pith-pipeline@v0.9.0 · 5547 in / 1516 out tokens · 82325 ms · 2026-05-08T19:18:44.431479+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Foundation.AlexanderDuality alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We compute all decays of massive bosonic open string states up to level 8 into non tachyonic states, amounting to approximately 2·10^6 processes... We work in the critical dimension, and our results may depend sensitively on the number of non-compact dimensions.
Cost.FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We perform this analysis in the critical dimension using the DDF/lightcone formalism... ⟨V3| = product of standard DDF vertex operator with N[r]m,[s]n combinatorics

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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swap canonical decay

At the same time we have to count properly the degeneration by considering all inequivalent decays. The set of decays on which we im- pose an equivalence relation in order to determine the inequivalent decays are the decays generated by a permutation on the canonical decay and the decays generated by a permutation on the “swap canonical decay” which is ge...

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