Pith. sign in

REVIEW 3 major objections 4 minor 65 references

Worldsheet for Generalized Veneziano Amplitudes

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims that one worldsheet action reproduces Mandelstam's generalized Veneziano amplitudes and generates new higher-point open and closed-string amplitudes with partial crossing symmetry.

desk verdict First worldsheet realization of Mandelstam's generalized Veneziano family, with a plausible dictionary but a load-bearing analytic-continuation step that needs a branch prescription. read the letter →

arxiv 2511.16280 v2 pith:DBLKUVEF submitted 2025-11-20 hep-th

Shota Komatsu , Pronobesh Maity This is my paper · ORCID
classification hep-th MSC 81T3081T4081U20 PACS 11.25.-w11.55.-m
keywords generalizedVenezianoamplitudeworldsheetchiralcompositelineardilatonMandelstammapdualresonancecrossingsymmetryopenstringclosed
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 attempts to establish that a specific two-dimensional worldsheet action — the chiral composite linear dilaton theory with a new boundary term — gives a first-principles description of Mandelstam's three-parameter family of generalized Veneziano amplitudes. The authors identify a dictionary relating the amplitude parameters to worldsheet data: δ = q/2, b = 1 − q − p², and λ = 1/(1+r)², with q the dilaton background charge, p² a dressing momentum, and r a ratio of winding numbers. They then use the same action to compute n-point open-string amplitudes and a four-point closed-string amplitude, and show that these inherit only partial crossing symmetry because winding-number conservation places external strings on unequal footing. A sympathetic reader would care because this is the first worldsheet formulation for these generalized amplitudes, opening the door to studying unitarity, factorization, and no-ghost questions from the worldsheet.

What carries the argument

The central object is the chiral composite linear dilaton (CLD) action, a βγ-system with a specific background-charge term and a newly added geodesic-curvature boundary term that restores Weyl invariance on worldsheets with boundaries. Combined with D0-brane boundary conditions on the γ-field and vertex operators carrying winding numbers {w_k}, the path integral localizes γ to the Mandelstam map, whose interaction points Z_I control the discriminant ∆(P_n) that enters the amplitude. The boundary term and the winding-number dictionary are what carry the argument: they turn a previously known CFT into a source of generalized Veneziano amplitudes.

What would settle it

Numerically evaluate the four-point worldsheet integral (17) for a non-integer q/2, say q=1/2 with r=1 (so a±=1/2 ± i√3/2), using a fixed branch convention, and compare with the Mandelstam integrand (1) for δ=1/4, b=1/2, λ=1/4; any difference at generic s,t shows the analytic continuation is not the branch-preserving one needed for the dictionary.

Watch

Extended reading notes

Core claim

The central claim is that the worldsheet action (4), with the D0-brane boundary conditions (7) and vertex operators (11), reproduces the generalized Veneziano amplitude (1) for generic parameters. The path integral over the γ-field localizes to the Mandelstam map ρ(z)=∑ w_k log(z−x_k), and evaluating the chiral composite dilaton action on this map yields a four-point integrand which, after tuning w1=w3 and dressing two vertex operators by a free boson, matches Mandelstam's formula with δ=q/2, b=1−q−p², λ=1/(1+r)². The same construction produces higher-point open-string amplitudes and a four-point closed-string amplitude, both of which are partially crossing-symmetric but not fully so.

Load-bearing premise

The load-bearing premise is that the worldsheet path integral, evaluated for real interaction points, can be analytically continued in the winding numbers without crossing branch ambiguities; if that continuation is not branch-safe for the complex branch points that appear when q/2 is not an integer, the four-point integrand is not Mandelstam's.

Editorial extensions

If this is right

  • The generalized Veneziano amplitudes now have a worldsheet origin, so questions about their unitarity, factorization, and Regge behavior can be addressed with worldsheet techniques.
  • The construction yields n-point open-string amplitudes (e.g., five-point formula (29)) that are new and can be studied systematically.
  • The closed-string analog (33) is a new partially crossing-symmetric amplitude whose KLT-like factorization expresses it as a sum of products of Appell hypergeometric functions.
  • Partial crossing symmetry is traced to winding-number conservation, explaining why full crossing symmetry is incompatible with the present worldsheet realization.
  • The dictionary δ=q/2, b=1−q−p², λ=1/(1+r)² makes clear how to engineer the parameters of the generalized Veneziano family at will.

Reading between the lines

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

  • The analytic continuation in winding numbers, asserted in the supplemental material, is the step that lets the four-point integrand (17) be identified with Mandelstam's formula for non-integer q/2; a branch-prescription check would be needed to fully settle the match.
  • If the worldsheet description is robust, it suggests a route to construct worldsheet actions for other generalized amplitudes (e.g., hypergeometric amplitudes) by suitably modifying the CLD action or its boundary conditions.
  • The winding-conservation obstruction to full crossing symmetry may point to a general no-go: any worldsheet theory whose vertex operators carry conserved winding numbers will produce only partially crossing-symmetric amplitudes, unless the localization mechanism is changed.
  • Testing the higher-point amplitudes numerically for small n could reveal whether they satisfy the expected duality properties (channel factorization) that the four-point case inherits from Mandelstam's formula.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript proposes a bosonic worldsheet action that is claimed to reproduce Mandelstam's three-parameter generalization of the Veneziano amplitude. The construction uses the chiral composite linear dilaton (CLD) beta-gamma system with a new boundary term, D0-brane boundary conditions, and vertex operators carrying fractional winding. After localization to the Mandelstam map, the four-point amplitude is matched to the target formula with the dictionary delta = q/2, b = 1 - q - p^2, lambda = 1/(1+r)^2. The same action is then used to compute n-point open-string amplitudes and a four-point closed-string amplitude with partial crossing symmetry. The supplemental material contains the Weyl-invariance check of the boundary term and the detailed evaluation of the localized CLD action.

Significance. If the central derivation is completed, this is a significant step: it would be the first worldsheet description of generalized Veneziano amplitudes beyond the standard Veneziano case, and it provides explicit higher-point and closed-string predictions. The paper's strengths include a detailed Weyl-anomaly computation for the new boundary term, a careful treatment of the Mandelstam map and its PSL(2,R) transformation, and explicit degenerate cross-ratio checks of the identities a_+ + a_- = 1 and lambda = 1/(1+r)^2. The target amplitude is an external benchmark, so the construction is not circular in the sense of input-equals-output. The main obstacle is not the final matching formula itself, but the analytic continuation used to obtain it from the path integral.

major comments (3)
  1. [SM B.1; Eqs. (B15), (17), (19)] The evaluation is performed for real interaction points Z_I and yields e^{-Gamma_ren} = |Delta(P_n) prod w_k|^{q/2} prod |x_i-x_j|^{-q}. This modulus is not analytic in w_k, so the statement in SM B1 that the result is analytic in {w_k} and can therefore be analytically continued is not valid as written. Eq. (17) contains (x-a_+)^{q/2}(x-a_-)^{q/2} with complex a_+ and a_- for the tuned four-point case and generically non-integer q/2, so a branch prescription is required. Without specifying the branch selected by the OPE/contour of the path integral, Eq. (17) is not established as the amplitude of action (4), and the dictionary (18)-(20)/(24) is not established. Please supply an explicit branch prescription and show that it follows from the localization, or alternatively prove that the relevant combination is branch-independent in the physical region.
  2. [SM C; Eqs. (C1)-(C18), (33)] The closed-string computation explicitly assumes a_+ and a_- are real and uses the ordering 0 < a_+ < a_- < 1. However, the s-t symmetric choice w_1 = w_3 used in the main text gives complex a_+ and a_- for generic r (Eq. (19)). The Appell-function expression (C18) is therefore derived only for real a_+; its analytic continuation to the complex case is not described. Since Eq. (33) is presented for a_+ and a_- given by (16), the closed-string prediction has the same branch gap as the open-string case. The higher-point open-string result (29) inherits the same issue through |Delta(Q_5)|^{q/2} in Eq. (25).
  3. [Sec. II, Eqs. (18), (24), (6)] The statement that the construction reproduces '(1) with general parameters' is stronger than what is established. The dictionary gives delta = q/2, b = 1 - q - p^2, and q = 1 - (d+c_chi)/24. For a real dressing momentum p^2 >= 0, this imposes b <= 1 - 2 delta. The allowed region of the Mandelstam parameters (b, delta, lambda) covered by the worldsheet construction should be stated explicitly, including any reality conditions on r. This does not invalidate the construction, but it qualifies the generality claim.
minor comments (4)
  1. [SM C, heading] Typo: 'Kawai-Lwewllen-Tye' should be 'Kawai-Lewellen-Tye'.
  2. [Sec. IV] Typo: 'connectons' should be 'connections'.
  3. [Eqs. (15), (25)] Eq. (15) writes the factor without absolute values, while Eq. (25) has |...|^{q/2}. This notational inconsistency should be resolved, especially because the absolute value is central to the analytic-continuation question.
  4. [Ref. [56]] Reference [56] is cited as 'to appear'. Since the present construction relies on the CLD framework developed there, an arXiv number or a more complete citation would help the reader verify the background.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Mandelstam amplitude is an external benchmark; the worldsheet matching is parameter identification, not input-equals-output.

full rationale

The claimed derivation chain is: action (4) with the new boundary term and D0-brane boundary conditions (7), vertex operators (11), path-integral localization to the Mandelstam map (14), evaluation of the matter correlator as (15)/(17), and finally comparison with Mandelstam's generalized Veneziano amplitude (1). None of these steps reduces to its own input. Mandelstam's amplitude is an external 1968 target, not a quantity defined by the worldsheet action. The dictionary (delta=q/2, b=1-q-p^2, lambda=1/(1+r)^2) is obtained by explicit algebra after setting w1=w3, so that (x-a+)(x-a-) is proportional to (1-4 lambda x(1-x)); this is parameter identification, not a fitted parameter renamed as a prediction. The paper does not claim to predict the Mandelstam parameters from first principles without tuning; it claims to reproduce the family of generalized Veneziano amplitudes, and the path-integral computation is the nontrivial content. The self-citations to [54,56] provide the CLD machinery and localization technique, but the present derivation of the boundary term, the higher-point integrands, and the closed-string amplitude is carried out here and tested against an external benchmark, so the argument does not collapse into a self-citation chain. The flagged weakness in SM B1 (analytic continuation from real interaction points Z_I to complex a+ and a-; Eq. (B15) contains absolute values and is not literally analytic in {w_k}) is a potential gap in the derivation of Eq. (17), not a circular equivalence. If the continuation is invalid, the worldsheet match fails, but it does not fail because the output was inserted as an input. Therefore the circularity score is 0.

Assumptions & free parameters 4 free parameters · 4 assumptions · 3 invented entities

The central claim rests on a small number of tuning knobs — q (via d and c_χ), the dressing momentum p, and the winding ratio r — which are chosen by hand to map onto the target parameters (δ, b, λ) of Mandelstam's amplitude. The framework itself (CLD, D0-branes, winding vertex operators, dressing boson) is entirely authored by the same group; none of these entities has an external falsifiable handle. The axioms are a mix of standard CFT background and paper-specific analytic-continuation and regularization choices.

free parameters (4)
  • q (CLD background charge, set by d and c_χ) = δ = q/2 (target δ)
    The dilatonic charge in (4); the four-point exponent (x−a±)^{q/2} fixes the target's δ. Chosen through the spacetime dimension d and internal central charge c_χ per (6).
  • p (momentum of the dressing boson y) = p² = 1 − q − b (target b)
    Dressing (22) shifts the Mandelstam 'b' via (24): b = 1 − q − p².
  • r = w2/w1 (winding ratio, with w1 = w3, Σw = 0) = λ = 1/(1+r)² (target λ)
    Tuning w's so that a+ + a− = 1 and a+a− = (1+r)²/4, giving the target λ via (20).
  • Internal CFT central charge c_χ (equivalently spacetime dimension d) = c_χ = 24(1−q) − d
    A knob for q and hence δ; the paper does not characterize which (b, λ, δ) triples are reachable.
assumptions (4)
  • domain assumption The CLD βγ path integral localizes to holomorphic maps (∂̄γ = ∂γ̄ = 0), with the renormalized on-shell action given by (B13)/(B15).
    Load-bearing input inherited from the authors' own [54, 56]; used to get (15), (17), (25), (33). Section II and SM B.
  • ad hoc to paper The analytic continuation of the Mandelstam-map evaluation from real interaction points Z_I to complex w_k is valid and branch-safe for the non-integer powers q/2.
    SM B: 'we will work in a parameter regime ... real ... analytically continued'; the four-point dictionary uses complex a±.
  • domain assumption Weyl invariance of the action with the new boundary term (second line of (4)); the SM A variation computation yielding the CLD central charge c = 24q is the correct boundary treatment.
    Appendix A; needed for the disk amplitude to be well defined; the boundary term is introduced ad hoc for this purpose.
  • domain assumption The vertex operators (11) have conformal weights k²+q, the on-shell condition is M² = q − 1 (13), and winding conservation Σw = 0 is exactly enforced (δ in (15)).
    Section II; used in the localization to the Mandelstam map and in the crossing-symmetry discussion.
invented entities (3)
  • Chiral composite linear dilaton (CLD) βγ-system with the new boundary term
    purpose: The matter sector whose path integral produces the non-Veneziano factor (x−a+)(x−a−) in the four-point amplitude and the discriminant factors in higher-point amplitudes.
    Framework invented by the same authors [54, 56]; the boundary term is added here ad hoc to restore Weyl invariance on the disk. No observable outside these papers tests it.
  • D0-branes localized in γ1 and extended in γ2, with fractional winding numbers w_a
    purpose: Boundary conditions (7); windings are the tuning parameters (r = w2/w1) that set λ.
    Internal device of the construction; no independent falsifiable handle.
  • Dressing free boson y in the internal CFT
    purpose: Shifts b to b = 1 − q − p² (24) and can make the lightest exchanged particle massless (p² = 1 − q).
    Added per-construction; its momentum p is tuned to the target amplitude's b parameter.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Worldsheet for Generalized Veneziano Amplitudes." pith.science (2026). https://pith.science/paper/DBLKUVEF

@misc{pith2026251116280,
  author       = {Pith},
  title        = {Pith review of: Worldsheet for Generalized Veneziano Amplitudes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DBLKUVEF}},
  note         = {Machine review of arXiv:2511.16280}
}
read the original abstract

We present a worldsheet action that reproduces a class of dual resonance amplitudes discussed in the literature, which generalize the Veneziano amplitude for open strings. Our proposal builds on the chiral composite linear dilaton introduced recently. We further compute higher-point extensions and closed-string analogs, which exhibit partial crossing symmetry.

Figures

Figures reproduced from arXiv: 2511.16280 by the authors.

Figure 1
Figure 1. FIG. 1: Open strings between D0-branes placed along [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: To regularise CLD action, we excise semi-circular discs from the upper half plane around insertion points [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Branch points of the [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Branch points of the [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Branch points of the [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Branch points of the [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Branch points of the [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

65 extracted references · 43 linked inside Pith

  1. [1]

    (17) is nots, t-crossing symmetric in general be- cause of the extra factor (x−a +) q 2 (x−a −) q 2

  2. [2]

    Witten-type

    The parametersbandδare both fixed byqand are not independent in (17): δ= q 2 , b= 1−q .(18) The first issue can be resolved by tuningw a’s. A par- ticularly convenient choice isw 1 =w 3, under whicha ± satisfya + = 1−a − and are given by a± = 1 2 ± i 2 p r(2 +r),(19) withr=w 2/w1. With this choice, the parameterλin (1) is given by λ= 1 (1 +r) 2 .(20) Gene...

  3. [3]

    Veneziano, Nuovo Cim

    G. Veneziano, Nuovo Cim. A57, 190 (1968)

    1968

  4. [4]

    Mandelstam, Phys

    S. Mandelstam, Phys. Rev. Lett.21, 1724 (1968)

    1968

  5. [5]

    Bardakci and H

    K. Bardakci and H. Ruegg, Phys. Lett. B28, 342 (1968)

    1968

  6. [6]

    D. D. Coon, Phys. Lett. B29, 669 (1969)

    1969

  7. [7]

    Matsuda,MODEL FOR INFINITE VENEZIANO SE- RIES, Report UCRL-19266 (Lawrence Radiation Labora- 5 tory, University of California, Berkeley, California, 1969)

    S. Matsuda,MODEL FOR INFINITE VENEZIANO SE- RIES, Report UCRL-19266 (Lawrence Radiation Labora- 5 tory, University of California, Berkeley, California, 1969)

    1969

  8. [8]

    Chan and S

    H.-M. Chan and S. T. Tsou, Phys. Lett. B28, 485 (1969)

    1969

Show all 65 references
  1. [9]

    M. A. Virasoro, Phys. Rev.177, 2309 (1969)

    1969

  2. [10]

    D. J. Gross, Nucl. Phys. B13, 467 (1969)

    1969

  3. [11]

    J. A. Shapiro, Phys. Lett. B33, 361 (1970)

    1970

  4. [12]

    L. J. Romans, Conf. Proc. C8903131, 57 (1989)

    1989

  5. [13]

    D. B. Fairlie and J. Nuyts, Nucl. Phys. B433, 26 (1995), arXiv:hep-th/9406043

    1995 arXiv

  6. [14]

    Huang, J.-Y

    Y.-t. Huang, J.-Y. Liu, L. Rodina, and Y. Wang, JHEP 04, 195 (2021), arXiv:2008.02293 [hep-th]

    2021 arXiv

  7. [15]

    Ridkokasha, Mod

    I. Ridkokasha, Mod. Phys. Lett. A36, 2150031 (2021), arXiv:2008.11418 [hep-ph]

    2021 arXiv

  8. [16]

    Maity, JHEP04, 064 (2022), arXiv:2110.01578 [hep- th]

    P. Maity, JHEP04, 064 (2022), arXiv:2110.01578 [hep- th]

    2022 arXiv

  9. [17]

    Arkani-Hamed, L

    N. Arkani-Hamed, L. Eberhardt, Y.-t. Huang, and S. Mizera, JHEP02, 197 (2022), arXiv:2201.11575 [hep- th]

    2022 arXiv

  10. [18]

    Figueroa and P

    F. Figueroa and P. Tourkine, Phys. Rev. Lett.129, 121602 (2022), arXiv:2201.12331 [hep-th]

    2022 arXiv

  11. [19]

    Huang and G

    Y.-t. Huang and G. N. Remmen, Phys. Rev. D106, L021902 (2022), arXiv:2203.00696 [hep-th]

    2022 arXiv

  12. [20]

    Geiser and L

    N. Geiser and L. W. Lindwasser, JHEP12, 112 (2022), arXiv:2207.08855 [hep-th]

    2022 arXiv

  13. [21]

    Chakravarty, P

    J. Chakravarty, P. Maity, and A. Mishra, JHEP10, 043 (2022), arXiv:2208.02735 [hep-th]

    2022 arXiv

  14. [22]

    Geiser and L

    N. Geiser and L. W. Lindwasser, JHEP04, 031 (2023), arXiv:2210.14920 [hep-th]

    2023 arXiv

  15. [23]

    Bhardwaj, S

    R. Bhardwaj, S. De, M. Spradlin, and A. Volovich, JHEP 08, 082 (2023), arXiv:2212.00764 [hep-th]

    2023 arXiv

  16. [24]

    A. S.-K. Chen, H. Elvang, and A. Herderschee, Phys. Rev. Lett.133, 091601 (2024), arXiv:2212.13998 [hep- th]

    2024 arXiv

  17. [25]

    Cheung and G

    C. Cheung and G. N. Remmen, Phys. Rev. D108, 026011 (2023), arXiv:2302.12263 [hep-th]

    2023 arXiv

  18. [26]

    A. S.-K. Chen, H. Elvang, and A. Herderschee, Phys. Rev. Lett.131, 031602 (2023), arXiv:2302.04895 [hep- th]

    2023 arXiv

  19. [27]

    C. B. Jepsen, JHEP06, 114 (2023), arXiv:2303.02149 [hep-th]

    2023 arXiv

  20. [28]

    Cheung and G

    C. Cheung and G. N. Remmen, Phys. Rev. D108, 086009 (2023), arXiv:2308.03833 [hep-th]

    2023 arXiv

  21. [29]

    Chiang, Y.-t

    L.-Y. Chiang, Y.-t. Huang, and H.-C. Weng, JHEP05, 289 (2024), arXiv:2310.10710 [hep-th]

    2024 arXiv

  22. [30]

    H¨ aring and A

    K. H¨ aring and A. Zhiboedov, JHEP10, 075 (2024), arXiv:2311.13631 [hep-th]

    2024 arXiv

  23. [31]

    Bhardwaj and S

    R. Bhardwaj and S. De, (2023), arXiv:2309.07214 [hep- th]

    2023 arXiv

  24. [32]

    K. C. Rigatos, Phys. Rev. D109, 086008 (2024), arXiv:2310.12207 [hep-th]

    2024 arXiv

  25. [33]

    Berman, H

    J. Berman, H. Elvang, and A. Herderschee, JHEP03, 021 (2024), arXiv:2310.10729 [hep-th]

    2024 arXiv

  26. [34]

    Geiser, JHEP10, 010 (2024), arXiv:2311.04130 [hep- th]

    N. Geiser, JHEP10, 010 (2024), arXiv:2311.04130 [hep- th]

    2024 arXiv

  27. [35]

    Eckner, F

    C. Eckner, F. Figueroa, and P. Tourkine, Phys. Rev. D 111, 126005 (2025), arXiv:2401.08736 [hep-th]

    2025 arXiv

  28. [36]

    K. C. Rigatos and B. Wang, Phys. Rev. D110, 126024 (2024), arXiv:2401.13031 [hep-th]

    2024 arXiv

  29. [37]

    Eckner, F

    C. Eckner, F. Figueroa, and P. Tourkine, JHEP02, 103 (2025), arXiv:2405.21057 [hep-th]

    2025 arXiv

  30. [38]

    Wang, JHEP04, 143 (2024), arXiv:2403.00906 [hep- th]

    B. Wang, JHEP04, 143 (2024), arXiv:2403.00906 [hep- th]

    2024 arXiv

  31. [39]

    Berman and H

    J. Berman and H. Elvang, JHEP09, 076 (2024), arXiv:2406.03543 [hep-th]

    2024 arXiv

  32. [40]

    Arkani-Hamed, C

    N. Arkani-Hamed, C. Figueiredo, and G. N. Remmen, JHEP04, 039 (2025), arXiv:2412.20639 [hep-th]

    2025 arXiv

  33. [41]

    Bhardwaj, M

    R. Bhardwaj, M. Spradlin, A. Volovich, and H.-C. Weng, Phys. Rev. D110, 106016 (2024), arXiv:2406.04410 [hep- th]

    2024 arXiv

  34. [42]

    Mansfield and M

    G. Mansfield and M. Spradlin, JHEP02, 145 (2025), arXiv:2409.09561 [hep-th]

    2025 arXiv

  35. [43]

    Gadde and S

    A. Gadde and S. Jain, (2025), arXiv:2502.18033 [hep-th]

    2025 arXiv

  36. [44]

    Caron-Huot, Z

    S. Caron-Huot, Z. Komargodski, A. Sever, and A. Zhi- boedov, JHEP10, 026 (2017), arXiv:1607.04253 [hep-th]

    2017 arXiv

  37. [45]

    Cheung and G

    C. Cheung and G. N. Remmen, JHEP01, 122 (2023), arXiv:2210.12163 [hep-th]

    2023 arXiv

  38. [46]

    Arkani-Hamed, C

    N. Arkani-Hamed, C. Cheung, C. Figueiredo, and G. N. Remmen, Phys. Rev. Lett.132, 091601 (2024), arXiv:2312.07652 [hep-th]

    2024 arXiv

  39. [47]

    Albert, W

    J. Albert, W. Knop, and L. Rastelli, JHEP02, 157 (2025), arXiv:2406.12959 [hep-th]

    2025 arXiv

  40. [48]

    Cheung, A

    C. Cheung, A. Hillman, and G. N. Remmen, Phys. Rev. Lett.133, 251601 (2024), arXiv:2406.02665 [hep-th]

    2024 arXiv

  41. [49]

    Cheung, A

    C. Cheung, A. Hillman, and G. N. Remmen, Phys. Rev. D111, 086034 (2025), arXiv:2408.03362 [hep-th]

    2025 arXiv

  42. [50]

    Cheung, G

    C. Cheung, G. N. Remmen, F. Sciotti, and M. Tarquini, (2025), arXiv:2508.09246 [hep-th]

    2025 arXiv

  43. [51]

    Albert and L

    J. Albert and L. Rastelli, JHEP08, 151 (2022), arXiv:2203.11950 [hep-th]

    2022 arXiv

  44. [52]

    Albert and L

    J. Albert and L. Rastelli, JHEP09, 039 (2024), arXiv:2307.01246 [hep-th]

    2024 arXiv

  45. [53]

    Fernandez, A

    C. Fernandez, A. Pomarol, F. Riva, and F. Sciotti, JHEP 06, 094 (2023), arXiv:2211.12488 [hep-th]

    2023 arXiv

  46. [54]

    Albert, J

    J. Albert, J. Henriksson, L. Rastelli, and A. Vichi, JHEP 09, 172 (2024), arXiv:2312.15013 [hep-th]

    2024 arXiv

  47. [55]

    We also introduced a new parameterλfollowing [28]

    We set the parameterain [2] to 1 by rescalingsandt. We also introduced a new parameterλfollowing [28]

  48. [56]

    Komatsu and P

    S. Komatsu and P. Maity, (2025), arXiv:2506.21663 [hep- th]

    2025

  49. [57]

    Throughout this letter, we setα ′ = 1

  50. [58]

    Komatsu and P

    S. Komatsu and P. Maity, to appear (2025)

    2025

  51. [59]

    Mandelstam, inWorkshop on Unified String Theories (1985)

    S. Mandelstam, inWorkshop on Unified String Theories (1985)

    1985

  52. [60]

    Kawai, D

    H. Kawai, D. C. Lewellen, and S. H. H. Tye, Nucl. Phys. B269, 1 (1986)

    1986

  53. [61]

    Witten, Nucl

    E. Witten, Nucl. Phys. B268, 253 (1986)

    1986

  54. [62]

    Rastelli and B

    L. Rastelli and B. Zwiebach, JHEP01, 018 (2008), arXiv:0708.2591 [hep-th]. 6 Supplemental Material Appendix A: Conformal invariance of our worldsheet action In this appendix, we find how our CLD worldsheet action with the boundary term (4) transforms under the Weyl transformat...

    2008 arXiv

  55. [63]

    In what follows, we will work in a parameter regime of{w k}n k=1 in which the interaction pointsZ I , I= 1,· · ·, n−2 are real

    Derivation of the Mandelstam formula. In what follows, we will work in a parameter regime of{w k}n k=1 in which the interaction pointsZ I , I= 1,· · ·, n−2 are real. The final result thus obtained is analytic in{w k}, and therefore can be analytically continued to arbitrary pa...

  56. [64]

    Conformal property We now study howe −Γren[ρ] transforms under the PSL(2,R) transformationz→(az+b)/(cz+d) withad−bc= 1. For this purpose, it is useful to recall that (x 1 −x 2) transforms under the PSL(2,R),x 1,2 →(ax 1,2 +b)/(cx 1,2 +d), as (x1 −x 2)→ x1 −x 2 (cx1 +d)(cx 2 +d...

  57. [65]

    7: Branch points of theξ-integrand for 1< ζ <∞and our choice of theξ-contour where the explicit forms ofI ζ andI ξ in each integration domain are given in Eqs

    F ull answer The resulting expression for the four-point closed-string amplitude (33) takes the form Aclosed 4 / ˜N closed 4 =i h I (0,a+) ζ I (0,a+) ξ +I (a+,a−) ζ I (a+,a−) ξ +I (a−,1) ζ I (a−,1) ξ i (C18) 15 iδ (a+ +iδ) (a− +iδ) (1 +iδ) ⌊ξ FIG. 7: Branch points of theξ-inte...

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.