SciPost Submission Page
Superstring amplitudes meet surfaceology
by Qu Cao, Jin Dong, Song He, Fan Zhu
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Qu Cao |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2504.21676v1 (pdf) |
| Date submitted: | July 7, 2025, 7:11 a.m. |
| Submitted by: | Qu Cao |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We reformulate tree-level amplitudes in open superstring theory (type-I) in terms of stringy Tr$(\phi^3)$ amplitudes with various kinematical shifts in the "curve-integral" formulation: while the bosonic-string amplitude with $n$ pairs of "scaffolding" scalars comes from a particularly simple shift of the Tr$(\phi^3)$ one (corresponding to $n$ length-$2$ cycles), the analogous superstring amplitude requires "correction" terms given by bosonic-string amplitudes with longer, even-length "cycles", which are also Tr$(\phi^3)$ ones at shifted kinematics dictated by the cycles; in total it is expressed as a sum of $(2n{-}3)!!$ shifted amplitudes originated from the expansion of a reduced Pfaffian. Upon taking $n$ scaffolding residues, this leads to a new formula of the $n$-gluon superstring amplitude, which is manifestly symmetric in $n{-}1$ legs, as a gauge-invariant combination of mixed bosonic string amplitudes with gluons and scalars, which come from length-$2$ cycles and longer ones respectively (the total sum is associated with the expansion a $n\times n$ symmetrical determinant); the corresponding prefactors are nested commutators of $2n$-gon kinematical variables, which nicely become traces of field-strengths for those legs corresponding to scalars in the mixed amplitudes. These interesting linear combinations of bosonic string amplitudes must guarantee the cancellation of tachyon poles and $F^3$ vertices ${\it etc.}$, and they give new relations between the superstring amplitude and its bosonic-string building blocks to all orders in the $\alpha'$ expansion (the first order gives a new formula for gluon amplitudes with a single $F^3$ insertion in terms of Yang-Mills-scalar amplitudes). We provide both the worldsheet and "curve-integral" derivations, and discuss applications to heterotic and type II cases.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2025-9-23 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2504.21676v1, delivered 2025-09-23, doi: 10.21468/SciPost.Report.11990
Report
In this paper, the authors present a new formula for superstring amplitudes with $n$-gluons. The formula is based on the ``surface'' description of scattering amplitudes that was recently introduced. Via the surface description, bosonic string amplitudes were constructed prior to this paper via a shift of Tr $\phi^3$ amplitudes. This paper extends this to a new formulation of $n$-gluon superstring amplitudes and their new formula has some very nice features:
- It is manifestly gauge-invariant.
- It is manifestly permutation invariant in $n{-}1$ labels.
- It is a sum over bosonic string amplitudes, giving a new relation between super and bosonic string amplitudes.
The paper is well-written and provides a lot of examples that make it easy to understand notation in formulae.
In addition to the points raised in the other report, I was wondering if the authors could add a few comments on the following:
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Under equation (2.33) the authors mention that there is a dimensional reduction that would give amplitudes of a ``superstring-induced'' NLSM. Since the supersymmetrizations of NLSMs are quite tightly constrained, could the authors explain what their proposed model is?
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The relation between these double-ordered bosonic amplitudes and single ordered superstring amplitudes is interesting. Can these new relations between bosonic and superstring amplitudes be seen as a result of monodromy relations or as coming from a KLT or KLT-like relation?
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The discussion in the first paragraph on page 10 about ghosts is a bit confusingly worded. From the text, it seems that if one fixes the last two legs to be the two (-1) ghost charge operators, then leg $n$ is clearly special. Could the authors clarify why leg $n$ is special if one is interested in the second case i.e. $(2i,2j)$ which on the scaffolding residue gives $n$ gluons?
These minor comments aside, I believe this paper is a great addition to the surfaceology literature and recommend it for publication on SciPost.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Report #1 by Anonymous (Referee 1) on 2025-8-26 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2504.21676v1, delivered 2025-08-26, doi: 10.21468/SciPost.Report.11808
Report
This paper derives a new formula for the tree-level gluon scattering amplitudes in type-I superstring theory. The final expression is presented in terms of mixed gluon–scalar amplitudes in bosonic string theory. The formula has several nice features, such as manifest symmetry and gauge invariance. It is obtained by exploiting recent advances in the “curve-integral” formulation of Tr\phi^3 amplitudes and their intriguing connections with gluon amplitudes.
The authors further study various non-trivial properties of the new formula, including the cancellation of the tachyon pole in bosonic string amplitudes and the vanishing of contributions from higher-derivative terms F^3; these are some basic properties of superstring amplitudes. Finally, they extend the discussion to closed superstring amplitudes by applying the double-copy construction.
Overall, I find the paper novel, interesting, and suitable for publication in SciPost. That said, there are some small typos and points where the presentation could be improved for clarity. My comments/questions are as follows:
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On page 1, first sentence: “... to all loops and to all orders in the ’t Hooft coupling...” is somewhat confusing, since loop expansion usually coincides with the expansion in the ’t Hooft coupling. Does “all loops” here instead refer to the genus expansion in color?
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Page 4: introducing the abbreviation “Yang-Mills (YM)” may not be necessary since it was introduced on page 1 already. Similar comments apply to " leading singularity (LS)" on page 13, the word "leading singularity" appears a few times before introducing "LS".
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Three lines after equation (2.2): the sentence containing “... with the usual.” seems incomplete and should be revised.
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After equation (2.2): the notation [I] is defined as a Parke-Taylor factor, but since this is used frequently later, it would be helpful to present it explicitly as an equation. Related to this, a different notation for Parke-Taylor factors was used in equation (A.2). It may be better to use a single notation.
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After equation (2.15): the phrase “A_8^{bos.} with a factor (s−1)(s−1)(s−1)...” might be better phrased as “with a factor of the form (s−1)(s−1)(s−1).” The same applies to the following sentence.
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After equation (2.20): “... selected from [n−1] (so leg n is fixed).”: It is not clear whether [n−1] has been defined. I assume it does not denote a Parke-Taylor factor, which I mentioned earlier.
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Equation (2.32): the period at the end should be a comma. Similar punctuation issues occur elsewhere and should be checked.
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Finally, it is known that in the \alpha'-expansion of superstring amplitudes, each order contains numbers of uniform transcendentality, whereas bosonic string amplitudes include numbers of lower transcendentality. Since the new formula expresses superstring amplitudes in terms of bosonic amplitudes, can this uniform transcendentality property be seen here? Presumably this is related to the cancellation of F^3 discussed in the paper?
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Thank you for the comments. Let me answer these comments. 1. For real Riemann surfaces, the all-loop expansion does not coincide with the expansion in the ’t Hooft coupling. We recommend the paper [arXiv:2309.15913] for details. 2. We have updated these abbreviations in the revised version. 3. This sentence has been revised accordingly. 4. The PT factor has been revised and is now presented explicitly in the form of an equation. 5. These sentences have been phrased accordingly. 6. The definition of $[n-1]$ has been revised. 7. The comma in the equation has been corrected. 8. As you mentioned, the bosonic string amplitudes include terms of lower transcendentality. In the literature, we have $A^{\text{super}} = A^{\text{YM}} \times F, \quad A^{\text{bos}} = B \times F$, where B is a function of $\alpha'$. In our new relation eq.(2.33), we find $A^{\text{YM}} = \sum_\rho D_\rho T_\rho B$. This means that all combinations of these B functions reproduce the Yang-Mills field-theory amplitudes, which indicates the uniform transcendentality. This result arises from the cancellation of the F^3 terms and tachyon poles.
Author: Qu Cao on 2025-09-30 [id 5873]
(in reply to Report 2 on 2025-09-23)Thank you for the comments. Let me answer these questions. 1. The NLSM model we suggest here is obtained from the superstring after a special dimensional reduction (e.g., imposing $\epsilon_i \cdot \epsilon_j = p_i \cdot p_j$ and $\epsilon_i \cdot k_j = 0$). We refer to this as the superstring-induced NLSM model, which, however, is well-defined only in the NS (bosonic) sector. For more details, we recommend [arXiv:2404.11648]. 2. As far as we know, we have no clue how this new relation could be derived from monodromy or KLT relations. 3. From the structure of vertex operators in different ghost pictures, if one chooses two adjacent vertex operators both in the (-1) picture, then through the OPE (or equivalently by taking the scaffolding residue), these two operators combine into a new vertex operator in the (-2) picture. In the second case, say for (2i,2j), the vertex operator at position 2i in the (-1) picture together with the vertex operator at position 2i-1 in the (0) picture combine into a vertex operator at position i in the (-1) picture. The same holds for (2j). Thus, in this case, the resulting integrand still involves two vertex operators in the (-1) picture, which is precisely the standard integrand discussed in the GSW textbook. In contrast, our new formula involves a single vertex operator in the (-2) picture, chosen at position n. We recommend the paper [arXiv:2506.15299] for the explict expression of the vertex operator in the (-2) picture.