Learning the Simplicity of Scattering Amplitudes

We thank the referee for their time and valuable insight. Please find below a detail response to each of their comments:

1) We include the references mentioned by the referee in the introduction, along with the related work of arXiv:1912.11055. We also include the more recent work of arXiv:2112.09145, which discusses the application of novel ML techniques for numerically evaluating scattering amplitudes.

2) We thank the referee for pointing this out and implement the suggested rewording. We also include the reference to the relevant arXiv:2203.04269.

3) We include arXiv:0905.1473 and arXiv:1408.2459 as references for the momentum twistor variables.

4) As pointed out by the referee, when doing a numerical comparison one could in principle have numerically similar but not analytically similar amplitudes. In our implementation this is mitigated by the fact that we numerically evaluate the amplitudes on two independent sets of phase space points (we add a comment on this point in the manuscript). We have tested the numerical checks on our validation and test sets and observed that all of the amplitudes were correctly matched following that procedure. Evidently, we could make the procedure even more robust by picking out additional independent phase space points. We also note that when using the iterative simplification algorithm the numerical checks are done at each simplification step (typically only involving expressions with roughly 20 numerator terms) so that the precision loss isn't as concerning. As mentioned by the referee however, when doing a check on a full amplitude with 100's or 1000's of terms this issue would become important and we would need additional checks - for instance using the lips package, as indicated by the referee, which we will look to implement in a future version of the code.

5) We thank the referee for pointing us to relevant literature and we adapt our discussion regarding uniqueness in footnote 4. When generating $N$ solutions with beam search or nucleus sampling, the model can indeed generate different valid solutions. We choose to return the valid solution with the smallest number of distinct numerator terms. When different solutions have the same number of numerator terms (such as $-\frac{\langle 1 2 \rangle [34]^2}{\langle 3 4 \rangle [14][23]}$ and $\frac{\langle 1 2 \rangle^3}{\langle 1 4 \rangle \langle 2 3 \rangle \langle 3 4 \rangle}$) we return the solution generated by the model which is associated with the best score, that is, the smallest average value of the log probability $-\log(p(x_i))$ for each token $x_i$. We do not manually restrict the denominator factors or implement additional checks on those. The amplitude of Eq 4.8 has indeed been compactly rewritten, where the raw output form is

6) In our current implementation we have chosen to always feed input amplitudes by first putting them under a common denominator (following the $\textit{cancel}()$ routine in $\textit{sympy}$) and training has only been done on such amplitudes. One could naturally have trained on amplitudes in a partial fraction form but we have not explored that data representation. We note that this choice of representation is mainly for convenience purposes and that indeed other alternatives could be considered - there is no difference from an architecture point of view. It would also be interesting to learn a similarity metric that looks directly at factors coming from a partial fraction decomposition, as opposed to numerator terms only. But, from a practical point of view (and due to our implementation choices), generating relevant training data was more straightforward when focusing only on numerator terms.

7) To go beyond our current setup and include other variables, $s_{ijk}, \langle 1| 2 + 5|1 ],\cdots$, we do not need to drastically modify the model's architecture. At this level, the only change that would need to be implemented is with respect to the size of the total word vocabulary (which in turn impacts the dimension of the initial embedding layer), that is, the total number of symbolic tokens required to represent an amplitude. However, one would need to generate appropriate training data using these new variables. In particular, one has to define a procedure for sampling over a set of relevant simple amplitudes using these variables and coding the desired shuffling moves. The rest of the pipeline (training and iterative simplification) can proceed almost identically. We add a sentence to specify this in the conclusion.

8) Referees 2 and 3 both raise an important question about the role of constants. Indeed, the application of our method to amplitudes with different numerical constants poses a challenge. In the footnote 14 we comment on our algorithmic choice for dealing with this complication, which involves blinding the constants and performing the simplification which reduces the norm of the coefficients as much as possible. In that way we are still able to reduce amplitudes with non trivial coefficients - though we have not performed a detailed analysis. Also, as the referee suggests, one could in principle add another check on top, for instance prioritizing terms whose coefficients are multiple of each other. Alternatively, one can also consider training on a dataset where non trivial coefficients are considered - though generalizing a transformer for dealing with arbitrary numbers is not easy (see the robustness issue highlighted in arXiv:2109.13986 when dealing with integers sparsely covered in the training set) and so we did not consider this path. In cases where distinct coefficients are informative, say coming from expanding $2([12] [34]+ [14][23])^4$ , one could also consider attempting to first recover the factored form (focusing only on high similarity terms for instance) and then simplifying the factored expression , blinding the overall power or numerical factors. We added a comment on these alternative approaches in the footnote 14, but leave their implementation for future work.

We thank the referee for their time and valuable insight. Please find below a detail response to each of their comments:

1) The referee raises a valid point about the potential bias introduced by the forwards vs backward generation. As pointed out in arXiv:1912.01412 (or the more recent arXiv:2410.08304), one can bias the model when the training data is generated by sampling from the target (backward generation) vs from the input (forward generation). In our implementation we have a backward generation since we start from the target amplitude and generate complex expressions from there. Any bias introduced will be on the form of the complicated amplitudes (for instance there could be complicated amplitudes that are not captured by our data generation procedure). However, our final test cases in Section 4.4 are on amplitudes coming from Feynman diagrams directly, so more closely matching real world applications and not directly generated as part of our training data. Since our models perform reasonably well on those we believe that our backward data generation procedure (although not perfect !) should generalize reasonably well to actual applications. Clearly, when one knows the form of the complicated amplitude (say if focusing on a particular theory/problem) it would be helpful to have training data generated in a forward fashion - starting from the complicated amplitudes and reducing them with some alternative software to get the simplified form. This comes at the cost of 1) having to sample over some distribution of complicated amplitudes that is to be determined and 2) the requirement of having some alternate software or tool to perform the simplification. We add comments to that effect in sections 2.2 and 2.3. Regarding the structure of the very simple forms we have considered in our training data (at most 3 terms), we don't believe this to be a strong limiting factor when doing an iterative simplification. For example, if we consider an input amplitude with 10 terms that simplifies to a "simpler" form with 9 terms, then the hope for our simplification algorithm is to identify different groups that can reduce independently. For instance, 3 terms of the original 10 terms amplitude might combine to give 2 terms via a Schouten identity. In that case our models would only predict the 3 $\rightarrow$ 2 simplification, which is captured by our training data (and overall we have 10 $\rightarrow$ 9 terms). However, a failure mode could arise if we need an intermediate increase in complexity to see a simplification (for example, from combining multiple identities to get 10 $\rightarrow$ 9 terms). We comment on this point further in the footnote 16 in the main text.

2) Referees 2 and 3 both raise an important question about the role of constants. Indeed, the application of our method to amplitudes with different numerical constants poses a challenge. In the footnote 14 we comment on our algorithmic choice for dealing with this complication, which involves blinding the constants and performing the simplification which reduces the norm of the coefficients as much as possible. In that way we are still able to reduce amplitudes with non trivial coefficients - though we have not performed a detailed analysis. Also, as the referee suggests, one could in principle add another check on top, for instance prioritizing terms whose coefficients are multiple of each other. Alternatively, one can also consider training on a dataset where non trivial coefficients are considered - though generalizing a transformer for dealing with arbitrary numbers is not easy (see the robustness issue highlighted in arXiv:2109.13986 when dealing with integers sparsely covered in the training set) and so we did not consider this path. In cases where distinct coefficients are informative, say coming from expanding $2([12] [34]+ [14][23])^4$ , one could also consider attempting to first recover the factored form (focusing only on high similarity terms for instance) and then simplifying the factored expression , blinding the overall power or numerical factors. We added a comment on these alternative approaches in the footnote 14, but leave their implementation for future work.

3) We refer to our response in point 1) regarding the restriction $N_{\text{terms}}<3$. In short, we expect our approach to be reasonable when intermediate simplification steps (a single identity application for instance) can have a simple target state, which is a subset of the entire amplitude.

4) The traditional simplification of spinor-helicity amplitudes does not have a canonical approach. One of the routes that is considered in practice to this day is simple guesswork. Akin to how Parke-Taylor obtained their simple formula for 6-pt gluon scattering, much simplifications still require some educated guess and a numerical check. Another approach is the explicit algebraic simplification through the packages referred to in [13,14]. For instance in SpinorHelicity4D one can make use of the SchoutenSimplify routine which uses Mathematica's internal FullSimplify routine. The simplification power there is tied to FullSimplify's performance and does not always guarantee returning the simplest possible form. The final route to traditional simplification is to build a smart Ansatz for the answer (satisfying little group and mass dimension scalings) and to fix it via numerical evaluation, for instance leveraging singular kinematic limits (see arXiv:2010.14525 or arXiv:1904.04067). This method yields drastic simplifications but will also not guarantee the "simplest" possible form (according to our own working defintion of what is "simple").

5) Following referee's 3 comment we decide to broadly state in the introduction the fact that we do not have access to a clear algorithmic way for performing a standard analytical simplification of spinor-helicity amplitudes. We delay the technical details to section 2.4 as they might obscure the reading of the introduction.

6) Following the referee's recommendation, we move the last few sentences discussing the applications of contrastive learning in physical contexts closer to the start of section 4.1.

7) We have updated the figure's caption to more clearly reflect the difference between the circle and triangle markers.

8) We refer to our answer for point 4)

9) We add a few sentences in the conclusion to emphasize the point raised by the referee, which is that model hallucinations can be systematically avoided in a mathematical context, where numerical evaluations and cross-checks are available.

We thank the referee for their time and valuable insight. Please find below a detail response to each of their comments:

1) We add information in footnote 1 to refer to the presence of an interactive demonstration and highlight the possibility of a simple local download via GitHub. The interactive demonstration linked in the abstract is hosted online via Streamlit, making it much slower to use in practice, so we prefer to direct the reader to the more stable version hosted on Github. This app is able to showcase the abilities of both the one-shot simplifier and the full iterative simplification pipeline.

2) Following the referee's recommendation, we emphasize in the introduction the fact that we do not have access to a canonical algorithm for performing a standard analytical simplification of spinor-helicity amplitudes.

3) Referees 2 and 3 both correctly point out that accesses to numerical evaluations are crucial to avoid model hallucinations. We highlight this point in the conclusion and also bring it up in the introduction.

4) The traditional simplification of spinor-helicity amplitudes does not have a canonical approach. One of the routes that is considered in practice to this day is simple guesswork. Akin to how Parke-Taylor obtained their simple formula for 6-pt gluon scattering, much simplifications still require some educated guess and a numerical check. Another approach is the explicit algebraic simplification through the packages referred to in [13,14]. For instance in SpinorHelicity4D one can make use of the SchoutenSimplify routine which uses Mathematica's internal FullSimplify routine. The simplification power there is tied to FullSimplify's performance and does not always guarantee returning the simplest possible form. The final route to traditional simplification is to build a smart Ansatz for the answer (satisfying little group and mass dimension scalings) and to fix it via numerical evaluation, for instance leveraging singular kinematic limits (see arXiv:2010.14525 or arXiv:1904.04067). This method yields drastic simplifications but will also not guarantee the "simplest" possible form (according to our own working defintion of what is "simple").

5) The training of the simplifier models on one A100 GPU took around 45h, 65h and 80h of running time for 4,5,6 points respectively. The contrastive models are much faster to train, taking around 2h each. We include some of these time metrics in the manuscript.

6) 4-pt amplitudes are easier to learn as they are typically much shorter in length. As we describe in Appendix B, the average expression that the networks will see contain much fewer numerator terms than at 5 or 6pts. The apparent difference between the 5pt and 6pt is to be taken carefully. Indeed, it seems like the models perform better on 6pts with 3 scrambles than on 5 pts with 3 scrambles. However, as we described in Appendix B, our restriction to amplitudes with less than 1k tokens in the training set is more restrictive for 6pt. Indeed, we discard 2% of all 5-pts generated against 8% of all 6-pts. This means that we do not include potentially harder amplitudes at 6-pt in Fig.3, which can explain the apparent gap in performance. We note that those amplitudes can be appropriately simplified in the full iterative setup since they are typically associated with either 3 scrambling moves or more than 2 numerator terms.

7) For completeness, we now report the standard deviations for the cosine similarities in Figure 8. Since one of the identities that we have considered (momentum squared) can be recovered by applying different successions of momentum conservation and Schouten we see an uncertainty band of $\pm 0.3$ around the average cosine similarities for more than 3 identities away (irrespective of whether we are considering 4,5 or 6-pt amplitudes), which we believe is attributed to this relationship. For a single identity away, in the masked comparison, we instead have a tighter spread of under $\pm 0.1$ around the cosine similarities displayed on the plot, while for the full comparison it is around $\pm 0.15$. We decide to comment on these points explicitly in Appendix F (and guide the reader to the Appendix in the main text) as they are not crucial to the main story - which is that we can correlate high cosine similarity values with terms that are $<3$ identities away.

8) Indeed, this is a typo, thank you for pointing it out !

9) In general, we expect that the overall framework that we have developed can be extended to other non-trivial cases. For instance, if we focus on the spinor-helicity formalism for massive particles (say in arXiv:1709.04891 or arXiv:2304.01589), then we can anticipate that we would require additional "words" (symbolic tokens) for describing the amplitudes. Changing an amplitude's representation in this way does not critically impact the entire model's architecture (it just requires adapting the initial embedding layer), but imposes the need for a dedicated data generation procedure. That is, when tackling a new task, one needs to create training pairs $( \overline{\mathcal{M}} ,\mathcal{M})$ by defining relevant identities (or scrambling steps) that can be applied to $\overline{\mathcal{M}}$ to scramble it. Provided these training pairs are available, the rest of the training pipeline we have presented can be applied without further modifications.

10) Thank you for spotting these typos !