Scalar subleading soft theorems from an infinite tower of charges

1 - Well organized, easy to follow
2 - Solid computations
3 - Fills a gap on the topic of scalar soft theorems
4 - Opens a new pathway in an existing research direction, with clear potential for follow-up work

1 - Validity of the assumptions and applicability of the results is not discussed
2 - Ignores important recent developments regarding logarithms and infrared divergences, in relation with long range interactions and tail effects

The submitted paper presents a study of scalar soft theorems in relation with conserved asymptotic charges. On the one hand, they use standard methods based on Feynman diagrams to establish an infinite tower of soft scalar theorems in a Yukawa theory featuring one massless scalar field and one massive scalar field. Considering a generic scattering amplitude where one of the scatterer is a massless scalar particle, and expanding the amplitude with respect to the (small) energy of this particle, the tower of soft theorems constrain the form of this expansion at every order. The authors then provide an alternative derivation of these soft theorems, based on conservation of asymptotic charges across spatial infinity. I wish to congratulate the authors, as the paper is clearly written and the results are novel. The computations are solid and well organized. Overall it constitutes a good addition to the literature on the correspondence between soft theorems and asymptotic charges, a tradition initiated by Strominger more than ten years ago. Some aspects of the paper deserve clarifications, which would likely make it suitable for publication in SciPost.

The main clarifications needed concern the assumptions and applicability of their results. On the one hand, their derivation of soft theorems in Section 2 assumes that amplitudes admit Laurent series representations with respect to the energy of the soft scalar particle. This is a strategy borrowed from [34] in the context of gluon and graviton amplitudes, by now known to be unwarranted. Indeed, the presence of logarithms of the energy has been established in [47]. This restricts the validity of [34] to tree-level diagrams, or equivalently, to leading order in the interaction coupling. While [47] concerns electromagnetism and gravity, the findings in [51] show that logarithms also appear in massless scalar theories, as expected in any theory with long-range interaction. This likely limitation finds its counterpart in the alternative derivation proposed in Sections 3 and 4. To start with, the computations in Section 3 are set up as if the theory was that of a massless scalar field sourced by some external current, while the equations motion of the Yukawa theory (3.48) are nonlinear coupled PDEs. In other words, the current J in (3.1) is a functional of the massless scalar through its backreaction on the massive scalar. This point is not at all discussed in the paper. I believe that the computations presented in Section 3 are in fact only valid to leading order in the interaction coupling, which amounts to the system described in (3.49)-(3.50). Beyond this leading order, we can expect that several assumptions made in Section 3 are invalid. For instance, log(u) terms appear in the expansion of the massless scalar field near future null infinity [51], invalidating (3.9). This in turn implies the existence of log(r) terms when expanding near past null infinity [37]. These logarithms would yield divergent charge expressions, a manifestation of infrared divergences in relation to tail effects. In addition, we can expect log(tau) terms in (3.34). All these aspects have been discussed thoroughly in the context of QED in [22,31]. (Note that [22] presented incorrect antipodal matching relations and a flawed derivation of the logarithmic soft photon theorem, which where first corrected in JHEP 10 (2020) 110 and JHEP 02 (2021) 82.)

While the remarks above suggest that the results of the paper only apply to tree-level amplitudes, they also suggest that they apply in any theory where tree-level is classically encoded by the system of equations (3.49)-(3.50). Indeed, nowhere in the computations does the form of the backreaction enter (the RHS of the second equation in (3.48)), explaining the simplicity of the results.

In addition, I believe that the authors have missed some modern references on the subject which, although they are set up for gravity, are precursor in a number of aspects relevant to this work. Together with [22,23] should be cited JHEP 09 (2022) 063 and JHEP 02 (2024) 120. The analysis of equations of motion near spatial infinity and their resolution in terms of Legendre functions, in relation with charge conservation across spatial infinity, have appeared in Class. Quantum Grav. 35 (2018) 074003 and SciPost Phys. 14 (2023) 014. In the latter, nonlinearities have been taken into account, which may inform the authors of a strategy to extend their results beyond tree-level (as an alternative to the methodology of [31]).

1 - I suggest that the authors rewrite the paper in a way that makes this restricted applicability transparent, or that they provide detailed proof that their results actually hold more generally, addressing the concerns stated above.

2 - Add references and discuss their relevance

Ask for major revision