Scalar subleading soft theorems from an infinite tower of charges

Dear Editor,

We thank the referee for highlighting that the results in our manuscript are novel and that our computations are solid and well-organized. We are fully aligned with the referee’s concerns, and although we mentioned in some parts of the article that this is a leading-order contribution, we recognize that we had not been sufficiently explicit and clear about this point.

Therefore, following the referee’s suggestions, we have introduced several clarifications throughout the manuscript to emphasize that our analysis is restricted to the tree-level regime and that logarithmic corrections are absent in this approximation. In addition, we have expanded the Discussion to comment on the possible origin of retarded-time logarithms as found in [55], and we have added the missing references suggested by the referee.

We believe these changes fully address the referee’s concerns by making the assumptions of our work explicit, clarifying the treatment of the source term, and discussing the role of logarithmic corrections. We hope that with these clarifications, the manuscript will now be suitable for publication in SciPost Physics.
Sincerely,

Matías Briceño, Hernán A. González, Alfredo Pérez.

We summarize the main changes:

1. Abstract: The final sentence we have changed the phrase: “valid at leading order” to “valid at tree-level”.

2. Introduction: at the end of Section 1, we now explicitly state:
“We focus on the tree-level regime, corresponding to leading order in the coupling. In this approximation, the asymptotic expansion of the scalar field does not develop logarithmic terms, so the tower of charges remains finite and well defined. Beyond tree level, logarithmic corrections are expected to appear and would modify the structure of the charges, a point to which we return in the Discussion.”

We also added references [23,24,41,42] to connect with related analyses in the gravitational context, as suggested by the referee.

1. Section 2 (between Eqs. (2.2) and (2.3)): we added a paragraph clarifying the assumption of a Laurent expansion:
“At this point, we assume that the amplitude admits a Laurent expansion in the soft energy $\omega$. This is the standard procedure used in the derivation of soft theorems at tree level, where amplitudes are meromorphic functions of external momenta. Beyond tree level, loop corrections are known to generate logarithmic terms in $\omega$ (see e.g. [43] , which would obstruct such an expansion. Since our analysis is restricted to tree level, the Laurent expansion adopted here is consistent and sufficient for deriving the tower of subleading soft relations.”

2. Section 3.1 (after Eq. (3.1)): we added a clarification about the external source:
“We model the effect of the interaction by introducing a generic external source $J$. This simplification is sufficient to capture the tree-level dynamics relevant for the derivation of the subleading soft charges. Indeed, in the Yukawa theory considered here, the current $J$ is generated by the massive field, but to leading order in the coupling, its backreaction does not alter the free propagation of the massless scalar near null infinity. In this approximation, the field equations reduce to those of a free scalar sourced linearly, which is precisely the regime where our construction of conserved charges applies. Nonlinear effects, such as those responsible for logarithmic terms in the asymptotic expansion, only appear beyond tree level and therefore lie outside the scope of the present work.”

3. Discussion (Section 5): At the end of the second paragraph on page 21, we added: 
 “Specifically, in [55] it has been shown that the logarithmic dependence on the retarded time shows up in the waveform of a massless scalar due to its interaction with classical point particles. One would therefore expect that relaxing condition (3.10) in our construction would naturally allow such terms to appear in the derivation of the charges, altering their structure compared to the tree-level case.”