One-Loop Soft Approximation in Scalar EFTs

• One-Loop Soft Approximation is defined as the study of soft momentum behavior in scalar EFTs at one-loop order, emphasizing the interplay between soft limits and IR-divergent loop integrals.
• It distinguishes derivatively-coupled theories, where soft limits commute with loop integrations, from scenarios with potential interactions that incur universal quantum corrections due to non-commutativity.
• The analysis provides explicit formulations and examples, offering a universal framework that links soft theorem modifications in scalar EFTs to similar phenomena in gauge theories.

The one-loop soft approximation refers to the leading and subleading behavior of scattering amplitudes in scalar effective field theories (EFTs) as the momentum $q$ of one external scalar particle becomes soft ($q\to 0$), specifically at one-loop order in perturbation theory. At tree level, soft theorems are governed by the geometry of field space and are universal for derivatively-coupled scalar EFTs. At one loop, however, the interplay between the soft limit and infrared (IR) divergences in loop integrals can lead to modifications—generically absent in derivatively coupled theories, but present and universal when potential (non-derivative) interactions exist. This subject organizes the quantum corrections to soft theorems and delineates their regime of validity and breaking in modern scalar EFTs.

1. Geometric Soft Theorem and Its Loop Extension

The tree-level soft theorem for scalar EFTs with multiple flavors and possible curved field space geometry asserts

$\lim_{q\rightarrow 0} A_{n+1} = \nabla_i A_n$

where $A_{n+1}$ is the $(n+1)$-point amplitude with a soft scalar of flavor $i$, $A_n$ is the $n$-point amplitude, and $\nabla_i$ is the covariant derivative on field space. This characterization is geometric and remains valid at tree level for any scalar EFT with derivative couplings. The principal question at one loop is under what conditions this geometric soft theorem remains unmodified. The analysis must account for possible IR divergences in loop integrals, and whether the process of taking the soft limit and performing the loop integral commutes.

2. Non-commutativity of the Soft Limit and Loop Integration

A central finding is that the soft limit $q\to 0$ does not necessarily commute with evaluation of IR-divergent loop integrals. Specifically, when loop integrals are IR divergent, evaluating the soft limit at the integrand level and then performing the integration differs from integrating first and then taking the limit. The difference is termed a "discontinuity," which manifests as universal logarithmic terms in four dimensions. For example, in a three-mass triangle diagram,

$$I^{3m}_{3}(k_1^2,k_2^2,k_3^2) \xrightarrow{k^2_3 \to 0} I^{2m}_{3}(k_1^2,k_2^2) - d_2(k_3^2; k_1^2, k_2^2) + d_2(k_3^2; k_2^2, k_1^2)$$

where $d_2$ quantifies the universal "discontinuity" arising from non-commutativity. Equations (3.2), (3.4), (3.5) in the source specify the analytic form of these terms for bubbles, triangles, and boxes.

3. Criteria for Unmodified One-Loop Soft Theorems

The geometric soft theorem does remain valid at one loop for any derivatively-coupled scalar EFT, i.e., when the scalar potential $V(\phi) = 0$ and all vertices contain derivatives of the fields. Under this condition, every diagram with a soft leg is suppressed by powers of $q$ sufficient to render all relevant loop integrals IR finite. In this scenario, the loop integral and soft limit commute, and

$$\lim_{q\to 0}A_{n+1,i_1\dots i_n i}^{(1)} = \nabla_{i} A_{n,i_1\dots i_n}^{(1)}$$

This property is conjectured to hold to all loop orders, as the field space geometry enforces sufficient powers of soft momenta on every vertex, preventing IR divergences from appearing as $q \to 0$.

4. Universal Quantum Corrections for Potential Interactions

When potential (non-derivative) interactions are present—i.e., for interactions such as $\phi^3$ or $\phi^4$—the soft theorem is modified at one loop by universal quantum corrections, even for massless theories. The key mechanism is that diagrams where the soft scalar attaches to a potential vertex may contain IR-divergent loop subgraphs (notably bubbles, triangles, and boxes with massless internal lines). The universal correction to the soft theorem is encoded in a factorized operator $K^{(1)}(q)$ acting on the tree-level $n$-point amplitude,

$$\lim_{q\to 0}A^{(1)}_{n+1} = K^{(1)}(q) A^{(0)}_n + \mathcal{O}\left(\frac{1}{p_a\cdot q}\right)$$

with $K^{(1)}(q)$ comprised of contributions scaling as $1/(p_a\cdot q)^2$ in the soft limit, constructed from explicit bubble, triangle, and box integrals. The universal structure is specified by the interaction vertices and is detailed in equations (5.4)–(5.9) of the source, distinguishing contributions from single hard lines and pairs of hard lines.

5. Explicit Formulation and Example Calculations

The explicit components of the correction term $K^{(1)}(q)$ are realized as follows for cubic interactions:

• Single-hard leg: $$\sum_{a} K^{(1)}_{a}(q) A^{(0)}_{n} = \sum_{a} I_{2}((p_a+q)^2) \frac{i V_{i_a i}^{~j_1} V_{j_1 }^{~j_2j_3} V_{j_2j_3}^{~j_a}}{2[2(p_a \cdot q)]^2} e^{q\cdot \partial_{p_a}}A^{(0)}_{n, ..., j_a, ...} + \cdots$$
• Double-hard leg: $$\sum_{a,b} K^{(1)}_{a,b}(q)A^{(0)}_{n} = \sum_{a<b} I^{(1)}_{\Box}i V_{i}^{~j_1 j_2} V_{i_a j_1}^{~j_a} V_{i_b j_2}^{~j_b}A^{(0)}_{n, ..., j_a, ..., j_b, ...}$$ where $I_{2}$, $I_{3}$, $I_{\Box}$ are the standard scalar integrals (bubble, triangle, box). All integrals and their soft limits are tabulated in Appendix A of the paper.

A concrete five-point amplitude example in a theory with both derivative and potential cubic interactions demonstrates that only diagrams with the soft leg attached to a cubic vertex in an IR-divergent loop generate the universal correction, in alignment with the general expressions given above.

6. Relation to Soft Theorems in Gauge Theories and Broader Implications

The universal, IR-induced quantum corrections described for scalar EFTs with potential interactions closely parallel one-loop soft theorem violations in non-abelian gauge theories, where non-commuting limits and IR divergences generate analogous universal factors. The precise analogy and mathematical similarity are highlighted in the original work, pointing to a wide universality class for such corrections in soft limits of quantum field theories.

The overall impact is that for purely derivatively-coupled scalar EFTs, soft theorems retain their geometric form unchanged at one loop and, conjecturally, to all loops. In contrast, inclusion of potential interactions activates a universal, model-independent structure of quantum corrections, whose magnitude and detailed form are set by the IR-divergent content of scalar loop integrals. This underpins the necessity of precise IR control in higher-loop calculations of soft-momentum limits for both phenomenological model building and precision quantum field theory.

7. Summary Table: Regimes of Scalar EFT One-Loop Soft Theorems

Scalar EFT Type	One-Loop Soft Limit Column	Quantum Correction Present?	Commutativity of Limits
Derivatively-coupled ($V(\phi)=0$)	$\lim_{q\to 0}A^{(1)}_{n+1} = \nabla_i A^{(1)}_n$	No	Yes (soft limit commutes)
Potential interactions ($V(\phi)\neq 0$)	$$\lim_{q\to 0}A^{(1)}_{n+1} = K^{(1)}(q)A_n^{(0)} + ...$$	Yes, universal	No (discontinuity appears)

The one-loop soft approximation in scalar EFTs is thus governed by the interplay between the coupling structure (derivative vs potential), the IR behavior of loop integrals, and the subtle commutation of limits. These principles are pivotal for the consistent application of the soft theorems in modern amplitude-based approaches to scalar quantum field theories (Cohen et al., 16 Apr 2025).