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REVIEW 2 major objections 4 minor 144 references

Soft scalar limits are exact field-space derivatives at all loops

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2026-08-01 09:30 UTC pith:PK6JMROS

load-bearing objection Strong new EFT framework and clean soft-theorem derivations, with an all-orders exactness claim that is plausible but depends on an unproven mode-completeness assumption—worth referee time. the 2 major comments →

arxiv 2607.27322 v1 pith:PK6JMROS submitted 2026-07-29 hep-th hep-ph

geoSCET: Soft Theorems from Power Counting

Timothy Cohen , Patrick Hager , Andreas Helset This is my paper
classification hep-th hep-ph
keywords geometric soft theoremsoft-collinear effective theoryfield-space geometryscalar effective field theorypower countingtree-level exactnessmultiple soft emissionsnon-linear sigma model
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper constructs an effective field theory, geoSCET, that separates the soft and collinear degrees of freedom of a general scalar field theory while keeping its field-space geometry manifest. Using power counting in this EFT, it rederives the geometric soft theorem — the statement that the soft limit of an amplitude equals a covariant derivative acting on the non-radiative amplitude — and extends it to multiple soft emissions and loop corrections. For theories without a potential, the paper claims exactness: every single and multiple soft theorem holds at tree level to all orders in perturbation theory and receives no loop corrections. For theories with a potential, the leading soft factor remains universal and, for massive soft scalars, factorization is robust against loops. The reason to care is that this identifies field-space diffeomorphism invariance, rather than an internal symmetry, as the origin of soft scalar universality.

Core claim

The paper's central claim is that the geometric soft theorem is not a tree-level accident but a consequence of the power-counting structure of the low-energy EFT. In geoSCET, soft emissions are governed by an emergent connection built from the field-space Christoffel symbols, and the soft limit is obtained by expanding the hard-scattering operator around a soft background field. For vanishing potential, normal coordinates eliminate three-point vertices; any loop then requires at least one extra interaction vertex suppressed by O(λ²), so no topology can modify the leading soft term. Hence the single-soft formula lim_{k_s→0} M_{N+1} = ∇_I M_N, together with the double-soft formula involving th

What carries the argument

The key object is geoSCET, built from the exponential-map decomposition ϕ = exp_φ[ξ], which makes the collinear fluctuation ξ a tangent vector and the soft mode φ a dynamical background. The central mechanism is the pullback covariant derivative D_µ = ∂_µ + Γ^I_JK ∂_µφ^K, which plays the role of a soft-covariant connection, and the parallel-transport Wilson line R(x,x_-) that moves collinear fields to a fixed line to make soft transformations homogeneous. The proof of tree-level exactness rests on power counting in normal coordinates: with no potential, three-point vertices vanish and every loop is suppressed by at least O(λ²), so no leading-power correction can arise.

Load-bearing premise

The all-orders proof relies on the premise that in normal coordinates every interaction vertex is suppressed by at least O(λ²) and that no hidden loop topology, such as those involving Glauber-like modes or multipole-expansion subtleties, regenerates a leading-power contribution.

What would settle it

Compute the two-loop soft limit in a derivative-only scalar theory with a curved field-space metric, such as a sphere or hyperbolic target space, in dimensional regularization, and check for any O((p·k)^0) correction to the covariant-derivative formula; a nonzero correction at two loops would refute the all-orders claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The single-soft geometric soft theorem is exact to all orders for derivative-only scalar EFTs: every amplitude's soft limit is the covariant derivative of the lower-point amplitude, regardless of loop order.
  • All multi-soft theorems, including the double-soft Riemann-curvature term and the infinite tower of higher-multiplicity soft theorems, are tree-level exact in the same theories.
  • The V=0 geometric scalar EFT is infrared finite at leading power, with no soft or collinear divergences, a fact that follows directly from the EFT power counting.
  • For theories with a cubic potential, the leading soft factor is universal; the one-loop bubble, triangle, and soft-box topologies reproduce known corrections, and for massive soft scalars the factorized soft factor receives no loop corrections at leading power.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the all-orders exactness holds, the geometric soft theorem likely has a deeper topological or symmetry origin, such as a higher-form symmetry; the paper hints at this connection but does not prove it.
  • The framework suggests a concrete route to proving tree-level exactness for soft scalars coupled to fermions or gauge bosons, by combining field-space geometry with collinear-superspace or frame methods.
  • A testable extension is a two-loop soft-limit computation in a nontrivial sigma model (for instance a sphere or hyperbolic target space): the EFT predicts the O((p·k)^0) correction vanishes identically at every finite loop order.
  • For the massive case, the paper's factorization statement could be checked by computing the leading soft factor for a single self-interacting scalar with m² ~ λ⁴ and verifying that no loop correction appears at leading power.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper constructs a Soft-Collinear Effective Theory for geometric scalar field theories ('geoSCET'). It separates fields into soft and collinear modes and shows that an emergent field-space connection appears in the soft sector, mirroring the emergent soft gauge symmetry of QCD SCET. For theories without a potential, the paper derives the geometric single- and double-soft theorems from the EFT Lagrangian plus the shift invariance of the hard matching coefficient, and argues by power counting that all single- and multiple-soft theorems are tree-level exact to all orders in perturbation theory. For theories with a potential, it derives the potential-dependent soft factor under a fine-tuned massless renormalization condition and analyzes the leading one-loop corrections, reproducing results of an earlier one-loop computation. The central claims are that the geometric soft theorem is a consequence of EFT power counting and that, for derivatively-coupled theories, it is not modified by low-energy loop corrections at leading power.

Significance. If the central claims hold, the paper provides a systematic and conceptually clean origin for the geometric soft theorems, unifying them with the SCET approach to gauge-theory soft theorems. It also extends the known results to multiple soft emissions and makes an all-orders statement that goes beyond previous tree-level derivations. The explicit tree-level derivations in Secs. 4.1--4.3 are detailed, the matching of the one-loop bubble, triangle, and box integrals in Sec. 4.4 to prior results is a concrete cross-check, and the geometric R-Wilson-line construction is elegant. The paper contains reproducible, explicit computations and is transparent about its assumptions, including the fine-tuned massless limit for potential theories. If the all-orders exactness claim survives scrutiny, the paper is a significant advance for both the SCET and soft-theorem literatures.

major comments (2)
  1. [Sec. 4.4; Sec. 2.4 footnote; App. B] The all-orders tree-level exactness proof for V=0 rests on the assertion that no loop topology can modify the soft theorems at leading power. This is argued from normal-coordinate power counting alone, but the argument presupposes that the soft and collinear sectors exhaust the relevant momentum regions. The paper explicitly states that Glauber contributions are neglected throughout (Sec. 2.4 footnote), and App. B's method-of-regions discussion treats only hard and collinear regions of a two-mass triangle. In gauge theories, Glauber modes connect different collinear sectors at the same virtuality and are known to generate factorization-breaking terms (citations [82,94]). For geoSCET, a Glauber-like mode could in principle attach to a soft emission or connect collinear legs, and no argument is given that its couplings vanish or are power-suppressed in normal coordinates. This is load-bear
  2. [Sec. 4.4, potential theories] For the case V≠0, after computing the leading one-loop bubble, triangle, and box contributions in the massless fine-tuned theory, the paper concludes for massive theories that the factorization of the soft factor in Eq. (4.22) receives no loop corrections. This conclusion is supported by a schematic power-counting discussion and by Eq. (4.49), but it is not backed by an exhaustive enumeration of the massive-theory topologies. The same mode-completeness caveat applies, and the paper notes that with a mass regulator, single-leg soft topologies that vanish in the massless theory become non-zero. Since the abstract promises that a potential can be turned on 'without spoiling the factorization of the soft physics,' the support for this statement should be strengthened or the claim should be phrased as a leading-power statement within the soft/collinear sector expansion.
minor comments (4)
  1. [Eq. (4.31)] The notation s_{AI_a} in the double-soft prefactor is used before the Mandelstam invariant is explicitly defined for the two-soft-kinematics case. Please define it, e.g., s_{AI_a} = (k_A + p_{I_a})^2, to avoid confusion with the soft flavor indices.
  2. [Sec. 4.4, title of subsection] The phrase 'tree-level exact' may be misread as excluding loop corrections to the non-radiative amplitude. In fact, hard loops in the matching coefficient are allowed. Consider phrasing such as 'exact at tree level in the low-energy EFT, up to hard matching' or 'no soft/collinear loop corrections at leading power.'
  3. [Sec. 3.2, footnote 8] The IR-finiteness statement for V=0 is made before the caveat about relevant interactions is introduced. The caveat is important and should be in the main text, not only in a footnote.
  4. [App. D] The appendix relies on a Lagrangian from [56,57] and states that a detailed derivation is 'in preparation [143].' If the current paper relies on this unpublished work, it would help to specify exactly which identities are being used and which are deferred.

Circularity Check

0 steps flagged

No material circularity: the geoSCET derivation of the geometric soft theorems does not use the theorems as input; self-citations are benchmarks, not load-bearing.

full rationale

The paper's central chain is: (i) expand the geometric scalar Lagrangian in soft/collinear modes using the exponential map; (ii) construct the geoSCET Lagrangian and N-jet operator from power counting and shift invariance; (iii) compute single/double soft emissions from Lagrangian insertions; (iv) argue by λ-power-counting that for V=0 no loop topology contributes at leading power. The soft theorems emerge as the result of these calculations (e.g., Eq. (4.16) assembles ∂_A C and -Γ C into ∇_A M), rather than being assumed. The all-orders exactness claim in Sec. 4.4 rests on the stated power-counting premises ('in normal coordinates there are no three-point interactions', 'any loop one can attach costs at least O(λ²)') which are internal to the EFT and not derived from the soft theorem. The paper explicitly neglects Glauber modes (Sec. 2.2 footnote) and assumes m²_ph=0 for the V≠0 massless case; these are assumptions/limitations that could affect the validity of the conclusions but are not cases where a prediction is equivalent by construction to an input. Citations to [10], [43] and [46] are used as benchmarks or methodological references ('reproduces', 'closely follow'), and the calculations are performed in the paper, so the self-citations are not load-bearing. No fitted parameters or renamings of known results are used as predictions. Accordingly the circularity score is low.

Axiom & Free-Parameter Ledger

0 free parameters · 8 axioms · 2 invented entities

The paper introduces no fitted parameters; it is a pure theory derivation. The ledger lists the structural assumptions (power counting, mode decomposition, shift-invariance of the matching coefficient, Glauber neglect, fine-tuned massless limit, and the all-orders normal-coordinate argument) and the standard mathematics (Riemannian geometry, SCET, RPI) that the central claims rest on.

axioms (8)
  • domain assumption Two-derivative non-linear sigma model with metric g_IJ(φ) and potential V(φ) (Eq. 3.1) is the full theory; higher-derivative terms are excluded.
    Defines the class of theories under study; extension to higher-derivative operators is stated but not worked out.
  • domain assumption SCET mode decomposition with exponential-map background-field split ϕ = exp_φ[ξ], power counting ξ ∼ λ, φ ∼ λ², and soft multipole expansion around x−.
    Sec. 3.1-3.2; the entire geoSCET construction rests on this split and counting.
  • domain assumption The hard matching coefficient depends on the soft background only through C(v+φ_s) and satisfies shift invariance Eq. (3.43); collinear N-jet operators are covariant under soft field redefinitions.
    Sec. 3.3; this is what fixes the hard-emission contribution ∂_A C in the single-soft theorem, Eq. (4.13).
  • domain assumption No Glauber contributions; only soft and collinear momentum regions are included.
    Footnote 2 in Sec. 2.2; on-shell soft theorems are insensitive to Glauber, but factorization proofs would need them.
  • ad hoc to paper For potential theories, the SCET I power counting m² ∼ λ⁴Q², g₃ ∼ λ²Q, and the fine-tuned renormalization condition m²_ph=0 for the massless soft limit.
    Secs. 3.2, 4.2; these are chosen to keep the massless k_s→0 limit accessible; the paper acknowledges the fine-tuning and discusses the massive alternative.
  • ad hoc to paper The all-orders loop argument: in normal coordinates, every interaction vertex is suppressed by at least O(λ²), so any loop correction to the leading soft theorem is power-suppressed.
    Sec. 4.4; this is the premise for tree-level exactness to all orders. It is argued from the normal-coordinate expansion, not proven by exhaustive case analysis.
  • standard math Standard SCET results: the tree-level SCET Lagrangian is exact with unit matching to all orders; RPI fixes A-type subleading N-jet coefficient towers; collinear EOM removes n·D operators.
    Used throughout Secs. 2 and 4, and App. C; taken from QCD SCET literature [55,56,64,99,106].
  • standard math Riemannian geometry: Levi-Civita connection, exponential map, pullback covariant derivative, normal coordinates, and RPI (Lorentz invariance recovery) are assumed as background.
    Appendix A defines conventions; these are standard mathematical facts.
invented entities (2)
  • Emergent soft connection (A_μ)^I_J = Γ^I_{JK} ∂_μ φ^K no independent evidence
    purpose: Plays the role of the soft gauge field in geoSCET, enabling parallel transport and the geometric R Wilson line.
    A derived object from the field-space metric; it introduces no new physical degrees of freedom or falsifiable predictions.
  • Geometric R Wilson line R(x,x−) no independent evidence
    purpose: Parallel-transports collinear fields from x to x−, the analog of the QCD Wilson line and the tool that makes the soft gauge transformation homogeneous.
    A mathematical construct within the framework; it has no independent physical handle outside the paper.

pith-pipeline@v1.3.0-daily-deepseek · 48700 in / 26554 out tokens · 233125 ms · 2026-08-01T09:30:14.294379+00:00 · methodology

0 comments
read the original abstract

We apply the framework of Soft Collinear Effective Theory (SCET) to the theory of the geometric scalar field. The resulting ``geoSCET'' manifests an emergent geometry in the soft sector, mirroring the emergent soft gauge invariance of QCD SCET. We use geoSCET to derive the geometric soft theorems as straightforward consequences of effective-field-theory power counting, including extensions to multiple soft emissions and loop corrections. We demonstrate that theories without a potential satisfy universal geometric soft theorems for any number of soft emissions to all orders in perturbation theory. We also show that we can turn on a potential at the soft scale without spoiling the factorization of the soft physics. This work demonstrates the underlying field-space diffeomorphism origin of the geometric soft theorem.

Figures

Figures reproduced from arXiv: 2607.27322 by Andreas Helset, Patrick Hager, Timothy Cohen.

Figure 1
Figure 1. Figure 1: A diagram depicting the non-radiative matching for an [PITH_FULL_IMAGE:figures/full_fig_p028_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Diagrams depicting (a) soft emissions from external leg [PITH_FULL_IMAGE:figures/full_fig_p030_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Diagram depicting the non-vanishing topology for double-soft emission in the () [PITH_FULL_IMAGE:figures/full_fig_p035_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Collinear one-loop topologies modifying the soft theorem with a cubic interaction: [PITH_FULL_IMAGE:figures/full_fig_p039_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Soft one-loop topologies modifying the soft theorem with a cubic interaction: the [PITH_FULL_IMAGE:figures/full_fig_p041_5.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

144 extracted references · 1 canonical work pages

  1. [1]

    Collins, D.E

    J.C. Collins, D.E. Soper and G.F. Sterman,Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production,Nucl. Phys. B250(1985) 199

    1985

  2. [2]

    Collins, D.E

    J.C. Collins, D.E. Soper and G.F. Sterman,Factorization of Hard Processes in QCD,Adv. Ser. Direct. High Energy Phys.5(1989) 1 [hep-ph/0409313]

    Pith/arXiv arXiv 1989

  3. [3]

    Low,Bremsstrahlung of very low-energy quanta in elementary particle collisions,Phys

    F.E. Low,Bremsstrahlung of very low-energy quanta in elementary particle collisions,Phys. Rev.110(1958) 974

    1958

  4. [4]

    Weinberg,Infrared photons and gravitons,Phys

    S. Weinberg,Infrared photons and gravitons,Phys. Rev.140(1965) B516

    1965

  5. [5]

    Burnett and N.M

    T.H. Burnett and N.M. Kroll,Extension of the low soft photon theorem,Phys. Rev. Lett.20(1968) 86

    1968

  6. [6]

    Adler,Consistency conditions on the strong interactions implied by a partially conserved axial vector current,Phys

    S.L. Adler,Consistency conditions on the strong interactions implied by a partially conserved axial vector current,Phys. Rev.137(1965) B1022

    1965

  7. [7]

    Cheung, M

    C. Cheung, M. Derda, A. Helset and J. Parra-Martinez,Soft phonon theorems, JHEP08(2023) 103 [2301.11363]

    Pith/arXiv arXiv 2023

  8. [8]

    Green, Y

    D. Green, Y. Huang and C.-H. Shen,Inflationary Adler conditions,Phys. Rev. D 107(2023) 043534 [2208.14544]

    Pith/arXiv arXiv 2023

  9. [9]

    Novotny,Geometry of special Galileons,Phys

    J. Novotny,Geometry of special Galileons,Phys. Rev. D95(2017) 065019 [1612.01738]

    Pith/arXiv arXiv 2017

  10. [10]

    Cheung, A

    C. Cheung, A. Helset and J. Parra-Martinez,Geometric soft theorems,JHEP04 (2022) 011 [2111.03045]

    Pith/arXiv arXiv 2022

  11. [11]

    Chisholm,Change of variables in quantum field theories,Nucl

    J.S.R. Chisholm,Change of variables in quantum field theories,Nucl. Phys.26 (1961) 469. 60

    1961

  12. [12]

    Kamefuchi, L

    S. Kamefuchi, L. O’Raifeartaigh and A. Salam,Change of variables and equivalence theorems in quantum field theories,Nucl. Phys.28(1961) 529

    1961

  13. [13]

    ’t Hooft and M.J.G

    G. ’t Hooft and M.J.G. Veltman,Combinatorics of gauge fields,Nucl. Phys. B50 (1972) 318

    1972

  14. [14]

    ’t Hooft and M.J.G

    G. ’t Hooft and M.J.G. Veltman,Diagrammar,NATO Sci. Ser. B4(1974) 177

    1974

  15. [15]

    Coleman, J

    S.R. Coleman, J. Wess and B. Zumino,Structure of phenomenological Lagrangians. 1.,Phys. Rev.177(1969) 2239

    1969

  16. [16]

    Callan, Jr., S.R

    C.G. Callan, Jr., S.R. Coleman, J. Wess and B. Zumino,Structure of phenomenological Lagrangians. 2.,Phys. Rev.177(1969) 2247

    1969

  17. [17]

    Deans and J.A

    W.S. Deans and J.A. Dixon,Theory of Gauge Invariant Operators: Their Renormalization and S Matrix Elements,Phys. Rev. D18(1978) 1113

    1978

  18. [18]

    Politzer,Power Corrections at Short Distances,Nucl

    H.D. Politzer,Power Corrections at Short Distances,Nucl. Phys. B172(1980) 349

    1980

  19. [19]

    Arzt,Reduced effective Lagrangians,Phys

    C. Arzt,Reduced effective Lagrangians,Phys. Lett. B342(1995) 189 [hep-ph/9304230]

    Pith/arXiv arXiv 1995

  20. [20]

    Passarino,Field reparametrization in effective field theories,Eur

    G. Passarino,Field reparametrization in effective field theories,Eur. Phys. J. Plus 132(2017) 16 [1610.09618]

    Pith/arXiv arXiv 2017

  21. [21]

    Manohar,Introduction to Effective Field Theories,1804.05863

    A.V. Manohar,Introduction to Effective Field Theories,1804.05863

    Pith/arXiv arXiv

  22. [22]

    Criado and M

    J.C. Criado and M. P´ erez-Victoria,Field redefinitions in effective theories at higher orders,JHEP03(2019) 038 [1811.09413]

    Pith/arXiv arXiv 2019

  23. [23]

    Cohen, M

    T. Cohen, M. Forslund and A. Helset,Field redefinitions can be nonlocal,JHEP10 (2025) 019 [2412.12247]

    arXiv 2025

  24. [24]

    Volkov,Phenomenological Lagrangians,Fiz

    D.V. Volkov,Phenomenological Lagrangians,Fiz. Elem. Chast. Atom. Yadra4 (1973) 3

    1973

  25. [25]

    Alonso, E.E

    R. Alonso, E.E. Jenkins and A.V. Manohar,A Geometric Formulation of Higgs Effective Field Theory: Measuring the Curvature of Scalar Field Space,Phys. Lett. B754(2016) 335 [1511.00724]

    Pith/arXiv arXiv 2016

  26. [26]

    Alonso, E.E

    R. Alonso, E.E. Jenkins and A.V. Manohar,Sigma Models with Negative Curvature, Phys. Lett. B756(2016) 358 [1602.00706]

    Pith/arXiv arXiv 2016

  27. [27]

    Alonso, E.E

    R. Alonso, E.E. Jenkins and A.V. Manohar,Geometry of the Scalar Sector,JHEP 08(2016) 101 [1605.03602]

    Pith/arXiv arXiv 2016

  28. [28]

    Helset, M

    A. Helset, M. Paraskevas and M. Trott,Gauge fixing the Standard Model Effective Field Theory,Phys. Rev. Lett.120(2018) 251801 [1803.08001]. 61

    Pith/arXiv arXiv 2018

  29. [29]

    Nagai, M

    R. Nagai, M. Tanabashi, K. Tsumura and Y. Uchida,Symmetry and geometry in a generalized Higgs effective field theory: Finiteness of oblique corrections versus perturbative unitarity,Phys. Rev. D100(2019) 075020 [1904.07618]

    Pith/arXiv arXiv 2019

  30. [30]

    Helset, A

    A. Helset, A. Martin and M. Trott,The Geometric Standard Model Effective Field Theory,JHEP03(2020) 163 [2001.01453]

    Pith/arXiv arXiv 2020

  31. [31]

    Cohen, N

    T. Cohen, N. Craig, X. Lu and D. Sutherland,Is SMEFT Enough?,JHEP03 (2021) 237 [2008.08597]

    Pith/arXiv arXiv 2021

  32. [32]

    Cohen, N

    T. Cohen, N. Craig, X. Lu and D. Sutherland,Unitarity violation and the geometry of Higgs EFTs,JHEP12(2021) 003 [2108.03240]

    Pith/arXiv arXiv 2021

  33. [33]

    Alonso and M

    R. Alonso and M. West,Roads to the Standard Model,Phys. Rev. D105(2022) 096028 [2109.13290]

    Pith/arXiv arXiv 2022

  34. [34]

    Talbert,The geometricνSMEFT: operators and connections,JHEP01(2023) 069 [2208.11139]

    J. Talbert,The geometricνSMEFT: operators and connections,JHEP01(2023) 069 [2208.11139]

    Pith/arXiv arXiv 2023

  35. [35]

    Alonso, J.C

    R. Alonso, J.C. Criado, R. Houtz and M. West,Walls, bubbles and doom — the cosmology of HEFT,JHEP05(2024) 049 [2312.00881]

    Pith/arXiv arXiv 2024

  36. [36]

    Helset, E.E

    A. Helset, E.E. Jenkins and A.V. Manohar,Renormalization of the Standard Model Effective Field Theory from geometry,JHEP02(2023) 063 [2212.03253]

    Pith/arXiv arXiv 2023

  37. [37]

    B. Assi, A. Helset, A.V. Manohar, J. Pag` es and C.-H. Shen,Fermion geometry and the renormalization of the Standard Model Effective Field Theory,JHEP11(2023) 201 [2307.03187]

    Pith/arXiv arXiv 2023

  38. [38]

    Jenkins, A.V

    E.E. Jenkins, A.V. Manohar, L. Naterop and J. Pag` es,Two loop renormalization of scalar theories using a geometric approach,JHEP02(2024) 131 [2310.19883]

    Pith/arXiv arXiv 2024

  39. [39]

    B. Assi, A. Helset, J. Pag` es and C.-H. Shen,Renormalizing two-fermion operators in the SMEFT via supergeometry,JHEP12(2025) 082 [2504.18537]

    Pith/arXiv arXiv 2025

  40. [40]

    Pinol,Multifield inflation beyondN field = 2: non-Gaussianities and single-field effective theory,JCAP04(2021) 002 [2011.05930]

    L. Pinol,Multifield inflation beyondN field = 2: non-Gaussianities and single-field effective theory,JCAP04(2021) 002 [2011.05930]

    Pith/arXiv arXiv 2021

  41. [41]

    Pinol,New exact bispectrum shapes in multifield inflation,2607.15251

    L. Pinol,New exact bispectrum shapes in multifield inflation,2607.15251

    Pith/arXiv arXiv

  42. [42]

    Derda, A

    M. Derda, A. Helset and J. Parra-Martinez,Soft scalars in effective field theory, JHEP06(2024) 133 [2403.12142]

    Pith/arXiv arXiv 2024

  43. [43]

    Cohen, I

    T. Cohen, I. Fadakar, A. Helset and F. Nardi,Geometry of soft scalars at one loop, JHEP08(2025) 140 [2504.12371]

    Pith/arXiv arXiv 2025

  44. [44]

    Cachazo and A

    F. Cachazo and A. Strominger,Evidence for a New Soft Graviton Theorem, 1404.4091. 62

    Pith/arXiv arXiv

  45. [45]

    Z. Bern, S. Davies and J. Nohle,On Loop Corrections to Subleading Soft Behavior of Gluons and Gravitons,Phys. Rev. D90(2014) 085015 [1405.1015]

    Pith/arXiv arXiv 2014

  46. [46]

    Beneke, P

    M. Beneke, P. Hager and R. Szafron,Gravitational soft theorem from emergent soft gauge symmetries,JHEP03(2022) 199 [2110.02969]

    Pith/arXiv arXiv 2022

  47. [47]

    Beneke, P

    M. Beneke, P. Hager and R. Szafron,Soft-Collinear Gravity and Soft Theorems, 2210.09336

    Pith/arXiv arXiv

  48. [48]

    Czakon, F

    M. Czakon, F. Eschment and T. Schellenberger,Subleading effects in soft-gluon emission at one-loop in massless QCD,JHEP12(2023) 126 [2307.02286]

    Pith/arXiv arXiv 2023

  49. [49]

    Czakon, K.E

    M. Czakon, K.E. Minguez and F. Eschment,Subleading effects in soft-gluon emission at one-loop in massive QCD,JHEP02(2026) 110 [2510.21655]

    arXiv 2026

  50. [50]

    Cachazo, S

    F. Cachazo, S. He and E.Y. Yuan,New Double Soft Emission Theorems,Phys. Rev. D92(2015) 065030 [1503.04816]

    Pith/arXiv arXiv 2015

  51. [51]

    Arkani-Hamed, F

    N. Arkani-Hamed, F. Cachazo and J. Kaplan,What is the Simplest Quantum Field Theory?,JHEP09(2010) 016 [0808.1446]

    Pith/arXiv arXiv 2010

  52. [52]

    Bauer, S

    C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart,An Effective field theory for collinear and soft gluons: Heavy to light decays,Phys. Rev. D63(2001) 114020 [hep-ph/0011336]

    Pith/arXiv arXiv 2001

  53. [53]

    Bauer, D

    C.W. Bauer, D. Pirjol and I.W. Stewart,Soft collinear factorization in effective field theory,Phys. Rev. D65(2002) 054022 [hep-ph/0109045]

    Pith/arXiv arXiv 2002

  54. [54]

    Bauer, S

    C.W. Bauer, S. Fleming, D. Pirjol, I.Z. Rothstein and I.W. Stewart,Hard scattering factorization from effective field theory,Phys. Rev. D66(2002) 014017 [hep-ph/0202088]

    Pith/arXiv arXiv 2002

  55. [55]

    Beneke, A.P

    M. Beneke, A.P. Chapovsky, M. Diehl and T. Feldmann,Soft collinear effective theory and heavy to light currents beyond leading power,Nucl. Phys. B643(2002) 431 [hep-ph/0206152]

    Pith/arXiv arXiv 2002

  56. [56]

    Beneke and T

    M. Beneke and T. Feldmann,Multipole expanded soft collinear effective theory with nonAbelian gauge symmetry,Phys. Lett. B553(2003) 267 [hep-ph/0211358]

    Pith/arXiv arXiv 2003

  57. [57]

    Beneke, P

    M. Beneke, P. Hager and R. Szafron,Soft-collinear gravity beyond the leading power, JHEP03(2022) 080 [2112.04983]

    Pith/arXiv arXiv 2022

  58. [58]

    Beneke, P

    M. Beneke, P. Hager and D. Schwienbacher,Soft-collinear gravity with fermionic matter,JHEP03(2023) 076 [2212.02525]

    Pith/arXiv arXiv 2023

  59. [59]

    Biswas and G.W

    S. Biswas and G.W. Semenoff,Soft scalars do not decouple,Phys. Rev. D106 (2022) 105023 [2208.05023]. 63

    Pith/arXiv arXiv 2022

  60. [60]

    Strominger,Lectures on the Infrared Structure of Gravity and Gauge Theory, Princeton University Press (2018), [1703.05448]

    A. Strominger,Lectures on the Infrared Structure of Gravity and Gauge Theory, Princeton University Press (2018), [1703.05448]

    Pith/arXiv arXiv 2018

  61. [61]

    Pasterski, M

    S. Pasterski, M. Pate and A.-M. Raclariu,Celestial Holography, inSnowmass 2021, 11, 2021 [2111.11392]

    Pith/arXiv arXiv 2021

  62. [62]

    Raclariu,Lectures on Celestial Holography,2107.02075

    A.-M. Raclariu,Lectures on Celestial Holography,2107.02075

    Pith/arXiv arXiv

  63. [63]

    Donnay,Celestial holography: An asymptotic symmetry perspective,Phys

    L. Donnay,Celestial holography: An asymptotic symmetry perspective,Phys. Rept. 1073(2024) 1 [2310.12922]

    Pith/arXiv arXiv 2024

  64. [64]

    B¨ oer and P

    P. B¨ oer and P. Hager,On the gauge-invariance of SCET beyond leading power, JHEP08(2023) 197 [2306.12412]

    Pith/arXiv arXiv 2023

  65. [65]

    Becher, A

    T. Becher, A. Broggio and A. Ferroglia,Introduction to Soft-Collinear Effective Theory, vol. 896, Springer (2015), 10.1007/978-3-319-14848-9, [1410.1892]

    Pith/arXiv arXiv 2015

  66. [66]

    Becher,Soft-Collinear Effective Theory,1803.04310

    T. Becher,Soft-Collinear Effective Theory,1803.04310

    Pith/arXiv arXiv

  67. [67]

    Cohen,As Scales Become Separated: Lectures on Effective Field Theory,PoS TASI2018(2019) 011 [1903.03622]

    T. Cohen,As Scales Become Separated: Lectures on Effective Field Theory,PoS TASI2018(2019) 011 [1903.03622]

    Pith/arXiv arXiv 2019

  68. [68]

    Georgi,An Effective Field Theory for Heavy Quarks at Low-energies,Phys

    H. Georgi,An Effective Field Theory for Heavy Quarks at Low-energies,Phys. Lett. B240(1990) 447

    1990

  69. [69]

    Caswell and G.P

    W.E. Caswell and G.P. Lepage,Effective Lagrangians for Bound State Problems in QED, QCD, and Other Field Theories,Phys. Lett. B167(1986) 437

    1986

  70. [70]

    Cohen and D

    T. Cohen and D. Green,Soft de Sitter Effective Theory,JHEP12(2020) 041 [2007.03693]

    Pith/arXiv arXiv 2020

  71. [71]

    Goldberger and I.Z

    W.D. Goldberger and I.Z. Rothstein,An Effective field theory of gravity for extended objects,Phys. Rev. D73(2006) 104029 [hep-th/0409156]

    Pith/arXiv arXiv 2006

  72. [72]

    Beneke and V.A

    M. Beneke and V.A. Smirnov,Asymptotic expansion of Feynman integrals near threshold,Nucl. Phys. B522(1998) 321 [hep-ph/9711391]

    Pith/arXiv arXiv 1998

  73. [73]

    Gardi, F

    E. Gardi, F. Herzog, S. Jones, Y. Ma and J. Schlenk,The on-shell expansion: from Landau equations to the Newton polytope,JHEP07(2023) 197 [2211.14845]

    Pith/arXiv arXiv 2023

  74. [74]

    Beneke, P

    M. Beneke, P. Hager and A.F. Sanfilippo,Cosmological correlators in massless ϕ4-theory and the method of regions,JHEP04(2024) 006 [2312.06766]

    Pith/arXiv arXiv 2024

  75. [75]

    Ma,Identifying regions in wide-angle scattering via graph-theoretical approaches, JHEP09(2024) 197 [2312.14012]

    Y. Ma,Identifying regions in wide-angle scattering via graph-theoretical approaches, JHEP09(2024) 197 [2312.14012]

    Pith/arXiv arXiv 2024

  76. [76]

    Gardi, F

    E. Gardi, F. Herzog, S. Jones and Y. Ma,Dissecting polytopes: Landau singularities and asymptotic expansions in 2→2 scattering,JHEP08(2024) 127 [2407.13738]. 64

    Pith/arXiv arXiv 2024

  77. [77]

    X. Guan, F. Herzog, Y. Ma, B. Mistlberger and A. Suresh,Splitting amplitudes at N3LO in QCD,JHEP01(2025) 090 [2408.03019]

    Pith/arXiv arXiv 2025

  78. [78]

    Ma,Identifying regions for asymptotic expansions of amplitudes: Fundamentals and recent advances,Int

    Y. Ma,Identifying regions for asymptotic expansions of amplitudes: Fundamentals and recent advances,Int. J. Mod. Phys. A40(2025) 2530013 [2505.01368]

    Pith/arXiv arXiv 2025

  79. [79]

    Ma,All-order prescription for facet regions in massless wide-angle scattering, 2601.22144

    Y. Ma,All-order prescription for facet regions in massless wide-angle scattering, 2601.22144

    arXiv

  80. [80]

    W. Chen, E. Gardi, R. Ma, Y. Ma, Y. Zhang and Z. Zhu,Spacelike-Collinear Scattering by the Method of Regions,2607.15126

    Pith/arXiv arXiv

Showing first 80 references.