Soft Theorem Research Articles

Soft Weakly Quasi-Continuous Functions Between Soft Topological Spaces

As an extension of quasi-continuity in general topology, we define soft quasi-continuity. We show that this notion is equivalent to the known notion of soft semi-continuity. Next, we define soft weak quasi-continuity. With the help of examples, we prove that soft weak quasi-continuity is strictly weaker than both soft semi-continuity and soft weak continuity. We introduce many characterizations of soft weak quasi-continuity. Moreover, we study the relationship between soft quasi-continuity and weak quasi-continuity with their analogous notions in general topology. Furthermore, we show that soft regularity of the co-domain of a soft function is a sufficient condition for equivalence between soft semi-continuity and soft weakly quasi-continuity. Furthermore, we provide several results of soft composition, restrictions, preservation, and soft graph theorems in terms of soft weak quasi-continuity.

On soft factors and transmutation operators

The well known soft theorems state the specific factorizations of tree level gravitational (GR) amplitudes at leading, sub-leading and sub-sub-leading orders, with universal soft factors. For Yang-Mills (YM) amplitudes, similar factorizations and universal soft factors are found at leading and sub-leading orders. Then it is natural to ask if the similar factorizations and soft factors exist at higher orders. In this note, by using transformation operators proposed by Cheung, Shen and Wen, we reconstruct the known soft factors of YM and GR amplitudes, and prove the nonexistence of higher order soft factor of YM or GR amplitude which satisfies the universality.

Soft limits of gluon and graviton correlators in Anti-de Sitter space

We derive formulae for the soft limit of tree-level gluon and graviton correlators in Anti-de Sitter space, which arise from Feynman diagrams encoding the Weinberg soft theorems in flat space. Other types of diagrams can also contribute to the soft limit at leading order in the soft momentum, but have a different pole structure. We derive these results at four points using explicit formulae recently obtained from the cosmological bootstrap and double copy, and extend them to any multiplicity using bootstrap techniques in Mellin-momentum space.

BRST covariant phase space and holographic Ward identities

This paper develops an enlarged BRST framework to treat the large gauge transformations of a given quantum field theory. It determines the associated infinitely many Noether charges stemming from a gauge fixed and BRST invariant Lagrangian, a result that cannot be obtained from Noether’s second theorem. The geometrical significance of this result is highlighted by the construction of a trigraded BRST covariant phase space, allowing a BRST invariant gauge fixing procedure. This provides an appropriate framework for determining the conserved BRST Noether current of the global BRST symmetry and the associated global Noether charges. The latter are found to be equivalent with the usual classical corner charges of large gauge transformations. It allows one to prove the gauge independence of their physical effects at the perturbative quantum level. In particular, the underlying BRST fundamental canonical relation provides the same graded symplectic brackets as in the classical covariant phase space. A unified Lagrangian Ward identity for small and large gauge transformations is built. It consistently decouples into a bulk part for small gauge transformations, which is the standard BRST-BV quantum master equation, and a boundary part for large gauge transformations. The boundary part provides a perturbation theory origin for the invariance of the Hamiltonian physical -matrix under asymptotic symmetries. Holographic anomalies for the boundary Ward identity are studied and found to be solutions of a codimension one Wess-Zumino consistency condition. Such solutions are studied in the context of extended BMS symmetry. Their existence clarifies the status of the 1-loop correction to the subleading soft graviton theorem.

On universal splittings of tree-level particle and string scattering amplitudes

In this paper, we study the newly discovered universal splitting behavior for tree-level scattering amplitudes of particles and strings [1]: when a set of Mandelstam variables (and Lorentz products involving polarizations for gluons/gravitons) vanish, the n-point amplitude factorizes as the product of two lower-point currents with n+3 external legs in total. We refer to any such subspace of the kinematic space of n massless momenta as “2-split kinematics”, where the scattering potential for string amplitudes and the corresponding scattering equations for particle amplitudes nicely split into two parts. Based on these, we provide a systematic and detailed study of the splitting behavior for essentially all ingredients which appear as integrands for open- and closed-string amplitudes as well as Cachazo-He-Yuan (CHY) formulas, including Parke-Taylor factors, correlators in superstring and bosonic string theories, and CHY integrands for a variety of amplitudes of scalars, gluons and gravitons. These results then immediately lead to the splitting behavior of string and particle amplitudes in a wide range of theories, including bi-adjoint ϕ3 (with string extension known as Z and J integrals), non-linear sigma model, Dirac-Born-Infeld, the special Galileon, etc., as well as Yang-Mills and Einstein gravity (with bosonic and superstring extensions). Our results imply and extend some other factorization behavior of tree amplitudes considered recently, including smooth splittings [2] and factorizations near zeros [3], to all these theories. A special case of splitting also yields soft theorems for gluons/gravitons as well as analogous soft behavior for Goldstone particles near their Adler zeros.

Linearized gravity and soft graviton theorem in de Sitter spacetime

We study the linearized gravity theory in the Newman-Unti gauge in the near horizon region of the de Sitter spacetime. The linearized Einstein equation involves the cosmological constant. The near horizon symmetry consists of near horizon supertranslation and near horizon superrotation. We compute the near horizon supertranslation charge and find the proper near horizon falloff conditions that uncover a soft graviton theorem from the Ward identity of the near horizon supertranslation. Published by the American Physical Society 2024

All-loop soft theorem for pions

In this paper we discuss a generalization of the Adler zero to loop integrands in the planar limit of the SU(N) nonlinear sigma model (NLSM). The Adler zero for integrands is violated starting at the two-loop order and is only recovered after integration. Here we propose a soft theorem satisfied by loop integrands with any number of loops and legs. This requires a generalization of NLSM integrands to an off shell framework with certain deformed kinematics. Defining an , we identify a simple nonvanishing soft behavior of integrands, which we call the . We find that the proposed soft theorem is satisfied by the “surface” integrand of Arkani-Hamed and Figueiredo [], which is obtained from the shifted Trφ3 surfacehedron integrand. Finally, we derive an on shell version of the algebraic soft theorem that takes an interesting form in terms of self-energy factors and lower-loop integrands in a mixed theory of pions and scalars. Published by the American Physical Society 2024

Axialgravisolitons at infinite corner

Abstract Gravitational solitons (gravisolitons) are particular exact solutions of Einstein field equation in vacuum build on a given background solution. Their interpretation is not yet fully clear but they contain many of the physically relevant solutions low N-solitons solutions. 
However, a systematic study and characterization of gravisolitons solution for every N is lacking and their relevance in a theory of quantum gravity is not fully understood. This work aims to investigate and characterize some properties of N-axialsoliton solutions such as their asymptotically behaviour and asymptotic symmetries given minimal assumptions on the background metric. We develop an explicit systematic asymptotically expansion for the N-axialsoliton solution and we compute the leading order of the asymptotic killing vectors. Moreover, in the perspective to better understand the role of gravisolitons in quantum gravity we make a link, and a one of the first explicit test, to the corner symmetry proposal deriving which subalgebra of the universal corner symmetry algebra is generated by the asymptotic Killing vectors of N-axialsoliton solution. In the spirit of the corner proposal, the axialgravisoliton corner symmetry algebra (agcsa) can be useful for the quantization of the non-asymptotically flat sector of gravity while, in the spirit of IR triangle, new soft theorems and memory effects could emerge.

QCD with an infrared fixed point and a dilaton

Following previous work we further explore the possibility that the chirally broken phase of gauge theories admits an infrared fixed point interpretation. The slope of the β function, β*′, is found to vanish at the infrared fixed point which has several attractive features such as logarithmic running. We provide a more in-depth analysis of our previous result that the mass anomalous assumes γ*=1 at the fixed point. The results are found to be consistent with N=1 supersymmetric gauge theories. In a second part the specific properties of a dilaton, the (pseudo-) Goldstone, due to spontaneous symmetry breaking are investigated. Dilaton soft theorems indicate that a soft dilaton mass can only originate from an operator of scaling dimension two. In the gauge theory this role is taken on by the q¯q-operator. The quantum chromodynamics (QCD) dilaton candidate, the σ=f0(500) meson is investigated and the singlet-octet mixing is found to be relevant. We briefly discuss the dilaton as a candidate for the Higgs boson, which relies on the ratio of dilaton to pion decay constant being close to unity. In QCD this is approximately satisfied but it is remains unclear if this is accidental or whether there is yet to be uncovered principle behind it. Published by the American Physical Society 2024

W1+∞ in 4D gravitational scattering

In four-dimensional asymptotically flat spacetimes, an infinite tower of soft graviton modes is known to generate the symmetry algebra of w1+∞ at tree-level. Here we demonstrate that the symmetry action follows from soft graviton theorems and acts non-trivially on massive scalar particles. By generalizing previous analyses that were specifically tailored to the scattering of massless particles, our results clarify that w1+∞ symmetry is a universal feature of tree-level gravitational scattering in four-dimensional asymptotically flat spacetimes and originates from minimally-coupled gravitational interactions. In addition, we show that the w1+∞ symmetry acts non-diagonally on massive states by mixing an infinite number of conformal families. We also present a concrete example of non-local behavior on the celestial sphere in the presence of massive scattering states.

Quasiclassical Gluon Fields and Low's Soft Theorem at Small Momentum-Fraction x.

In the high-energy limit, soft gluons can be approximately described by quasiclassical gluon fields. It is well known that the gluon field is a pure gauge field on the transverse plane at eikonal order. We derived the complete next-to-eikonal order solutions of the classical Yang-Mills equations for soft gluons in the dense nuclear regime. Utilizing these solutions, it is shown that Low's soft theorem at small x can be obtained by considering off-diagonal matrix elements of quasiclassical chromoelectric field between single-gluon states in the dilute regime. We further propose on extending Low's soft theorem at small x to incorporate the effects of gluon saturation in the dense regime.

Fracton Infrared Triangle.

In theories with conserved dipole moment, isolated charged particles (fractons) are immobile, but dipoles can move. We couple these dipoles to the fracton gauge theory and analyze the universal infrared structure. This uncovers an observable double kick memory effect which we relate to a novel dipole soft theorem. Together with their asymptotic symmetries this constitutes the first realization of an infrared triangle beyond Lorentz symmetry. This demonstrates the robustness of these IR structures and paves the way for their investigation in condensed matter systems and beyond.

Massive carrollian fields at timelike infinity

Motivated by flat space holography, we demonstrate that massive spin-s fields in Minkowski space near timelike infinity are massive carrollian fields on the carrollian counterpart of anti-de Sitter space called Ti. Its isometries form the Poincaré group, and we construct the carrollian spin-s fields using the method of induced representations. We provide a dictionary between massive carrollian fields on Ti and massive fields in Minkowski space, as well as to fields in the conformal primary basis used in celestial holography. We show that the symmetries of the carrollian structure naturally account for the BMS charges underlying the soft graviton theorem. Finally, we initiate a discussion of the correspondence between massive scattering amplitudes and carrollian correlation functions on Ti, and introduce physical definitions of detector operators using a suitable notion of conserved carrollian energy-momentum tensor.

Celestial soft currents at one-loop and their OPEs

Conformally soft operators and their associated soft theorems on the celestial sphere encode the low energy behaviour of bulk scattering amplitudes. They lead to an infinite dimensional symmetry algebra of the celestial CFT at tree-level. In this paper, focusing our attention to Yang-Mills theory, we introduce new operators in the boundary celestial CFT in order to extend the definition of conformally soft currents to include one-loop effects. We then compute their OPEs with other operators in the theory. We also examine new subtleties that arise in defining OPEs of two conformally soft operators. We elucidate the connection between the new operators and loop corrected soft theorems in the bulk. Finally, we conclude by demonstrating how these operators fit into the framework of a logarithmic CFT.

Soft theorems for boostless amplitudes

We consider effective field theories (EFTs) of scalar fields with broken Lorentz boosts, which arise by taking the decoupling and flat-space limits of the EFT of inflation, and derive constraints that must be satisfied by the corresponding scattering amplitudes if there is an underlying non-linearly realised symmetry. We primarily concentrate on extended shift symmetries which depend on the space-time coordinates, and find that combinations of scattering amplitudes obey enhanced Adler zeros. That is, such combinations vanish as one external momentum is taken soft, with the rate at which they vanish dictated by the corresponding symmetry. In our soft theorem derivation, we pay particular care to the energy and momentum-conserving delta functions that arise due to space-time translations, and show that when acted upon by derivatives with respect to spatial momenta, they yield a tower of soft theorems which are ultimately required for closure of the underlying symmetry algebra. All of our soft theorems correspond to constraints that must be satisfied by on-shell amplitudes and, even for symmetries that depend on the time coordinate, our soft theorems only require derivatives to be taken with respect to spatial momenta. We perform a soft bootstrap procedure to find solutions to our soft theorems, and compare these solutions to what we find from an off-shell analysis using the coset construction.

Celestial sector in CFT: Conformally soft symmetries

We show that time intervals of width \Delta \tauΔτ in 3-dimensional conformal field theories (CFT_33) on the Lorentzian cylinder admit an infinite dimensional symmetry enhancement in the limit \Delta \tau → 0Δτ→0. The associated vector fields are approximate solutions to the conformal Killing equations in the strip labelled by a function and a conformal Killing vector on the sphere. An Inonu-Wigner contraction yields a set of symmetry generators obeying the extended BMS_44 algebra. We analyze the shadow stress tensor Ward identities in CFT_dd on the Lorentzian cylinder with all operator insertions in infinitesimal time intervals separated by \piπ. We demonstrate that both the leading and subleading conformally soft graviton theorems in (d-1)(d−1)-dimensional celestial CFT (CCFT_{d-1}d−1) can be recovered from the transverse traceless components of these Ward identities in the limit \Delta \tau → 0Δτ→0. A similar construction allows for the leading conformally soft gluon theorem in CCFT_{d-1}d−1 to be recovered from shadow current Ward identities in CFT_dd.

Soft scalars in effective field theory

We derive a soft theorem for a massless scalar in an effective field theory with generic field content using the geometry of field space. This result extends the geometric soft theorem for scalar effective field theories by allowing the massless scalar to couple to other scalars, fermions, and gauge bosons. The soft theorem keeps its geometric form, but where the field-space geometry now involves the full field content of the theory. As a bonus, we also present novel double soft theorems with fermions, which mimic the geometric structure of the double soft theorem for scalars.

Asymptotic structure of higher dimensional Yang-Mills theory

Using the covariant phase space formalism, we construct the phase space for non-Abelian gauge theories in (d+2)-dimensional Minkowski spacetime for any d ≥ 2, including the edge modes that symplectically pair to the low energy degrees of freedom of the gauge field. Despite the fact that the symplectic form in odd and even-dimensional spacetimes appear ostensibly different, we demonstrate that both cases can be treated in a unified manner by utilizing the shadow transform. Upon quantization, we recover the algebra of the vacuum sector of the Hilbert space and derive a Ward identity that implies the leading soft gluon theorem in (d+2)-dimensional spacetime.

W1+∞ and Carrollian holography

In a 1 + 2D Carrollian conformal field theory, the Ward identities of the two local fields S0+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_0^{+} $$\\end{document} and S1+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_1^{+} $$\\end{document}, entirely built out of the Carrollian conformal stress-tensor, contain respectively up to the leading and the subleading positive helicity soft graviton theorems in the 1 + 3D asymptotically flat space-time. This work investigates how the subsubleading soft graviton theorem can be encoded into the Ward identity of a Carrollian conformal field S2+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_2^{+} $$\\end{document}. The operator product expansion (OPE) S2+S2+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_2^{+}{S}_2^{+} $$\\end{document} is constructed using general Carrollian conformal symmetry principles and the OPE commutativity property, under the assumption that any time-independent, non-Identity field that is mutually local with S0+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_0^{+} $$\\end{document}, S1+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_1^{+} $$\\end{document}, S2+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_2^{+} $$\\end{document} has positive Carrollian scaling dimension. It is found that, for this OPE to be consistent, another local field S3+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_3^{+} $$\\end{document} must automatically exist in the theory. The presence of an infinite tower of local fields Sk≥3+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_{k\\ge 3}^{+} $$\\end{document} is then revealed iteratively as a consistency condition for the S2+Sk−1+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_2^{+}{S}_{k-1}^{+} $$\\end{document} OPE. The general Sk+Sl+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_k^{+}{S}_l^{+} $$\\end{document} OPE is similarly obtained and the symmetry algebra manifest in this OPE is found to be the Kac-Moody algebra of the wedge sub-algebra of w1+∞. The Carrollian time-coordinate plays the central role in this purely holographic construction. The 2D Celestial conformally soft graviton primary Hkzz¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {H}^k\\left(z,\\overline{z}\\right) $$\\end{document} is realized to be contained in the Carrollian conformal primary S1−k+tzz¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_{1-k}^{+}\\left(t,z,\\overline{z}\\right) $$\\end{document}. Finally, the existence of the infinite tower of fields Sk+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {S}_k^{+} $$\\end{document} is shown to be directly related to an infinity of positive helicity soft graviton theorems.

Deriving interaction vertices in higher derivative theories

We derive cubic interaction vertices for a class of higher-derivative theories involving three arbitrary integer spin fields. This derivation uses the requirement of closure of the Poincaré algebra in four-dimensional flat spacetime. We find two varieties of permitted structures at the cubic level and eliminate one variety, which is proportional to the equations of motion, using suitable field redefinitions. We then consider soft theorems for field theories with these higher-derivative interactions and construct amplitudes in these theories using the inverse-soft approach.