Scalar Amplitudes Research Articles

ϕp amplitudes from the positive tropical Grassmannian: Triangulations of extended diagrams

The global Schwinger formula, introduced by Cachazo and Early as a single integral over the positive tropical Grassmannian, provides a way to uncover properties of scattering amplitudes which are hard to see in their standard Feynman diagram formulation. In a recent work, Cachazo and one of the authors extended the global Schwinger formula to general ϕp theories. When p=4, it was conjectured that the integral decomposes as a sum over cones which are in bijection with noncrossing chord diagrams, and further that these can be obtained by finding the zeroes of a piecewise linear function, H(x). In this note we give a proof of this conjecture. We also present a purely combinatorial way of computing ϕp amplitudes by triangulating a trivial extended version of noncrossing (p−2)-chord diagrams, called extended diagrams, and present a proof of the bijection between triangulated extended diagrams and Feynman diagrams when p=4. This is reminiscent of recent constructions using Stokes polytopes and accordiohedra. However, the ϕp amplitude is now partitioned by a new collection of objects, each of which characterizes a polyhedral cone in the positive tropical Grassmannian in the form of an associahedron or of an intersection of two associahedra. Moreover, we comment on the bijection between extended diagrams and double-ordered biadjoint scalar amplitudes. We also conjecture the form of the general piecewise linear function, Hϕp(x), whose zeroes generate the regions in which the ϕp global Schwinger formula decomposes into. Published by the American Physical Society 2024

Connecting scalar amplitudes using the positive tropical Grassmannian

The biadjoint scalar partial amplitude, mnII, can be expressed as a single integral over the positive tropical Grassmannian thus producing a Global Schwinger Parameterization. The first result in this work is an extension to all partial amplitudes mn(α, β) using a limiting procedure on kinematic invariants that produces indicator functions in the integrand. The same limiting procedure leads to an integral representation of ϕ4 amplitudes where indicator functions turn into Dirac delta functions. Their support decomposes into Cn/2−1 regions, with Cq the qth-Catalan number. The contribution from each region is identified with a mn/2+1(α, I) amplitude. We provide a combinatorial description of the regions in terms of non-crossing chord diagrams and propose a general formula for ϕ4 amplitudes using the Lagrange inversion construction. We start the exploration of ϕp theories, finding that their regions are encoded in non-crossing (p – 2)-chord diagrams. The structure of the expansion of ϕp amplitudes in terms of ϕ3 amplitudes is the same as that of Green functions in terms of connected Green functions in the planar limit of Φp−1 matrix models. We also discuss possible connections to recent constructions based on Stokes polytopes and accordiohedra.

Cosmological amplitudes in power-law FRW universe

The correlators of large-scale fluctuations belong to the most important observables in modern cosmology. Recently, there have been considerable efforts in analytically understanding the cosmological correlators and the related wavefunction coefficients, which we collectively call cosmological amplitudes. In this work, we provide a set of simple rules to directly write down analytical answers for arbitrary tree-level amplitudes of conformal scalars with time-dependent interactions in power-law FRW universe. With the recently proposed family-tree decomposition method, we identify an over-complete set of multivariate hypergeometric functions, called family trees, to which all tree-level conformal scalar amplitudes can be easily reduced. Our method yields series expansions and monodromies of family trees in various kinematic limits, together with a large number of functional identities. The family trees are in a sense generalizations of polylogarithms and do reduce to polylogarithmic expressions for the cubic coupling in inflationary limit. We further show that all family trees can be decomposed into linear chains by taking shuffle products of all subfamilies, with which we find simple connection between bulk time integrals and boundary energy integrals.

Einstein’s spacetime curvature and gravitational waves: Elucidating the gradient flow of scalar invariants, variations in the Ricci scalar, and gravitational wave propagation

In the general theory of relativity initially rationalized by Einstein, the gravity is considered as an occurrence ensuing from the spacetime curvature, which is in turn triggered by the presence of mass. This article provides a detailed analysis of spacetime curvature and gravitational waves, enhancing our understanding of general relativity and astrophysics. Using both analytical and numerical methods, we examined the gradient flow of scalar invariants, variations in the Ricci scalar, and gravitational wave propagation. New findings include the identification of unique curvature patterns around rotating black holes, a detailed comparison of gradient flow behavior in different spacetime geometries, and the discovery of a correlation between scalar invariant fluctuations and gravitational wave amplitudes. These findings validate numerical techniques and reveal how different parameters affect gravitational wave characteristics. While simplified models are used, the present study offers a robust framework for future research. Our work aligns with previous studies but also reveals unique aspects of wave behavior, with implications for quantum gravity and cosmology.

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.

Higher-derivative relations between scalars and gluons

We extend the covariant color-kinematics duality introduced by Cheung and Mangan to effective field theories. We focus in particular on relations between the effective field theories of gluons only and of gluons coupled to bi-adjoint scalars. Maps are established between their respective equations of motion and between their tree-level scattering amplitudes. An additional rule for the replacement of flavor structures by kinematic factors realizes the map between higher-derivative amplitudes. As an example of new relations, the pure-gluon amplitudes of mass dimension up to eight, featuring insertions of the F3 and F4 operators which satisfy the traditional color-kinematics duality, can be generated at all multiplicities from just renormalizable amplitudes of gluons and bi-adjoint scalars. We also obtain closed-form expressions for the kinematic numerators of the dimension-six gluon effective field theory, which are valid in D space-time dimensions. Finally, we find strong evidence that this extended covariant color-kinematics duality relates the (DF)2+YM(+ϕ3) theories which, at low energies, generate infinite towers of operators satisfying the traditional color-kinematics duality, beyond aforementioned F3 and F4 ones.

Constructibility of AdS Supergluon Amplitudes.

We prove that all tree-level n-point supergluon (scalar) amplitudes in AdS_{5} can be recursively constructed, using factorization and flat-space limit. Our method is greatly facilitated by a natural R-symmetry basis for planar color-ordered amplitudes, which reduces the latter to "partial amplitudes" with simpler pole structures and factorization properties. Given the n-point scalar amplitude, we first extract spinning amplitudes with n-2 scalars and one gluon by imposing "gauge invariance," and then use a special "no-gluon kinematics" to determine the (n+1)-point scalar amplitude completely (which in turn contains the n-point single-gluon amplitude). Explicit results of up to 8-point scalar amplitudes and up to 6-point single-gluon amplitudes are included as Supplemental Material.

Generalized permutohedra in the kinematic space

In this note, we study the permutohedral geometry of the singularities of a certain differential form introduced in recent work of Arkani-Hamed, Bai, He and Yan. There it was observed that the poles of the form determine a family of polyhedra which have the same face lattice as that of the permutohedron. We realize that family explicitly, proving that it in fact fills out the configuration space of a particularly well-behaved family of generalized permutohedra, the zonotopal generalized permutohedra, that are obtained as the Minkowski sums of line segments parallel to the root directions ei − ej.Finally we interpret Mizera’s formula for the biadjoint scalar amplitude m(\U0001d540n, \U0001d540n), restricted to a certain dimension n − 2 subspace of the kinematic space, as a sum over the boundary components of the standard root cone, which is the conical hull of the roots e1 − e2, … , en−2 − en−1.

Charmonium χc0 and χc2 resonances in coupled-channel scattering from lattice QCD

In order to explore the spectrum of hidden-charm scalar and tensor resonances, we study meson-meson scattering with JPC=0++,2++ in the charmonium energy region using lattice QCD. Employing a light-quark mass corresponding to mπ≈391 MeV, we determine coupled-channel scattering amplitudes up to around 4100 MeV considering all kinematically relevant channels consisting of a pair of open-charm mesons or a charmonium meson with a light meson. A single isolated scalar resonance near 4000 MeV is found with large couplings to DD¯, DsD¯s and the kinematically closed D*D¯* channel. A single tensor resonance at a similar mass couples strongly to DD¯, DD¯* and D*D¯*. We compare the extracted resonances to contemporary experimental candidate states, previous lattice results and theoretical modeling. In contrast to several other studies, we do not find any significant feature in the scalar amplitudes between the ground state χc0(1P) and the resonance found around 4000 MeV. Published by the American Physical Society 2024

Testing the scalar weak gravity conjecture in no-scale supergravity

We explore possible extensions of the Weak Gravity Conjecture (WGC) to scalar field theories. To avoid charged black hole remnants, the WGC requires the existence of a particle with a mass m ≤ gqMP, with charge q and U(1) gauge coupling g, allowing the decay to shed the black hole charge. Although there is no obvious problem that arises in the absence of a U(1) charge, it has been postulated that gravity must remain the weakest force even when extended to scalar interactions. Quantifying this conjecture may be done by comparing scalar and gravitational amplitudes, or as we advocate here by comparing scattering cross sections. In theories with non-trivial field space geometries, by working out examples with perturbation theory around arbitrary field values and performing tadpole resummations, we argue that the conjecture must be applied only at extrema of the scalar potential (when expressed in locally canonical coordinates). We consider several toy models in the context of no-scale supergravity and also consider examples of inflationary models.

Classical observables using exponentiated spin factors: electromagnetic scattering

In [1], the authors argued that the Newman-Janis algorithm on the space of classical solutions in general relativity and electromagnetism could be used in the space of scattering amplitudes to map an amplitude with external scalar states to an amplitude associated to the scattering of “infinite spin particles”. The minimal coupling of these particles to the gravitational or Maxwell field is equivalent to the classical coupling of the Kerr blackhole with linearized gravity or the so-called Kerr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{\ extrm{Kerr}} $$\\end{document} charged state with the electromagnetic field. The action of the Newman-Janis mapping on scattering amplitudes was then used to compute the linear impulse at first post-Minkowskian (1PM) order, via the Kosower, Maybee, O’Connell (KMOC) formalism. In this paper, we continue with the idea of using the Newman-Janis mapping on the space of scalar QED amplitudes to compute classical observables such as the radiative gauge field and the angular impulse. We show that for tree-level amplitudes, the Newman-Janis action can be reinterpreted as a dressing of the photon propagator. This turns out to be an efficient way to compute these classical observables. Along the way, we highlight a subtlety that arises in proving the conservation of angular momentum for scalar −Kerr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ -\\sqrt{\ extrm{Kerr}} $$\\end{document} scattering.

Bayesian analysis of a generalized Starobinsky model with reheating constraints

We study a generalization of the Starobinsky model adding a term of the form R 2p to the Einstien-Hilbert action. we take the power p as a parameter of the model and explore the constraints from CMB plus BAO data through a Bayesian analysis, thus exploring a range of values for the exponent parameter. We incorporate a reheating phase to the model through the background matter content (equation of state) and the duration of this period (number of e-foldings of reheating). We find that incorporating information from reheating imposes constraints on cosmological quantities, more stringent than the case of no reheating when tested with the Planck+BAO data. The inferred value of the exponent parameter is statistically consitent with p = 1, favoring the original Starobinsky potential. Moreover, we report tighter constraints on p and the number of e-folds in comparison with previous works. The obtained values for other inflationary observational parameters, such as the scalar spectral index ns and the scalar amplitude of perturbations As , are consistent with prior measurements. Finally we present the alternative use of consistency relations in order to simplify the parameter space and test the generalized Starobinsky potential even more efficiently.

Multi-trace YMS amplitudes from soft behavior

Tree level multi-trace Yang-Mills-scalar (YMS) amplitudes have been shown to satisfy a recursive expansion formula, which expresses any YMS amplitude by those with fewer gluons and/or scalar traces. In an earlier work, the single-trace expansion formula has been shown to be determined by the universality of soft behavior. This approach is nevertheless not extended to multi-trace case in a straightforward way. In this paper, we derive the expansion formula of tree-level multi-trace YMS amplitudes in a bottom-up way: we first determine the simplest amplitude, the double-trace pure scalar amplitude which involves two scalars in each trace. Then insert more scalars to one of the traces. Based on this amplitude, we further obtain the double-soft behavior when the trace contains only two scalars is soft. The multi-trace amplitudes with more scalars and more gluons finally follow from the double-soft behavior as well as the single-soft behaviors which has been derived before.

Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity

The biadjoint scalar theory has cubic interactions and fields transforming in the biadjoint representation of ${\rm SU}(N)\times {\rm SU}\big({\tilde N}\big)$. Amplitudes are ''color'' decomposed in terms of partial amplitudes computed using Feynman diagrams which are simultaneously planar with respect to two orderings. In 2019, a generalization of biadjoint scalar amplitudes based on generalized Feynman diagrams (GFDs) was introduced. GFDs are collections of Feynman diagrams derived by incorporating an additional constraint of ''local planarity'' into the construction of the arrangements of metric trees in combinatorics. In this work, we propose a natural generalization of color orderings which leads to color-dressed amplitudes. A generalized color ordering (GCO) is defined as a collection of standard color orderings that is induced, in a precise sense, from an arrangement of projective lines on $\mathbb{RP}^2$. We present results for $n\leq 9$ generalized color orderings and GFDs, uncovering new phenomena in each case. We discover generalized decoupling identities and propose a definition of the ''colorless'' generalized scalar amplitude. We also propose a notion of GCOs for arbitrary $\mathbb{RP}^{k-1}$, discuss some of their properties, and comment on their GFDs. In a companion paper, we explore the definition of partial amplitudes using CEGM integral formulas.

Scattering amplitudes for self-force

The self-force expansion allows the study of deviations from geodesic motion due to the emission of radiation and its consequent back-reaction. We investigate this scheme within the on-shell framework of semiclassical scattering amplitudes for particles emitting photons or gravitons on a static, spherically symmetric background. We first present the exact scalar two-point amplitudes for Coulomb and Schwarzschild, from which one can extract classical observables such as the change in momentum due to geodesic motion. We then present, for the first time, the three-point semiclassical amplitudes for a scalar emitting a photon in Coulomb and a graviton on linearised Schwarzschild, outlining how the latter calculation can be generalized to the fully non-linear Schwarzschild metric. Our results are proper resummations of perturbative amplitudes in vacuum but, notably, are expressed in terms of Hamilton’s principal function for the backgrounds, rather than the radial action.

Planar kinematics: cyclic fixed points, mirror superpotential, $k$-dimensional Catalan numbers, and root polytopes

In this paper, we prove that points in the space X(k,n) of configurations of n points in \mathbb{CP}^{k-1} which are fixed under a certain cyclic action are the solutions to the generalized scattering equations on planar kinematics (PK). In the first part, we give a constructive upper bound: we show that these solutions inject into certain aperiodic k -element subsets of \{1,\dotsc,n\} , and consequently that their number is bounded above by the number of Lyndon words with k ones and n-k zeros. The proof uses a somewhat surprising connection between the superpotential of the mirror of G(n-k,n) and the generalized CHY potential on X(k,n) . We also check the recent conjecture that generalized biadjoint amplitudes evaluate to k -dimensional Catalan numbers on PK for several examples including k=3 and n\leq 40 and (k,n)=(6,13) . We then reformulate the CEGM generalized biadjoint scalar amplitude directly as a Laplace transform-type integral over \mathrm{Trop}^{+} G(k,n) , and we use it to evaluate the amplitude on PK with the purpose of exhibiting how generalized Feynman diagrams glue together. We initiate the study of two minimal lattice polytopal neighborhoods of the planar kinematics point. One of these, the rank-graded root polytope \mathcal{R}_{k,n} , in the case k=2 , is a projection of the standard type A root polytope. The other, denoted by \prod_{k,n} , in the case k=2 , is a degeneration of the associahedron. We check up to and including \mathcal{R}_{3,9} and \mathcal{R}_{4,9} that the relative volume of \mathcal{R}_{k,n} is the multi-dimensional Catalan number C^{(k)}_{n-k} , hinting towards the possibility of deeper geometric and combinatorial interpretations of m^{(k)}(\mathbb{I}_{n},\mathbb{I}_{n}) near the PK point.

Celestial conformal blocks of massless scalars and analytic continuation of the Appell function F1

In celestial conformal field theory (CCFT), the 4d massless scalars are represented by 2d conformal operators with conformal dimensions h = h¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{h} $$\\end{document} = (1 + iλ)/2. The Mellin transform of 4d massless scalar amplitudes gives the conformal correlators of CCFT. We study the conformal block decomposition of these celestial correlators in CCFT and obtain the explicit blocks. This conformal block decomposition is highly nontrivial, even for the simplest 4d massless scalar amplitude. We use the analytic continuation of the Appell hypergeometric function F1 and the method of monodromy projection of conformal blocks, to achieve this block decomposition. This procedure is consistent with the crossing symmetry, in both the correlator-level and each explicit block-level. We also investigate its behavior in the conformal soft limit and find that the Appell hypergeometric function F1 does not reduce to the Gauss hypergeometric function. This is different from the block decomposition of celestial gluons we studied before, where the Appell hypergeometric function F1 reduces to the Gauss hypergeometric function. This difference comes from the shift of conformal dimensions and is the reason why we adopt the new method here for the block decomposition of celestial massless scalars.

Note on NLSM tree amplitudes and soft theorems

AbstractThis note provides a new point of view for bootstrapping the tree amplitudes of the nonlinear sigma model (NLSM). We use the universality of single soft behavior, together with the double copy structure, to completely determine the tree amplitudes of the NLSM. We first observe Adler’s zero for four-point NLSM amplitudes, by considering kinematics. Then we assume the universality of Adler’s zero and use this requirement to construct general tree amplitudes of the NLSM in the expanded formula, i.e., the formula of expanding NLSM amplitudes to bi-adjoint scalar amplitudes, which allows us to give explicit expressions of amplitudes with arbitrary numbers of external legs. The construction does not require the assumption of quartic diagrams. We also derive double soft factors for NLSM tree amplitudes based on the resulting expanded formula, and the results are consistent with those in the literature.

Holonomic representation of biadjoint scalar amplitudes

We study tree-level biadjoint scalar amplitudes in the language of D-modules. We construct left ideals in the Weyl algebra D that allow a holonomic representation of n-point amplitudes in terms of the linear partial differential equations they satisfy. The resulting representation encodes the simple pole and recursive properties of the amplitude.

Perturbative soft photon theorems in de Sitter spacetime

We define a perturbative S-matrix in a local patch of de Sitter background in the limit when the curvature length scale (ℓ) is large and study the ‘soft’ behavior of the scalar QED amplitudes in de Sitter spacetime in generic dimensions. We obtain the leading and subleading perturbative corrections to flat space soft photon theorems in the large ℓ limit, and comment on the universality of these corrections. We compare our results with the electromagnetic memory tails obtained earlier in d = 4 using classical radiation analysis.