本文在任意维度构造 p-形式场的共形初级波函数基，计算标量积与影子变换，并利用区域方法给出靠近 \mathscr I 的大 r 渐近极限，从而在 D=4 时用共形初级语言重新表述标量与对偶二形式的联系。
摘要：
我们在任意维度中为 p-形式场构造了一组共形初级波函数（CPWs）基，计算了它们的标量积，并展示了传统平面波与 CPW 模式展开之间的基变换。我们还对相应的影子变换进行了分析。对于每一族 p-形式 CPWs，我们观察到存在共形维度为 Δ=p 的纯规范波函数，而该权重的影子 p-形式仅在临界时空维度值 D=2p+2 时才是纯规范的。随后，我们基于区域方法提供了一种系统化技术来获得靠近 \mathscr I 的大 r 渐近极限，该方法自然地考虑了天球上普通项与接触项的同时存在。在 D=4 时，这使我们能够以共形初级语言重新表述标量与对偶二形式之间的联系。
英文摘要：
We construct a basis of conformal primary wavefunctions (CPWs) for $p$-form fields in any dimension, calculating their scalar products and exhibiting the change of basis between conventional plane wave and CPW mode expansions. We also perform the analysis of the associated shadow transforms. For each family of $p$-form CPWs, we observe the existence of pure gauge wavefunctions of conformal dimension $Δ=p$, while shadow $p$-forms of this weight are only pure gauge in the critical spacetime dimension value $D=2p+2$. We then provide a systematic technique to obtain the large-$r$ asymptotic limit near $\mathscr I$ based on the method of regions, which naturally takes into account the presence of both ordinary and contact terms on the celestial sphere. In $D=4$, this allows us to reformulate in a conformal primary language the links between scalars and dual two-forms.
We study the effect of loop corrections to conformal correlators on the celestial sphere at null infinity. We first analyze finite one-loop celestial amplitudes in pure Yang-Mills theory and Einstein gravity. We then turn to our main focus: infrared divergent loop amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory. We compute the celestial one-loop amplitude in dimensional regularization and show that it can be recast as an operator acting on the celestial tree-level amplitude. This extends to any loop order and the re-summation of all planar loops enables us to write down an express…
It is shown that the infinite tower of tree-level soft graviton symmetries in asymptotically flat 4D quantum gravity can be organized into a single chiral 2D Kac-Moody symmetry based on the wedge algebra of w(1+infinity). The infinite towers of soft photon or gluon symmetries also transform irreducibly under w(1+infinity).
We introduce a non-commutative deformation of the algebra of bipolar spherical harmonics supporting the action of the full Lorentz algebra. Our construction is close in spirit to the one of the non-commutative spherical harmonics associated to the fuzzy sphere and, as such, it leads to a maximal value of the angular momentum. We derive the action of Lorentz boost generators on such non-commutative spherical harmonics and show that it is compatible with the existence of a maximal angular momentum.
Conformally soft gluons are conserved currents of the Celestial Conformal Field Theory (CCFT) and generate a Kac-Moody algebra. We study celestial amplitudes of Yang-Mills theory, which are Mellin transforms of gluon amplitudes and take the double soft limit of a pair of gluons. In this manner we construct the Sugawara energy-momentum tensor of the CCFT. We verify that conformally soft gauge bosons are Virasoro primaries of the CCFT under the Sugawara energy-momentum tensor. The Sugawara tensor though does not generate the correct conformal transformations for hard states. In Einstein-Yang- Mi…
The analytic structures of scattering amplitudes in gauge theory and gravity are examined on the celestial sphere. The celestial amplitudes in the two theories - computed by employing a regulated Mellin transform - can be compared at low multiplicity. It is established by direct computation that up to five external particles, the double copy relations of Kawai, Lewellen and Tye continue to hold identically, modulo certain multiplicative factors which are explicitly determined. Supersymmetric representations of the amplitudes are utilized throughout, manifesting the double copy structure betwee…
We transform superstring scattering amplitudes into the correlation functions of primary conformal fields on two-dimensional celestial sphere. The points on celestial sphere are associated to the asymptotic directions of (light-like) momenta of external particles, with the Lorentz group realized as the SL(2,C) conformal symmetry of the sphere. The energies are dualized through Mellin transforms into the parameters that determine dimensions of the primaries. We focus on four-point amplitudes involving gauge bosons and gravitons in type I open superstring theory and in closed heterotic superstri…
Pasterski, Shao and Strominger have recently proposed that massless scattering amplitudes can be mapped to correlators on the celestial sphere at infinity via a Mellin transform. We apply this prescription to arbitrary $n$-point tree-level gluon amplitudes. The Mellin transforms of MHV amplitudes are given by generalized hypergeometric functions on the Grassmannian $Gr(4,n)$, while generic non-MHV amplitudes are given by more complicated Gelfand $A$-hypergeometric functions.
A natural extension of the Pasterski-Shao-Strominger (PSS) prescription is described, enabling the map of Minkowski space amplitudes with massive spinning external legs to the celestial sphere to be performed. An integral representation for the conformal primary wave function (CPW) of massive spinning bosons on the celestial sphere is derived explicitly for spin-one and -two. By analogy with the spin-zero case, the spinning bulk-to-boundary propagator on Euclidean AdS is employed to extend the massive CPW integral representation to arbitrary integer spin, and to describe the appropriate invers…
Celestial Holography, A Hitchhiker's Guide to the Celestial Sphere
LPTHE, Physics Department, Faculty of Sciences, Ibnou Zohr University（伊本·佐赫尔大学理学院物理系理论物理实验室）；CRMEF, Regional Center for Education and Training Professions Marrakesh（马拉喀什地区教育培训中心）
We generalize Zwanziger's pairwise little group to include a boost subgroup. We do so by working in the celestial sphere representation of scattering amplitudes. We propose that due to late time soft photon and graviton exchanges, matter particles in the asymptotic states in massless QED and gravity transform under the Poincare group with an additional pair of pairwise celestial representations for each pair of matter particles. We demonstrate that the massless abelian and gravitational exponentiation theorems are consistent with the proposed pairwise Poincare transformation properties. For ma…
In this note we determine the graviton-graviton OPE and the null states in any $w_{1+\infty}$ symmetric theory on the celestial sphere. Our analysis shows that there exists a discrete \textit{infinite} family of such theories. The MHV-sector and the quantum self dual gravity are two members of this infinite family. Although the Bulk Lagrangian description of this family of theories is not currently known to us, the graviton scattering amplitudes in these theories are heavily constrained due to the existence of null states. Presumably they are exactly solvable in the same way as the minimal mod…
We recast the soft $S$-matrices on the celestial sphere as correlation functions of certain $2$-dimensional models of topological defects. In pointing out the double copy structure between the soft photon and soft graviton cases, we arrive at a putative classical double copy between the corresponding topological models and a rederivation of gauge invariance and the equivalence principle as Ward identities of the $2$-dimensional theories.
We consider the infrared factorisation of non-abelian multi-particle scattering amplitudes, and we study the form of the universal colour operator responsible for infrared divergences, when expressed in terms of coordinates on the `celestial sphere' intersecting the future light-cone at asymptotic distances. We find that colour-dipole contributions to the infrared operator, to all orders in perturbation theory, have a remarkably simple expression in these coordinates, with scale and coupling dependence factorised from kinematics and colour. Generalising earlier suggestions in the abelian theor…
The basic ingredient of CCFT holography is to regard four-dimensional amplitudes describing conformal wave packets as two-dimensional conformal correlation functions of the operators associated to external particles. By construction, these operators transform as quasi-primary fields under SL(2,C) conformal symmetry group of the celestial sphere. We derive the OPE of the CCFT energy-momentum tensor with the operators representing gauge bosons and show that they transform as Virasoro primaries under diffeomorphisms of the celestial sphere.
The coadjoint representation of the BMS group in four dimensions is constructed in a formulation that covers both the sphere and the punctured plane. The structure constants are worked out for different choices of bases. The conserved current algebra of non-radiative asymptotically flat spacetimes is explicitly interpreted in these terms.
Celestial holography expresses $\mathcal{S}$-matrix elements as correlators in a CFT living on the night sky. Poincaré invariance imposes additional selection rules on the allowed positions of operators. As a consequence, $n$-point correlators are only supported on certain patches of the celestial sphere, depending on the labeling of each operator as incoming/outgoing. Here we initiate a study of the celestial geometry, examining the kinematic support of celestial amplitudes for different crossing channels. We give simple geometric rules for determining this support. For $n\ge 5$, we can view …
In these lecture notes we review the isomorphism between the (connected) Lorentz group and the set of conformal transformations of the sphere. More precisely, after establishing the main properties of the Lorentz group, we show that it is isomorphic to the group SL(2,C) of complex 2 by 2 matrices with unit determinant. We then classify conformal transformations of the sphere, define the notion of null infinity in Minkowski space-time, and show that the action of Lorentz transformations on the celestial spheres at null infinity is precisely that of conformal transformations. In particular, we d…
The Cachazo-Strominger subleading soft graviton theorem for a positive helicity soft graviton is equivalent to the Ward identities for $\overline{SL(2,\mathbb C)}$ currents. This naturally gives rise to a $\overline{SL(2,\mathbb C)}$ current algebra living on the celestial sphere. The generators of the $\overline{SL(2,\mathbb C)}$ current algebra and the supertranslations, coming from a positive helicity leading soft graviton, form a closed algebra. We find that the OPE of two graviton primaries in the Celestial CFT, extracted from MHV amplitudes, is completely determined in terms of this alge…
The bulk-to-boundary dictionary for 4D celestial holography is given a new entry defining 2D boundary states living on oriented circles on the celestial sphere. The states are constructed using the 2D CFT state-operator correspondence from operator insertions corresponding to either incoming or outgoing particles which cross the celestial sphere inside the circle. The BPZ construction is applied to give an inner product on such states whose associated bulk adjoints are shown to involve a shadow transform. Scattering amplitudes are then given by BPZ inner products between states living on the s…
Celestial amplitudes provide holographic imprints of four-dimensional scattering processes in terms of conformal correlation functions on a two-dimensional sphere describing Minkowski space at null infinity. We construct the generators of Poincare and conformal groups in the celestial representation and discuss how these symmetries are manifest in the amplitudes.
Massless scattering amplitudes in four-dimensional Minkowski spacetime can be Mellin transformed to correlation functions on the celestial sphere at null infinity called celestial amplitudes. We study various properties of massless four-point scalar and gluon celestial amplitudes such as conformal partial wave decomposition, crossing relations and optical theorem. As a byproduct, we derive the analog of the single and double soft limits for all gluon celestial amplitudes.
We elaborate on the proposal of flat holography in which four-dimensional physics is encoded in two-dimensional celestial conformal field theory (CCFT). The symmetry underlying CCFT is the extended BMS symmetry of (asymptotically) flat spacetime. We use soft and collinear theorems of Einstein-Yang-Mills theory to derive the OPEs of BMS field operators generating superrotations and supertranslations. The energy-momentum tensor, given by a shadow transform of a soft graviton operator, implements superrotations in the Virasoro subalgebra of $\mathfrak{bms_4}$. Supertranslations can be obtained fr…
All loop soft photon theorems and higher spin currents on the celestial sphere
National Institute of Science Education and Research (NISER)（印度科学教育研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Indian Institute of Technology Ropar（印度技术学院罗帕尔分校）；King’s College London（伦敦国王学院）
We explore conformal primary wavefunctions for all half integer spins up to the graviton. Half steps are related by supersymmetry, integer steps by the classical double copy. The main results are as follows: we 1) introduce a convenient spin frame and null tetrad to organize all radiative modes of varying spin; 2) identify the massless spin-3/2 conformal primary wavefunction as well as the conformally soft Goldstone mode corresponding to large supersymmetry transformations; 3) indicate how to express a conformal primary of arbitrary spin in terms of differential operators acting on a scalar pr…
We develop two approaches to the problem of soft fragmentation of hadrons in a gauge theory for high energy processes. The first approach directly adapts the standard resummation of the parton distribution function's anomalous dimension (that of twist-two local operators) in the forward scattering regime, using $k_T$-factorization and BFKL theory, to the case of the fragmentation function by exploiting the mapping between the dynamics of eikonal lines on transverse-plane to the celestial-sphere. Critically, to correctly resum the anomalous dimension of the fragmentation function under this map…
Off-shell celestial amplitudes with both time-like and space-like external legs are defined. The Feynman rules for scalar amplitudes, viewed as a set of recursion relations for off-shell momentum space amplitudes, are transformed to the celestial sphere using the split representation. For four-point celestial amplitudes, the Feynman expansion is shown to be equivalent to a conformal partial wave decomposition, providing an interpretation of conformal partial wave expansion coefficients as integrals over off-shell three-point structures. A conformal partial wave decomposition for a simple four-…
The soft limits of scattering amplitudes have been extensively studied due to their essential role in the computation of physical observables in collider physics. The universal factorisation that occurs in these kinematic limits has been shown to be related to conservation laws associated with asymptotic, or large, gauge symmetries. This connection has led to a deeper understanding of the symmetries of gauge and gravitational theories and to a reformulation of scattering amplitudes in a basis of boost eigenstates which makes manifest the two-dimensional global conformal symmetry of the celesti…
The operator product expansion (OPE) on the celestial sphere of conformal primary gluons and gravitons is studied. Asymptotic symmetries imply recursion relations between products of operators whose conformal weights differ by half-integers. It is shown, for tree-level Einstein-Yang-Mills theory, that these recursion relations are so constraining that they completely fix the leading celestial OPE coefficients in terms of the Euler beta function. The poles in the beta functions are associated with conformally soft currents.
We study scattering amplitudes in the shadow conformal primary basis, which satisfies the same defining properties as the original conformal primary basis and has many advantages over it. The shadow celestial amplitudes exhibit locality manifestly on the celestial sphere, and behave like correlation functions in conformal field theory under the operator product expansion (OPE) limit. We study the OPE limits for three-point shadow celestial amplitude, and general $2\to n-2$ shadow celestial amplitudes from a large class of Feynman diagrams. In particular, we compute the conformal block expansio…
Starting from an analysis of four-dimensional asymptotically flat gravity in first order formulation, we show that superrotation reparametrization modes are governed by an Alekseev--Shatashvili action on the celestial sphere. This two-dimensional conformal theory describes spontaneous symmetry breaking of Virasoro superrotations together with the explicit symmetry breaking of more general Diff$(\mathcal{S}^2)$ superrotations. We arrive at this result by first reformulating the asymptotic field equations and symmetries of the radiative vacuum sector in terms of a Chern--Simons theory at null in…
We study celestial amplitudes in (super) Yang-Mills theory using a parameterisation of the spinor helicity variables where their overall phase is not fixed by the little group action. In this approach the spin constraint $h-\bar{h}=J$ for celestial conformal primaries emerges naturally from a new Mellin transform, and the action of conformal transformations on celestial amplitudes is derived. Applying this approach to $\mathcal{N}\!=\!4$ super Yang-Mills, we show how the appropriate definition of on-shell superspace coordinates leads naturally to a formulation of chiral celestial superamplitud…
We provide a unified treatment of conformally soft Goldstone modes which arise when spin-one or spin-two conformal primary wavefunctions become pure gauge for certain integer values of the conformal dimension $Δ$. This effort lands us at the crossroads of two ongoing debates about what the appropriate conformal basis for celestial CFT is and what the asymptotic symmetry group of Einstein gravity at null infinity should be. Finite energy wavefunctions are captured by the principal continuous series $Δ\in 1+i\mathbb{R}$ and form a complete basis. We show that conformal primaries with analyticall…
Soft-operators, loosely speaking, are operators which create or annihilate zero energy massless particles on the celestial sphere in Minkowski space. The Lorentz group acts on the celestial sphere by conformal transformation and the soft-operators transform as conformal primary operators of various dimension and spin. Working in space-time dimensions $D=4$ and $6$, we study some properties of the conformal representations with the (leading) soft photon and graviton as the highest weight vectors. Typically these representations contain null-vectors. We argue, from the $S$-matrix point of view, …
The four-dimensional $S$-matrix is reconsidered as a correlator on the celestial sphere at null infinity. Asymptotic particle states can be characterized by the point at which they enter or exit the celestial sphere as well as their $SL(2,\mathbb C)$ Lorentz quantum numbers: namely their conformal scaling dimension and spin $h\pm \bar h$ instead of the energy and momentum. This characterization precludes the notion of a soft particle whose energy is taken to zero. We propose it should be replaced by the notion of a conformally soft particle with $h=0$ or $\bar h=0$. For photons we explicitly c…
Superrotations arise from singular vector fields on the celestial sphere in asymptotically flat space, and their finite integrated versions have been argued by Strominger and Zhiboedov to insert cosmic strings into the spacetime. In this work, we argue for an alternative definition of the action of superrotations on Minkowski space that avoids introducing any defects. This involves realizing the finite superrotation not as a diffeomorphism between spaces, but as a mapping of Minkowski space to itself that may be multivalued or non-surjective. This eliminates any defects in the bulk spacetime a…
We identify an eikonal regime in celestial CFT$_2$ in which massless 2-2 scattering is dominated by t-channel exchange. We derive a formula for the celestial amplitude that resums exchanges of arbitrary integer spin to all orders in the coupling. The resulting eikonal phase takes the same form as in flat space with the powers of center-of-mass energy replaced by weight-shifting operators on the celestial sphere. We independently compute the celestial two-point function for a scalar propagating in a shockwave background and show that to leading order in the gravitational coupling and for a suit…
We discuss the properties of recently constructed "single-valued" celestial four-gluon amplitudes. We show that the amplitude factorizes into the "current" part and the "scalar" part. The current factor is given by the group-dependent part of the Wess-Zumino-Witten correlator of four holomorphic currents with a non-vanishing level of Kač-Moody algebra. The scalar factor can be expressed in terms of a complex integral of the Koba-Nielsen form, similar to the integrals describing four-point correlators in Coulomb gas models and, more generally, in the infinite central charge limit of Liouville t…
Celestial amplitudes are flat-space amplitudes which are Mellin-transformed to correlators living on the celestial sphere. In this note we present a recursion relation, based on a tree-level BCFW recursion, for gravitational celestial amplitudes and use it to explore the notion of conformal softness. As the BCFW formula exponentiates in the soft energy, it leads directly to conformal soft theorems in an exponential form. These appear from a soft piece of the amplitude characterized by a discrete family of singularities with weights $Δ=1-\mathbb{Z}_+$. As a byproduct, in the case of the MHV sec…
Scattering amplitudes in $d+2$ dimensions can be expressed in terms of a conformal basis, for which the S-matrix behaves as a CFT correlation function on the celestial $d$-sphere. We explain how compact expressions for the full tree-level S-matrix of gauge theory, gravity and other QFTs extend to this conformal basis, and are easily derived from ambitwistor strings. Using these formulae and their worldsheet origins, we prove various tree-level 'conformal soft theorems' in gauge theory and gravity in any dimension; these arise from limits where the scaling dimension of an external state in the …
