Stephan Stieberger, Tomasz R. Taylor
We express all tree-level graviton amplitudes in Einstein's gravity as the collinear limits of a linear combination of pure Yang-Mills amplitudes in which each graviton is represented by two gauge bosons, each of them carrying exactly one half of graviton's momentum and helicity.
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We show that the mixed gravitational/gauge superstring amplitudes describing decays of massless closed strings - gravitons or dilatons - into a number of gauge bosons, can be written at the tree (disk) level as linear combinations of pure open string amplitudes in which the graviton (or dilaton) is replaced by a pair of collinear gauge bosons. Each of the constituent gauge bosons carry exactly one half of the original closed string momentum, while their +/- 1 helicities add up to +/- 2 for the graviton or to 0 for the dilaton.
Multi-collinear factorization limits provide a window to study how locality and unitarity of scattering amplitudes can emerge dynamically from celestial CFT, the conjectured holographic dual to gauge and gravitational theories in flat space. To this end, we first use asymptotic symmetries to commence a systematic study of conformal and Kac-Moody descendants in the OPE of celestial gluons. Recursive application of these OPEs then equips us with a novel holographic method of computing the multi-collinear limits of gluon amplitudes. We perform this computation for some of the simplest helicity as…
Gravitational scattering amplitudes in collinear and high-energy limits
本论文研究引力散射振幅在高能和共线极限下的行为，提取次多Regge冲击因子，构建弹性H核及其辐射扩展，并联系经典散射与辐射可观测量。
Gauge-invariant charges of the dual graviton
该论文针对对偶引力子理论中的守恒荷在拓扑非平凡情形下定义不良的问题，提出添加改进项以构造规范不变的守恒荷，这些荷在两种表述中局域表达，并作为理论真正对称性的生成元。
Graviton propagator in de Sitter space in a simple one-parameter gauge
本文研究了在德西特空间中使用一种简单一参数规范下的格拉维通传播子，该规范扩展了以往大多数德西特空间格拉维通环计算所使用的规范，所得传播子形式较为简单，有助于检验一阶计算中规范依赖性及所提观测量。
In this follow up paper, we calculate higher point tree level graviton Witten diagrams in AdS$_4$ via bulk perturbation theory. We show that by rearranging the bulk to bulk graviton propagators, the calculations effectively reduce to the computation of a scalar factor. Analogous to the amplitudes for vector boson interactions we computed in the previous paper, scalar factors for the graviton exchange diagrams also become relatively simple when written in momentum space. We explicitly calculate higher point correlators and discuss how this momentum space formalism makes flat space and collinear…
We formulate the graviton propagator on de Sitter background in a 2-parameter family of simple gauges which break de Sitter invariance. Explicit results are derived for the first order perturbations in each parameter. These results should be useful in computations to check for gauge dependence of graviton loop corrections.
When graviton loops are taken into account, the background metric obtained as a solution to the one-loop corrected Einstein equations turns out to be gauge fixing dependent. Therefore it is of no physical relevance. Instead we consider a physical observable, namely the trajectory of a test particle in the presence of gravitons. We derive a quantum corrected geodesic equation that includes backreaction effects and is explicitly independent of any gauge fixing parameter.
Collinear and soft divergences in perturbative quantum gravity are investigated to arbitrary orders in amplitudes for wide-angle scattering, using methods developed for gauge theories. We show that collinear singularities cancel when all such divergent diagrams are summed over, by using the gravitational Ward identity that decouples the unphysical polarizations from the S-matrix. This analysis generalizes a result previously demonstrated in the eikonal approximation. We also confirm that the only virtual graviton corrections that give soft logarithmic divergences are of the ladder and crossed …
Gauge Independent Logarithms from Inflationary Gravitons
本文将消除平直空间规范依赖性的技术推广至de Sitter背景，研究了无质量最小耦合标量交换势的1-loop引力子修正，从而解决了传统有效场方程中的引力子规范依赖问题。
For two massless particles i and j, the collinear limit is a special kinematic configuration in which the particles propagate with parallel four-momentum vectors, with the total momentum P distributed as p_i=xP and p_j=(1-x)P, so that s_{ij}=(p_i+p_j)^2=P^2=0. In Yang-Mills theory, if i and j are among N gauge bosons participating in a scattering process, it is well known that the partial amplitudes associated to the (single trace) group factors with adjacent i and j are singular in the collinear limit and factorize at the leading order into N-1-particle amplitudes times the universal, x-depen…
We construct the gauge field and graviton propagators in Euclidean AdS(d+1) space-time by two different methods. In the first method the gauge invariant Maxwell or linearized Ricci operator is applied directly to bitensor ansatze for the propagators which reflect their gauge structure. This leads to a rapid determination of the physical part of the propagators in terms of elementary functions. The second method is a more traditional approach using covariant gauge fixing which leads to a solution for both physical and gauge parts of the propagators. The gauge invariant parts agree in both metho…
It is known that the covariant graviton two-point function in de Sitter spacetime is infrared divergent for some choices of gauge parameters. On the other hand it is also known that there are no infrared divergences requiring an infrared cutoff for the physical graviton two-point function for this spacetime in the transverse-traceless-synchronous gauge in the global coordinate system. We show in this paper that the covariant graviton Wightman two-point function with two gauge parameters is equivalent to the physical one in the global coordinate system in the sense that they produce the same tw…
Cancellation of one-parameter graviton gauge dependence in the effective scalar field equation in de Sitter
研究了德西特空间中无质量最小耦合标量有效场方程的一重引力子环修正的规范依赖性，通过考虑源和观察者修正来修正方程的自质量。通过计算德西特破坏引力子传播子的Δα变化，证明了所有图类贡献的收集可使规范依赖性抵消，支持从量子修正的有效方程中构建引力子规范无关的宇宙量子引力观测量。
Single-minus graviton tree amplitudes are nonzero
研究证明在特定条件下单减重子树幅非零，通过递归关系和解析性假设生成n重子幅。
The two-point Wightman function of the free photon field defined in a gauge-invariant manner is known to be equivalent to the field-strength two-point function in any spacetime that is topologically trivial. We show that the gauge-invariant graviton two-point function defined in a similar manner is equivalent to the Weyl-tensor two-point function in Minkowski space and in the Poincare patch of de Sitter space. This implies that in the Poincare patch of de Sitter space the gauge-invariant graviton two-point function decays like (distance)^(-4) as a function of coordinate distance for spacelike …
The structure of tree-level scattering amplitudes for collinear massless bosons is studied beyond their leading splitting function behavior. These near-collinear limits at sub-leading order are best studied using the Cachazo-He-Yuan (CHY) formulation of the S-matrix based on the scattering equations. We compute the collinear limits for gluons, gravitons and scalars. It is shown that the CHY integrand for an n-particle gluon scattering amplitude in the collinear limit at sub-leading order is expressed as a convolution of an (n-1)-particle gluon integrand and a collinear kernel integrand, which …
We present how to construct a Soft Collinear Effective Theory (SCET) for gravity at the leading and next-to-leading powers from the ground up. The soft graviton theorem and decoupling of collinear gravitons at the leading power are manifest from the outset in the effective symmetries of the theory. At the next-to-leading power, certain simple structures of amplitudes, which are completely obscure in Feynman diagrams of the full theory, are also revealed, which greatly simplifies calculations. The effective lagrangian is highly constrained by effectively multiple copies of diffeomorphism invari…
The leading term in the gauge coupling beta function comes due to interaction of gauge field with gravitons. It is shown to be a universal quantity for all gauge theories. At one-loop it is found to be zero in four dimensions. This is independent of the gravity action with metric as the field variable, gauge fixing condition and regularization scheme. This term being universally same for all gauge groups is further studied in the case of abelian gauge theories, where due to self-duality this term is shown to be zero to all loops, on-shell. Consequences of this are discussed.
We construct the graviton propagator on de Sitter background in exact de Donder gauge. We prove that it must break de Sitter invariance, just like the propagator of the massless, minimally coupled scalar. Our explicit solutions for its two scalar structure functions preserve spatial homogeneity and isotropy so that the propagator can be used within the larger context of inflationary cosmology, however, it is simple to alter the residual symmetry. Because our gauge condition is de Sitter invariant (although no solution for the propagator can be) renormalization should be simpler using this prop…
We investigate asymptotic symmetries which preserve the Bondi gauge conditions but do not preserve the asymptotic falloff conditions for the metric near the null boundary, and their connection to soft graviton theorems for scattering amplitudes. These include generalized superrotation symmetries parameterized by a smooth vector field $Y^{A}$ obeying $D_{A}Y^{A} = 0$, for which we show that the associated conserved charge can be derived by applying the Noether procedure to the Einstein-Katz action. We also discuss the connection between asymptotic symmetries and the conserved charge associated …
It is known that the graviton two-point function for the de Sitter invariant "Euclidean" vacuum in a physical gauge grows logarithmically with distance in spatially-flat de Sitter spacetime. We show that this logarithmic behaviour is a gauge artifact by explicitly demonstrating that the same behaviour can be reproduced by a pure-gauge two-point function.
We evaluate the four-closed-string scattering amplitude, using the Polyakov string path integral in the proper-time gauge. By identifying the Fock space representation of the four-closed-string-vertex, we obtain a field theoretic expression of the closed string scattering amplitudes. In the zero-slope limit, the four-closed-string scattering amplitude reduces to the four-graviton-scattering amplitude of Einstein's gravity. However, at a finite slope, the four-graviton scattering amplitude in the proper-time gauge differs not only from that of Einstein gravity, but also significantly differs fr…
Collinear Corrections to the Cachazo-Strominger Soft Theorem
本文针对Cachazo-Strominger软定理，指出为与渐近电荷结果一致，需仔细处理振幅中的共线奇点，从而引入共线修正。
We identify in Einstein gravity an asymptotic spin-$2$ charge aspect whose conservation equation gives rise, after quantization, to the sub-subleading soft theorem. Our treatment reveals that this spin-$2$ charge generates a non-local spacetime symmetry represented at null infinity by pseudo-vector fields. Moreover, we demonstrate that the non-linear nature of Einstein's equations is reflected in the Ward identity through collinear corrections to the sub-subleading soft theorem. Our analysis also provides a unified treatment of the universal soft theorems as conservation equations for the spin…
We study the constraint structure of Fierz -Pauli action in both flat and curved space in the framework of Hamiltonian formalism. We observe an abrupt change in the constraint algebra and the characteristics of the constraints when the mass term is turned off. As is well-known, for de Sitter background with a special tuning of the mass, we will have a gauge symmetry and one degree of freedom less. We will show how this abrupt change in the behavior of the system is reflected on the properties of the constraint algebra. We also obtain the generating functional of gauge transformations both for …
We examine the scalar sector of the covariant graviton two-point function in de Sitter spacetime. This sector consists of the pure-trace part and another part described by a scalar field. We show that it does not contribute to two-point functions of gauge-invariant quantities. We also demonstrate that the long-distance growth present in some gauges is absent in this sector for a wide range of gauge parameters.
Note on single-trace EYM amplitudes with MHV configuration
本文将含一个和两个引力子的树图阶单迹 EYM 振幅的分裂公式推广至一般情况，提出了用纯胶子振幅表示树图阶单迹 MHV 振幅的通用公式，其中每个引力子转化为一对共线胶子。
Gauge theory of gravity is formulated based on principle of local gauge invariance. Because the model has strict local gravitational gauge symmetry, gauge theory of gravity is a perturbatively renormalizable quantum model. However, in the original model, all gauge gravitons are massless. We want to ask that whether there exists massive gravitons in Nature? In this paper, we will propose a gauge model with massive gravitons. The mass term of gravitational gauge field is introduced into the theory without violating the strict local gravitational gauge symmetry. Massive gravitons can be considere…
Gauge anomalies on shell and collinear factorization
该研究从在壳视角探究规范反常，指出反常抵消条件违反会导致共线因子化破缺，通过构造带引力子交换的单圈5点振幅验证了相关奇异行为，提出引力可作为两类反常的通用探针。
We evaluate the one-graviton loop contribution to the vacuum polarization on de Sitter background in a 1-parameter family of exact, de Sitter invariant gauges. Our result is computed using dimensional regularization and fully renormalized with BPHZ counterterms, which must include a noninvariant owing to the time-ordered interactions. Because the graviton propagator engenders a physical breaking of de Sitter invariance two structure functions are needed to express the result. In addition to its relevance for the gauge issue this is the first time a covariant gauge graviton propagator has been …
We construct "soft-collinear gravity", the effective field theory which describes the interaction of collinear and soft gravitons with matter (and themselves), to all orders in the soft-collinear power expansion. Despite the absence of collinear divergences in gravity at leading power, the construction exhibits remarkable similarities with soft-collinear effective theory of QCD (gauge fields). It reveals an emergent soft background gauge symmetry, which allows for a manifestly gauge-invariant representation of the interactions in terms of a soft covariant derivative, the soft Riemann tensor, a…
Manifest Duality Invariance for the Partially Massless Graviton
本文通过构建规范不变的场强，展示了部分无质量引力子在de Sitter空间中的电磁对偶类似物，并以显式局域协变形式呈现该对偶性，该对偶性交换一阶场方程和Bianchi恒等式。
We extend our investigation of soft graviton effects on the microscopic dynamics of matter fields in de Sitter space. We evaluate the quantum equation of motion in generic gauge theories. We find that the Lorentz invariance can be respected and the velocity of light is not renormalized at the one-loop level. The gauge coupling constant is universally screened by soft gravitons and diminishes with time. These features are in common with other four dimensional field theories with dimensionless couplings. In particular the couplings scale with time with definite scaling exponents. Although indivi…
The two-point function of linearized gravitons on de Sitter space is infrared divergent in the standard transverse traceless synchronous gauge defined by $k=0$ cosmological coordinates (also called conformal or Poincare coordinates). We show that this divergence can be removed by adding a linearized diffeomorphism to each mode function; i.e., by an explicit change of gauge. It follows that the graviton vacuum state is well-defined and de Sitter invariant in agreement with various earlier arguments.
We consider the covariant graviton propagator in de Sitter spacetime in a gauge with two parameters, alpha and beta, in the Euclidean approach. We give an explicit form of the propagator with a particular choice of beta but with arbitrary value of alpha. We confirm that two-point functions of local gauge-invariant quantities do not increase as the separation of the two points becomes large.
We explicitly work out the de Sitter breaking contributions to the recent solution for the de Donder gauge graviton propagator on de Sitter. We also provide explicit power series expansions for the two structure functions, which are suitable for implementing dimensional regularization. And we evaluate the coincidence limit of the propagator.
A remarkably simple covariant graviton propagator in Anti-de Sitter spacetime
本文提出在任意维反德西特时空中，通过特殊规范选择得到极其简单的协变引力子和鬼传播子表达式，该规范改善了红外行为且平坦时空无法实现。
A recent proposal for gauge-invariant observables in inflation [R. Brunetti et al., JHEP 1608 (2016) 032] is examined. We give a generalisation of their construction to general background spacetimes. In flat space, we calculate one-loop graviton corrections to a scalar two-point function in a general gauge for the graviton. We explicitely show how the gauge-dependent terms cancel between the usual self-energy contributions and the additional corrections inherent in these observables. The one-loop corrections have the expected functional form, contrary to another recently studied proposal for g…
Small mass graviton propagator via finite-field-dependent BRST transformations in the critical dimension Siegel-Zwiebach action from string theory
针对D=26中引力子传播子在无质量极限下的发散问题，通过有限场相关BRST变换连接不同的规范固定Siegel-Zwiebach生成泛函，在广义Lorenz型规范下实现了有限的连续极限传播子。
We use the spinor helicity formalism to study KLT-like relations for the inflationary graviton four-point correlation function. New features are observed in this correlation function compared to the graviton scattering amplitude in flat spacetime. After obtaining the general momentum dependence, collinear, squeezed and collapsed limits are considered to further study the features of the correlation function, and the relation to the corresponding flat space scattering amplitude.
The thermal partition function of photons in any covariant gauge and gravitons in the harmonic gauge, propagating in a Rindler wedge, are computed using a local $ζ$-function regularization approach. The correct Planckian leading order temperature dependence $T^4$ is obtained in both cases. For the photons, the existence of a surface term giving a negative contribution to the entropy is confirmed, as earlier obtained by Kabat, but this term is shown to be gauge dependent in the four-dimensional case and, therefore is discarded. It is argued that similar terms could appear dealing with any integ…
We show that at tree level, on-shell gauge invariance can be used to fully determine the first subleading soft-gluon behavior and the first two subleading soft-graviton behaviors. Our proofs of the behaviors for n-gluon and n-graviton tree amplitudes are valid in D dimensions and are similar to Low's proof of universality of the first subleading behavior of photons. In contrast to photons coupling to massive particles, in four dimensions the soft behaviors of gluons and gravitons are corrected by loop effects. We comment on how such corrections arise from this perspective. We also show that lo…
This chapter reviews the construction of ``soft-collinear gravity'', the effective field theory which describes the interaction of collinear and soft gravitons with matter (and themselves), to all orders in the soft-collinear power expansion, focusing on the essential concepts. Among them are an emergent soft background gauge symmetry, which lives on the light-like trajectories of energetic particles and allows for a manifestly gauge-invariant representation of the interactions in terms of a soft covariant derivative and the soft Riemann tensor, and a systematic treatment of collinear interact…
The extended-BMS algebra of asymptotically flat spacetime contains an SO(3,1) subgroup that acts by conformal transformations on the celestial sphere. It is of interest to study the representations of this subgroup associated with gravitons. To reduce the equation of motion to a Schrodinger-like equation it is necessary to impose a non-covariant gauge condition. Using these solutions, leading-order gauge invariant Weyl scalars are then computed and decomposed into families of unitary principal series representations. An invertible holographic mapping is constructed between these unitary princi…
We show that the trace anomaly for gravitons calculated using the usual effective action formalism depends on the choice of gauge when the background spacetime is not a solution of the classical equation of motion, that is, when off-shell. We then use the gauge independent Vilkovisky-DeWitt effective action to restore gauge independence to the off-shell case. Additionally we explicitly evaluate trace anomalies for some N-sphere background spacetimes.
Using relationships between open and closed strings, we present a construction of tree-level scattering amplitudes for gravitons minimally coupled to matter in terms of gauge theory partial amplitudes. In particular, we present examples of amplitudes with gravitons coupled to vectors or to a single fermion pair. We also present two examples with massive graviton exchange, as would arise in the presence of large compact dimensions. The gauge charges are represented by flavors of dynamical scalars or fermions. This also leads to an unconventional decomposition of color and kinematics in gauge th…
We investigate the graviton propagator in the type IIB supergravity background which is dual to 4 dimensional noncommutative gauge theory. We assume that the boundary is located not at the infinity but at the noncommutative scale where the string frame metric exhibits the maximum. We argue that the Neumann boundary condition is the appropriate boundary condition to be adopted at the boundary. We find that the graviton propagator behaves just as that of the 4 dimensional massless graviton. On the other hand, the non-analytic behaviors of the other Kaluza-Klein modes are not significantly affect…
We employ a recent, general gauge computation of the one loop graviton contribution to the vacuum polarization on de Sitter to solve for one loop corrections to the photon mode function. The vacuum polarization takes the form of a gauge independent, spin 2 contribution and a gauge dependent, spin 0 contribution. We show that the leading secular corrections derive entirely from the spin 2 contribution.
In this paper we show that the one-loop graviton self-energy contribution is ultraviolet finite, without introducing counterterms, and cutoff-free in the framework of causal perturbation theory. In addition, it satisfies the gravitational Slavnov-Ward identities for the two-point connected Green function. The condition of perturbative gauge invariance to second order for loop graphs is proved. Corrections to the Newtonian potential are also derived.
The graviton self-energy at finite temperature depends on fourteen structure functions. We show that, in the absence of tadpoles, the gauge invariance of the effective action imposes three non-linear relations among these functions. The consequences of such constraints, which must be satisfied by the thermal graviton self-energy to all orders, are explicitly verified in general linear gauges to one loop order.
The quantum field theoretic description of general relativity is a modern approach to gravity where gravitational force is carried by spin-2 gravitons. In the classical limit of this theory, general relativity as described by the Einstein field equations is obtained. This limit, where classical general relativity is derived from quantum field theory is the topic of this thesis. The Schwarzschild-Tangherlini metric, which describes the gravitational field of an inertial point particle in arbitrary space-time dimensions, $D$, is analyzed. The metric is related to the three-point vertex function …
It has been claimed that the super Hubble modes of the graviton generated during inflation can make loop corrections diverge. Even if we introduce an infrared (IR) cutoff at a comoving scale as an ad hoc but a practical way for the regularization, we encounter the secular growth, which may lead to the breakdown of perturbative expansion for a sufficiently long lasting inflation. In this paper, we show that the IR pathology concerning the graviton can be attributed to the presence of residual gauge degrees of freedom in the local observable universe as in the case of the adiabatic curvature per…
Structure of Chern-Simons Graviton Scattering Amplitudes from Topological Graviton Equivalence Theorem and Double Copy
本文基于拓扑引力等价定理和双重复制方法，研究了Chern-Simons引力子散射幅度的结构，揭示了高能极限下引力子与稀释子散射幅度之间的等价关系，并证明了引力子散射幅度中能量抵消的规律。
We study the graviton self-energy function in a general gauge, using a hard thermal loop expansion which includes terms proportional to T^4, T^2 and log(T). We verify explicitly the gauge independence of the leading T^4 term and obtain a compact expression for the sub-leading T^2 contribution. It is shown that the logarithmic term has the same structure as the ultraviolet pole part of the T=0 self-energy function. We argue that the gauge-dependent part of the T^2 contribution is effectively canceled in the dispersion relations of the graviton plasma, and present the solutions of these equation…
It has been suggested that infrared gravitons in de Sitter space may lead to a secular screening of the effective cosmological constant. This seems to clash with the naive expectation that the curvature scalar should stay constant due to the Heisenberg equation of motion. Here, we show that the tadpole correction to the local expansion rate, which has been used in earlier analyses as an indicator of a decaying effective $Λ$, is not gauge invariant. On the other hand, we construct a gauge invariant operator which measures the renormalized curvature scalar smeared over an arbitrary window functi…
In this paper we point out the inevitability of an explosive production of gauge field and gravity wave during an open string tachyon condensation in a cosmological setting. We will be particularly studying an example of brane-anti-brane inflation in a warped throat where inflation ends via tachyon condensation. We point out that a tachyonic instability helps fragmenting the homogeneous tachyon and excites gauge field and contributes to the stress energy tensor which also feeds into the gravity waves.
The double copy is a much-studied relationship between scattering amplitudes in gauge and gravity theories, that has subsequently been extended to classical field solutions. In nearly all previous examples, the graviton field is defined around Minkowski space. Recently, it has been suggested that one may set up a double copy for gravitons defined around a non-trivial background. We investigate this idea from the point of view of the classical double copy. First, we use Kerr-Schild spacetimes to construct graviton solutions in curved space, as double copies of gauge fields on non-zero gauge bac…
The S-matrix, its unitarity and the graviton self-energy at the one-loop level are discussed on the basis of quantum Yang-Mills gravity with the translational gauge symmetry in flat space-time. The unitarity and gauge invariance of the S-matrix in a class of gauge conditions is preserved by massless ghost vector particles, called `Feynman-DeWitt-Mandelstam' (FDM) ghosts, in quantum Yang-Mills gravity. Using dimensional regularization, the graviton self-energy are explicitly calculated with a general gauge condition. The resultant divergence of graviton self-energy at the one-loop level resembl…
We study five-dimensional solutions to Einstein equations coupled to a scalar field. Bounce-type solutions for the scalar field are associated with AdS_5 spaces with smooth warp functions. Gravitons are dynamically localized in this framework in analogy to the Randall-Sundrum solution whereas, a bulk fermion gives rise to a single chiral zero mode localized at the bounce. Additional bulk scalar fields are incorporated in this picture. The dilaton, as a bulk scalar leads, through its coupling, to localized gauge boson fields, something that holds also in the case that the bounce system is repla…
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