Gluon amplitudes as 2 d conformal correlators
TL;DR: In this paper, the authors compute low-point, tree-level gluon scattering amplitudes in the space of spin-one primary wave functions in the Lorentz group.
Abstract: Recently, spin-one wave functions in four dimensions that are conformal primaries of the Lorentz group $SL(2,\mathbb{C})$ were constructed. We compute low-point, tree-level gluon scattering amplitudes in the space of these conformal primary wave functions. The answers have the same conformal covariance as correlators of spin-one primaries in a $2d$ CFT. The Britto--Cachazo--Feng--Witten (BCFW) recursion relation between three- and four-point gluon amplitudes is recast into this conformal basis.
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TL;DR: A transcript of a course given by Strominger at Harvard in spring semester 2016 as discussed by the authors contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes.
Abstract: This is a redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. The lectures may be viewed online at this https URL. Please send typos or corrections to strominger@physics.this http URL.
599 citations
TL;DR: In this paper, the authors study slow-roll inflation with weak couplings to extra massive particles, for which all correlation functions are controlled by an approximate conformal symmetry on the boundary of the spacetime.
Abstract: Scattering amplitudes at weak coupling are highly constrained by Lorentz invariance, locality and unitarity, and depend on model details only through coupling constants and the particle content of the theory. For example, four-particle amplitudes are analytic for contact interactions and have simple poles with appropriately positive residues for tree-level exchange. In this paper, we develop an understanding of inflationary correlators which parallels that of flat-space scattering amplitudes. Specifically, we study slow-roll inflation with weak couplings to extra massive particles, for which all correlation functions are controlled by an approximate conformal symmetry on the boundary of the spacetime. After systematically classifying all possible contact terms in de Sitter space, we derive an analytic expression for the four-point function of conformally coupled scalars mediated by the tree-level exchange of massive scalars. Conformal symmetry implies that the correlator satisfies a pair of differential equations with respect to spatial momenta, encoding bulk time evolution in purely boundary terms. The absence of unphysical singularities (and the correct normalization of physical ones) completely fixes this correlator. Moreover, a “spin-raising” operator relates it to the correlators associated with the exchange of particles with spin, while “weight-shifting” operators map it to the four-point function of massless scalars. We explain how these de Sitter four-point functions can be perturbed to obtain inflationary three-point functions. Using our formalism, we reproduce many classic results in the literature, such as the three-point function of slow-roll inflation, and provide a complete classification of all inflationary three- and four-point functions arising from weakly broken conformal symmetry. Remarkably, the inflationary bispectrum associated with the exchange of particles with arbitrary spin is completely characterized by the soft limit of the simplest scalar-exchange four-point function of conformally coupled scalars and a series of contact terms. Finally, we demonstrate that the inflationary correlators contain flat-space scattering amplitudes via a suitable analytic continuation of the external momenta, which can also be directly connected with the signals for particle production seen in the squeezed limit.
300 citations
Cites background from "Gluon amplitudes as 2 d conformal c..."
...Indeed, even some first hints for such a description, seen in perturbation theory, involve much more alien combinatorial, geometric and number-theoretic ideas [9–16], from which the physics of spacetime and quantum mechanics—locality and unitarity—emerge as derivative notions, with a huge amount left to be understood....
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TL;DR: In this paper, the soft limit of tree-level Minkowskian scattering amplitudes in a conformal group on the celestial two-sphere at null infinity is considered.
Abstract: Four-dimensional Minkowskian scattering amplitudes can also be presented in a basis of a different group ($S\phantom{\rule{0}{0ex}}L(2,C)$), which acts as the conformal group on the celestial two-sphere at null infinity. This type of representation is useful for understanding the flat-space holography. The authors propose a procedure for taking the (conformally) soft limit of tree-level scattering amplitudes in this basis.
157 citations
TL;DR: In this article, the four-dimensional S-matrix is reconsidered as a correlator on the celestial sphere at null infinity and the notion of a soft particle whose energy is taken to zero is replaced by a conformally soft particle with h = 0 or $$ \overline{h} = 0.
Abstract: The four-dimensional S-matrix is reconsidered as a correlator on the celestial sphere at null infinity. Asymptotic particle states can be characterized by the point at which they enter or exit the celestial sphere as well as their SL(2, ℂ) Lorentz quantum numbers: namely their conformal scaling dimension and spin $$ h\pm \overline{h} $$
instead of the energy and momentum. This characterization precludes the notion of a soft particle whose energy is taken to zero. We propose it should be replaced by the notion of a conformally soft particle with h = 0 or $$ \overline{h} $$
= 0. For photons we explicitly construct conformally soft SL(2, ℂ) currents with dimensions (1, 0) and identify them with the generator of a U(1) Kac-Moody symmetry on the celestial sphere. For gravity the generator of celestial conformal symmetry is constructed from a (2, 0) SL(2, ℂ) primary wavefunction. Interestingly, BMS supertranslations are generated by a spin-one weight (
$$ \frac{3}{2} $$
, $$ \frac{1}{2} $$
) operator, which nevertheless shares holomorphic characteristics of a conformally soft operator. This is because the right hand side of its OPE with a weight (h, $$ \overline{h} $$
) operator $$ {\mathcal{O}}_{h,\overline{h}} $$
involves the shifted operator $$ {\mathcal{O}}_{h+\frac{1}{2},\overline{h}+\frac{1}{2}} $$
. This OPE relation looks quite unusual from the celestial CFT2 perspective but is equivalent to the leading soft graviton theorem and may usefully constrain celestial correlators in quantum gravity.
157 citations
TL;DR: In this paper, 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 ∆ is provided.
Abstract: 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 ∆. 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 ∆ ∈ 1 + iℝ and form a complete basis. We show that conformal primaries with analytically continued conformal dimension can be understood as certain contour integrals on the principal series. This clarifies how conformally soft Goldstone modes fit in but do not augment this basis. Conformally soft gravitons of dimension two and zero which are related by a shadow transform are shown to generate superrotations and non-meromorphic diffeomorphisms of the celestial sphere which we refer to as shadow superrotations. This dovetails the Virasoro and Diff(S2) asymptotic symmetry proposals and puts on equal footing the discussion of their associated soft charges, which correspond to the stress tensor and its shadow in the two-dimensional celestial CFT.
147 citations
References
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TL;DR: In this paper, Fourier transform the scattering amplitudes from momentum space to twistor space, and argue that the transformed amplitudes are supported on certain holomorphic curves, which is a consequence of an equivalence between the perturbative expansion of = 4 super Yang-Mills theory and the D-instanton expansion of a certain string theory.
Abstract: Perturbative scattering amplitudes in Yang-Mills theory have many unexpected properties, such as holomorphy of the maximally helicity violating amplitudes. To interpret these results, we Fourier transform the scattering amplitudes from momentum space to twistor space, and argue that the transformed amplitudes are supported on certain holomorphic curves. This in turn is apparently a consequence of an equivalence between the perturbative expansion of = 4 super Yang-Mills theory and the D-instanton expansion of a certain string theory, namely the topological B model whose target space is the Calabi-Yau supermanifold
1,626 citations
TL;DR: A short and direct proof of this recursion relation for tree-level scattering amplitudes based on properties of tree- level amplitudes only is given.
Abstract: Recently, by using the known structure of one-loop scattering amplitudes for gluons in Yang-Mills theory, a recursion relation for tree-level scattering amplitudes has been deduced. Here, we give a short and direct proof of this recursion relation based on properties of tree-level amplitudes only.
1,605 citations
"Gluon amplitudes as 2 d conformal c..." refers methods in this paper
...In this section we transform the BCFW recursion relation [31,32] from momentum space to the space of conformal primary wavefunctions....
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TL;DR: In this article, the authors presented new recursion relations for tree amplitudes in gauge theory that give very compact formulas, in which all particles are on-shell and momentum conservation is preserved.
1,267 citations
"Gluon amplitudes as 2 d conformal c..." refers methods in this paper
...In this section we transform the BCFW recursion relation [31,32] from momentum space to the space of conformal primary wavefunctions....
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...A3(1 −, 2−, 3) = 〈12〉(3) 〈23〉〈31〉 , A3(1 , 2, 3−) = [12](3) [23][31] ....
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...A++−(ωi, zi, z̄i) = [12](3) [23][31] δ((4))(pμ1 + p μ 2 + p μ 3) = 2 ω1ω2 ω3 z̄(3) 12 z̄23z̄31 δ((4))( ∑...
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TL;DR: A nontrivial squared helicity amplitude is given for the scattering of an arbitrary number of gluons to lowest order in the coupling constant and to leading order inThe number of colors.
Abstract: A nontrivial squared helicity amplitude is given for the scattering of an arbitrary number of gluons to lowest order in the coupling constant and to leading order in the number of colors.
963 citations
"Gluon amplitudes as 2 d conformal c..." refers background in this paper
...We focus on the amplitude with color order (1234) [29],...
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TL;DR: A compact formula for the complete tree-level S-matrix of pure Yang-Mills and gravity theories in arbitrary spacetime dimensions is presented and Gauge invariance is completely manifest as it follows from a simple property of the Pfaffian.
Abstract: A new formula for the scattering of massless particles may simplify predictions and analyses of LHC experiments and shed new light on quantum gravity theories.
828 citations
"Gluon amplitudes as 2 d conformal c..." refers background in this paper
...Indeed several constructions including the scattering equations [17] and twistor space [18–20] seem close to achieving this goal....
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