On differential operators and unifying relations for 1-loop Feynman integrands

Abstract We generalize the unifying relations for tree amplitudes to the 1-loop Feynman integrands. By employing the 1-loop CHY formula, we construct differential operators which transmute the 1-loop gravitational Feynman integrand to Feynman integrands for a wide range of theories, including Einstein-Yang-Mills theory, Einstein-Maxwell theory, pure Yang-Mills theory, Yang-Mills-scalar theory, Born-Infeld theory, Dirac-Born-Infeld theory, bi-adjoint scalar theory, non-linear sigma model, as well as special Galileon theory. The unified web at 1-loop level is established. Under the well known unitarity cut, the 1-loop level operators will factorize into two tree level operators. Such factorization is also discussed.

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Expansions of tree amplitudes for Einstein–Maxwell and other theories

Progress of Theoretical and Experimental Physics ◽

10.1093/ptep/ptaa095 ◽

2020 ◽

Vol 2020 (7) ◽

Author(s):

Kang Zhou ◽

Shi-Qian Hu

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Scattering Amplitudes ◽

Recent Work ◽

Sigma Model ◽

Differential Operators ◽

Linear Sigma Model ◽

Tree Level ◽

Quantum Field ◽

Yang Mills

Abstract The expansions of tree-level scattering amplitudes for one theory into amplitudes for another theory, which have been studied in recent work, exhibit hidden connections between different theories that are invisible in the traditional Lagrangian formulism of quantum field theory. In this paper, the general expansion of tree Einstein–Maxwell amplitudes into the Kleiss–Kuijf basis of tree Yang–Mills amplitudes has been derived by applying a method based on differential operators. The obtained coefficients are shared by the expansion of tree $\phi^4$ amplitudes into tree BS (bi-adjoint scalar) amplitudes and the expansion of tree special Yang–Mills scalar amplitudes into tree BS amplitudes, as well the expansion of tree Dirac–Born–Infeld amplitudes into tree non-linear sigma model amplitudes.

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YANG-MILLS THEORY AND NON-LINEAR SIGMA MODEL FROM HIGHER DIMENSIONAL GRAVITY

The Ninth Marcel Grossmann Meeting ◽

10.1142/9789812777386_0161 ◽

2002 ◽

pp. 1059-1060

Author(s):

KAZUO GHOROKU

Keyword(s):

Sigma Model ◽

Linear Sigma Model ◽

Yang Mills ◽

Higher Dimensional ◽

Non Linear ◽

Dimensional Gravity

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Kinematic numerators from the worldsheet: cubic trees from labelled trees

Journal of High Energy Physics ◽

10.1007/jhep08(2021)118 ◽

2021 ◽

Vol 2021 (8) ◽

Author(s):

Song He ◽

Linghui Hou ◽

Jintian Tian ◽

Yong Zhang

Keyword(s):

Sigma Model ◽

Building Blocks ◽

Linear Sigma Model ◽

Tree Level ◽

Natural Class ◽

Yang Mills ◽

Tree Expansion ◽

Labelled Trees ◽

Logarithmic Functions ◽

Special Case

Abstract In this note we revisit the problem of explicitly computing tree-level scattering amplitudes in various theories in any dimension from worldsheet formulas. The latter are known to produce cubic-tree expansion of tree amplitudes with kinematic numerators automatically satisfying Jacobi-identities, once any half-integrand on the worldsheet is reduced to logarithmic functions. We review a natural class of worldsheet functions called “Cayley functions”, which are in one-to-one correspondence with labelled trees, and natural expansions of known half-integrands onto them with coefficients that are particularly compact building blocks of kinematic numerators. We present a general formula expressing kinematic numerators of all cubic trees as linear combinations of coefficients of labelled trees, which satisfy Jacobi identities by construction and include the usual combinations in terms of master numerators as a special case. Our results provide an efficient algorithm, which is implemented in a Mathematica package, for computing all tree amplitudes in theories including non-linear sigma model, special Galileon, Yang-Mills-scalar, Einstein-Yang-Mills and Dirac-Born-Infeld.

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The special Galileon as Goldstone of diffeomorphisms

Journal of High Energy Physics ◽

10.1007/jhep01(2021)096 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Diederik Roest

Keyword(s):

General Relativity ◽

Scalar Field ◽

Sigma Model ◽

Field Theories ◽

Linear Sigma Model ◽

Goldstone Mode ◽

Coordinate Transformations ◽

Yang Mills ◽

Non Linear ◽

Conformal Symmetries

Abstract The special Galileon stands out amongst scalar field theories due to its soft limits, non-linear symmetries and scattering amplitudes. This prompts the question what the origin of its underlying symmetry is. We show that it is intimately connected to general relativity: the special Galileon is the Goldstone mode of the affine group, consisting of linear coordinate transformations, analogous to the dilaton for conformal symmetries. We construct the corresponding metric, and discuss various relations to gravity, Yang-Mills and the non-linear sigma-model.

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