scholarly journals Kinematic numerators from the worldsheet: cubic trees from labelled trees

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.

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

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Kang Zhou
Keyword(s):  
Sigma Model ◽  
Differential Operators ◽  
Linear Sigma Model ◽  
Tree Level ◽  
Maxwell Theory ◽  
Scalar Theory ◽  
Yang Mills ◽  
Loop Level ◽  
Wide Range ◽  
Non Linear

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.

scholarly journals Extended DBI and its generalizations from graded soft theorems

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Karol Kampf ◽  
Jiří Novotný ◽  
Petr Vaško
Keyword(s):  
Sigma Model ◽  
Power Counting ◽  
Linear Sigma Model ◽  
Tree Level ◽  
Bottom Up ◽  
Slight Generalization ◽  
S Matrix ◽  
New Type ◽  
Mixed Power ◽  
Special Case

Abstract We analyze a theory known as extended DBI, which interpolates between DBI and the U(N) × U(N)/U(N) non-linear sigma model and represents a nontrivial example of theories with mixed power counting. We discuss symmetries of the action and their geometrical origin; the special case of SU(2) extended DBI theory is treated in great detail. The revealed symmetries lead to a new type of graded soft theorem that allows us to prove on-shell constructibility of the tree-level S-matrix. It turns out that the on-shell constructibility of the full extended DBI remains valid, even if its DBI sub-theory is modified in such a way to preserve its own on-shell constructibility. We thus propose a slight generalization of the DBI sub-theory, which we call 2-scale DBI theory. Gluing it back to the rest of the extended DBI theory gives a new set of on-shell reconstructible theories — the 2-scale extended DBI theory and its descendants. The uniqueness of the parent theory is confirmed by the bottom-up approach that uses on-shell amplitude methods exclusively.

scholarly journals Expansions of tree amplitudes for Einstein–Maxwell and other theories

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.

CHIRAL SYMMETRY RESTORATION IN THE LINEAR SIGMA MODEL

1992 ◽  
Vol 03 (05) ◽  
pp. 993-1009 ◽  
Author(s):  
H. MEYER-ORTMANNS ◽  
H.-J. PIRNER ◽  
A. PATKÓS
Keyword(s):  
Chiral Symmetry ◽  
Sigma Model ◽  
Chiral Limit ◽  
Linear Sigma Model ◽  
Tree Level ◽  
Gradual Change ◽  
Chiral Symmetry Restoration ◽  
Meson Octet ◽  
First Order Transition ◽  
Experimental Values

We report on results about the mass sensitivity of chiral symmetry restoration in the linear sigma model. For masses of the pseudoscalar meson octet which are close to the experimental values, we observed only a gradual change in the order parameters, when the temperature was changed. To estimate the size of the first order transition region around the chiral limit, we have varied the mass input for the tree level parametrization in several ways. The point with realistic meson masses turned out to lie well inside the crossover region.

scholarly journals Undoing decomposition

2019 ◽  
Vol 34 (35) ◽  
pp. 1950233 ◽  
Author(s):  
Eric Sharpe
Keyword(s):  
Sigma Model ◽  
Gauge Theories ◽  
Disjoint Union ◽  
Path Integrals ◽  
Two Dimensions ◽  
Linear Sigma Model ◽  
Two Dimensional ◽  
Yang Mills ◽  
Witten Theory ◽  
Supersymmetric Gauge

In this paper we discuss gauging one-form symmetries in two-dimensional theories. The existence of a global one-form symmetry in two dimensions typically signals a violation of cluster decomposition — an issue resolved by the observation that such theories decompose into disjoint unions, a result that has been applied to, for example, Gromov–Witten theory and gauged linear sigma model phases. In this paper we describe how gauging one-form symmetries in two-dimensional theories can be used to select particular elements of that disjoint union, effectively undoing decomposition. We examine such gaugings explicitly in examples involving orbifolds, nonsupersymmetric pure Yang–Mills theories, and supersymmetric gauge theories in two dimensions. Along the way, we learn explicit concrete details of the topological configurations that path integrals sum over when gauging a one-form symmetry, and we also uncover “hidden” one-form symmetries.

scholarly journals The special Galileon as Goldstone of diffeomorphisms

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.

scholarly journals Can a Linear Sigma Model Describe Walking Gauge Theories at Low Energies?

2018 ◽  
Vol 175 ◽  
pp. 08024 ◽  
Author(s):  
Andrew Gasbarro
Keyword(s):  
Sigma Model ◽  
Gauge Theories ◽  
Pion Mass ◽  
Effective Field ◽  
Linear Sigma Model ◽  
Tree Level ◽  
Lattice Data ◽  
Conformal Window ◽  
Quark Masses ◽  
Low Energies

In recent years, many investigations of confining Yang Mills gauge theories near the edge of the conformal window have been carried out using lattice techniques. These studies have revealed that the spectrum of hadrons in nearly conformal ("walking") gauge theories differs significantly from the QCD spectrum. In particular, a light singlet scalar appears in the spectrum which is nearly degenerate with the PNGBs at the lightest currently accessible quark masses. This state is a viable candidate for a composite Higgs boson. Presently, an acceptable effective field theory (EFT) description of the light states in walking theories has not been established. Such an EFT would be useful for performing chiral extrapolations of lattice data and for serving as a bridge between lattice calculations and phenomenology. It has been shown that the chiral Lagrangian fails to describe the IR dynamics of a theory near the edge of the conformal window. Here we assess a linear sigma model as an alternate EFT description by performing explicit chiral fits to lattice data. In a combined fit to the Goldstone (pion) mass and decay constant, a tree level linear sigma model has a Χ2/d.o.f. = 0.5 compared to Χ2/d.o.f. = 29.6 from fitting nextto-leading order chiral perturbation theory. When the 0++ (σ) mass is included in the fit, Χ2/d.o.f. = 4.9. We remark on future directions for providing better fits to the σ mass.

scholarly journals PION-NUCLEON SCATTERING WITHIN A GAUGED LINEAR SIGMA MODEL WITH PARITY-DOUBLED NUCLEONS

2007 ◽  
Vol 16 (07n08) ◽  
pp. 2388-2393 ◽  
Author(s):  
SUSANNA WILMS ◽  
FRANCESCO GIACOSA ◽  
DIRK H. RISCHKE
Keyword(s):  
Sigma Model ◽  
Chiral Condensate ◽  
Linear Sigma Model ◽  
Tree Level ◽  
Model Parameters ◽  
Nucleon Scattering ◽  
Pion Nucleon ◽  
Scattering Lengths ◽  
Chiral Partner ◽  
Gauged Linear Sigma Model

We compute pion-nucleon scattering at tree-level within a gauged linear sigma model which contains the nucleon and its chiral partner. Such an investigation in principle allows to make definite predictions as to whether the main contribution to the nucleon mass comes from the chiral condensate or from the mixing with its chiral partner. We find that there seems to be no set of model parameters that allows for a simultaneous description of all experimentally measured scattering lengths and range parameters. This indicates the need to improve the dynamical ingredients of the model.

scholarly journals Covariant color-kinematics duality

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Clifford Cheung ◽  
James Mangan
Keyword(s):  
Sigma Model ◽  
Equations Of Motion ◽  
Explicit Construction ◽  
Tree Level ◽  
Nonlinear Sigma Model ◽  
Yang Mills ◽  
Novel Structure ◽  
Double Copy ◽  
Number Of Particles ◽  
Field Strengths

Abstract We show that color-kinematics duality is a manifest property of the equations of motion governing currents and field strengths. For the nonlinear sigma model (NLSM), this insight enables an implementation of the double copy at the level of fields, as well as an explicit construction of the kinematic algebra and associated kinematic current. As a byproduct, we also derive new formulations of the special Galileon (SG) and Born-Infeld (BI) theory.For Yang-Mills (YM) theory, this same approach reveals a novel structure — covariant color-kinematics duality — whose only difference from the conventional duality is that 1/□ is replaced with covariant 1/D2. Remarkably, this structure implies that YM theory is itself the covariant double copy of gauged biadjoint scalar (GBAS) theory and an F3 theory of field strengths encoding a corresponding kinematic algebra and current. Directly applying the double copy to equations of motion, we derive general relativity (GR) from the product of Einstein-YM and F3 theory. This exercise reveals a trivial variant of the classical double copy that recasts any solution of GR as a solution of YM theory in a curved background.Covariant color-kinematics duality also implies a new decomposition of tree-level amplitudes in YM theory into those of GBAS theory. Using this representation we derive a closed-form, analytic expression for all BCJ numerators in YM theory and the NLSM for any number of particles in any spacetime dimension. By virtue of the double copy, this constitutes an explicit formula for all tree-level scattering amplitudes in YM, GR, NLSM, SG, and BI.

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