This article derives and presents the Feynman rules for (effective) Quantum General Relativity coupled to the Standard Model for any vertex valence and with general gauge parameter $ζ$. The results are worked out for the metric decomposition $g_{μν} = η_{μν} + \varkappa h_{μν}$, a linearized de Donder gauge fixing and four dimensions of spacetime. To this end, we calculate the Feynman rules for gravitons, graviton-ghosts and for the couplings of gravitons to scalars, spinors, gauge bosons and gauge ghosts.
In this paper, we generalize the integration rules for scattering equations to situations where higher-order poles are present. We describe the strategy to deduce the Feynman rules of higher-order poles from known analytic results of simple CHY-integrands, and propose the Feynman rules for single double pole and triple pole as well as duplex-double pole and triplex-double pole structures. We demonstrate the validation and strength of these rules by ample non-trivial examples.
We complete the proof of "Feynman rules" for constructing $M$-point conformal blocks with external and internal scalars in any topology for arbitrary $M$ in any spacetime dimension by combining the rules for the blocks (based on their Witten diagram interpretation) with the rules for the construction of conformal cross ratios (based on OPE flow diagrams). The full set of Feynman rules leads to blocks as power series of the hypergeometric type in the conformal cross ratios. We then provide a proof by recursion of the Feynman rules which relies heavily on the first Barnes lemma and the decomposi…
We elaborate on the functional integral describing the stochastic dynamics of a spectator field during inflation, comparing its diagrammatic expansion to that obtained directly from a perturbative solution of the corresponding Langevin equation. We state Feynman rules for computing arbitrary temporal $n$-point functions and perform some illustrative computations for a $λϕ^4$ interaction, paying attention to the role played by a functional Jacobian determinant in the path integral. We also briefly consider the case when the field contributes to the expansion rate, making the noise multiplicativ…
Environmental Effects in Post-Minkowskian Dynamics: Effective Field Theory, Feynman Rules, and Ward Identities for Compact Objects in Relativistic Fluids
Mani L. Bhaumik Institute for Theoretical Physics, University of California at Los Angeles（加州大学洛杉矶分校马尼·L·巴乌米克理论物理研究所）
We investigate Feynman graphs and their Feynman rules from the viewpoint of graph complexes. We focus on graph homology and on the appearance of cubical complexes when either reducing internal edges or when removing them by putting them on the massshell.
In this article functorial Feynman rules are introduced as large generalizations of physicists Feynman rules, in the sense that they can be applied to arbitrary classes of hypergraphs, possibly endowed with any kind of structure on their vertices and hyperedges. We show that the reconstruction conjecture for classes of (possibly structured) hypergraphs admit a sheaf-theoretic characterization, allowing us to consider analogous conjectures. We propose an axiomatization for the notion of superposition principle and prove that the functorial Feynman rules work as a bridge between reconstruction c…
We determine the Feynman rules for the minimal type A higher-spin gauge theory on AdS$_{d+1}$ at cubic order. In particular, we establish the quantum action at cubic order in de Donder gauge, including ghosts. We also give the full de Donder gauge propagators of higher-spin gauge fields and their ghosts. This provides all ingredients needed to quantise the theory at cubic order.
We conjecture a simple set of "Feynman rules" for constructing $n$-point global conformal blocks in any channel in $d$ spacetime dimensions, for external and exchanged scalar operators for arbitrary $n$ and $d$. The vertex factors are given in terms of Lauricella hypergeometric functions of one, two or three variables, and the Feynman rules furnish an explicit power-series expansion in powers of cross-ratios. These rules are conjectured based on previously known results in the literature, which include four-, five- and six-point examples as well as the $n$-point comb channel blocks. We prove t…
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-…
We examine the electroweak gauge sector of noncommutative standard model and in particular, obtain the $\mathcal{O} \,(θ)$ Feynman rules for all quadrilinear gauge boson couplings. Surprisingly, an electroweak-chromodynamics mixing appears in the gauge sector of noncommutative standard model, where photon as well as the neutral weak boson are coupled directly to three gluons. The phenomenological perspectives of the model in $W^-W^+ \rightarrow Z Z$ scattering are studied and it is shown that there is a characteristic oscillatory behaviour in azimuthal distribution of scattering cross sections…
We present a series of new gauge invariant quantities in Witten's open string field theory. They are defined for a given set of open string states which satisfy the physical state condition around a classical solution. For known classical solutions, we show that these gauge invariant quantities compute on shell tree-level scattering amplitudes around the correspondent D-brane configuration.
I work on a set of Feynman rules that were derived in order to incorporate the constraint of Gauss's law in the perturbation expansion of gauge field theories and calculate the interaction energy of two static sources. The constraint is implemented via a Lagrange multiplier field, $λ$, which, in the case of the non-Abelian theory, develops a radiatively generated effective potential term. After analysing the contributions of various solutions for $λ$, the confining properties and the various phases of the theory are discussed.
A compact method for amplitude calculations in theories with Dirac and Majorana effective operators is discussed. Using the renormalizable formalism of Denner et al., [1,2] for propagators, vertices and fermion (number) flow and introducing new "reading-rules", it is shown that fermions can be treated as scalars in the diagrams. The effect of Fermi-statistics appears only in overall signs and is determined once for whole classes of diagrams. Each vertex in this method corresponds to two or more vertices in the standard treatment of effective theories. As such, the advantages of this approach g…
In order to generalize the integration rules to general CHY integrands which include higher order poles, algorithms are proposed in two directions. One is to conjecture new rules, and the other is to use the cross-ratio identity method. In this paper,we use the cross-ratio identity approach to re-derive the conjectured integration rules involving higher order poles for several special cases: the single double pole, single triple pole and duplex-double pole. The equivalence between the present formulas and the previously conjectured ones is discussed for the first two situations.
We study relation between the gauge invariant quantity obtained in [arXiv:1908.09784] and the Feynman diagrams in the dressed $ \mathcal B_0 $ gauge in the open cubic string field theory. We derive a set of recurrence relations that hold among the terms of this gauge invariant quantity. By using these relations, we prove that this gauge invariant quantity equals the $S$-matrix at the tree level. We also present a proof that a set of new Feynman rules proposed in [arXiv:2003.05021 [hep-th]] reproduces the onshell disc amplitudes correctly by using the same combinatorial identities.
Two-component spinors are the basic ingredients for describing fermions in quantum field theory in four space-time dimensions. We develop and review the techniques of the two-component spinor formalism and provide a complete set of Feynman rules for fermions using two-component spinor notation. These rules are suitable for practical calculations of cross-sections, decay rates, and radiative corrections in the Standard Model and its extensions, including supersymmetry, and many explicit examples are provided. The unified treatment presented in this review applies to massless Weyl fermions and m…
An Algorithm for the Symbolic Reduction of Multi-loop Feynman Integrals via Generating Functions
State Key Laboratory of Nuclear Physics and Technology, Institute of Quantum Matter, South China Normal University（华南师范大学量子物质研究所核物理与技术国家重点实验室）；Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, Guangdong Provincial Key Laboratory of Nuclear Science（广东省核科学重点实验室物质结构与基本相互作用卓越基础研究广东基地）；Beijing Computational Science Research Center（北京计算科学研究中心）；Peng Huanwu Center for Fundamental Theory（彭桓武理论物理中心）；School of Physics, Peking University（北京大学物理学院）；Interdisciplinary Center for Theoretical Study, University of Science and Technology of China（中国科学技术大学理论交叉研究中心）；Center for High Energy Physics, Peking University（北京大学高能物理中心）
The Einstein-Hilbert Lagrangian for gravity is non-renormalizable at loop level. However, it can be treated in the effective field theory framework which means that gravity as an effective theory can be renormalized when a proper expansion of the effective Lagrangian is made. At the same time, the Feynman rules for gravity are very complicated, although the resulting amplitudes do not have the same complications. Therefore, in this thesis we want to simplify the Feynman rules as much as possible by using the most general parameterized gauge condition, adding all possible parameterized total de…
We analyze quantum Yang-Mills theory on $\mathbb{R}^2$ using a novel discretization method based on an algebraic analogue of stochastic calculus. Such an analogue involves working with "Gaussian" free fields whose covariance matrix is indefinite rather than positive definite. Specifically, we work with Lie-algebra valued fields on a lattice and exploit an approximate gauge-invariance that is restored when taking the continuum limit. This analysis is applied to show the equivalence between Wilson loop expectations computed using partial axial-gauge, complete axial-gauge, and the Migdal-Witten l…
