arXiv subjects

Rijun Huang

Publications and source records attributed to Rijun Huang.

At least 19 recordsLinked to original sources

arXiv2026

Four-loop Anomalous Dimensions of Scalar-QED Theory from Operator Product Expansion

We apply the Operator Product Expansion (OPE) algorithm to the renormalization of scalar-QED theory, with a specific focus on the fixed-charge operator $\phi^Q$. Within the OPE framework, the anomalous dimension of the $\phi^Q$ operator is perturbatively computed to four-loop order in the modified minimal subtraction scheme, extending beyond the previously available three-loop result. The beta functions, as well as the mass and field anomalous dimensions, are also computed at this order. An alternative loop-integrand construction method is proposed, based on graph decomposition and skeleton expansion techniques, for deriving the integrands of one-Particle-Irreducible correlation functions. This work represents the first non-trivial validation of the OPE algorithm for higher-loop renormalization beyond pure scalar theories. The present successful computations further confirm the efficiency and versatility of the OPE algorithm in renormalization analysis.

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arXiv2025

Five-loop Anomalous Dimensions of Cubic Scalar Theory from Operator Product Expansion

In this work, we compute the anomalous dimensions of the $\phi^Q$ operator in six-dimensional cubic scalar theory. The renormalization analysis is carried out within the framework of the Operator Product Expansion method, while the ultraviolet divergences of Feynman integrals are evaluated using the graphical function method. Inspired by the intrinsic connection between Wilson coefficients and anomalous dimensions, an algorithm was proposed recently, which provides a practical and systematic framework for calculating the anomalous dimensions of masses, fields, and composite operators, with broad potential applicability to generic quantum field theories. Notably, the HyperlogProcedures package, developed based on the graphical function method, enables the computation of two-point propagator-type integrals, derived herein for capturing ultraviolet divergences, to very high loop orders. With these advanced techniques, we have successfully computed the anomalous dimensions of the $\phi^Q$ operator up to five loops. Furthermore, we present a large $N$ expansion of the scaling dimensions at the Wilson-Fisher fixed point, extended to the $1/N^5$ order. This computation sets a new loop-order record for the anomalous dimension of the $\phi^Q$ operator in cubic scalar theory, while further validating the efficiency and versatility of the proposed algorithm in renormalization analyses.

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arXiv2025

On the Seven-loop Renormalization of Gross-Neveu Model

The presence of an infinite number of marginal four-fermion operators is a key characteristic of the two-dimensional Gross-Neveu model. In this study, we investigate the structure of UV divergences in this model, and by symmetry argument we found that the renormalizability only requires a subset of evanescent operators. We perform a 7-loop renormalization computation of beta function for the corresponding evanescent operator, and confirm its non-trivial contribution to UV divergences. By integrating infrared rearrangement, dimensional shifting, and large momentum expansion techniques, we systematically reduce the two-dimensional tensor integrals in the four-fermion correlation functions into four-dimensional scalar integrals. These scalar integrals are subsequently evaluated using the graphical function method, which marks the first application of the method to models with fermionic fields. Our result represents the first time that beta functions have been computed analytically beyond 5-loop in a model with spinning particles.

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arXiv2024

From Operator Product Expansion to Anomalous Dimensions

We propose a new method for computing the renormalization functions, which is based on the ideas of operator product expansion and large momentum expansion. In this method, the renormalization $Z$-factors are determined by the ultraviolet finiteness of Wilson coefficients in the dimensional regularization scheme. The ultraviolet divergence is extracted solely from two-point integrals at the large momentum limit. We develop this method in scalar field theories and establish a general framework for computing anomalous dimensions of fields, mass, couplings and composite operators. In particular, it is applied to the 6-dimensional cubic scalar theory and the 4-dimensional quartic scalar theory. We demonstrate this method by computing the anomalous dimension of the $\phi^Q$ operator in cubic theory up to four loops for arbitrary $Q$, which is in agreement with the known result in the large $N$ limit. The idea of computing anomalous dimensions from the operator production expansion is general and can be extended beyond scalar theories. This is demonstrated through examples of the Gross-Neveu-Yukawa model with generic operators.

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arXiv2022

Wilson Lines and Boundary Operators of BCFW Shifts

Boundary operators are gauge invariant operators whose form factors correspond to boundary contributions of BCFW shifts. In gauge theory, the boundary operators contain infinite series, which are constrained by gauge symmetry. We compute the boundary operators of all possible BCFW shifts in Yang-Mills theory and QCD, and show that the infinite series can be elegantly organized into Wilson lines, which are natural building blocks for non-local gauge invariant operators. We comment on their connection to jet functions and gauge invariant off-shell amplitudes. We also verify our results by studying various BCFW shifts of four and five-point amplitudes.

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arXiv2020

Expansion of EYM Amplitudes in Gauge Invariant Vector Space

Motivated by the problem of expanding single-trace tree-level amplitude of Einstein-Yang-Mills theory to the BCJ basis of Yang-Mills amplitudes, we present an alternative expansion formula in the gauge invariant vector space. Starting from a generic vector space consisting of polynomials of momenta and polarization vectors, we define a new sub-space as gauge invariant vector space by imposing constraints of gauge invariant conditions. To characterize this sub-space, we compute its dimension and construct an explicit gauge invariant basis from it. We propose an expansion formula in the gauge invariant basis with expansion coefficients being linear combinations of Yang-Mills amplitude, manifesting the gauge invariance of both expansion basis and coefficients. With help of quivers, we compute the expansion coefficients via differential operators and demonstrate the general expansion algorithm by several examples.

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arXiv2018

Permutation in the CHY-Formulation

The CHY-integrand of bi-adjoint cubic scalar theory is a product of two PT-factors. This pair of PT-factors can be interpreted as defining a permutation. We introduce the cycle representation of permutation in this paper for the understanding of cubic scalar amplitude. We show that, given a permutation related to the pair of PT-factors, the pole and vertex information of Feynman diagrams of corresponding CHY-integrand is completely characterized by the cycle representation of permutation. Inversely, we also show that, given a set of Feynman diagrams, the cycle representation of corresponding PT-factor can be recursively constructed. In this sense, there exists a deep connection between cycles of a permutation and amplitude. Based on these results, we have investigated the relations among different independent pairs of PT-factors in the context of cycle representation as well as the multiplication of cross-ratio factors.

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arXiv2017

Expansion of Einstein-Yang-Mills Amplitude

In this paper, we provide a thorough study on the expansion of single trace Einstein-Yang-Mills amplitudes into linear combination of color-ordered Yang-Mills amplitudes, from various different perspectives. Using the gauge invariance principle, we propose a recursive construction, where EYM amplitude with any number of gravitons could be expanded into EYM amplitudes with less number of gravitons. Through this construction, we can write down the complete expansion of EYM amplitude in the basis of color-ordered Yang-Mills amplitudes. As a byproduct, we are able to write down the polynomial form of BCJ numerator, i.e., numerators satisfying the color-kinematic duality, for Yang-Mills amplitude. After the discussion of gauge invariance, we move to the BCFW on-shell recursion relation and discuss how the expansion can be understood from the on-shell picture. Finally, we show how to interpret the expansion from the aspect of KLT relation and the way of evaluating the expansion coefficients efficiently.

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arXiv2017

Understanding the Cancelation of Double Poles in the Pfaffian of CHY-formulism

For a physical field theory, the tree-level amplitudes should possess only single poles. However, when computing amplitudes with Cachazo-He-Yuan (CHY) formulation, individual terms in the intermediate steps will contribute higher-order poles. In this paper, we investigate the cancelation of higher-order poles in CHY formula with Pfaffian as the building block. We develop a diagrammatic rule for expanding the reduced Pfaffian. Then by organizing diagrams in appropriate groups and applying the cross-ratio identities, we show that all potential contributions to higher-order poles in the reduced Pfaffian are canceled out, i.e., only single poles survive in Yang-Mills theory and gravity. Furthermore, we show the cancelations of higher-order poles in other field theories by introducing appropriate truncations, based on the single pole structure of Pfaffian.

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arXiv2016

Note on recursion relations for the $\mathcal{Q}$-cut representation

In this note, we study the $\mathcal{Q}$-cut representation by combining it with BCFW deformation. As a consequence, the one-loop integrand is expressed in terms of a recursion relation, i.e., $n$-point one-loop integrand is constructed using tree-level amplitudes and $m$-point one-loop integrands with $m\leq n-1$. By giving explicit examples, we show that the integrand from the recursion relation is equivalent to that from Feynman diagrams or the original $\mathcal{Q}$-cut construction, up to scale free terms.

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arXiv2016

Cross-ratio Identities and Higher-order Poles of CHY-integrand

The evaluation of generic Cachazo-He-Yuan(CHY)-integrands is a big challenge and efficient computational methods are in demand for practical evaluation. In this paper, we propose a systematic decomposition algorithm by using cross-ratio identities, which provides an analytic and easy to implement method for the evaluation of any CHY-integrand. This algorithm aims to decompose a given CHY-integrand containing higher-order poles as a linear combination of CHY-integrands with only simple poles in a finite number of steps, which ultimately can be trivially evaluated by integration rules of simple poles. To make the method even more efficient for CHY-integrands with large number of particles and complicated higher-order pole structures, we combine the $Λ$-algorithm and the cross-ratio identities, and as a by-product it provides us a way to deal with CHY-integrands where the $Λ$-algorithm was not applicable in its original formulation.

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arXiv2016

The Q-cut Representation of One-loop Integrands and Unitarity Cut Method

Recently, a new construction for complete loop integrands of massless field theories has been proposed, with on-shell tree-level amplitudes delicately incorporated into its algorithm. This new approach reinterprets integrands in a novel form, namely the Q-cut representation. In this paper, by deriving one-loop integrands as examples, we elaborate in details the technique of this new representation, e.g., the summation over all possible Q-cuts as well as helicity states for the non-scalar internal particle in the loop. Moreover, we show that the integrand in the Q-cut representation naturally reduces to the integrand in the traditional unitarity cut method for each given cut channel, providing a cross-check for the new approach.

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arXiv2016

Feynman Rules of Higher-order Poles in CHY Construction

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.

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arXiv2016

Form Factor and Boundary Contribution of Amplitude

The boundary contribution of an amplitude in the BCFW recursion relation can be considered as a form factor involving boundary operator and unshifted particles. At the tree-level, we show that by suitable construction of Lagrangian, one can relate the leading order term of boundary operators to some composite operators of N=4 super-Yang-Mills theory, then the computation of form factors is translated to the computation of amplitudes. We compute the form factors of these composite operators through the computation of corresponding double trace amplitudes.

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arXiv2015

An Algebraic Approach to the Scattering Equations

We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.

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arXiv2015

Global Structure of Curves from Generalized Unitarity Cut of Three-loop Diagrams

This paper studies the global structure of algebraic curves defined by generalized unitarity cut of four-dimensional three-loop diagrams with eleven propagators. The global structure is a topological invariant that is characterized by the geometric genus of the algebraic curve. We use the Riemann-Hurwitz formula to compute the geometric genus of algebraic curves with the help of techniques involving convex hull polytopes and numerical algebraic geometry. Some interesting properties of genus for arbitrary loop orders are also explored where computing the genus serves as an initial step for integral or integrand reduction of three-loop amplitudes via an algebraic geometric approach.

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arXiv2014

Integral Reduction by Unitarity Method for Two-loop Amplitudes: A Case Study

In this paper, we generalize the unitarity method to two-loop diagrams and use it to discuss the integral bases of reduction. To test out method, we focus on the four-point double-box diagram as well as its related daughter diagrams, i.e., the double-triangle diagram and the triangle-box diagram. For later two kinds of diagrams, we have given complete analytical results in general (4-2\eps)-dimension.

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arXiv2013

The classification of two-loop integrand basis in pure four-dimension

In this paper, we have made the attempt to classify the integrand basis of all two-loop diagrams in pure four-dimension space-time. Our classification includes the topology of two-loop diagrams which determines the structure of denominators, and the set of numerators under different kinematic configurations of external momenta by using Gröbner basis method. In our study, the variety defined by setting all propagators to on-shell has played an important role. We discuss the structure of variety and how it splits to various irreducible branches when external momenta at each corner of diagrams satisfy some special kinematic conditions. This information is crucial to the numerical or analytical fitting of coefficients for integrand basis in reduction process.

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