M. Yu. Kalmykov, V. V. Bytev, Bernd A. Kniehl, B. F. L. Ward, S. A. Yost
We will present some (formal) arguments that any Feynman diagram can be understood as a particular case of a Horn-type multivariable hypergeometric function. The advantages and disadvantages of this type of approach to the evaluation of Feynman diagrams is discussed.
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We describe the application of differential reduction algorithms for Feynman Diagram calculation. We illustrate the procedure in the context of generalized hypergeometric functions, and give an example for a type of q-loop bubble diagram.
We present a new methodology, suitable for implementation on computer, to perform the $ε$-expansion of hypergeometric functions with linear $ε$ dependent Pochhammer parameters in any number of variables. Our approach allows one to perform Taylor as well as Laurent series expansion of multivariable hypergeometric functions. Each of the coefficients of $ε$ in the series expansion is expressed as a linear combination of multivariable hypergeometric functions with the same domain of convergence as that of the original hypergeometric function. We present illustrative examples of hypergeometric func…
Higher-order diagrams required for radiative corrections to mixed electroweak and QCD processes at the LHC and anticipated future colliders will require numerically stable representations of the associated Feynman diagrams. The hypergeometric representation supplies an analytic framework that is useful for deriving such stable representations. We discuss the reduction of Feynman diagrams to master integrals, and compare integration-by-parts methods to differential reduction of hypergeometric functions. We describe the problem of constructing higher-order terms in the epsilon expansion, and cha…
We briefly discuss the transcendental constants generated through the epsilon-expansion of generalized hypergeometric functions and their interrelation with the "sixth root of unity."
GKZ hypergeometric systems of the three-loop vacuum Feynman integrals
本文基于Mellin-Barnes表示和Miller变换，为任意质量三圈真空图费曼积分构造GKZ超几何系统，并利用其级数解在参数空间子集上构建基本解系，从而得到真空积分的解析表达式。
We study Feynman integrals in the framework of Gel'fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems. The latter defines a class of functions wherein Feynman integrals arise as special cases, for any number of loops and kinematic scales. Utilizing the GKZ system and its relation to $D$-module theory, we propose a novel method for obtaining differential equations for master integrals. This note is based on the longer manuscript arXiv:2204.12983.
The relationship between Feynman diagrams and hypergeometric functions is discussed. Special attention is devoted to existing techniques for the construction of the $ε$-expansion. As an example, we present a detailed discussion of the construction of the epsilon-expansion of the Appell function $F_3$ around rational values of parameters via an iterative solution of differential equations. As a by-product, we have found that the one-loop massless pentagon diagram in dimension $d=3-2ε$ is not expressible in terms of multiple polylogarithms. Another interesting example is the Puiseux-type solutio…
In the Lee-Pomeransky representation, Feynman integrals can be identified as a subset of Euler-Mellin integrals, which are known to satisfy Gel'fand-Kapranov-Zelevinsky (GKZ) system of partial differential equations. Here we present an automated package to derive the associated GKZ system for a given Feynman diagram and solve it in terms of hypergeometric functions using two equivalent algorithms, namely the triangulation method and the Gröbner deformation method. We present our code in the form of a Mathematica package FeynGKZ.wl which requires the softwares polymake, Macaulay2 and TOPCOM, an…
We show that the Lee-Pomeransky parametric representation of Feynman integrals can be understood as a solution of a certain Gel'fand-Kapranov-Zelevinsky (GKZ) system. In order to define such GKZ system, we consider the polynomial obtained from the Symanzik polynomials $g=\mathcal{U}+\mathcal{F}$ as having indeterminate coefficients. Noncompact integration cycles can be determined from the coamoeba---the argument mapping---of the algebraic variety associated with $g$. In general, we add a deformation to $g$ in order to define integrals of generic graphs as linear combinations of their canonical…
In this thesis we will study Feynman integrals from the perspective of A-hypergeometric functions, a generalization of hypergeometric functions which goes back to Gelfand, Kapranov, Zelevinsky (GKZ) and their collaborators. This point of view was recently initiated by the works [74] and [150]. Inter alia, we want to provide here a concise summary of the mathematical foundations of A-hypergeometric theory in order to substantiate this viewpoint. This overview will concern aspects of polytopal geometry, multivariate discriminants as well as holonomic D-modules. As we will subsequently show, ever…
We review the hypergeometric function approach to Feynman diagrams. Special consideration is given to the construction of the Laurent expansion. As an illustration, we describe a collection of physically important one-loop vertex diagrams for which this approach is useful.
Several new relationships between hypergeometric functions are found by comparing results for Feynman integrals calculated using different methods. A new expression for the one-loop propagator-type integral with arbitrary masses and arbitrary powers of propagators is derived in terms of only one Appell hypergeometric function F_1. From the comparison of this expression with a previously known one, a new relation between the Appell functions F_1 and F_4 is found. By comparing this new expression for the case of equal masses with another known result, a new formula for reducing the F_1 function …
$\mathcal{A}$-hypergeometric functions and creation operators for Feynman and Witten diagrams
本文利用 GKZ/$\mathcal{A}$-超几何函数框架，从 Newton 多胞体谱奇点出发推导 Feynman 图与 Witten 图的产生算子，并给出仅含非导数顶点的新型全息权重转移关系。
The differential-reduction algorithm, which allows one to express generalized hypergeometric functions with parameters of arbitrary values in terms of such functions with parameters whose values differ from the original ones by integers, is discussed in the context of evaluating Feynman diagrams. Where this is possible, we compare our results with those obtained using standard techniques. It is shown that the criterion of reducibility of multiloop Feynman integrals can be reformulated in terms of the criterion of reducibility of hypergeometric functions. The relation between the numbers of mas…
GKZ hypergeometric systems of the four-loop vacuum Feynman integrals
本文基于Mellin-Barnes表示和Miller变换，构建了任意质量4-loop真空Feynman积分的GKZ超几何系统，从而在原点邻域内获得了其解析超几何解，并将其解析表达式表述为基本解系统的线性组合。
We show that almost all Feynman integrals as well as their coefficients in a Laurent series in dimensional regularization can be written in terms of Horn hypergeometric functions. By applying the results of Gelfand-Kapranov-Zelevinsky (GKZ) we derive a formula for a class of hypergeometric series representations of Feynman integrals, which can be obtained by triangulations of the Newton polytope $Δ_G$ corresponding to the Lee-Pomeransky polynomial $G$. Those series can be of higher dimension, but converge fast for convenient kinematics, which also allows numerical applications. Further, we dis…
The computational technique of $N$-fold Mellin-Barnes (MB) integrals, presented in a companion paper by the same authors, is used to derive sets of series representations of the massive one-loop conformal 3-point Feynman integral in various configurations. This shows the great simplicity and efficiency of the method in nonresonant cases (generic propagator powers) as well as some of its subtleties in the resonant ones (for unit propagator powers). We confirm certain results in the physics and mathematics literature and provide many new results, some of them dealing with the more general massiv…
Feynman integrals that have been evaluated in dimensional regularization can be written in terms of generalized hypergeometric functions. It is well known that properties of these functions are revealed in the framework of intersection theory. We propose a new application of intersection theory to construct a coaction on generalized hypergeometric functions. When applied to dimensionally regularized Feynman integrals, this coaction reproduces the coaction on multiple polylogarithms order by order in the parameter of dimensional regularization.
Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package {\tt Si…
Polytope symmetries of Feynman integrals
本文利用 A-超几何系统与 Newton 多胞体对称性，系统计算并分析 Lee-Pomeransky 表示中多种费曼积分的对称性。
Feynman integrals in two dimensions and single-valued hypergeometric functions
本文证明二维无质量传播子费曼积分可用单值 Aomoto-Gelfand 超几何函数表示，并将其应用于单圈积分与梯形积分，建立与 Lauricella 函数及广义超几何函数的联系。
The Gauss hypergeometric functions 2F1 with arbitrary values of parameters are reduced to two functions with fixed values of parameters, which differ from the original ones by integers. It is shown that in the case of integer and/or half-integer values of parameters there are only three types of algebraically independent Gauss hypergeometric functions. The epsilon-expansion of functions of one of this type (type F in our classification) demands the introduction of new functions related to generalizations of elliptic functions. For the five other types of functions the higher-order epsilon-expa…
Basing on the systems of linear partial differential equations derived from Mellin-Barnes representations and Miller's transformation, we obtain GKZ-hypergeometric systems of one-loop self energy, one-loop triangle, two-loop vacuum, and two-loop sunset diagrams, respectively. The codimension of derived GKZ-hypergeometric system equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. Taking GKZ-hypergeometric systems of one-loop self energy, massless one-loop triangle, and two-loop vacuum diagrams as examples, we present in detail how …
It is well known that Feynman integrals in dimensional regularization often evaluate to functions of hypergeometric type. Inspired by a recent proposal for a coaction on one-loop Feynman integrals in dimensional regularization, we use intersection numbers and twisted homology theory to define a coaction on certain hypergeometric functions. The functions we consider admit an integral representation where both the integrand and the contour of integration are associated with positive geometries. As in dimensionally-regularized Feynman integrals, endpoint singularities are regularized by means of …
We extend Gegenbauer Polynomials technique to evaluate a class of complicated Feynman diagrams. New results in the form of $_3F_2$-hypergeometrical series of unit argument, are presented. As a by-product, we present a new transformation rule for $_3F_2$-hypergeometric series with argument $-1$.
Gauss Relations in Feynman Integrals
本文提出利用格拉斯曼流形嵌入与高斯关系体系，系统延拓费曼积分的超几何表达式并导出其 $D=4$ 附近洛朗级数，通过单圈与双圈图例验证方法有效性。
Canonical differential equations for Feynman integrals from $\mathcal{A}$-hypergeometric systems in the Schwinger representation
本文提出从Schwinger表示中的GKZ超几何系统构造Feynman积分正则微分方程的方法，利用广义Euler积分和Frobenius基，所得正则基维数不超过master积分数量，实现微分系统的约化。
Quantum de Sitter Entropy and Sphere Partition Functions: A-Hypergeometric Approach to Higher Loop Corrections
本文提出用 A-超几何级数/GKZ 系统方法将球面背景上的费曼积分化为广义欧拉积分，从而计算德西特熵的高阶圈量子修正，并推广到矢量和一般自旋场。
The connection between Feynman integrals and GKZ $A$-hypergeometric systems has been a topic of recent interest with advances in mathematical techniques and computational tools opening new possibilities; in this paper we continue to explore this connection. To each such hypergeometric system there is an associated toric ideal, we prove that the latter has the Cohen-Macaulay property for two large families of Feynman integrals. This implies, for example, that both the number of independent solutions and dynamical singularities are independent of space-time dimension and generalized propagator p…
We explore the large set of linear transformations of Srivastava's $H_C$ triple hypergeometric function. This function has been recently linked to the massive one-loop conformal scalar 3-point Feynman integral. We focus here on the class of linear transformations of $H_C$ that can be obtained from linear transformations of the Gauss $_2F_1$ hypergeometric function and, as $H_C$ is also a three variable generalization of the Appell $F_1$ double hypergeometric function, from the particular linear transformation of $F_1$ known as Carlson's identity and some of its generalizations. These transform…
We calculate 3-loop master integrals for heavy quark correlators and the 3-loop QCD corrections to the $ρ$-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-$N$ space either. The solution of the homogeneous equations is possible in terms of convergent close integer power series as $_2F_1$ Gauß hypergeometric functions at rational argument. In some cases, integrals of this type can be mapped to complete elliptic integrals at rational argument. This class of functions appears to be the next one ari…
By symbolic summation methods based on difference fields we present a general strategy that transforms definite multi-sums, e.g., in terms of hypergeometric terms and harmonic sums, to indefinite nested sums and products. We succeeded in this task with all our concrete calculations of 2--loop and 3--loop massive single scale Feynman diagrams with local operator insertion.
It is known that one-loop Feynman integrals possess an algebraic structure encoding some of their analytic properties called the coaction, which can be written in terms of Feynman integrals and their cuts. This diagrammatic coaction, and the coaction on other classes of integrals such as hypergeometric functions, may be expressed using suitable bases of differential forms and integration contours. This provides a useful framework for computing coactions of Feynman integrals expressed using the hypergeometric functions. We will discuss examples where this technique has been used in the calculat…
HYPERDIRE is a project devoted to the creation of a set of Mathematica based programs for the differential reduction of hypergeometric functions. The current version includes two parts: the first one, FdFunction, for manipulations with Appell hypergeometric functions $F_D$ of $r$ variables; and the second one, FsFunction, for manipulations with Lauricella-Saran hypergeometric functions $F_S$ of three variables. Both functions are related with one-loop Feynman diagrams. The published version includes also Chapter 5 with two theorems about structure of coefficients of epsilon-expansion of the Ho…
A geometrical approach to the calculation of N-point Feynman diagrams is reviewed. It is shown that the geometrical splitting yields useful connections between Feynman integrals with different momenta and masses. It is demonstrated how these results can be used to reduce the number of variables in the occurring functions.
We consider the derivatives of Horn hypergeometric functions of any number variables with respect to their parameters. The derivative of the function in $n$ variables is expressed as a Horn hypergeometric series of $n+1$ infinite summations depending on the same variables and with the same region of convergence as for original Horn function. The derivatives of Appell functions, generalized hypergeometric functions, confluent and non-confluent Lauricella series and generalized Lauricella series are explicitly presented. Applications to the calculation of Feynman diagrams are discussed, especial…
Stable reduction methods will be important in the evaluation of high-order perturbative diagrams appearing in QCD and mixed QCD-electroweak radiative corrections at the LHC. Differential reduction techniques are useful for relating hypergeometric functions with shifted values of the parameters. We present a proposition relating the number of master integrals in the expansion of a Feynman diagram to the number of derivatives in a differential reduction.
We consider the two-loop massless propagator-type Feynman diagram with an arbitrary (non-integer) index on the central line. We analytically prove the equality of the two well-known results existing in the literature which express this diagram in terms of ${}_3F_2$-hypergeometric functions of argument $-1$ and $1$, respectively. We also derive new representations for this diagram which may be of importance in practical calculations.
We investigate a system of differential equations for the beta function of massless scalar $ϕ^4$ theory and continue the combinatorial investigation of the cut structure of Feynman diagrams.
We discuss the symmetry factors of Feynman diagrams of scalar field theories with polynomial potential. After giving a concise general formula for them, we present an elementary and direct proof that when computing scattering amplitudes using the homological perturbation lemma, each contributing Feynman diagram is indeed included with the correct symmetry factor.
Hypergeometric functions provide a useful representation of Feynman diagrams occuring in precision phenomenology. In dimension regularization, the epsilon-expansion of these functions about d=4 is required. We discuss the current status of differential reduction algorithms. As an illustration, we consider the construction of the all-order epsilon-expansion of the Appell hypergeometric function about integer values of the parameters and present an explicit evaluations of the first few terms.
We determine closed and compact expressions for the epsilon-expansion of certain Gaussian hypergeometric functions expanded around half-integer values by explicitly solving for their recurrence relations. This epsilon-expansion is identified with the normalized solution of the underlying Fuchs system of four regular singular points. We compute its regularized zeta series (giving rise to two independent associators) whose ratio gives the epsilon-expansion at a specific value. Furthermore, we use the well known one-loop massive bubble integral as an example to demonstrate how to obtain all-order…
Ramanujan Master Theorem is a technique developed by the indian mathematician S. Ramanujan to evaluate a class of definite integrals. This technique is used here to calculate the values of integrals associated with specific Feynman diagrams.
Results are presented for some infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-, two- and three-dimensional series. The sums of these series can be evaluated with the help of various integral representations for hypergeometric functions, and expressed in terms of $ζ(2)$, $ζ(3)$, the Catalan constant $G$ and ${\rm Cl}_2(π/3)$ where ${\rm Cl}_2(θ)$ is Clausen's function.
A number of irreducible master integrals for L-loop sunrise-type and bubble Feynman diagrams with generic values of masses and external momenta are explicitly evaluated via Mellin-Barnes representation.
We continue our study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we apply the approach of obtaining iteratated solutions to the differential equations associated with hypergeometric functions to prove the following result (Theorem 1): The epsilon-expansion of a generalized hypergeometric function with integer values of parameters is expressible in terms of generalized polylogarithms with coefficients that are ratios of polynomials. The method used in this proof provides an effici…
Application of the Meijer theorem in calculation of three-loop massive vacuum Feynman integrals and beyond
本文提出基于Mellin-Barnes表示并利用Meijer定理的解析方法，用于计算任意维数下的三圈有质量Feynman积分，并可推广至其他多圈积分。
We present the lattice structure of Feynman diagram renormalization in physical QFTs from the viewpoint of Dyson-Schwinger-Equations and the core Hopf algebra of Feynman diagrams. The lattice structure encapsules the nestedness of diagrams. This structure can be used to give explicit expressions for the counterterms in zero-dimensional QFTs using the lattice-Moebius function. Different applications for the tadpole-free quotient, in which all appearing elements correspond to semimodular lattices, are discussed.
Two-loop massive Feynman integrals for $ϕ^4$ field-theoretical model with long-range correlated disorder are considered. Massive integrals for the vertex function $Γ^{(4)}$ including two or three massless propagators for generic space dimension and for any value of the correlation parameter are evaluated analytically applying Mellin-Barnes method as well as familiar representation for one-loop integrals. Obtained expressions are presented in the form of hypergeometric functions.
We argue that the Mellin-Barnes representations of Feynman diagrams can be used for obtaining linear systems of homogeneous differential equations for the original Feynman diagrams with arbitrary powers of propagators without recourse to the integration-by-parts technique. These systems of differential equations can be used (i) for the differential reductions to sets of basic functions and (ii) for counting the numbers of master integrals.
Starting from the Mellin-Barnes integral representation of a Feynman integral depending on set of kinematic variables $z_i$, we derive a system of partial differential equations w.r.t.\ new variables $x_j$, which parameterize the differentiable constraints $z_i=y_i(x_j)$. In our algorithm, the powers of propagators can be considered as arbitrary parameters. Our algorithm can also be used for the reduction of multiple hypergeometric sums to sums of lower dimension, finding special values and reduction equations of hypergeometric functions in a singular locus of continuous variables, or finding …
Feynman Diagrams from Conformal Integrals
本文提出将含内部无质量场的动量空间费曼图表示为共形积分，从而把费曼图分类为共形族，并利用已知共形积分精确结果给出同族其他费曼图的全阶解。
This is a historical note. In 1979 we wrote a paper in a Russian Journal called Vestnik Leingradskogo Gosudarstvennogo Universiteta. We considered massive scalar quantum filed theory. One loop Feynman diagrams were evaluated. Theorem was proved that one loop diagram with many internal lines [more then dimension of space-time] can be expressed in terms of one loop diagram with number of internal lines equal to the dimension of space-time [multiplied by tree diagrams]. This is translation in English.
We find a Lax pair equation corresponding to the Connes-Kreimer Birkhoff factorization of the character group of a Hopf algebra. This flow preserves the locality of counterterms. In particular, we obtain a flow for the character given by Feynman rules, and relate this flow to the Renormalization Group Flow.
New explicit expressions are derived for the one-loop two-point Feynman integral with arbitrary external momentum and masses $m_1^2$ and $m_2^2$ in D dimensions. The results are given in terms of Appell functions, manifestly symmetric with respect to the masses $m_i^2$. Equating our expressions with previously known results in terms of Gauss hypergeometric functions yields reduction relations for the involved Appell functions that are apparently new mathematical results.
A new topological field theory is constructed, which is characterized by cubic interactions similar to those of non-abelian Chern-Simons field theories, but still retains the simplicity of the abelian case. The perturbative expansion of this theory contains in fact only two connected Feynman diagrams, the propagator and a three vertex. Apart from the Gauss linking number, the Wilson loop amplitudes generate a further topological invariant, whose physical and mathematical meaning is investigated.
We completely generalize previous results related to the counting of connected Feynman diagrams. We use a generating function approach, which encodes the Wick contraction combinatorics of the respective connected diagrams. Exact solutions are found for an arbitrary number of external legs, and a general algorithm is implemented for this calculus. From these solutions, we calculate many asymptotics expansion terms for a simple analytical tool (Taylor expansion theorem). Our approach offers new perspectives in the realm of Feynman diagrams enumeration (or zero-dimensional quantum field theory).
We show how to apply the BCFW recursion relation to Feynman loop integrals with the help of the Feynman-tree theorem. We deconstruct in this way all Feynman diagrams in terms of on-shell subamplitudes. Every cut originating from the Feynman-tree theorem corresponds to an integration over the phase space of an unobserved particle pair. We argue that we can calculate scattering amplitudes alternatively by the construction of on-shell and gauge-invariant subamplitudes.
Explicit formulae for asymptotic expansions of Feynman diagrams in typical limits of momenta and masses with external legs on the mass shell are presented.
We propose a general coaction for families of integrals appearing in the evaluation of Feynman diagrams, such as multiple polylogarithms and generalized hypergeometric functions. We further conjecture a link between this coaction and graphical operations on Feynman diagrams. At one-loop order, there is a basis of integrals for which this correspondence is fully explicit. We discuss features and present examples of the diagrammatic coaction on two-loop integrals. We also present the coaction for the functions ${}_{p+1}F_p$ and Appell $F_1$.
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