Gabriel Baditoiu, Steven Rosenberg
We find a Lax pair equation corresponding to the Connes-Kreimer Birkhoff factorization of the character group of the Hopf algebra of Feynman diagrams. In particular, we obtain a flow for the character given by Feynman rules, and present a worked example.
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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.
This paper gives a review of Connes-Kreimer formulation of perturbative renormalization in Quantum Field Theory. We begin with the derivation of the Feynman calculus, the Hopf algebra structure on Feynman diagrams and we show the natural occurence of Birkhoff decomposition in perturbative renormalization. Most of proofs is added in the present paper.
For a given base space $M$ (spacetime), we consider the Guichardet space over the Guichardet space over $M$. Here we develop a ''field calculus'' based on the Guichardet integral. This is the natural setting in which to describe Green function relations for Boson systems. Here we can follow the suggestion of Schwinger and develop a differential (local field) approach rather than the integral one pioneered by Feynman. This is helped by a DEFG (Dyson-Einstein-Feynman-Guichardet) shorthand which greatly simplifies expressions. This gives a convenient framework for the formal approach of Schwinger…
We obtain Hurwitz numbers as the number of Feynman diagrams of a certain type divided by the order of the automorphism group of the diagram.
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.
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.
From the perturbative expansion of the exact Green function, an exact counting formula is derived to determine the number of different types of connected Feynman diagrams. This formula coincides with the Arquès-Walsh sequence formula in the rooted map theory, supporting the topological connection between Feynman diagrams and rooted maps. A classificatory summing-terms approach is used, in connection to discrete mathematical theory.
Electron-photon interaction: Feynman diagrams and contact points
本文为费曼图定义了正问题、反问题及接触点的概念，针对电子-光子相互作用给出了正反问题的解，并讨论了费曼图与经典着色图之间的关系。
We look at a graph property called reducibility which is closely related to a condition developed by Brown to evaluate Feynman integrals. We show for graphs with a fixed number of external momenta, that reducibility with respect to both Symanzik polynomials is graph minor closed. We also survey the known forbidden minors and the known structural results. This gives some structural information on those Feynman diagrams which are reducible.
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.
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.
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.
The generating series of the intersection numbers of the stable cohomology classes on moduli spaces of curves satisfies the string equation and a KdV hierarchy. Kontsevich's original proof of this result uses a matrix model and the matrix Airy equation. Witten then recasted Kontsevich's results in terms of Virasoro algebras, by means of an ingenious mixture of Feynman diagrams techniques and integrations by parts with some "rather formidable choice" of the integrands. In this note we show how Witten's formidable choices can be bartered for standard Feynman diagram manipulations, as soon as one…
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.
This paper recalls the development of gauge theory culminating in Yang-Mills theory, and the application of differential geometry including connections on fiber bundles to field theory. Finally, we see how the preceding is used to explain the Feynman diagrams appearing on the Feynman postage stamp released in May 2005. Version 2 included the Feynman diagrams, Version 3 corrected typos and Version 4 included an appendix for the derivation of the Yang-Mills transformation and field strength. Version 5 indicates that the article has been published in the Notices of the AMS and in July 2009 appear…
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 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 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).
Many-body perturbation theory (MBPT) is widely used in quantum physics, chemistry, and materials science. At the heart of MBPT is the Feynman diagrammatic expansion, which is, simply speaking, an elegant way of organizing the combinatorially growing number of terms of a certain Taylor expansion. In particular, the construction of the `bold Feynman diagrammatic expansion' involves the partial resummation to infinite order of possibly divergent series of diagrams. This procedure demands investigation from both the combinatorial (perturbative) and the analytical (non-perturbative) viewpoints. In …
This is a simple mathematical introduction into Feynman diagram technique, which is a standard physical tool to write perturbative expansions of path integrals near a critical point of the action. I start from a rigorous treatment of a finite dimensional case (which actually belongs more to multivariable calculus than to physics), and then use a simple "dictionary" to translate these results to an infinite dimensional case. The standard methods such as gauge-fixing and Faddeev-Popov ghosts are also included. Resulting Feynman diagram series often may be used rigorously without any references t…
The Rooted Maps Theory, a branch of the Theory of Homology, is shown to be a powerful tool for investigating the topological properties of Feynman diagrams, related to the single particle propagator in the quantum many-body systems. The numerical correspondence between the number of this class of Feynman diagrams as a function of perturbative order and the number of rooted maps as a function of the number of edges is studied. A graphical procedure to associate Feynman diagrams and rooted maps is then stated. Finally, starting from rooted maps principles, an original definition of the genus of …
It is well-known that the symmetry group of a Feynman diagram can give important information on possible strategies for its evaluation, and the mathematical objects that will be involved. Motivated by ongoing work on multi-loop multi-photon amplitudes in quantum electrodynamics, here I will discuss the usefulness of introducing a polynomial basis of invariants of the symmetry group of a diagram in Feynman-Schwinger parameter space.
We study differential equations for Feynman amplitudes and we show that the corresponding D-module is isomorphic to a GKZ D-modules. We show that the sheaf of solutions to the D-module is isomorphic to a certain relative homology and the amplitudes are periods of a relative motive. Using these ideas, we develop a method of regularization which specializes to dimensional regularization and analytic regularization.
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.
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.
In this work, we generalize a recursive enumerative formula for connected Feynman diagrams with two external legs. The Feynman diagrams are defined from a fermionic gas with a two-body interaction. The generalized recurrence is valid for connected Feynman diagrams with an arbitrary number of external legs and an arbitrary order. The recurrence formula terms are expressed in function of weak compositions of non-negative integers and partitions of positive integers in such a way that to each term of the recurrence correspond a partition and a weak composition. The foundation of this enumeration …
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.
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…
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 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.
Vanishing cycles and analysis of singularities of Feynman diagrams
本研究利用Mayer-Vietoris谱序列分析费曼圈积分的消失圈并完成分类，推导了奇点附近渐近展开的显式系数，为轻子-强子散射实验辐射修正中的高阶费曼图精确计算提供了框架基础。
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.
Some problems related to construction of the epsilon-expansion of dimensionally regulated Feynman integrals are discussed. For certain classes of diagrams, an arbitrary term of the epsilon-expansion can be expressed in terms of log-sine integrals related to the polylogarithms. It is shown how the analytic continuation of these functions can be constructed in terms of the generalized Nielsen polylogarithms.
Gauge Theory Amplitudes from Cubic Scalar Feynman diagrams
本文通过Witten开弦场论在场论极限中揭示了规范理论振幅与标量量子场论三价费曼图重整化积分元的联系。
New algebraic approach to analytical calculations of D-dimensional integrals for multi-loop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of multi-loop Feynman integrals, such as integration by parts and star-triangle relation methods, can be drastically simplified by using this algebraic approach. To demonstrate the advantages of the algebraic method of analytical evaluation of multi-loop Feynman diagrams, we calculate ladder diagrams for the massless $ϕ^3$ theory. Using our algebraic approach we show that the problem of evaluation of special classes of…
Numerators in Parametric Representations of Feynman Diagrams
本文利用 Dodgson 恒等式简化费曼图参数表示中分子的处理，并在无质量情形下通过约化为真空图补全，显著降低分母中 Symanzik 多项式的最大幂次。
Renormalising Feynman diagrams with multi-indices
本文提出用多重指标构造新的 Hopf 代数，直接对原始测度展开式进行 BPHZ 重整化以获得量子场论重整化测度，并以 Φ⁴ 测度为例验证方法。
We describe the "Feynman diagram" approach to nonrelativistic quantum mechanics on R^n, with magnetic and potential terms. In particular, for each classical path γconnecting points q_0 and q_1 in time t, we define a formal power series V_γ(t,q_0,q_1) in \hbar, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V_γ) satisfies Schrödinger's equation, and explain in what sense the t\to 0 limit approaches the δdistribution. As such, our construction gives explicitly the full \hbar\to 0 asymptotics of the fundamental solution to…
A classical longstanding open problem in statistics is finding an explicit expression for the probability measure which maximizes entropy with respect to given constraints. In this paper a solution to this problem is found, using perturbative Feynman calculus. The explicit expression is given as a sum over weighted trees.
We briefly discuss the transcendental constants generated through the epsilon-expansion of generalized hypergeometric functions and their interrelation with the "sixth root of unity."
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.
Matroidal polynomials, their singularities, and applications to Feynman diagrams
该研究针对拟阵与拟阵旗引入满足删除-收缩恒等式的三类多项式族，证明了正特征下齐次拟阵多项式的强$F$-正则性，以及零特征域上相关jet概形的不可约性，得出这类多项式均有有理奇点或光滑的结论，可应用于费曼图研究。
Combinatorial summation of Feynman diagrams: Equation of state of the 2D SU(N) Hubbard model
本研究提出基于动态规划的费曼图高效组合求和通用框架，经典计算成本为图阶指数级、量子计算有望达多项式级，并通过二维SU(N) Hubbard模型物态方程的无偏图蒙特卡罗计算验证了其有效性。
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