Analytical expressions of 3 and 4-loop sunrise Feynman integrals and 4-dimensional lattice integrals
S. Laporta
备注：
13 pages, 1 figure, LaTex; v2: some rephrasing of text; v3: reference added, minor modifications; v4: checks of lattice integrals up to 2400 digits; some modifications of text; version accepted for publication in IJMPA, needs the document class ws-ijmpa.cls
In this paper we continue the work begun in 2002 on the identification of the analytical expressions of Feynman integrals which require the evaluation of multiple elliptic integrals. We rewrite and simplify the analytical expression of the 3-loop self-mass integral with three equal masses and on-shell external momentum. We collect and analyze a number of results on double and triple elliptic integrals. By using very high-precision numerical fits, for the first time we are able to identify a very compact analytical expression for the 4-loop on-shell self-mass integral with 4 equal masses, that is one of the master integrals of the 4-loop electron g-2. Moreover, we fit the analytical expressions of some integrals which appear in lattice perturbation theory, and in particular the 4-dimensional generalized Watson integral.
A new class of identities for Feynman graph amplitudes, dubbed Schouten identities, valid at fixed integer value of the dimension d is proposed. The identities are then used in the case of the two loop sunrise graph with arbitrary masses for recovering the second order differential equation for the scalar amplitude in d=2 dimensions, as well as a chained sets of equations for all the coefficients of the expansions in (d-2). The shift from $d\approx2$ to $d\approx4$ dimensions is then discussed.
It is shown that the study of the imaginary part and of the corresponding dispersion relations of Feynman graph amplitudes within the differential equations method can provide a powerful tool for the solution of the equations, especially in the massive case. The main features of the approach are illustrated by discussing the simple cases of the 1-loop self-mass and of a particular vertex amplitude, and then used for the evaluation of the two-loop massive sunrise and the QED kite graph (the problem studied by Sabry in 1962), up to first order in the (d-4) expansion.
We review recently developed new powerful techniques to compute a class of Feynman diagrams at any loop order, known as sunrise-type diagrams. These sunrise-type topologies have many important applications in many different fields of physics and we believe it to be timely to discuss their evaluation from a unified point of view. The method is based on the analysis of the diagrams directly in configuration space which, in the case of the sunrise-type diagrams and diagrams related to them, leads to enormous simplifications as compared to the traditional evaluation of loops in momentum space. We …
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.
An explicit investigation about the equal-mass two-loop sunrise Feynman graph is performed. Such perturbative amplitude is related with many important physical process treated in the standard model context. The background of this investigation is an alternative strategy to handle with the divergences typical of perturbative solutions of quantum field theory. Since its proposition, the mentioned method was exhaustively used to calculate and manipulate one-loop Feynman integrals with a great success. However, the great advances in precision of experimental data collected in particle physics coll…
The system of 4 differential equations in the external invariant satisfied by the 4 master integrals of the general massive 2-loop sunrise self-mass diagram is solved by the Runge-Kutta method in the complex plane. The method, whose features are discussed in details, offers a reliable and robust approach to the direct and precise numerical evaluation of Feynman graph integrals.
The analytical values of the sunrise master amplitudes were found for one mass equal to zero, two other arbitrary masses and an arbitrary external four momentum. Differential equations in the square of the external four momentum and masses were used to obtain the master integrals.
We discuss the analytical solution of the two-loop sunrise graph with arbitrary non-zero masses in two space-time dimensions. The analytical result is obtained by solving a second-order differential equation. The solution involves elliptic integrals and in particular the solutions of the corresponding homogeneous differential equation are given by periods of an elliptic curve.
We review the main steps of the differential equation approach to the analytic evaluation of Feynman graphs, showing at the same time its application to the 3-loop sunrise graph in a particular kinematical configuration.
We study the analytic continuation of Feynman integrals from the kite family, expressed in terms of elliptic generalisations of (multiple) polylogarithms. Expressed in this way, the Feynman integrals are functions of two periods of an elliptic curve. We show that all what is required is just the analytic continuation of these two periods. We present an explicit formula for the two periods for all values of $t \in {\mathbb R}$. Furthermore, the nome $q$ of the elliptic curve satisfies over the complete range in $t$ the inequality $|q|\le 1$, where $|q|=1$ is attained only at the singular points…
In this paper we show that certain Feynman integrals can be expressed as linear combinations of iterated integrals of modular forms to all orders in the dimensional regularisation parameter $\varepsilon$ . We discuss explicitly the equal mass sunrise integral and the kite integral. For both cases we give the alphabet of letters occurring in the iterated integrals. For the sunrise integral we present a compact formula, expressing this integral to all orders in $\varepsilon$ as iterated integrals of modular forms.
The threshold behavior of the master amplitudes for two loop sunrise self-mass graph is studied by solving the system of differential equations, which they satisfy. The expansion at the threshold of the master amplitudes is obtained analytically for arbitrary masses.
In this talk, we discuss recent progress in the application of generalizations of polylogarithms in the symbolic computation of multi-loop integrals. We briefly review the Maple program MPL which supports a certain approach for the computation of Feynman integrals in terms of multiple polylogarithms. Furthermore we discuss elliptic generalizations of polylogarithms which have shown to be useful in the computation of the massive two-loop sunrise integral.
We show that configuration space techniques can be used to efficiently calculate the complete Laurent series \eps-expansion of sunrise-type diagrams to any loop order in D-dimensional space-time for any external momentum and for arbitrary mass configurations. For negative powers of \eps the results are obtained in analytical form. For positive powers of \eps including the finite \eps^0 contribution the result is obtained numerically in terms of low-dimensional integrals. We present general features of the calculation and provide exemplary results up to five loop order which are compared to ava…
We present a loop-by-loop method for computing the differential equations of Feynman integrals using the recently developed dual form formalism. We give explicit prescriptions for the loop-by-loop fibration of multi-loop dual forms. Then, we test our formalism on a simple, but non-trivial, example: the two-loop three-mass elliptic sunrise family of integrals. We obtain an epsilon-form differential equation within the correct function space in a sequence of relatively simple algebraic steps. In particular, none of these steps relies on the analysis of $q$-series. Then, we discuss interesting pr…
We introduce a class of iterated integrals that generalize multiple polylogarithms to elliptic curves. These elliptic multiple polylogarithms are closely related to similar functions defined in pure math- ematics and string theory. We then focus on the equal-mass and non-equal-mass sunrise integrals, and we develop a formalism that enables us to compute these Feynman integrals in terms of our iterated integrals on elliptic curves. The key idea is to use integration-by-parts identities to identify a set of integral kernels, whose precise form is determined by the branch points of the integral i…
We derive a second-order differential equation for the two-loop sunrise graph in two dimensions with arbitrary masses. The differential equation is obtained by viewing the Feynman integral as a period of a variation of a mixed Hodge structure, where the variation is with respect to the external momentum squared. The fibre is the complement of an elliptic curve. From the fact that the first cohomology group of this elliptic curve is two-dimensional we obtain a second-order differential equation. This is an improvement compared to the usual way of deriving differential equations: Integration-by-…
In these proceedings we discuss a representation for modular forms that is more suitable for their application to the calculation of Feynman integrals in the context of iterated integrals and the differential equation method. In particular, we show that for every modular form we can find a representation in terms of powers of complete elliptic integrals of the first kind multiplied by algebraic functions. We illustrate this result on several examples. In particular, we show how to explicitly rewrite elliptic multiple zeta values as iterated integrals over powers of complete elliptic integrals …
The system of 4 differential equations in the external invariant satisfied by the 4 master integrals of the general massive 2-loop sunrise self-mass diagram is solved by the Runge-Kutta method in the complex plane. The method offers a reliable and robust approach to the direct and precise numerical evaluation of Feynman graph integrals.
In this talk we discuss how ideas from the theory of mixed Hodge structures can be used to find differential equations for Feynman integrals. In particular we discuss the two-loop sunrise graph in two dimensions and show that these methods lead to a differential equation which is simpler than the ones obtained from integration-by-parts.
Resonance and Differential Reduction of Feynman Integrals
School of Mathematics and Hamilton Mathematics Institute, Trinity College（三一学院数学与汉密尔顿数学研究所）；Institute for Theoretical Physics, Utrecht University（乌得勒支大学理论物理研究所）；Center of Mathematical Sciences and Applications Harvard University（哈佛大学数学科学与应用中心）；Jefferson Physical Laboratory, Harvard University（哈佛大学杰斐逊物理实验室）
We present compact, fully analytical expressions for singular parts of a class of three-loop diagrams which cannot be factorized into lower-loop integrals. As a result of the calculations we obtain the analytical expression for the three-loop effective potential of the massive O(N) ϕ^4 model presented recently by J.-M.Chung and B.K.Chung, Phys.Rev. D56, 6508 (1997).
We review the Laporta algorithm for the reduction of scalar integrals to the master integrals and the differential equations technique for their evaluation. We discuss the use of the basis of harmonic polylogarithms for the analytical expression of the results and some generalization of this basis to wider sets of transcendental functions.
A framework to represent and compute two-loop $N$-point Feynman diagrams as double-integrals is discussed. The integrands are 'generalised one-loop type" multi-point functions multiplied by simple weighting factors. The final integrations over these two variables are to be performed numerically, whereas the ingredients involved in the integrands, in particular the "generalised one-loop type" functions, are computed analytically. The idea is illustrated on a few examples of scalar three- and four-point functions.
The bilinear combination of Dirac spinors $u(p_1,n_1)\bar u(p_2,n_2)$ is expressed in terms of Lorentz vectors in an explicit covariant form. The fact that the obtained expression involves only one auxiliary vector makes it very convenient for analytical computations with REDUCE (or FORM) package in the helicity formalism. The other advantage of the proposed formulas is that they are applicable to massive fermions as well as to massless fermions. The proposed approach is employed for the computation of one-loop Feynman diagrams and it is demonstrated that it considerably reduces the time of co…
The non-planar Feynman diagram with seven massless, scalar propagators and four on-shell legs (the crossed double box) is calculated analytically in dimensional regularization. The non-planar diagram with six propagators is also discussed.
We determine the numerical values of scalar multi-loop two-vertex Feynman diagrams, the generalized sunset diagrams, by integrating all but the longitudinal momenta analytically. For the longitudinal momenta we introduce one collective coordinate, which allows us to determine the numerical value of the diagram efficiently and to an arbitrary accuracy. The imaginary part and the threshold behavior of the diagram are also handled within this framework.
A method for the evaluation of the epsilon expansion of multi-loop massless Feynman integrals is introduced. This method is based on the Gegenbauer polynomial technique and the expansion of the Gamma function in terms of harmonic sums. Algorithms for the evaluation of nested and harmonic sums are used to reduce the expressions to get analytical or numerical results for the expansion coefficients. Methods to increase the precision of numerical results are discussed.
An asymptotic expansion of the two-loop two-point ``master'' diagram with two masses $m$ and $M$, on the mass shell $Q^2=M^2$, is presented. The treatment of the non-analytical terms arising in the expansion around the branching point is discussed. Some details of the calculation of a new class of two-loop integrals are given.
The status of analytical evaluation of double and triple box diagrams is characterized. The method of Mellin-Barnes representation as a tool to evaluate master integrals in these problems is advocated. New MB representations for massive on-shell double boxes with general powers of propagators are presented.
The dimensionally regularized massless on-shell planar triple box Feynman diagram with powers of propagators equal to one is analytically evaluated for general values of the Mandelstam variables s and t in a Laurent expansion in the parameter \ep=(4-d)/2 of dimensional regularization up to a finite part. An explicit result is expressed in terms of harmonic polylogarithms, with parameters 0 and 1, up to the sixth order. The evaluation is based on the method of Feynman parameters and multiple Mellin-Barnes representation. The same technique can be quite similarly applied to planar triple boxes w…
An algorithm for the systematic analytical approximation of multi-scale Feynman integrals is presented. The algorithm produces algebraic expressions as functions of the kinematical parameters and mass scales appearing in the Feynman integrals, allowing for fast numerical evaluation. The results are valid in all kinematical regions, both above and below thresholds, up to in principle arbitrary orders in the dimensional regulator. The scope of the algorithm is demonstrated by presenting results for selected two-loop three-point and four-point integrals with an internal mass scale that appear in …
We analytically evaluate the three-loop Feynman integral which was the last missing ingredient for the analytical evaluation of the three-loop quark static potential. To evaluate the integral we introduce an auxiliary parameter $y$, which corresponds to the residual energy in some of the HQET propagators. We construct a differential system for 109 master integrals depending on $y$ and fix boundary conditions from the asymptotic behaviour in the limit $y\to \infty$. The original integral is recovered from the limit $y\to 0$. To solve these linear differential equations we try to find an $ε$-for…
We present an algorithm to evaluate multiloop Feynman integrals with an arbitrary number of internal massive lines, with the masses being in general complex-valued, and its implementation in the \textsc{Mathematica} package \textsc{SeaSyde}. The implementation solves by series expansions the system of differential equations satisfied by the Master Integrals. At variance with respect to other existing codes, the analytical continuation of the solution is performed in the complex plane associated to each kinematical invariant. We present the results of the evaluation of the Master Integrals rele…
In this paper, we study systematically scalar one-loop two-, three-, and four-point Feynman integrals with complex internal masses. Our analytic results presented in this report are valid for both real and complex internal masses. The calculations are then implemented into a {\tt Mathematica} (version $9$) package and {\tt FORTRAN} program. Our program is cross-checked numerically with {\tt LoopTools} (version $2.14$) in real as well as complex internal masses. We find a perfect agreement between our results and {\tt LoopTools} for all cases. Additionally, this work is applied for evaluating s…
The differential equation in the external invariant p^2 satisfied by the master integral of the general massive 2-loop 4-denominator self-mass diagram is exploited and the expansion of the master integral at p^2=0 is obtained analytically. The system composed by this differential equation with those of the master integrals related to the general massive 2-loop sunrise diagram is numerically solved by the Runge-Kutta method in the complex p^2 plane. A numerical method to obtain results for values of p^2 at and close to thresholds and pseudo-thresholds is discussed in details.
A FORM based package (ON-SHELL2) for the calculation of two loop self-energy diagrams with one nonzero mass in internal lines and the external momentum on the same mass shell is elaborated. The algorithm, based on recurrence relations obtained from the integration-by-parts method, allows us to reduce diagrams with arbitrary indices (powers of scalar propagators) to a set of master integrals. The SHELL2 package is used for the calculation of special types of diagrams. Analytical results for master integrals are collected.
Elliptic multiple polylogarithms occur in Feynman integrals and in particular in scattering amplitudes. They can be characterized by their symbol, a tensor product in the so-called symbol letters. In contrast to the non-elliptic case, the elliptic letters themselves satisfy highly non-trivial identities, which we discuss in this paper. Moreover, we introduce the symbol prime, an analog of the symbol for elliptic symbol letters, which makes these identities manifest. We demonstrate its use in two concrete examples at two-loop order: the unequal-mass sunrise integral in two dimensions and the te…
Based on the method in Refs.~{\tt [D.~Kreimer, Z.\ Phys.\ C {\bf 54} (1992) 667} and {\tt Int.\ J.\ Mod.\ Phys.\ A {\bf 8} (1993) 1797]}, we present analytic results for scalar one-loop four-point Feynman integrals with complex internal masses. The results are not only valid for complex internal masses, but also for real internal mass cases. Different from the traditional approach proposed by G. 't Hooft and M. Veltman in the paper {\tt[Nucl.\ Phys.\ B {\bf 153} (1979) 365]}, this method can be extended to evaluate tensor integrals directly. Therefore, it may open a new approach to cure the in…
Based on the Baikov representation, we present a systematic approach to compute cuts of Feynman Integrals, appropriately defined in $d$ dimensions. The information provided by these computations may be used to determine the class of functions needed to analytically express the full integrals.
The method of the large mass expansion (LME) has the technical advantage that two-loop integrals occur only as bubbles with large masses. In many cases only one large mass occurs. In such cases these integrals are expressible in terms of $Γ$-functions, i.e. they can be handled completely analytically avoiding even recursions and therefore this approach may find a wide field of application. We consider it necessary to investigate the precision of this method and test it for several two-loop vertex functions ocurring in the $Z \to b\bar{b}$ decay by comparing it with the small momentum expansion…
Collisions at the LHC produce many-particle final states, and for precise predictions the one-loop $N$-point corrections are needed. We study here the tensor reduction for Feynman integrals with $N \ge 6$. A general, recursive solution by Binoth et al. expresses $N$-point Feynman integrals of rank $R$ in terms of $(N-1)$-point Feynman integrals of rank $(R-1)$ (for $N\ge 6$). We show that the coefficients can be obtained analytically from suitable representations of the metric tensor. Contractions of the tensor integrals with external momenta can be efficiently expressed as well. We consider o…
By introducing an auxiliary parameter, we find a new representation for Feynman integrals, which defines a Feynman integral by analytical continuation of a series containing only vacuum integrals. The new representation therefore conceptually translates the problem of computing Feynman integrals to the problem of performing analytical continuations. As an application of the new representation, we use it to construct a novel reduction method for multi-loop Feynman integrals, which is expected to be more efficient than known integration-by-parts reduction method. Using the new method, we success…
Problems occurring in physically important non-trivial examples of loop calculations are discussed. A procedure of deriving expansions of two-loop self-energy diagrams with different masses is constructed. The cases of small and large external momentum are considered. The coefficients of the expansions are calculated analytically. Comparison with numerical results shows good agreement below the first threshold and above the last threshold of the diagram. An approach to evaluating three- and four-point ladder diagrams with massless internal particles and off-shell external momenta is presented.…
We define linearly reducible elliptic Feynman integrals, and we show that they can be algorithmically solved up to arbitrary order of the dimensional regulator in terms of a 1-dimensional integral over a polylogarithmic integrand, which we call the inner polylogarithmic part (IPP). The solution is obtained by direct integration of the Feynman parametric representation. When the IPP depends on one elliptic curve (and no other algebraic functions), this class of Feynman integrals can be algorithmically solved in terms of elliptic multiple polylogarithms (eMPLs) by using integration by parts iden…
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…
