Journal articles: 'Multi-loop Feynman integrals'

1

Smirnov, Vladimir A., and Matthias Steinhauser. "Solving recurrence relations for multi-loop Feynman integrals." Nuclear Physics B 672, no. 1-2 (November 2003): 199–221. http://dx.doi.org/10.1016/j.nuclphysb.2003.09.003.

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Isaev, A. P. "Multi-loop Feynman integrals and conformal quantum mechanics." Nuclear Physics B 662, no. 3 (July 2003): 461–75. http://dx.doi.org/10.1016/s0550-3213(03)00393-6.

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3

Baikov, P. A. "Criterion of irreducibility of multi-loop Feynman integrals." Physics Letters B 474, no. 3-4 (February 2000): 385–88. http://dx.doi.org/10.1016/s0370-2693(00)00053-8.

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4

Zhou, Yajun. "Wick rotations, Eichler integrals, and multi-loop Feynman diagrams." Communications in Number Theory and Physics 12, no. 1 (2018): 127–92. http://dx.doi.org/10.4310/cntp.2018.v12.n1.a5.

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5

Aguilera-Verdugo, José de Jesús, Félix Driencourt-Mangin, Roger José Hernández-Pinto, Judith Plenter, Renato Maria Prisco, Norma Selomit Ramírez-Uribe, Andrés Ernesto Rentería-Olivo, et al. "A Stroll through the Loop-Tree Duality." Symmetry 13, no. 6 (June 8, 2021): 1029. http://dx.doi.org/10.3390/sym13061029.

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The Loop-Tree Duality (LTD) theorem is an innovative technique to deal with multi-loop scattering amplitudes, leading to integrand-level representations over a Euclidean space. In this article, we review the last developments concerning this framework, focusing on the manifestly causal representation of multi-loop Feynman integrals and scattering amplitudes, and the definition of dual local counter-terms to cancel infrared singularities.

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Kastening, Boris, and Hagen Kleinert. "Efficient algorithm for perturbative calculation of multi-loop Feynman integrals." Physics Letters A 269, no. 1 (April 2000): 50–54. http://dx.doi.org/10.1016/s0375-9601(00)00199-7.

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7

Baikov, P. A. "A practical criterion of irreducibility of multi-loop Feynman integrals." Physics Letters B 634, no. 2-3 (March 2006): 325–29. http://dx.doi.org/10.1016/j.physletb.2006.01.052.

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Preti, Michelangelo. "STR: A Mathematica package for the method of uniqueness." International Journal of Modern Physics C 31, no. 10 (September 16, 2020): 2050146. http://dx.doi.org/10.1142/s0129183120501466.

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We present Star–Triangle Relations (STRs), a Mathematica® package designed to solve Feynman diagrams by means of the method of uniqueness in any Euclidean space-time dimension. The method of uniqueness is a powerful technique to solve multi-loop Feynman integrals in theories with conformal symmetry imposing some relations between the powers of propagators and the space-time dimension. In our algorithm, we include both identities for scalar and Yukawa type integrals. The package provides a graphical environment in which it is possible to draw the desired diagram with the mouse input and a set of tools to modify and compute it. Throughout the use of a graphic interface, the package should be easily accessible to users with little or no previous experience on diagrams computation. This manual includes some pedagogical examples of computation of Feynman graphs as the scalar two-loop kite master integral and a fermionic diagram appearing in the computation of the spectrum of the [Formula: see text]-deformed [Formula: see text] SYM in the double scaling limit.

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Jurčišinová, E., and M. Jurčišin. "A general formula for analytic reduction of multi-loop tensor Feynman integrals." Physics Letters B 692, no. 1 (August 2010): 57–60. http://dx.doi.org/10.1016/j.physletb.2010.07.018.

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10

Doncker, E. de, F. Yuasa, and R. Assaf. "Multi-threaded adaptive extrapolation procedure for Feynman loop integrals in the physical region." Journal of Physics: Conference Series 454 (August 12, 2013): 012082. http://dx.doi.org/10.1088/1742-6596/454/1/012082.

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11

Motoki, S., H. Daisaka, N. Nakasato, T. Ishikawa, F. Yuasa, T. Fukushige, A. Kawai, and J. Makino. "A development of an accelerator board dedicated for multi-precision arithmetic operations and its application to Feynman loop integrals." Journal of Physics: Conference Series 608 (May 22, 2015): 012011. http://dx.doi.org/10.1088/1742-6596/608/1/012011.

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Daisaka, H., N. Nakasato, T. Ishikawa, F. Yuasa, and K. Nitadori. "A development of an accelerator board dedicated for multi-precision arithmetic operations and its application to Feynman loop integrals II." Journal of Physics: Conference Series 1085 (September 2018): 052004. http://dx.doi.org/10.1088/1742-6596/1085/5/052004.

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13

Winterhalder, Ramon, Vitaly Magerya, Emilio Villa, Stephen Jones, Matthias Kerner, Anja Butter, Gudrun Heinrich, and Tilman Plehn. "Targeting multi-loop integrals with neural networks." SciPost Physics 12, no. 4 (April 13, 2022). http://dx.doi.org/10.21468/scipostphys.12.4.129.

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Numerical evaluations of Feynman integrals often proceed via a deformation of the integration contour into the complex plane. While valid contours are easy to construct, the numerical precision for a multi-loop integral can depend critically on the chosen contour. We present methods to optimize this contour using a combination of optimized, global complex shifts and a normalizing flow. They can lead to a significant gain in precision.

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14

He, Song, Zhenjie Li, Yichao Tang, and Qinglin Yang. "The Wilson-loop d log representation for Feynman integrals." Journal of High Energy Physics 2021, no. 5 (May 2021). http://dx.doi.org/10.1007/jhep05(2021)052.

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Abstract We introduce and study a so-called Wilson-loop d log representation of certain Feynman integrals for scattering amplitudes in $$ \mathcal{N} $$ N = 4 SYM and beyond, which makes their evaluation completely straightforward. Such a representation was motivated by the dual Wilson loop picture, and it can also be derived by partial Feynman parametrization of loop integrals. We first introduce it for the simplest one-loop examples, the chiral pentagon in four dimensions and the three-mass-easy hexagon in six dimensions, which are represented by two- and three-fold d log integrals that are nicely related to each other. For multi-loop examples, we write the L-loop generalized penta-ladders as 2(L − 1)-fold d log integrals of some one-loop integral, so that once the latter is known, the integration can be performed in a systematic way. In particular, we write the eight-point penta-ladder as a 2L-fold d log integral whose symbol can be computed without performing any integration; we also obtain the last entries and the symbol alphabet of these integrals. Similarly we study the symbol of the seven-point double-penta-ladder, which is represented by a 2(L − 1)-fold integral of a hexagon; the latter can be written as a two-fold d log integral plus a boundary term. We comment on the relation of our representation to differential equations and resumming the ladders by solving certain integral equations.

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Giroux, Mathieu, and Andrzej Pokraka. "Loop-by-loop differential equations for dual (elliptic) Feynman integrals." Journal of High Energy Physics 2023, no. 3 (March 21, 2023). http://dx.doi.org/10.1007/jhep03(2023)155.

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Abstract 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 ε-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 properties satisfied by our dual basis as well as its simple relation to the known ε-form basis of Feynman integrands. The underlying K3-geometry of the three-loop four-mass sunrise integral is also discussed. Finally, we speculate on how to construct a “good” loop-by-loop basis at three-loop.

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16

Bönisch, Kilian, Claude Duhr, Fabian Fischbach, Albrecht Klemm, and Christoph Nega. "Feynman integrals in dimensional regularization and extensions of Calabi-Yau motives." Journal of High Energy Physics 2022, no. 9 (September 20, 2022). http://dx.doi.org/10.1007/jhep09(2022)156.

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Abstract We provide a comprehensive summary of concepts from Calabi-Yau motives relevant to the computation of multi-loop Feynman integrals. From this we derive several consequences for multi-loop integrals in general, and we illustrate them on the example of multi-loop banana integrals. For example, we show how Griffiths transversality, known from the theory of variation of mixed Hodge structures, leads quite generically to a set of quadratic relations among maximal cut integrals associated to Calabi-Yau motives. These quadratic relations then naturally lead to a compact expression for l-loop banana integrals in D = 2 dimensions in terms of an integral over a period of a Calabi-Yau (l − 1)-fold. This new integral representation generalizes in a natural way the known representations for l ≤ 3 involving logarithms with square root arguments and iterated integrals of Eisenstein series. In a second part, we show how the results obtained by some of the authors in earlier work can be extended to dimensional regularization. We present a method to obtain the differential equations for banana integrals with an arbitrary number of loops in dimensional regularization without the need to solve integration-by-parts relations. We also present a compact formula for the leading asymptotics of banana integrals with an arbitrary number of loops in the large momentum limit. This generalizes the novel $$ \hat{\Gamma} $$ Γ ̂ -class introduced by some of the authors to dimensional regularization and provides a convenient boundary condition to solve the differential equations for the banana integrals. As an application, we present for the first time numerical results for equal-mass banana integrals with up to four loops and up to second order in the dimensional regulator.

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Mandal, Manoj K., and Xiaoran Zhao. "Evaluating multi-loop Feynman integrals numerically through differential equations." Journal of High Energy Physics 2019, no. 3 (March 2019). http://dx.doi.org/10.1007/jhep03(2019)190.

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18

von Manteuffel, Andreas, Erik Panzer, and Robert M. Schabinger. "A quasi-finite basis for multi-loop Feynman integrals." Journal of High Energy Physics 2015, no. 2 (February 2015). http://dx.doi.org/10.1007/jhep02(2015)120.

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19

Görges, Lennard, Christoph Nega, Lorenzo Tancredi, and Fabian J. Wagner. "On a procedure to derive ϵ-factorised differential equations beyond polylogarithms." Journal of High Energy Physics 2023, no. 7 (July 26, 2023). http://dx.doi.org/10.1007/jhep07(2023)206.

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Abstract In this manuscript, we elaborate on a procedure to derive ϵ-factorised differential equations for multi-scale, multi-loop classes of Feynman integrals that evaluate to special functions beyond multiple polylogarithms. We demonstrate the applicability of our approach to diverse classes of problems, by working out ϵ-factorised differential equations for single- and multi-scale problems of increasing complexity. To start we are reconsidering the well-studied equal-mass two-loop sunrise case, and move then to study other elliptic two-, three- and four-point problems depending on multiple different scales. Finally, we showcase how the same approach allows us to obtain ϵ-factorised differential equations also for Feynman integrals that involve geometries beyond a single elliptic curve.

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20

Frellesvig, Hjalte, Federico Gasparotto, Stefano Laporta, Manoj K. Mandal, Pierpaolo Mastrolia, Luca Mattiazzi, and Sebastian Mizera. "Decomposition of Feynman integrals by multivariate intersection numbers." Journal of High Energy Physics 2021, no. 3 (March 2021). http://dx.doi.org/10.1007/jhep03(2021)027.

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AbstractWe present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers, and provide three different approaches for a direct decomposition of Feynman integrals, which we dub thestraight decomposition, thebottom-up decomposition, and thetop-down decomposition. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.

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21

He, Song, Zhenjie Li, Rourou Ma, Zihao Wu, Qinglin Yang, and Yang Zhang. "A study of Feynman integrals with uniform transcendental weights and their symbology." Journal of High Energy Physics 2022, no. 10 (October 26, 2022). http://dx.doi.org/10.1007/jhep10(2022)165.

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Abstract Multi-loop Feynman integrals are key objects for the high-order correction computations in high energy phenomenology. These integrals with multiple scales may have complicated symbol structures, and we show that twistor geometries of closely related dual conformal integrals shed light on their alphabet and symbol structures. In this paper, first, as a cutting-edge example, we derive the two-loop four-external-mass Feynman integrals with uniform transcendental (UT) weights, based on the latest developments on UT integrals. Then we find that all the symbol letters of these integrals can be explained non-trivially by studying the so-called Schubert problem of certain dual conformal integrals with a point at infinity. Certain properties of the symbol such as first two entries and extended Steinmann relations are also studied from analogous properties of dual conformal integrals.

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22

Meyer, Christoph. "Transforming differential equations of multi-loop Feynman integrals into canonical form." Journal of High Energy Physics 2017, no. 4 (April 2017). http://dx.doi.org/10.1007/jhep04(2017)006.

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23

Aguilera-Verdugo, J. Jesús, Roger J. Hernández-Pinto, Germán Rodrigo, German F. R. Sborlini, and William J. Torres Bobadilla. "Causal representation of multi-loop Feynman integrands within the loop-tree duality." Journal of High Energy Physics 2021, no. 1 (January 2021). http://dx.doi.org/10.1007/jhep01(2021)069.

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Abstract The numerical evaluation of multi-loop scattering amplitudes in the Feynman representation usually requires to deal with both physical (causal) and unphysical (non-causal) singularities. The loop-tree duality (LTD) offers a powerful framework to easily characterise and distinguish these two types of singularities, and then simplify analytically the underling expressions. In this paper, we work explicitly on the dual representation of multi-loop Feynman integrals generated from three parent topologies, which we refer to as Maximal, Next-to-Maximal and Next-to-Next-to-Maximal loop topologies. In particular, we aim at expressing these dual contributions, independently of the number of loops and internal configurations, in terms of causal propagators only. Thus, providing very compact and causal integrand representations to all orders. In order to do so, we reconstruct their analytic expressions from numerical evaluation over finite fields. This procedure implicitly cancels out all unphysical singularities. We also interpret the result in terms of entangled causal thresholds. In view of the simple structure of the dual expressions, we integrate them numerically up to four loops in integer space-time dimensions, taking advantage of their smooth behaviour at integrand level.

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24

Blümlein, J., M. Saragnese, and C. Schneider. "Hypergeometric structures in Feynman integrals." Annals of Mathematics and Artificial Intelligence, April 3, 2023. http://dx.doi.org/10.1007/s10472-023-09831-8.

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AbstractFor the precision calculations in perturbative Quantum Chromodynamics (QCD) gigantic expressions (several GB in size) in terms of highly complicated divergent multi-loop Feynman integrals have to be calculated analytically to compact expressions in terms of special functions and constants. In this article we derive new symbolic tools to gain large-scale computer understanding in QCD. Here we exploit the fact that 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 in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton–type functions are considered. We illustrate the algorithms by examples.

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Guillet, J. Ph, E. Pilon, Y. Shimizu, and M. S. Zidi. "Framework for a novel mixed analytical/numerical approach for the computation of two-loop N-point Feynman diagrams." Progress of Theoretical and Experimental Physics 2020, no. 4 (April 1, 2020). http://dx.doi.org/10.1093/ptep/ptaa020.

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Abstract 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.

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Blümlein, Johannes, and Carsten Schneider. "The SAGEX Review on scattering amplitudes Chapter 4: Multi-loop Feynman integrals." Journal of Physics A: Mathematical and Theoretical, July 12, 2022. http://dx.doi.org/10.1088/1751-8121/ac8086.

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Abstract The analytic integration and simplification of multi-loop Feynman integrals to special functions and constants plays an important role to perform higher order perturbative calculations in the Standard Model of elementary particles. In this survey article the most recent and relevant computer algebra and special function algorithms are presented that are currently used or that may play an important role to perform such challenging precision calculations in the future. They are discussed in the context of analytic zero, single and double scale calculations in the Quantum Field Theories of the Standard Model and effective field theories, also with classical applications. These calculations play a central role in the analysis of precision measurements at present and future colliders to obtain ultimate information for fundamental physics.

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Dlapa, Christoph, Johannes M. Henn, and Fabian J. Wagner. "An algorithmic approach to finding canonical differential equations for elliptic Feynman integrals." Journal of High Energy Physics 2023, no. 8 (August 18, 2023). http://dx.doi.org/10.1007/jhep08(2023)120.

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Abstract In recent years, differential equations have become the method of choice to compute multi-loop Feynman integrals. Whenever they can be cast into canonical form, their solution in terms of special functions is straightforward. Recently, progress has been made in understanding the precise canonical form for Feynman integrals involving elliptic polylogarithms. In this article, we make use of an algorithmic approach that proves powerful to find canonical forms for these cases. To illustrate the method, we reproduce several known canonical forms from the literature and present examples where a canonical form is deduced for the first time. Together with this article, we also release an update for INITIAL, a publicly available Mathematica implementation of the algorithm.

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Derkachov, S. E., A. P. Isaev, and L. A. Shumilov. "Ladder and zig-zag Feynman diagrams, operator formalism and conformal triangles." Journal of High Energy Physics 2023, no. 6 (June 12, 2023). http://dx.doi.org/10.1007/jhep06(2023)059.

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Abstract We develop an operator approach to the evaluation of multiple integrals for multiloop Feynman massless diagrams. A commutative family of graph building operators Hα for ladder diagrams is constructed and investigated. The complete set of eigenfunctions and the corresponding eigenvalues for the operators Hα are found. This enables us to explicitly express a wide class of four-point ladder diagrams and a general two-loop propagator-type master diagram (with arbitrary indices on the lines) as Mellin-Barnes-type integrals. Special cases of these integrals are explicitly evaluated. A certain class of zig-zag four-point and two-point planar Feynman diagrams (relevant to the bi-scalar D-dimensional “fishnet” field theory and to the calculation of the β-function in ϕ4-theory) is considered. The graph building operators and convenient integral representations for these Feynman diagrams are obtained. The explicit form of the eigenfunctions for the graph building operators of the zig-zag diagrams is fixed by conformal symmetry and these eigenfunctions coincide with the 3-point correlation functions in D-dimensional conformal field theories. By means of this approach, we exactly evaluate the diagrams of the zig-zag series in special cases. In particular, we find a fairly simple derivation of the values for the zig-zag multi-loop two-point diagrams for D = 4. The role of conformal symmetry in this approach, especially a connection of the considered graph building operators with conformal invariant solutions of the Yang-Baxter equation is investigated in detail.

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Heckelbacher, Till, Ivo Sachs, Evgeny Skvortsov, and Pierre Vanhove. "Analytical evaluation of AdS4 Witten diagrams as flat space multi-loop Feynman integrals." Journal of High Energy Physics 2022, no. 8 (August 3, 2022). http://dx.doi.org/10.1007/jhep08(2022)052.

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Abstract We describe a systematic approach for the evaluation of Witten diagrams for multi-loop scattering amplitudes of a conformally coupled scalar ϕ4-theory in Euclidean AdS4, by recasting the Witten diagrams as flat space Feynman integrals. We derive closed form expressions for the anomalous dimensions for all double-trace operators up to the second order in the coupling constant. We explain the relation between the flat space unitarity methods and the discontinuities of the short distance expansion on the boundary of Witten diagrams.

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Jinno, Ryusuke, Gregor Kälin, Zhengwen Liu, and Henrique Rubira. "Machine learning Post-Minkowskian integrals." Journal of High Energy Physics 2023, no. 7 (July 24, 2023). http://dx.doi.org/10.1007/jhep07(2023)181.

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Abstract We study a neural network framework for the numerical evaluation of Feynman loop integrals that are fundamental building blocks for perturbative computations of physical observables in gauge and gravity theories. We show that such a machine learning approach improves the convergence of the Monte Carlo algorithm for high-precision evaluation of multi-dimensional integrals compared to traditional algorithms. In particular, we use a neural network to improve the importance sampling. For a set of representative integrals appearing in the computation of the conservative dynamics for a compact binary system in General Relativity, we perform a quantitative comparison between the Monte Carlo integrators VEGAS and i-flow, an integrator based on neural network sampling.

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Broedel, Johannes, Claude Duhr, and Nils Matthes. "Meromorphic modular forms and the three-loop equal-mass banana integral." Journal of High Energy Physics 2022, no. 2 (February 2022). http://dx.doi.org/10.1007/jhep02(2022)184.

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Abstract We consider a class of differential equations for multi-loop Feynman integrals which can be solved to all orders in dimensional regularisation in terms of iterated integrals of meromorphic modular forms. We show that the subgroup under which the modular forms transform can naturally be identified with the monodromy group of a certain second-order differential operator. We provide an explicit decomposition of the spaces of modular forms into a direct sum of total derivatives and a basis of modular forms that cannot be written as derivatives of other functions, thereby generalising a result by one of the authors form the full modular group to arbitrary finite-index subgroups of genus zero. Finally, we apply our results to the two- and three-loop equal-mass banana integrals, and we obtain in particular for the first time complete analytic results for the higher orders in dimensional regularisation for the three-loop case, which involves iterated integrals of meromorphic modular forms.

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32

Anselmi, Damiano. "Diagrammar of physical and fake particles and spectral optical theorem." Journal of High Energy Physics 2021, no. 11 (November 2021). http://dx.doi.org/10.1007/jhep11(2021)030.

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Abstract We prove spectral optical identities in quantum field theories of physical particles (defined by the Feynman iϵ prescription) and purely virtual particles (defined by the fakeon prescription). The identities are derived by means of purely algebraic operations and hold for every (multi)threshold separately and for arbitrary frequencies. Their major significance is that they offer a deeper understanding on the problem of unitarity in quantum field theory. In particular, they apply to “skeleton” diagrams, before integrating on the space components of the loop momenta and the phase spaces. In turn, the skeleton diagrams obey a spectral optical theorem, which gives the usual optical theorem for amplitudes, once the integrals on the space components of the loop momenta and the phase spaces are restored. The fakeon prescription/projection is implemented by dropping the thresholds that involve fakeon frequencies. We give examples at one loop (bubble, triangle, box, pentagon and hexagon), two loops (triangle with “diagonal”, box with diagonal) and arbitrarily many loops. We also derive formulas for the loop integrals with fakeons and relate them to the known formulas for the loop integrals with physical particles.

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Aguilera-Verdugo, J. Jesús, Roger J. Hernández-Pinto, Germán Rodrigo, German F. R. Sborlini, and William J. Torres Bobadilla. "Mathematical properties of nested residues and their application to multi-loop scattering amplitudes." Journal of High Energy Physics 2021, no. 2 (February 2021). http://dx.doi.org/10.1007/jhep02(2021)112.

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Abstract The computation of multi-loop multi-leg scattering amplitudes plays a key role to improve the precision of theoretical predictions for particle physics at high-energy colliders. In this work, we focus on the mathematical properties of the novel integrand-level representation of Feynman integrals, which is based on the Loop-Tree Duality (LTD). We explore the behaviour of the multi-loop iterated residues and explicitly show, by developing a general compact and elegant proof, that contributions associated to displaced poles are cancelled out. The remaining residues, called nested residues as originally introduced in ref. [1], encode the relevant physical information and are naturally mapped onto physical configurations associated to nondisjoint on-shell states. By going further on the mathematical structure of the nested residues, we prove that unphysical singularities vanish, and show how the final expressions can be written by using only causal denominators. In this way, we provide a mathematical proof for the all-loop formulae presented in ref. [2].

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Fael, Matteo, Fabian Lange, Kay Schönwald, and Matthias Steinhauser. "A semi-analytic method to compute Feynman integrals applied to four-loop corrections to the $$ \overline{\mathrm{MS}} $$-pole quark mass relation." Journal of High Energy Physics 2021, no. 9 (September 2021). http://dx.doi.org/10.1007/jhep09(2021)152.

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Abstract We describe a method to numerically compute multi-loop integrals, depending on one dimensionless parameter x and the dimension d, in the whole kinematic range of x. The method is based on differential equations, which, however, do not require any special form, and series expansions around singular and regular points. This method provides results well suited for fast numerical evaluation and sufficiently precise for phenomenological applications. We apply the approach to four-loop on-shell integrals and compute the coefficient function of eight colour structures in the relation between the mass of a heavy quark defined in the $$ \overline{\mathrm{MS}} $$ MS ¯ and the on-shell scheme allowing for a second non-zero quark mass. We also obtain analytic results for these eight coefficient functions in terms of harmonic polylogarithms and iterated integrals. This allows for a validation of the numerical accuracy.

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Slepian, Zachary. "On Decoupling the Integrals of Cosmological Perturbation Theory." Monthly Notices of the Royal Astronomical Society, June 20, 2020. http://dx.doi.org/10.1093/mnras/staa1789.

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Abstract Perturbation theory (PT) is often used to model statistical observables capturing the translation and rotation-invariant information in cosmological density fields. PT produces higher-order corrections by integration over linear statistics of the density fields weighted by kernels resulting from recursive solution of the fluid equations. These integrals quickly become high-dimensional and naively require increasing computational resources the higher the order of the corrections. Here we show how to decouple the integrands that often produce this issue, enabling PT corrections to be computed as a sum of products of independent 1-D integrals. Our approach is related to a commonly used method for calculating multi-loop Feynman integrals in Quantum Field Theory, the Gegenbauer Polynomial x-Space Technique (GPxT). We explicitly reduce the three terms entering the 2-loop power spectrum, formally requiring 9-D integrations, to sums over successive 1-D radial integrals. These 1-D integrals can further be performed as convolutions, rendering the scaling of this method Nglog Ng with Ng the number of grid points used for each Fast Fourier Transform. This method should be highly enabling for upcoming large-scale structure redshift surveys where model predictions at an enormous number of cosmological parameter combinations will be required by Monte Carlo Markov Chain searches for the best-fit values.

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He, Song, Zhenjie Li, Yichao Tang, and Qinglin Yang. "Bootstrapping octagons in reduced kinematics from A2 cluster algebras." Journal of High Energy Physics 2021, no. 10 (October 2021). http://dx.doi.org/10.1007/jhep10(2021)084.

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Abstract Multi-loop scattering amplitudes/null polygonal Wilson loops in $$ \mathcal{N} $$ N = 4 super-Yang-Mills are known to simplify significantly in reduced kinematics, where external legs/edges lie in an 1 + 1 dimensional subspace of Minkowski spacetime (or boundary of the AdS3 subspace). Since the edges of a 2n-gon with even and odd labels go along two different null directions, the kinematics is reduced to two copies of G(2, n)/T ∼ An−3. In the simplest octagon case, we conjecture that all loop amplitudes and Feynman integrals are given in terms of two overlapping A2 functions (a special case of two-dimensional harmonic polylogarithms): in addition to the letters v, 1 + v, w, 1 + w of A1 × A1, there are two letters v − w, 1 − vw mixing the two sectors but they never appear together in the same term; these are the reduced version of four-mass-box algebraic letters. Evidence supporting our conjecture includes all known octagon amplitudes as well as new computations of multi-loop integrals in reduced kinematics. By leveraging this alphabet and conditions on first and last entries, we initiate a bootstrap program in reduced kinematics: within the remarkably simple space of overlapping A2 functions, we easily obtain octagon amplitudes up to two-loop NMHV and three-loop MHV. We also briefly comment on the generalization to 2n-gons in terms of A2 functions and beyond.

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Boehm, Janko, Marcel Wittmann, Zihao Wu, Yingxuan Xu, and Yang Zhang. "IBP reduction coefficients made simple." Journal of High Energy Physics 2020, no. 12 (December 2020). http://dx.doi.org/10.1007/jhep12(2020)054.

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Abstract We present an efficient method to shorten the analytic integration-by-parts (IBP) reduction coefficients of multi-loop Feynman integrals. For our approach, we develop an improved version of Leinartas’ multivariate partial fraction algorithm, and provide a modern implementation based on the computer algebra system Singular. Furthermore, we observe that for an integral basis with uniform transcendental (UT) weights, the denominators of IBP reduction coefficients with respect to the UT basis are either symbol letters or polynomials purely in the spacetime dimension D. With a UT basis, the partial fraction algorithm is more efficient both with respect to its performance and the size reduction. We show that in complicated examples with existence of a UT basis, the IBP reduction coefficients size can be reduced by a factor of as large as ∼ 100. We observe that our algorithm also works well for settings without a UT basis.

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Niarchos, V., C. Papageorgakis, A. Pini, and E. Pomoni. "(Mis-)matching type-B anomalies on the Higgs branch." Journal of High Energy Physics 2021, no. 1 (January 2021). http://dx.doi.org/10.1007/jhep01(2021)106.

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Abstract Building on [1], we uncover new properties of type-B conformal anomalies for Coulomb-branch operators in continuous families of 4D $$ \mathcal{N} $$ N = 2 SCFTs. We study a large class of such anomalies on the Higgs branch, where conformal symmetry is spontaneously broken, and compare them with their counterpart in the CFT phase. In Lagrangian the- ories, the non-perturbative matching of the anomalies can be determined with a weak coupling Feynman diagram computation involving massive multi-loop banana integrals. We extract the part corresponding to the anomalies of interest. Our calculations support the general conjecture that the Coulomb-branch type-B conformal anomalies always match on the Higgs branch when the IR Coulomb-branch chiral ring is empty. In the opposite case, there are anomalies that do not match. An intriguing implication of the mismatch is the existence of a second covariantly constant metric on the conformal manifold (other than the Zamolodchikov metric), which imposes previously unknown restrictions on its holonomy group.

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Journal articles on the topic 'Multi-loop Feynman integrals'

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