*This Maplesoft guest blog post is from Prof. Dr. Johannes Blümlein from *__Deutsches Elektronen-Synchrotron__* (DESY), **one of the world’s leading particle accelerator centres *used by thousands of researchers from around the world to advance our knowledge of the microcosm. *Prof. Dr.* *Blümlein is a senior researcher in the Theory Group at DESY, where h**e and his team make significant use of Maple in their investigations of high energy physics, as do other groups working in Quantum Field Theory. In addition, he has been involved in EU programs that give PhD students opportunities to develop their Maple programming skills to support their own research and even expand Maple’s support for theoretical physics.*

**The use of Maple in solving frontier problems in theoretical high energy physics**

For several decades, progress in theoretical high energy physics relies on the use of efficient computer-algebra calculations. This applies both to so-called perturbative calculations, but also to non-perturbative computations in lattice field theory. In the former case, large classes of Feynman diagrams are calculated analytically and are represented in terms of classes of special functions. In early approaches started during the 1960s, packages like **Reduce** [1] and **Schoonship** [2] were used. In the late 1980s **FORM** [3] followed and later on more general packages like **Maple** and **Mathematica** became more and more important in the solution of these problems. Various of these problems are related to data amounts in computer-algebra of O(Tbyte) and computation times of several CPU years currently, cf. [4].

Initially one has to deal with huge amounts of integrals. An overwhelming part of them is related by Gauss’ divergence theorem up to a much smaller set of the so-called master integrals (MIs). One performs first the reduction to the MIs which are special multiple integrals. No general analytic integration procedures for these integrals exist. There are, however, several specific function spaces, which span these integrals. These are harmonic polylogarithms, generalized harmonic polylogarithms, root-valued iterated integrals and others. For physics problems having solutions in these function spaces codes were designed to compute the corresponding integrals. For generalized harmonic polylogarithms there is a **Maple** code **HyperInt** [5] and other codes [6], which have been applied in the solution of several large problems requiring storage of up to 30 Gbyte and running times of several days. In the systematic calculation of special numbers occurring in quantum field theory such as the so-called β-functions and anomalous dimensions to higher loop order, e.g. 7–loop order in Φ^{4} theory, the **Maple** package **HyperLogProcedures** [7] has been designed. Here the largest problems solved require storage of O(1 Tbyte) and run times of up to 8 months. Both these packages are available in **Maple** only.

A very central method to evaluate master integrals is the method of ordinary differential equations. In the case of first-order differential operators leading up to root-valued iterative integrals their solution is implemented in **Maple** ** **in [8] taking advantage of the very efficient differential equation solvers provided by **Maple**. Furthermore, the **Maple ** methods to deal with generating functions as e.g. **gfun,** has been most useful here. For non-first order factorizing differential equation systems one first would like to factorize the corresponding differential operators [9]. Here the most efficient algorithms are implemented in Maple only. A rather wide class of solutions is related to 2^{nd} order differential equations with more than three singularities. Also here **Maple** is the only software package which provides to compute the so-called _{2}F_{1} solutions, cf. [10], which play a central role in many massive 3-loop calculations

The **Maple**-package is intensely used also in other branches of part**i**cle physics, such as in the computation of next-to-next-to leading order jet cross sections at the Large Hadron Collider (LHC) with the package **NNLOJET** and double-parton distribution functions. **NNLOJET** uses **Maple** extensively to build the numerical code. There are several routines to first build the driver with automatic links to the matrix elements and subtraction terms, generating all of the partonic subprocesses with the correct factors. To build the antenna subtraction terms, a meta-language has been developed that is read by **Maple** and converted into calls to numerical routines for the momentum mappings, calls to antenna and to routines with experimental cuts and plotting routines, cf. [11].

In lattice gauge calculations there is a wide use of **Maple** too. An important example concerns the perturbative predictions in the renormalization of different quantities. Within different European training networks, PhD students out of theoretical high energy physics and mathematics took the opportunity to take internships at Maplesoft for several months to work on parts of the **Maple** package and to improve their programming skills. In some cases also new software solutions could be obtained. Here Maplesoft acted as industrial partner in these academic networks.

References

[1] A.C. Hearn, *Applications of Symbol Manipulation in Theoretical Physics*, Commun. ACM **14** No. 8, 1971.

[2] M. Veltman*, Schoonship (1963), a program for symbolic handling*, documentation, 1991, edited by D.N. Williams.

[3] J.A.M. Vermaseren, *New features of FORM*, math-ph/0010025.

[4] J. *Blümlein* and C. Schneider, *Analytic computing methods for precision calculations in quantum field theory*, Int. J. Mod. Phys. A **33** (2018) no.17, 1830015 [arXiv:1809.02889 [hep-ph]].

[5] E. Panzer, *Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals*, Comput. Phys. Commun. **188** (2015) 148–166 [arXiv:1403.3385 [hep-th]].

[6] J. Ablinger, J. *Blümlein,* C .Raab, C. Schneider and F. Wissbrock, *Calculating Massive 3-loop Graphs for Operator Matrix Elements by the Method of Hyperlogarithms*, Nucl. Phys. ** 885 **(2014) 409-447 [arXiv:1403.1137 [hep-ph]].

[7] O. Schnetz, *φ*^{4}* theory at seven loops*, Phys. Rev. D **107** (2023) no.3, 036002 [arXiv: 2212.03663 [hep-th]].

[8] J. Ablinger, J. *Blümlein*, C. G. Raab and C. Schneider, *Iterated Binomial Sums and their Associated Iterated Integrals*, J. Math. Phys. **55** (2014) 112301 [arXiv:1407.1822 [hep-th]].

[9] M. van Hoeij, *Factorization of Differential Operators with Rational Functions Coefficients*, Journal of Symbolic Computation, **24** (1997) 537–561.

[10] J. Ablinger, J. *Blümlein*, A. De Freitas, M. van Hoeij, E. Imamoglu, C. G. Raab, C. S. Radu and C. Schneider, *Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams*, J. Math. Phys. **59** (2018) no.6, 062305 [arXiv:1706.01299 [hep-th]].

[11] A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss and T.A. Morgan, *Precise QCD predictions for the production of a Z boson in association with a hadronic jet*, Phys. Rev. Lett. **117** (2016) no.2, 022001 [arXiv:1507.02850 [hep-ph]].