Iterated integrals over letters induced by quadratic forms

An automated treatment of iterated integrals based on letters induced by real-valued quadratic forms and Kummer – Poincar´e letters is presented. These quantities emerge in analytic single and multiscale Feynman diagram calculations. To compactify representations, one wishes to apply general properties of these quantities in computer-algebraic implementations. We provide the reduction to basis representations, expansions, analytic continuation and numerical evaluation of these quantities.


I. INTRODUCTION
In analytic calculations of single and multiscale Feynman integrals different principal structures have been revealed in particular during the last 30 years. 1 Beyond the multiple zeta values [2] and other special numbers for zeroscale quantities, there are the spaces of harmonic sums [3,4], harmonic polylogarithms [5], generalized harmonic sums [6,7] and Kummer-Poincaré iterated integrals [6][7][8], cyclotomic harmonic sums and iterated integrals [9], finite and infinite binomial sums and inverse binomial sums and the associated root-letter integrals [10,11], and iterative non-iterative integrals [12], including those containing complete elliptic integrals [12,13]. This list is expected still to extend in analytic calculations at even higher loops and for more contributing scales in the future.
In decomposing Feynman parameter representations in the general case, cf., e.g., [14], often real polynomials of higher degree have to be factored. According to the fundamental theorem of algebra [15] this leads to either linear and quadratic factors in real representations or to linear complex-valued factors with conjugated pairs. Real representations have often advantages in calculations. This is the main reason to extend the class of Kummer-Poincaré iterated integrals based on the alphabet There is an overlap with the cyclotomic iterated integrals [9] with respect to the letters 1=Φ k ðxÞ; k ¼ 3, 4, 6. The associated iterative integrals are given by H b; ⃗a ðzÞ ¼ which is instrumental for building the corresponding algebra, to be closed under differentiation. Sometimes more general iterated integrals are used, cf. e.g., [16], with letters outside A R . As we will show below this class of integrals can be cast into the class generated by A R . Using the alphabet (1.2) has the advantage to stay inside a real representation in calculating a real quantity. Complex decompositions [17] request the thorough observation of the pairing of complex conjugated letters. Both approaches can handle poles inside the integration region.
We will device an algorithm to transform the mentioned formal iterated integrals into real ones, also referring to one simple main variable. This has the advantage that the corresponding results can be iterated over in further integrations, which will be necessary for the use in a higher order calculation.
Using iterative integrals in the description of physical quantities it is required to give them a clear definition. In some cases it is possible that a first definition has singularities in subintegrals, which have to be dealt with to obtain a measurable quantity. Furthermore, the real and imaginary parts of the respective integrals have to be separated from the beginning, because they have a different physical meaning and it then allows to deal with real integrals only. One has also to observe that certain transformations in the main argument may effect the position of cuts chosen. In the case of singularities of the real integrals we will apply Cauchy's principal value for definiteness, as it is also the case in amplitudes referring to the Källén-Lehmann representation [18].
In Sec. II we will describe the different operations for the iterated integrals induced by quadratic forms in the package HARMONICSUMS [3][4][5]7,9,10,[19][20][21] and provide test examples. Section III deals with integrals of a recent physical application [16] which we reconsider in the present formalism, and Sec. IV contains the conclusion.

II. OPERATIONS FOR THE ITERATED INTEGRALS
In the following we describe a series of operations which allow to deal with iterated integrals containing letters of the alphabet A R . The statement represents the integral covering a number of iterated letters out of A R . The command ToHarmonicSumsIntegrate reveals the integral structure in explicit form. One may convert these integrals into GL-functions, cf. [10], by QLToGL and GL-functions with letters out of A R to QL-functions by GLToQL. It is allowed that the Kummer-Poincaré letters in A KP have poles in the integration region. The iterative integral is then defined taking Cauchy's principal value. However, the quadratic denominators are assumed to not factorize in real numbers.
The numerical evaluation of QL-functions is performed as in the following example Here not all letters corresponding to quadratic forms are yet in the standard form. There is no singularity, however, in the integration region since ð ffiffi ffi To be able to deal with properly defined letters, the mapping QLToStandardForm is used. A typical example is ð2:8Þ The algebraic reduction with respect to shuffle relations of a given expression is performed by the command ReduceToQLBasis, which will also transform the shuffled expression (2.8) into the corresponding product expression. Often one would like to remove trailing indices or leading indices of QL-functions. This is done by the commands RemoveTrailing0, RemoveTrailingIndex or RemoveLeading1, RemoveLeadingIndex. Here Trailing0 refers to the letter 1=x and Leading1 to 1=ð1 − xÞ. Examples are RemoveTrailingIndex½QL½fff1; 1; 1g; 0g; ff−1; 1; 0g; 0g; ff1; 1; 1g; 0g; ff1; 1; 1g; 0gg; z;ff1; 1; 1g; 0g Often a change in the argument of the QL-functions is desirable. The following transformations are implemented and are carried out by the command TransformQL It is furthermore useful to transform QL-functions in a representation in which they have a convergent Taylor series representation, which is also applied in their numerical evaluation. This is provided by the command QLToConvergentRegion. An example is QLToConvergentRegion½QL½fff1; 1; 0g; 0g; ff−1; 1; 0g; 0g; ff2; 1; 1g; 1gg; 4 for a i ∈ Rnf0g is convergent for jzj < ja i j. Hence given a QL-function H m ðzÞ with letters from A KP , i.e., with letters of the form (2.26), we remove all poles p at the real axis for which jpj < jzj using the strategy mentioned above. Now expanding the resulting functions about z ¼ 0 will lead to a convergent series at z. In the case that there are also quadratic forms present i.e., letters of the form ð2:27Þ we have to additionally treat the ffiffiffiffiffiffiffi ja i j p as poles and apply the method mentioned above also with p ¼ AE ffiffiffiffiffiffiffi ja i j p for ffiffiffiffiffiffiffi ja i j p < jzj. This is due to the fact that the series expansion of a QL-function with a letter of the type (2.27) is convergent for jzj < ffiffiffiffiffiffiffi ja i j p . The option PrincipalValue → False in QLEvaluate allows us to evaluate integrals sub-integrals containing poles yielding a complex result.
Here the regularization is performed in adding þiε to the denominator of the singular letters.
The QL-functions in the variable z ∈ ½0; 1 can be Mellin transformed by One obtains representations by harmonic sums [3,4] and (generalized) S-sums [7] at complex weights Instead of working with differential equations one may work with difference equations, which are solved by the package SIGMA [22,23]. The ground field of the package needs then to contain the corresponding special constants arising in f⃗ cg. An example is the Mellin transformation of which is given by where we dropped the argument N both for the harmonic and generalized harmonic sums. The constants T i are iterated integrals at x ¼ 1 and are given by ffiffi ffi 3 p ð2:34Þ where Li n ðxÞ denotes the classical polylogarithm [24] with the representation x k k n ; x∈ ½−1; 1; n∈ N: ð2:40Þ Not all of the above polylogarithms are independent, cf. [25,26], and the following relations hold, ð2:41Þ The basis of contributing constants is here lnð2Þ; ζ 2 ; ζ 3 ; π; lnð3Þ; Li 2 − 1 2 ; Li 3 − 1 2 :

III. PHYSICS EXAMPLES
As an application of the operations given in Sec. II we calculate a few examples of particular iterated integrals, which have emerged in the calculation of inclusive Compton scattering cross sections at next-to-leading order (NLO) [16] recently. The corresponding integrals were defined by where y ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi x=ð4 þ xÞ p ; x ¼ s=m 2 − 1, s the cms energy and m a mass, implying y ∈ ½0; 1, and y ¼ yðxÞ.
Iterative integrals obey shuffle algebras [27][28][29] and are first reduced to a corresponding basis representation by exploiting the relations implied by (2.7). The formal iterated integrals (3.1) are given by We now cast these integrals into a root-free form. Furthermore, we choose a simple main argument, to allow further iterated integration, needed in potential higher order calculations. For the following representations we choose the variable for the main argument and set H ⃗ a ðzÞ ≡ H ⃗ a for the harmonic polylogarithms and cyclotomic harmonic polylogarithms.
We now transform the integrals (3.3) into iterative integrals, see also [20,21], containing also letters generated by quadratic forms. and similar in the case of the other integrals. In general also QL-functions at argument x ¼ 1 will appear, inducing new constants. This set can be reduced to the algebraic basis by shuffle and stuffle relations [29]. Depending on the defining constants of the contributing quadratic forms, further sets of relations may be present. Often particular algebraic numbers occur in this context.

IV. CONCLUSIONS
In the analytic calculation of Feynman diagrams hierarchies of function spaces and algebras emerge both in the z space and N space representation consisting out of iterative integrals or nested sums of different kind. The simplest structures are harmonic polylogarithms, followed by Kummer-Poincaré iterated integrals and cyclotomic integrals, and also iterated integrals over square-root valued letters and the associated sums and special constants. Here we consider a real extension of the Kummer-Poincaré iterated integrals, allowing also for letters generated by general real quadratic forms without real factorization. Here the range of constants is not limited to c i ∈ Q, but general real numbers are allowed, which are usually implied by the values of different masses and virtualities in the processes to be considered. Quantities of this kind appear in higher order and multileg calculations. Since real representations have sometimes advantages compared to complex representations, we provide algorithms to build the associated algebra to a set of these letters, their basis representation, different mappings of the main argument, including analytic continuation in the case of the presence of cuts. Finally, also the expressions can be evaluated numerically. The different commands in HarmonicSums to provide these operations are described and illustrated by examples. We have applied the corresponding mappings to a class of functions which have emerged recently in the NLO calculation of the inclusive Compton cross section. In viewing physics results within this class, it is for structural reasons also interesting to see whether the result can be expressed by functions out of a particular function space. In the case of Ref. [16] it turns out to be the space of cyclotomic harmonic polylogarithms if the final expression is written using the variable z, (3.4).
The different commands to treat iterated integrals of the QL-type are implemented in the package HarmonicSums, which is available from https://risc.jku.at/sw/harmonicsums/. As well we attach the Supplemental Material Quadratic-Letters.nb to this paper [30].