Elliptic Integrals in Higher Loop Calculations - from IBPs to η -weighted elliptic polylogarithms - Johannes Bl¨ umlein (in collab. with: J. Ablinger, A. De Freitas, M. van Hoeij, E. Imamoglu, C. Raab, S. Radu, C. Schneider) DESY, Zeuthen (Johannes Kepler University, Linz, Florida State University, Tallahassee,FL, USA) DESY 16-147 Seminar, DESY Hamburg November 20th 2017 1/40
Introduction One of the main and difficult issues in high energy physics is the calculation of involved multi-dimensional integrals. In the following our attitude will be their analytic integration. For quite some classes of integrals, particularly at lower order in the coupling constant, quite a series of analytic computational methods exist. cf. e.g. [arXiv:1509.08324] for the algorithms. ◮ Hypergeometric functions. ◮ Summation methods based on difference fields, implemented in the Mathematica program Sigma [C. Schneider, 2005–] . ◮ Reduction of the sums to a small number of key sums. ◮ Expansion of the summands in ε . ◮ Simplification by symbolic summation algorithms based on ΠΣ-fields [Karr 1981 J. ACM, Schneider 2005–] . ◮ Harmonic sums, polylogarithms and their various generalizations are algebraically reduced using the package HarmonicSums [Ablinger 2010, 2013, Ablinger, Bl¨ umlein, Schneider 2011,2013] . 2/40
Introduction ◮ Mellin-Barnes representations. ◮ In the case of convergent massive 3-loop Feynman integrals, they can be performed in terms of Hyperlogarithms [Generalization of a method by F. Brown, 2008, to non-vanishing masses and local operators] . ◮ Systems of Differential Equations. ◮ Almkvist-Zeilberger Theorem as Integration Method. [Multi-Integration] In the following we will concentrate on the method of Differential Equations since these are automatically obtained from the integration-by-parts identities representing all integrals by the so-called master integrals. These may either be considered directly or in terms of difference equations obtained through a formal power-series ansatz or a Mellin transform. 3/40
Introduction Starting from the most simple cases and moving to gradually more and more involved (massive) topologies one observes: ◮ The lower order topologies correspond to differential or difference equation systems which are first order factorizable. ◮ Here, a wider class of solution methods exists. There are methods in both cases to constructively find all letters of the alphabet needed to express the solutions in terms of indefinitely nested sums or iterative integrals. ◮ Later also differential or difference equations occur which contain genuine higher than 1st order factors. ◮ The first example are 2 F 1 solutions. In special cases these are also elliptic solutions. ◮ In the latter case one may represent the solutions in terms of modular functions and in more special cases in terms of modular forms and therefore in polynomials of Lambert-Eisenstein series (elliptic polylogarithms). 4/40
Introduction Project: 3-loop massive OMEs and DIS structure functions for lager Q 2 . ◮ The 2 F 1 solutions there appear to be the same or very closely related to those of the ρ parameter. ◮ Perform a thorough study for the latter case first. ◮ There is a lot of particular order in all these structures (although they look accidentally very different). 5/40
Function Spaces Sums Integrals Special Numbers Harmonic Sums Harmonic Polylogarithms multiple zeta values � 1 � x � y N k ( − 1) l 1 dx Li 3 ( x ) dy dz � � 1 + x = − 2 Li 4 (1 / 2) + ... l 3 k y 1 + z 0 0 0 k =1 l =1 gen. Harmonic Sums gen. Harmonic Polylogarithms gen. multiple zeta values � x � y � 1 N k (1 / 2) k ( − 1) l dy dz dx ln( x + 2) � � x − 3 / 2 = Li 2 (1 / 3) + ... l 3 k y z − 3 0 0 0 k =1 l =1 Cycl. Harmonic Sums Cycl. Harmonic Polylogarithms cycl. multiple zeta values N k � x � y ∞ ( − 1) l ( − 1) k 1 dy dz � � � C = l 3 1 + y 2 1 − z + z 2 (2 k + 1) 2 (2 k + 1) 0 0 k =1 l =1 k =0 Binomial Sums root-valued iterated integrals associated numbers � x � y N 1 � 2 k dy dz √ � � ( − 1) k 7) 2 H 8 , w 3 = 2 arccot ( z √ 1 + z k 2 k y 0 0 k =1 iterated integrals on CIS fct. associated numbers � 4 � 4 � z � 1 3 , 5 ; x 2 ( x 2 − 9) 2 3 , 5 ; x 2 ( x 2 − 9) 2 dx ln( x ) � � 3 3 2 F 1 dx 2 F 1 ( x 2 + 3) 3 ( x 2 + 3) 3 1 + x 2 2 0 0 shuffle, stuffle, and various structural relations = ⇒ algebras Except the last line integrals, all other ones stem from 1st order factorizable equations. 6/40
Some historical aspects: Iterative Integrals Li 2 ( x ), 1696 shuffle, 1775 iter. integrals { 1 / ( x − a i ) } , int. over { 1 / x , 1 / (1 − x ) } , 1840 [1884] Nielsen, 1909 Indefinitely nested sums: km − 1 k 1 N � � � S ( N ) = s ( k 1 ) s ( k 2 ) ... s ( k m ) k 1=1 k 2=1 km =1 Iterated integrals: � x � y 1 � yl − 1 F ( x ) = dy 1 f 1 ( y 1 ) dy 2 f 2 ( y 2 ) ... dy l f l ( y l ) 0 0 0 Mellin transform: � 1 � dxx N − 1 F ( x ) c α S α ( N ) = 0 α ... much more to say about the historic development, cf. e.g. J. Ablinger, JB, C. Schneider, 1304.7071, 1310.5645 7/40
The generalized polylogarithms E.E. Kummer H. Poincar´ e A.I. Lappo- K.T. Chen A. Goncharov Danielevskij 1840 1884 1934/36 1977 1998 (posthumus) � x � y 1 � yl − 1 dy 1 dy 2 dy l ... , a l ∈ C y 1 − a 1 y 2 − a 2 y l − a l 0 0 0 See also: Chr. Kassel, Quantum Groups, (Springer, Berlin, 1995). 8/40
The world until ∼ 1997 Calculations up to ∼ 2 Loops massless and single mass: � z ◮ Express all results in terms of Li n ( z ) = 0 dxLi n − 1 ( x ) / x , Li 0 ( x ) = x / (1 − x ) . , � 1 ◮ and possibly S p , n ( z ) = ( − 1) n + p − 1 / ( p !( n − 1)!) 0 dx ln p − 1 ( x ) ln n (1 − xz ) / x . ◮ The argument z = z(x) becomes a more and more complicated function. ◮ covering algebras of wider function spaces were widely unknown in physics, despite they were known in mathematics ... ◮ The complexity of expressions grew significantly, calling urgently for mathematical extensions. ◮ More complex argument structures do not easily allow an analytic Mellin inversion. ◮ Extremely long expressions are obtained, which would be much more compact, using adequate mathematical functions. � z ◮ Somewhen, new functions appeared: 0 dx Li 3 ( x ) / (1 + x ) not fitting into this frame. 9/40
Spill-Off: New Mathematical Function Classes and Algebras ◮ 1998: Harmonic Sums [Vermaseren; JB] ◮ 1999: Harmonic Polylogarithms [Remiddi, Vermaseren] ◮ 2000,2003, 2009: Analytic continuation of harmonic sums, systematic algebraic reduction; structural relations [JB] ◮ 2001: Generalized Harmonic Sums [Moch, Uwer, Weinzierl] ◮ 2004: Infinite harmonic (inverse) binomial sums [Davydychev, Kalmykov; Weinzierl] ◮ 2011: (generalized) Cyclotomic Harmonic Sums, polylogarithms and numbers [Ablinger, JB, Schneider] ◮ 2013: Systematic Theory of Generalized Harmonic Sums, polylogarithms and numbers [Ablinger, JB, Schneider] ◮ 2014: Finite nested Generalized Cyclotomic Harmonic Sums with (inverse) Binomial Weights [Ablinger, JB, Raab, Schneider] ◮ 2016: Elliptic integrals with (involved) rational arguments appear in part of the functions of our project already as base cases. They stem from Heun equations. [ since April 2016.] [Ablinger, JB, De Freitas, van Hoeij, Raab, Radu, Schneider, DESY16-147]. Particle Physics Generates NEW Mathematics. 10/40
integral representation (inv. Mellin transform) S-Sums H-Sums C-Sums C-Logs H-Logs G-Logs � 1 � S − 1 , 2( n ) S(2 , 1 , − 1)( n ) H(4 , 1) , (0 , 0) ( x ) H − 1 , 1 ( x ) H2 , 3 ( x ) S1 , 2 2 , 1; n x → c ∈ R Mellin transform n → ∞ x → 1 x → 1 � 1 � S(2 , 1 , − 1)( ∞ ) H(4 , 1) , (0 , 0) (1) S1 , 2 2 , 1; ∞ S − 1 , 2( ∞ ) H − 1 , 1 (1) H2 , 3 ( c ) power series expansion � 2 i � square-root valued letters ⇐ ⇒ nested binomial sums i All these cases obey difference or differential equations, which factorize in 1 / 1 first order. 11/40
From Iterative Integrals to Non-Iterative Integrals Iterative integrals (nested sums) over whatsoever alphabet are: The World of Yesterday. All these cases are solved algorithmically for any basis, cf. J. Ablinger et al. (2015): Comp. Phys. Commun. 202 (2016) 33 . The current challenge is formed by systems factorizing not in first order, see also the talks by: JB, van Hoeij, Paule, Radu, Remiddi, Weinzierl @ RADCOR 2017 and more Review-Talks at the Workshop: “Elliptic Integrals, Elliptic Funct- ions and Modular Forms in Quantum Field Theory”, Oct. 23-26, Zeuthen, Germany. � S. Fischerverlage. c 12/40
Decoupling of Systems ◮ We consider linear systems of N inhomogeneous differential equations and decouple them into a single scalar equation + ( N − 1) other determining equations. ◮ Usually one may use a series ansatz (+ ln k ( x ) modulation) ∞ � a ( k ) x k f ( x ) = k =1 and obtain m � p k ( N ) F ( N + k ) = G ( N ) k =0 . ◮ The latter equation is now tried to be solved using difference-field techniques. ◮ If the equation has successive 1st order solutions one ends up with a nested sums solution. All these cases have been algorithmized. [arXiv:1509.08324 [hep-ph]]. ◮ This even applies for some cases ending up elliptic in x -space [arXiv:1310.5645 [math-ph]]. 13/40
A didactical example : 3 loop QCD corrections to the ρ -parameter 14/40
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