## Structural relations of harmonic sums and Mellin transforms at weight w=6, arXiv:0901.0837 [math-ph

Citations: | 9 - 4 self |

### BibTeX

@MISC{Blümlein_structuralrelations,

author = {Johannes Blümlein},

title = {Structural relations of harmonic sums and Mellin transforms at weight w=6, arXiv:0901.0837 [math-ph},

year = {}

}

### OpenURL

### Abstract

We derive the structural relations between the Mellin transforms of weighted Nielsen integrals emerging in the calculation of massless or massive single–scale quantities in QED and QCD, such as anomalous dimensions and Wilson coefficients, and other hard scattering cross sections depending on a single scale. The set of all multiple harmonic sums up to weight five cover the sums needed in the calculation of the 3–loop anomalous dimensions. The relations extend the set resulting from the quasi-shuffle product between harmonic sums studied earlier. Unlike the shuffle relations, they depend on the value of the quantities considered. Up to weight w = 5, 242 nested harmonic sums contribute. In the present physical applications it is sufficient to consider the sub-set of harmonic sums not containing an index i = −1, which consists out of 69 sums. The algebraic relations reduce this set to 30 sums. Due to the structural relations a final reduction of the number of harmonic sums to 15 basic functions is obtained. These functions can be represented in terms of factorial series, supplemented by harmonic sums which are algebraically reducible. Complete analytic representations are given for these 15 meromorphic functions in the complex plane

### Citations

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Citation Context ...ms with index i = {−1} do not contribute. One finds that the number of all harmonic sums of weight w, which do not contain any index i = −1, is obtained by expanding the following generating function =-=[51]-=- with 1 − x = 1 − 2x − x2 ∞∑ N¬(−1)(w)x w , (4.4) w=0 N¬{−1}(w) = 1 [( 1 − 2 √ 2 ) w + ( 1 + √ 2 ) w ] [w/2] ∑ = k=0 ( ) w 2 2k k . (4.5) N¬{−1}(w) obeys the recursion relation N¬{−1}(w) = 2 · N¬{−1}(... |

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Citation Context ... quantum field-theoretic calculations both denominators occur, one may apply this decomposition based on the first two cyclotomic polynomials, cf. [21], and the relation between Lik(x2 ) and Lik(±x), =-=[22]-=-. The corresponding Mellin transforms also require half– integer arguments. In more general situations other cyclotomic polynomials might emerge. The relation 1 M 2k−1 [( Lik(x2 ) x2 ) − 1 + ] (N ) − ... |

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Citation Context ...n of Γ-functions, with according replacement of the sum-index in place of the variables σa. For these multiple sum expressions one may seek representations which are generalized hypergeometric series =-=[7, 44]-=- and generalizations thereof. Going to ever higher orders not all of these functions may be known by now but have to be introduced newly. They are naturally generated by the above integrals. At this s... |

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Citation Context ...r applications, cf. [25] and [26]. 4Precise analytic continuations of the basic functions [2, 29] based on semi–numerical representations were given in [30,31]. Here we made use of the MINIMAX method =-=[32,33]-=-. This method has also been applied to derive the analytic continuation for the heavy flavor Wilson coefficients up to 2-loop order [34]. For another proposal for the analytic continuation of harmonic... |

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Citation Context ...) + ln(2) . (5.6) The latter relation can even be analytically continued in closed form since, see [2], −ψ ( ) N − γE = −ψ(N) − γE + β(N) + ln(2) , (5.7) 2 β(N) = 1 [ ψ 2 ( ) N + 1 − ψ 2 ( )] N 2 cf. =-=[27,54]-=-. The duplication relation for the ψ–function yields ψ(N) = 1 [ ( ) N ψ + ψ 2 2 , (5.8) ( )] N + 1 + ln(2) . (5.9) 2 Here fractional arguments in the ψ–function emerge quite naturally. For k > 1 the h... |

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Citation Context ...e zeta values emerge in higher order calculations of zero- and single scale quantities.1 Introduction Finite harmonic sums [1–3] and associated to them, by a Mellin transform, Nielsen–type integrals =-=[4]-=-, emerge in perturbative calculations of massless and massive single scale problems in Quantum Field-Theory. The Wilson coefficients and anomalous dimensions in deeply inelastic scattering to three lo... |

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4 |
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Citation Context ... the respective graph consisting out of k monomials raised to a real non-integer power. A possible next step consists in expressing the Feynman-parameter integrals in terms of Mellin-Barnes integrals =-=[43]-=- for each of these monomials, 1 1 = (A + B) q 2πi ∫ γ+i∞ γ−i∞ dσ A σ −q−σ Γ(−σ)Γ(q + σ) B Γ(q) . (2.4) The use of Mellin-Barnes integrals usually leads to rather large set of sums if compared to the s... |

4 |
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Citation Context ...tive generalizations in higher orders, as described above. 7 The according sums can be uniquely found investigating the corresponding recurrences in N over ΠΣ–fields, as possible with the code Sigma, =-=[46]-=-. The harmonic sums, which occur in the calculations of single scale quantities up to a given weight can be represented over a basis of functions, which is independent of the respective quantity being... |

3 |
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Citation Context ... Lyndon words as a consequence of Radford’s theorem [20]. In the limit N → ∞ harmonic sums turn into multiple zeta values (MZVs) [21] 2 . Since multiple zeta values obey many more relations, see e.g. =-=[22, 23]-=-, than the shuffle relations the question arises which other relations may hold for multiple harmonic sums. 3 A series of relations emerges due to the integral representations of the harmonic sums whi... |

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Citation Context ...cale problems in Quantum Field-Theory. The Wilson coefficients and anomalous dimensions in deeply inelastic scattering to three loops [5], the massive quark Wilson coefficients in the limit Q 2 ≫ m 2 =-=[6,7]-=-, as well as the Wilson coefficients for the Drell-Yan process, hadronic Higgs production in the heavy mass limit, the time-like coefficient functions for parton fragmentation into hadrons, cf. [8], t... |

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Citation Context ... process, hadronic Higgs production in the heavy mass limit, the time-like coefficient functions for parton fragmentation into hadrons, cf. [8], the soft– and virtual corrections to Bhabha scattering =-=[9]-=-, and various processes more belong to this class. At 2–loop order the respective expressions in terms of harmonic sums were given in [10] using the algebraic and structural relations between these qu... |

1 |
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Citation Context ...r applications, cf. [25] and [26]. 4Precise analytic continuations of the basic functions [2, 29] based on semi–numerical representations were given in [30,31]. Here we made use of the MINIMAX method =-=[32,33]-=-. This method has also been applied to derive the analytic continuation for the heavy flavor Wilson coefficients up to 2-loop order [34]. For another proposal for the analytic continuation of harmonic... |

1 |
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Citation Context ...), 1/(1 + x) and 5 The structural relations of w = 6 are presented in Ref. [37]. 4their Mellin transforms. Going to even higher orders, or including a scale more, this alphabet will be extended, cf. =-=[42]-=-, which leads to similar structures. The rôle of the harmonic polylogarithms [12] will be taken by the polylogarithms over the extended alphabet. The multiple ζ-values are then generalized by the valu... |

1 |
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Citation Context ...(−1) N , which emerge in the following, have therefore a definite meaning and are either equal to +1 or −1. Both branches may be continued analytically in N from N to C according to Carlson’s theorem =-=[47]-=-. The distributions f(x) considered in the following are differentiable functions in the class C ∞ (]0, 1[) or δ–distributions δ (k) (1 − x), k ≥ 0 in D ′ [0, 1] [14]. Furthermore, we consider functio... |

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et al., arXiv:hep-ph/0511119
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Citation Context ...–sets over the alphabet {−1, 0, 1}, since the letter 0 cannot appear first. The counting relation is given by Eq. (4.1). Investigating the structure of the single–scale quantities in QCD and QED, cf. =-=[6,7,10,29,31, 50]-=- to 2– and 3–loop order in more detail, it turns out that harmonic sums with index i = {−1} do not contribute. One finds that the number of all harmonic sums of weight w, which do not contain any inde... |

1 |
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Citation Context ... (−1) 1 + x N [S−1(N) + ln(2)] = β(N + 1) . (5.3) Here γE denotes the Euler–Mascheroni constant. The representation of the integral in Eq. (5.2) in terms of the ψ–function was first given by Legendre =-=[53]-=-. The Mellin transform for an integrable function x N f(x) obeys the identity M [f (x a )] (N) = 1 a M[f(x)] ( N + 1 − a a ) , a ∈ R, a > 0 . (5.4) This relation modifies in case of f(x) being a +-dis... |

1 |
private communication
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Citation Context ...differential relations there is only one basic single harmonic sum, S1(N). One may wonder whether differentiation relations can also be of importance for multiple zeta values. This is indeed the case =-=[55]-=-. The multiple zeta values can be generalized to multiple Hurwitz zeta values [56] defined by iteratively with ζ a c, ⃗ d = ∞∑ k1=1 (sign(c)) k1 (a + k1) SH,a |c| ⃗d (k1) , (5.12) S H,a ⃗d (k1) = k1 ∑... |

1 |
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Citation Context ... wonder whether differentiation relations can also be of importance for multiple zeta values. This is indeed the case [55]. The multiple zeta values can be generalized to multiple Hurwitz zeta values =-=[56]-=- defined by iteratively with ζ a c, ⃗ d = ∞∑ k1=1 (sign(c)) k1 (a + k1) SH,a |c| ⃗d (k1) , (5.12) S H,a ⃗d (k1) = k1 ∑ k2=1 (sign(d1)) k2 SH,a (a + k2) |d1| d2...dm (k2) . (5.13) Here a ∈ C so that th... |

1 |
KITP Workshop on Collider Physics Jan. 12
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(Show Context)
Citation Context ...[58] asymptotic relations for non-alternating harmonic sums to low orders in 1/N k were derived. Our algorithm given below is free of these restrictions. The main ideas were presented in January 2004 =-=[59]-=-, see also [29]. 24C 0 k C l k = 1 (B.7) = C l k+1 − kC l−1 k+1 . (B.8) Not for all harmonic sums the asymptotic representation can be found in the way outlined above using the associated Mellin tran... |

1 |
20 (1775) 140; D. Zagier, First European Congress of Mathematics
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(Show Context)
Citation Context ... functions. Zero scale quantities, like the loop-expansion coefficients for renormalized couplings and masses in massless filed theories, are given by special numbers, which are the multiple ζ-values =-=[1,2]-=- in the known orders. At higher orders and in the massive case other quantities more will contribute [3]. The next class of interest are the single scale quantities to which the anomalous dimensions a... |

1 | Inverting Exact Functions from - Blümlein, Kauers, et al. |

1 | 476; Nucl. Phys. B383 - B273 - 1991 |

1 |
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(Show Context)
Citation Context ...to a continuation of N ∈ Q. Harmonic sums can be represented in terms of Mellin-integrals of harmonic polylogarithms H⃗a(z) weighted by 1/(1 ± z) [13], which belong to the Poincaré–iterated integrals =-=[14]-=-. 3 The Mellin integrals are valid for N ∈ R, N ≥ N0. From these representations integration-by-parts relations can be derived. Furthermore, there is a large number of differentiation relations dl [ M... |