Graphical functions and single-valued multiple polylogarithms

Graphical functions are single-valued complex functions which arise from Feynman amplitudes. We study their properties and use their connection to multiple polylogarithms to calculate Feynman periods. For the zigzag and two more families of

\phi^4

periods we give exact results modulo products. These periods are proved to be expressible as integer linear combinations of single-valued multiple polylogarithms evaluated at one. For the larger family of 'constructible' graphs, we give an algorithm that allows one to calculate their periods by computer algebra. The theory of graphical functions is used in [19] to prove the zig-zag conjecture.

Note:

60 pages

computer: algebra
family
Feynman graph
field theory: scalar
graph theory

References(40)

Figures(0)

[1]

Orthogonal polynomials and special functions, Regional conference Series in Applied Mathematics, Vol. 21 Philadelphia, PA (1975)

R. Askey

[2]

On Motives associated to graph polynomials

Spencer Bloch
(
- Chicago U., Dept. Math.
)
,
Helene Esnault
(
- Essen U.
)
,
Dirk Kreimer
(
- IHES, Bures-sur-Yvette
)

- Commun.Math.Phys. 267 (2006) 181-225
•
e-Print:
- math/0510011
•
DOI:
- 10.1007/s00220-006-0040-2

[3]

The Multiple Zeta Value Data Mine

J. Blumlein
(
- DESY, Zeuthen
)
,
D.J. Broadhurst
(
- Open U., England
)
,
J.A.M. Vermaseren
(
- DESY, Zeuthen and
- NIKHEF, Amsterdam
)

- Comput.Phys.Commun. 181 (2010) 582-625
•
e-Print:
- 0907.2557
•
DOI:
- 10.1016/j.cpc.2009.11.007

[4]

Open University report OUT-4102-18 (1985)

D. Broadhurst

[5]

Evaluation of a Class of Feynman Diagrams for All Numbers of Loops and Dimensions

David J. Broadhurst
(
- Open U., England
)

- Phys.Lett.B 164 (1985) 356-360
•
DOI:
- 10.1016/0370-2693(85)90340-5

[6]

Multiple zeta values and other periods in quantum field theory, conference talk, Bristol, 4 May 2011

D. Broadhurst

[7]

Knots and numbers in Phi**4 theory to 7 loops and beyond

David J. Broadhurst
(
- Open U., England
)
,
D. Kreimer
(
- Tasmania U.
)

- Int.J.Mod.Phys.C 6 (1995) 519-524,
- In *Pisa 1995, Proceedings, New computing techniques in physics research* 127-132, and Milton Keynes Open U. - OUT-4102-57 (95,rec.May) 6 p. Mainz U. - MZ-TH-95-12 (95,rec.May) 6 p. Hobart Tasmania U. - UTAS-PHYS-95-12 (95,rec.May) 6 p
•
e-Print:
- hep-ph/9504352
•
DOI:
- 10.1142/S012918319500037X