52
52

Sep 20, 2013
09/13

by
Oliver Schnetz

texts

######
eye 52

######
favorite 0

######
comment 0

A careful analysis of differential renormalization shows that a distinguished choice of renormalization constants allows for a mathematically more fundamental interpretation of the scheme. With this set of a priori fixed integration constants differential renormalization is most closely related to the theory of generalized functions. The special properties of this scheme are illustrated by application to the toy example of a free massive bosonic theory. Then we apply the scheme to the...

Source: http://arxiv.org/abs/hep-th/9610025v1

5
5.0

Jun 29, 2018
06/18

by
Oliver Schnetz

texts

######
eye 5

######
favorite 0

######
comment 0

We review recent results in the theory of numbers and single-valued functions on the complex plane which arise in quantum field theory.

Topics: High Energy Physics - Theory, Number Theory, Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1606.08598

39
39

Sep 22, 2013
09/13

by
Oliver Schnetz

texts

######
eye 39

######
favorite 0

######
comment 0

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 zig-zag 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...

Source: http://arxiv.org/abs/1302.6445v1

50
50

Sep 20, 2013
09/13

by
Oliver Schnetz

texts

######
eye 50

######
favorite 0

######
comment 0

The spectral test of random number generators (R.R. Coveyou and R.D. McPherson, 1967) is generalized. The sequence of random numbers is analyzed explicitly, not just via their n-tupel distributions. We find that the mixed multiplicative generator with power of two modulus does not pass the extended test with an ideal result. Best qualities has a new generator with the recursion formula X(k+1)=a*X(k)+c*int(k/2) mod 2^d. We discuss the choice of the parameters a, c for very large moduli 2^d and...

Source: http://arxiv.org/abs/physics/9610004v1

45
45

Sep 23, 2013
09/13

by
Oliver Schnetz

texts

######
eye 45

######
favorite 0

######
comment 0

Perturbative quantum field theories frequently feature rational linear combinations of multiple zeta values (periods). In massless \phi^4-theory we show that the periods originate from certain `primitive' vacuum graphs. Graphs with vertex connectivity 3 are reducible in the sense that they lead to products of periods with lower loop order. A new `twist' identity amongst periods is proved and a list of graphs (the census) with their periods, if available, is given up to loop order 8.

Source: http://arxiv.org/abs/0801.2856v2

37
37

Sep 20, 2013
09/13

by
Oliver Schnetz

texts

######
eye 37

######
favorite 0

######
comment 0

The spectral test of random number generators (R.R. Coveyou and R.D. McPherson, 1967) is generalized. The sequence of random numbers is analyzed explicitly not just via their n-tupel distributions. The generalized analysis of many generators becomes possible due to a theorem on the harmonic analysis of multiplicative groups of residue class rings. We find that the mixed multiplicative generator with power of two modulus does not pass the extended test with an ideal result. Best qualities has a...

Source: http://arxiv.org/abs/physics/9610003v1

29
29

Sep 19, 2013
09/13

by
Oliver Schnetz

texts

######
eye 29

######
favorite 0

######
comment 0

We review a reduction formula by Petersson that reduces the calculation of a one-loop amplitude with N external lines in n

Source: http://arxiv.org/abs/1010.5334v1

48
48

Sep 19, 2013
09/13

by
Oliver Schnetz

texts

######
eye 48

######
favorite 0

######
comment 0

We present an analytic calculation of the first transcendental in phi^4-Theory that is not of the form zeta(2n+1). It is encountered at 6 loops and known to be a weight 8 double sum. Here it is obtained by reducing multiple zeta values of depth

Source: http://arxiv.org/abs/hep-th/9912149v1

28
28

Sep 22, 2013
09/13

by
Oliver Schnetz

texts

######
eye 28

######
favorite 0

######
comment 0

We prove that the period of a family of $n$ loop graphs with triangle and box ladders evaluates to $\frac{4}{n}\binom{2n-2}{n-1}\zeta(2n-3)$

Source: http://arxiv.org/abs/1210.5376v1

26
26

Sep 19, 2013
09/13

by
Oliver Schnetz

texts

######
eye 26

######
favorite 0

######
comment 0

We consider the number \bar N(q) of points in the projective complement of graph hypersurfaces over \F_q and show that the smallest graphs with non-polynomial \bar N(q) have 14 edges. We give six examples which fall into two classes. One class has an exceptional prime 2 whereas in the other class \bar N(q) depends on the number of cube roots of unity in \F_q. At graphs with 16 edges we find examples where \bar N(q) is given by a polynomial in q plus q^2 times the number of points in the...

Source: http://arxiv.org/abs/0909.0905v3

45
45

Sep 22, 2013
09/13

by
Oliver Schnetz

texts

######
eye 45

######
favorite 0

######
comment 0

A formula is derived that provides generating functions for any multi-j-symbol, such as the 3-j-symbol, the 6-j-symbol, the 9-j-symbol, etc. The result is completely determined by geometrical objects (loops and curves) in the graph of the the multi-j-symbol. A geometric-combinatorical interpretation for multi-j-symbols is given.

Source: http://arxiv.org/abs/math-ph/9805027v1

37
37

Sep 17, 2013
09/13

by
Michael Kastner; Oliver Schnetz

texts

######
eye 37

######
favorite 0

######
comment 0

Based on the study of saddle points of the potential energy landscapes of generic classical many-particle systems, we present a necessary criterion for the occurrence of a thermodynamic phase transition. Remarkably, this criterion imposes conditions on microscopic properties, namely curvatures at the saddle points of the potential, and links them to the macroscopic phenomenon of a phase transition. We apply our result to two exactly solvable models, corroborating that the criterion derived is...

Source: http://arxiv.org/abs/0711.0852v2

183
183

Jul 20, 2013
07/13

by
Francis Brown; Oliver Schnetz

texts

######
eye 183

######
favorite 0

######
comment 0

The amplitude of a Feynman graph in Quantum Field Theory is related to the point-count over finite fields of the corresponding graph hypersurface. This article reports on an experimental study of point counts over F_q modulo q^3, for graphs up to loop order 10. It is found that many of them are given by Fourier coefficients of modular forms of weights

Source: http://arxiv.org/abs/1304.5342v2

4
4.0

Jun 30, 2018
06/18

by
David Broadhurst; Oliver Schnetz

texts

######
eye 4

######
favorite 0

######
comment 0

Single-scale Feynman diagrams yield integrals that are periods, namely projective integrals of rational functions of Schwinger parameters. Algebraic geometry may therefore inform us of the types of number to which these integrals evaluate. We give examples at 3, 4 and 6 loops of massive Feynman diagrams that evaluate to Dirichlet $L$-series of modular forms and examples at 6, 7 and 8 loops of counterterms that evaluate to multiple zeta values or polylogarithms of the sixth root of unity. At 8...

Topics: High Energy Physics - Theory, Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1409.5570

47
47

Sep 23, 2013
09/13

by
Francis Brown; Oliver Schnetz

texts

######
eye 47

######
favorite 0

######
comment 0

Inspired by Feynman integral computations in quantum field theory, Kontsevich conjectured in 1997 that the number of points of graph hypersurfaces over a finite field $\F_q$ is a (quasi-) polynomial in $q$. Stembridge verified this for all graphs with $\leq12$ edges, but in 2003 Belkale and Brosnan showed that the counting functions are of general type for large graphs. In this paper we give a sufficient combinatorial criterion for a graph to have polynomial point-counts, and construct some...

Source: http://arxiv.org/abs/1006.4064v5

27
27

Sep 23, 2013
09/13

by
Oliver Schnetz; Michael Thies; Konrad Urlichs

texts

######
eye 27

######
favorite 0

######
comment 0

The massive Gross-Neveu model is solved in the large N limit at finite temperature and chemical potential. The scalar potential is given in terms of Jacobi elliptic functions. It contains three parameters which are determined by transcendental equations. Self-consistency of the scalar potential is proved. The phase diagram for non-zero bare quark mass is found to contain a kink-antikink crystal phase as well as a massive fermion gas phase featuring a cross-over from light to heavy effective...

Source: http://arxiv.org/abs/hep-th/0511206v2

40
40

Sep 18, 2013
09/13

by
Michael Kastner; Oliver Schnetz; Steffen Schreiber

texts

######
eye 40

######
favorite 0

######
comment 0

The relation between saddle points of the potential of a classical many-particle system and the analyticity properties of its Boltzmann entropy is studied. For finite systems, each saddle point is found to cause a nonanalyticity in the Boltzmann entropy, and the functional form of this nonanalytic term is derived for the generic case of potentials having the Morse property. With increasing system size the order of the nonanalytic term grows unboundedly, leading to an increasing...

Source: http://arxiv.org/abs/0803.1550v2

48
48

Sep 19, 2013
09/13

by
Michael Kastner; Oliver Schnetz; Steffen Schreiber

texts

######
eye 48

######
favorite 0

######
comment 0

The relation between saddle points of the potential of a classical many-particle system and the analyticity properties of its thermodynamic functions is studied. For finite systems, each saddle point is found to cause a nonanalyticity in the Boltzmann entropy, and the functional form of this nonanalytic term is derived. For large systems, the order of the nonanalytic term increases unboundedly, leading to an increasing differentiability of the entropy. Analyzing the contribution of the saddle...

Source: http://arxiv.org/abs/cond-mat/0703376v2

30
30

Sep 19, 2013
09/13

by
Oliver Schnetz; Michael Thies; Konrad Urlichs

texts

######
eye 30

######
favorite 0

######
comment 0

The massive Gross-Neveu model is solved in the large N limit at finite temperature and chemical potential. The phase diagram features a kink-antikink crystal phase which was missed in previous works. Translated into the framework of condensed matter physics our results generalize the bipolaron lattice in non-degenerate conducting polymers to finite temperature.

Source: http://arxiv.org/abs/hep-th/0507120v3

29
29

Jul 20, 2013
07/13

by
Francis Brown; Oliver Schnetz; Karen Yeats

texts

######
eye 29

######
favorite 0

######
comment 0

The c_2 invariant of a Feynman graph is an arithmetic invariant which detects many properties of the corresponding Feynman integral. In this paper, we define the c_2 invariant in momentum space and prove that it equals the c_2 invariant in parametric space for overall log-divergent graphs. Then we show that the c_2 invariant of a graph vanishes whenever it contains subdivergences. Finally, we investigate how the c_2 invariant relates to identities such as the four-term relation in knot theory.

Source: http://arxiv.org/abs/1203.0188v2

36
36

Sep 21, 2013
09/13

by
Oliver Schnetz; Michael Thies; Konrad Urlichs

texts

######
eye 36

######
favorite 0

######
comment 0

Recently the revised phase diagram of the (large N) Gross-Neveu model in 1+1 dimensions with discrete chiral symmetry has been determined numerically. It features three phases, a massless and a massive Fermi gas and a kink-antikink crystal. Here we investigate the phase diagram by analytical means, mapping the Dirac-Hartree-Fock equation onto the non-relativistic Schroedinger equation with the (single gap) Lame potential. It is pointed out that mathematically identical phase diagrams appeared...

Source: http://arxiv.org/abs/hep-th/0402014v4