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Oliver Schnetz

Publications and source records attributed to Oliver Schnetz.

At least 37 records · Page 2Linked to original sources
arXiv2018

Closed strings as single-valued open strings: A genus-zero derivation

Based on general mathematical assumptions we give an independent, elementary derivation of a theorem by Francis Brown and Clément Dupont which states that tree-level amplitudes of closed and open strings are related through the single-valued map `sv'. This relation can be traced back to the underlying moduli-space integrals over punctured Riemann surfaces of genus zero. The sphere integrals $J$ in closed-string amplitudes and the disk integrals $Z$ in open-string amplitudes are shown to obey $J = {\rm sv} \, Z$.

hep-th↗
arXiv2018

The Galois coaction on the electron anomalous magnetic moment

Recently S. Laporta published a partial result on the fourth order QED contribution to the electron anomalous magnetic moment $g-2$. This result contains explicit polylogarithmic parts with fourth and sixth roots of unity. In this note we convert Laporta's result into the motivic `$f$ alphabet'. This provides a much shorter expression which makes the Galois structure visible. We conjecture the $Q$ vector spaces of Galois conjugates of the QED $g-2$ up to weight four. The conversion into the $f$ alphabet relies on a conjecture by D. Broadhurst that iterated integrals in certain Lyndon words provide an algebra basis for the extension of multiple zeta values by sixth roots of unity. We prove this conjecture in the motivic setup.

math-ph↗
arXiv2017

Numbers and Functions in Quantum Field Theory

We review recent results in the theory of numbers and single-valued functions on the complex plane which arise in quantum field theory. We use the results to calculate the renormalization functions $β$, $γ$, $γ_m$ of dimensionally regularized $ϕ^4$ theory in the minimal subtraction scheme up to seven loops.

hep-th↗
arXiv2017

Graphical functions in parametric space

Graphical functions are positive functions on the punctured complex plane $\mathbb{C}\setminus\{0,1\}$ which arise in quantum field theory. We generalize a parametric integral representation for graphical functions due to Lam, Lebrun and Nakanishi, which implies the real analyticity of graphical functions. Moreover we prove a formula that relates graphical functions of planar dual graphs.

math-ph↗
arXiv2017

The Galois coaction on $ϕ^4$ periods

We report on calculations of Feynman periods of primitive log-divergent $ϕ^4$ graphs up to eleven loops. The structure of $ϕ^4$ periods is described by a series of conjectures. In particular, we discuss the possibility that $ϕ^4$ periods are a comodule under the Galois coaction. Finally, we compare the results with the periods of primitive log-divergent non-$ϕ^4$ graphs up to eight loops and find remarkable differences to $ϕ^4$ periods. Explicit results for all periods we could compute are provided in ancillary files.

hep-th↗
arXiv2014

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 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 that allows one to calculate their periods by computer algebra. The theory of graphical functions is used to prove the zig-zag conjecture.

math.NT↗
arXiv2014

Algebraic geometry informs perturbative quantum field theory

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 loops and beyond, algebraic geometry informs us that polylogs are insufficient for the evaluation of terms in the beta-function of $ϕ^4$ theory. Here, modular forms appear as obstructions to polylogarithmic evaluation.

hep-th↗
arXiv2013

Modular forms in Quantum Field Theory

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 <=8 and levels <=17.

math.AG↗
arXiv2013

Proof of the zig-zag conjecture

A long-standing conjecture in quantum field theory due to Broadhurst and Kreimer states that the amplitudes of the zig-zag graphs are a certain explicit rational multiple of the odd values of the Riemann zeta function. In this paper we prove this conjecture by constructing a certain family of single-valued multiple polylogarithms. The zig-zag graphs therefore provide the only infinite family of primitive graphs in $ϕ^4_4$ theory (in fact, in any renormalisable quantum field theory in four dimensions) whose amplitudes are now known.

math.NT↗
arXiv2012

Properties of c_2 invariants of Feynman graphs

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.

math.AG↗
arXiv2011

A K3 in phi4

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 explicit counter-examples to Kontsevich's conjecture which are in $ϕ^4$ theory. Their counting functions are given modulo $pq^2$ ($q=p^n$) by a modular form arising from a certain singular K3 surface.

math.AG↗
arXiv2011

Quantum field theory over F_q

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 projective complement of a singular K3 in ¶^3. In the second part of the paper we show that applying momentum space Feynman-rules over \F_q lets the perturbation series terminate for renormalizable and non-renormalizable bosonic quantum field theories.

math.CO↗
arXiv2010

The geometry of one-loop amplitudes

We review a reduction formula by Petersson that reduces the calculation of a one-loop amplitude with N external lines in n<N space-time dimensions to the case n=N and give it a geometric interpretation. In the case n=N the calculation of the euclidean amplitude is shown to be equivalent to the calculation of the volume of a tetrahedron spanned by the momenta in (n-1)-dimensional hyperbolic space. The underlying geometry is intimately linked to the geometry of the reduction formula.

hep-th↗
arXiv2009

Quantum periods: A census of ϕ^4-transcendentals

Perturbative quantum field theories frequently feature rational linear combinations of multiple zeta values (periods). In massless ϕ^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.

hep-th↗
arXiv2008

Nonanalyticities of the entropy induced by saddle points of the potential energy landscape

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 differentiability of the entropy. Nonetheless, a distribution of an unboundedly growing number of saddle points may cause a phase transition in the thermodynamic limit. Analyzing the contribution of the saddle points to the density of states in the thermodynamic limit, conditions on the distribution of saddle points and their curvatures are derived which are necessary for a phase transition to occur. With these results, the puzzling absence of topological signatures in the spherical model is elucidated. As further applications, the phase transitions of the mean-field XY model and the mean-field k-trigonometric model are shown to be induced by saddle points of vanishing curvature.

cond-mat.stat-mech↗
arXiv2008

Phase transitions induced by saddle points of vanishing curvature

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 not only valid, but also sharp and useful: For both models studied, the criterion excludes the occurrence of a phase transition for all values of the potential energy but the transition energy. This result adds a geometrical ingredient to an established topological condition for the occurrence of a phase transition, thereby providing an answer to the long standing question of which topology changes in configuration space can induce a phase transition.

cond-mat.stat-mech↗
arXiv2007

Phase Transitions from Saddles of the Potential Energy Landscape

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 points to the density of states in the thermodynamic limit, our results provide an explanation of how, and under which circumstances, saddle points of the potential energy landscape may (or may not) be at the origin of a phase transition in the thermodynamic limit. As an application, the puzzling observations by Risau-Gusman et al. on topological signatures of the spherical model are elucidated.

cond-mat.stat-mech↗