Modular forms in quantum field theory

Brown, Francis; Schnetz, Oliver

doi:10.4310/cntp.2013.v7.n2.a3

Cited by 40 publications

(110 citation statements)

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“…Additionally, these sorts of studies of the mathematical structure of high-order perturbation expansions are common in quantum field theory, where one finds deep connections to other mathematical structures: See, e.g., [25][26][27]. It would be interesting to see if similar structures exist in the post-Newtonian expansion, particularly since it can be written in terms of the same sorts of loop integrals studied in quantum field theory, as noted by Bini and Damour [28].…”

Section: Discussionmentioning

confidence: 99%

Taming the post-Newtonian expansion: Simplifying the modes of the gravitational wave energy flux at infinity for a point particle in a circular orbit around a Schwarzschild black hole

Johnson-McDaniel

2014

Phys. Rev. D

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High-order terms in the post-Newtonian (PN) expansions of various quantities for compact binaries exhibit a combinatorial increase in complexity, with an ever-increasing number of terms, including more transcendentals and logarithms of the velocity, higher powers of these transcendentals and logarithms, and larger and larger rational numbers as coefficients. Here we consider the gravitational wave energy flux at infinity from a point particle in a circular orbit around a Schwarzschild black hole, which is known to 22PN [O(v 44 ), where v is the particle's orbital velocity] beyond the lowest-order Newtonian prediction, at which point each order has over 1000 terms. We introduce a factorization that considerably simplifies the spherical harmonic modes of the energy flux (and thus also the amplitudes of the spherical harmonic modes of the gravitational waves); it is likely that much of the complexity this factorization removes is due to wave propagation on the Schwarzschild spacetime (e.g., tail effects). For the modes with azimuthal number ℓ ≥ 7, this factorization reduces the expressions for the modes that enter the 22PN total energy flux to pure integer PN series with rational coefficients, which amounts to a reduction of up to a factor of ∼ 150 in the total number of terms in a given mode (and also in the size of the entire expression for the mode). The reduction in complexity becomes less dramatic for smaller ℓ, due to the structure of the expansion, and one only obtains purely rational coefficients up to some order, though the factorization is still able to remove all the half-integer PN terms. For the 22PN ℓ = 2 modes, this factorization still reduces the total number of terms (and size) by a factor of ∼ 10 and gives purely rational coefficients through 8PN. This factorization also improves the convergence of the series, though we find the exponential resummation introduced for the full energy flux in [Isoyama et al., Phys. Rev. D 87, 024010 (2013)] to be even more effective at improving the convergence of the individual modes, producing improvements of over four orders of magnitude over the original series for some modes. However, the exponential resummation is not as effective at simplifying the series, particularly for the higher-order modes.

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Section: Discussionmentioning

confidence: 99%

Taming the post-Newtonian expansion: Simplifying the modes of the gravitational wave energy flux at infinity for a point particle in a circular orbit around a Schwarzschild black hole

Johnson-McDaniel

2014

Phys. Rev. D

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“…All c 2 invariants of primitive log-divergent graphs with ≤ 20 edges are listed for the first six primes in [12].…”

Section: Conjecturementioning

confidence: 99%

Properties of $c_2$ invariants of Feynman graphs

Brown

Schnetz

Yeats

2014

Advances in Theoretical and Mathematical Physics

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Abstract. The c2 invariant of a Feynman graph is an arithmetic invariant which detects many properties of the corresponding Feynman integral. In this paper, we define the c2 invariant in momentum space and prove that it equals the c2 invariant in parametric space for overall log-divergent graphs. Then we show that the c2 invariant of a graph vanishes whenever it contains subdivergences. Finally, we investigate how the c2 invariant relates to identities such as the four-term relation in knot theory. IntroductionLet G be a connected graph. The graph polynomial of G is defined by associating a variable x e to every edge e of G and settingwhere the sum is over all spanning trees T of G. These polynomials first appeared in Kirchhoff's work on currents in electrical networks [16]. Let N G denote the number of edges of G, and let h G denote the number of independent cycles in G (the first Betti number). Of particular interest is the case when G is primitive and overall logarithmically divergent:N γ > 2h γ for all strict non-trivial subgraphs γ G .For such graphs, the corresponding Feynman integral (or residue) is independent of the choice of renormalization scheme and can be defined by the following convergent integral in parametric spaceThe numbers I G are notoriously difficult to calculate, and have been investigated intensively from the numerical [5,20] and algebro-geometric points of view [4,9]. For graphs in φ 4 theory with subdivergences, the renormalised Karen Yeats is supported by an NSERC discovery grant and would like to thank Samson Black for explaining knots. Francis Brown is partially supported by ERC grant 257638. Francis Brown and Oliver Schnetz thank Humboldt University, Berlin, for support as visiting guest scientists. All three authors thank Humboldt University for hospitality.

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“…Note that in [3] they have the condition that the dimension of the cycle space of H is at most two less than the number of edges of H, however using Euler's formula this condition is equivalent to H having at least 3 vertices. In what follows we will restrict to c 2 invariants at primes p, not more general prime powers, since this is computationally accessible and corresponds to what has been calculated elsewhere [4].…”

Section: Set Upmentioning

confidence: 99%

“…The c 2 invariant is an arithmetic graph invariant introduced by Schnetz in [11] in order to better understand certain Feynman integrals. The c 2 invariant sees aspects of the same underlying geometry that the Feynman period sees [3,4] and consequently the c 2 invariant can predict things about what classes of numbers can show up in a given Feynman period. See Subsection 3.3 for some further comments in this direction.…”

Section: Introductionmentioning

confidence: 99%

“…As we will see, c (q) 2 can be defined for prime powers, not just primes, but we will stick to primes herein. Previous work of Brown and Schnetz [4] calculated c 2 invariants for graphs up to 10 loops at small primes (up to the first 100 primes in some cases). These calculations uncovered many interesting patterns, most notably coefficient sequences of q-expansions of modular forms.…”

Section: Introductionmentioning

confidence: 99%

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A study on prefixes of $c_2$ invariants

Yeats

2020

IRMA Lectures in Mathematics and Theoretical Physics

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This paper begins by reviewing recent progress that has been made by taking a combinatorial perspective on the c 2 invariant, an arithmetic graph invariant with connections to Feynman integrals. Then it proceeds to report on some recent calculations of c 2 invariants for two families of circulant graphs at small primes. These calculations support the idea that all possible finite sequences appear as initial segments of c 2 invariants, in contrast to their apparent sparsity on small graphs.Thanks to Oliver Schnetz for goading me into doing the prefix calculation, and for his continued excitement about the c 2 invariant. Thanks to Iain Crump and Freddy Cachazo for discussions. Thanks to NSERC and the Humboldt foundation for support. Thanks to the organizers of the Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) conference for their excellent conference.

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Modular forms in quantum field theory

Cited by 40 publications

References 13 publications

Taming the post-Newtonian expansion: Simplifying the modes of the gravitational wave energy flux at infinity for a point particle in a circular orbit around a Schwarzschild black hole

Taming the post-Newtonian expansion: Simplifying the modes of the gravitational wave energy flux at infinity for a point particle in a circular orbit around a Schwarzschild black hole

Properties of $c_2$ invariants of Feynman graphs

A study on prefixes of $c_2$ invariants

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