Simplifying one-loop amplitudes in superstring theory

Bianchi, Massimo; Consoli, Dario

doi:10.1007/jhep01(2016)043

Cited by 15 publications

(44 citation statements)

References 77 publications

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“…in case of n ≥ 8 external legs [19,52]. Similarly, one-loop superstring amplitudes with reduced supersymmetry additionally involve degree-η −1 and η −5 parts of (1.1) [53,54], while degree-η +1 parts only enter bosonicstring amplitudes or chiral halves of the heterotic string [55]. Higher orders in the η j are likely to be relevant for one-loop amplitudes involving massive states.…”

Section: Summary Of the Main Resultsmentioning

confidence: 99%

“…One-loop string amplitudes can be reduced to A-cycle integrals over products of f (m k ) ij at fixed overall weight k m k [19,[52][53][54][55] and do not involve the full η-dependent Ω(z ij , η, τ ) in the integrand. In order to apply the results on the generating functions Z τ η in (1.1) to the integrals in string amplitudes, one has to extract particular orders in η j .…”

Section: Extracting Component Integralsmentioning

confidence: 99%

“…It will be shown in appendix F that the coefficients in the q-expansion of A-cycle graph functions at any multiplicity are Q-linear as opposed to Q[(2πi) −1 ]-linear combinations of MZVs. Four-point one-loop amplitudes with half-or quarter-maximal spacetime supersymmetry involve moduli-space integrals over f (2) ij and f (1) ij f (1) pq [53,54]. Hence, their α -expansions will require the on-shell limit of Z τ (2,0,0) (1, 2, 3, 4|1, b 2 , b 3 , 4) and Z τ (1,0,1) (1, 2, 3, 4|1, b 2 , b 3 , 4) with (b 2 , b 3 ) ∈ S 2 , which address the cyclically inequivalent integrands f 6 Non-planar genus-one integrals at the cusp We shall now extend the above discussion of initial values Z i∞ η (A| * ) associated with planar integration cycles C(A) to non-planar ones C Q P .…”

Section: Component Integralsmentioning

confidence: 99%

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One-loop open-string integrals from differential equations: all-order α′-expansions at n points

Mafra

Schlotterer

2020

J. High Energ. Phys.

183

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We study generating functions of moduli-space integrals at genus one that are expected to form a basis for massless n-point one-loop amplitudes of open superstrings and open bosonic strings. These integrals are shown to satisfy the same type of linear and homogeneous first-order differential equation w.r.t. the modular parameter τ which is known from the A-elliptic Knizhnik-Zamolodchikov-Bernard associator. The expressions for their τderivatives take a universal form for the integration cycles in planar and non-planar one-loop open-string amplitudes. These differential equations manifest the uniformly transcendental appearance of iterated integrals over holomorphic Eisenstein series in the low-energy expansion w.r.t. the inverse string tension α . In fact, we are led to matrix representations of certain derivations dual to Eisenstein series. Like this, also the α -expansion of non-planar integrals is manifestly expressible in terms of iterated Eisenstein integrals without referring to twisted elliptic multiple zeta values. The degeneration of the moduli-space integrals at τ → i∞ is expressed in terms of their genus-zero analogues -(n+2)-point Parke-Taylor integrals over disk boundaries. Our results yield a compact formula for α -expansions of n-point integrals over boundaries of cylinder-or Möbius-strip worldsheets, where any desired order is accessible from elementary operations. Contents 7 Formal properties 59 7.1 Uniform transcendentality 59 7.2 Coaction 60 8 Conclusions 63 A Resolving cycles of Kronecker-Eisenstein series 65 A.1 Two points 65 A.2 Three points 66 A.3 Higher points 67 B The non-planar Green function at the cusp 67 C More on the τ -derivatives of A-cycle integrals 68 C.1 The 6 × 6 representation of the derivations at four points 68 C.2 The τ -derivative at five points in a 24-element basis 69 D Transformation matrices between twisted cycles 69 D.1 Four-point example 70 D.2 Weighted combinations 70 E Examples of α -expansions 71 E.1 Three points: integrating f (1) 12 f (3) 23 71 E.2 Four points: MZVs for the initial values 71 F Fourier expansion of A-cycle graph functions 72 1 The notion of single-valued MZVs was introduced in [6, 7]. 2 See for instance [13-16] for earlier work on tree-level α -expansions at n ≤ 7 points, in particular [14, 15, 17, 18] for synergies with hypergeometric-function representations. 3 This relies on the linear-independence result of [26] on iterated Eisenstein integrals.-2 -trices in [34]. As a genus-one generalization, the A-cycle integrals under investigation are shown to obey the same type of differential equation in τ as the elliptic Knizhnik-Zamolodchikov-Bernard (KZB) associator [35][36][37]. In particular, n-point A-cycle integrals induce (n−1)!×(n−1)! matrix representations of certain derivations dual to Eisenstein series [38] which accompany the iterated Eisenstein integrals in the α -expansions.• Third, only (n−3)! choices for n-point disk integrands are inequivalent under integration by parts, i.e. the so-called twisted cohomology at genus zero has dimensio...

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Section: Summary Of the Main Resultsmentioning

confidence: 99%

Section: Extracting Component Integralsmentioning

confidence: 99%

Section: Component Integralsmentioning

confidence: 99%

See 1 more Smart Citation

One-loop open-string integrals from differential equations: all-order α′-expansions at n points

Mafra

Schlotterer

2020

J. High Energ. Phys.

183

View full text Add to dashboard Cite

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“…, b n−2 , 0, b n ), respectively, then the only places where the points z n−1 and z n 6 Also see e.g. [65][66][67] for earlier work on RNS spin sums, [68][69][70][71][72] for g in one-loop amplitudes in the pure-spinor formalism and [73,74] for applications to RNS one-loop amplitudes with reduced supersymmetry. 7 String-theory integrals can also involve integrals over ∂z i f…”

Section: Relations Between Component Integralsmentioning

confidence: 99%

“…i k j k of weight k a k = n−2 or w+ k a k = n−2 [73,74]. The resulting component integrals W τ (A|B) (σ|ρ) in closedstring amplitudes with such chiral halves have holomorphic modular weights |A| ≤ n−2.…”

Section: Component Integrals Versus N-point String Amplitudesmentioning

confidence: 99%

All-order differential equations for one-loop closed-string integrals and modular graph forms

Gerken

Kleinschmidt

Schlotterer

2020

J. High Energ. Phys.

140

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We investigate generating functions for the integrals over world-sheet tori appearing in closed-string one-loop amplitudes of bosonic, heterotic and type-II theories. These closed-string integrals are shown to obey homogeneous and linear differential equations in the modular parameter of the torus. We spell out the first-order Cauchy-Riemann and secondorder Laplace equations for the generating functions for any number of external states. The low-energy expansion of such torus integrals introduces infinite families of non-holomorphic modular forms known as modular graph forms. Our results generate homogeneous first-and second-order differential equations for arbitrary such modular graph forms and can be viewed as a step towards all-order low-energy expansions of closed-string integrals. arXiv:1911.03476v2 [hep-th] 21 Jan 2020 F Derivation of component equations at three points 65 F.1 General Cauchy-Riemann component equations at three points 66 F.2 Examples of Cauchy-Riemann component equations at three points 67 F.3 Further examples for Laplace equations at three points 68 -ii -10The subscript of ∇DG aims to avoid confusion with the differential operators of Sections 3.1 and 3.2 and refers to the authors D'Hoker and Green of [9] which initiated the systematic study of relations between modular graph forms via repeated action of (2.59).

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String (Gravi)photons, “Dark Brane Photons”, Holography and the Hypercharge Portal

Anastasopoulos

Bianchi

Consoli

et al. 2021

Fortschritte der Physik

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The mixing of graviphotons and dark brane photons to the Standard Model hypercharge is analyzed in full generality, in weakly‐coupled string theory. Both the direct mixing as well as effective terms that provide mixing after inclusion of SM corrections are estimated to lowest order. The results are compared with Effective Field Theory (EFT) couplings, originating in a hidden large‐N theory coupled to the SM where the dark photons are composite. The string theory mixing terms are typically subleading compared with the generic EFT couplings. The case where the hidden theory is a holographic theory is also analyzed, providing also suppressed mixing terms to the SM hypercharge.

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Simplifying one-loop amplitudes in superstring theory

Cited by 15 publications

References 77 publications

One-loop open-string integrals from differential equations: all-order α′-expansions at n points

One-loop open-string integrals from differential equations: all-order α′-expansions at n points

All-order differential equations for one-loop closed-string integrals and modular graph forms

String (Gravi)photons, “Dark Brane Photons”, Holography and the Hypercharge Portal

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