New BCJ representations for one-loop amplitudes in gauge theories and gravity

He, Song; Schlotterer, Oliver; Zhang, Yong

doi:10.1016/j.nuclphysb.2018.03.003

Cited by 92 publications

(176 citation statements)

References 161 publications

(535 reference statements)

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“…21 See Section 5.3 of [98] for a recent account on the key challenges in setting up loop-level KLT relations among string amplitudes from the viewpoint of intersection theory [99]. A field-theory version of one-loop KLT relations among loop integrands in gauge theories and supergravity has been derived from ambitwistor strings [100,101]. A connection with the present results may be investigated by adapting the generating functions of this work to the chiral-splitting formalism [102][103][104], where a string-theory analogue of the loop momentum is introduced at the expense of obscuring modular invariance.…”

Section: Discussionmentioning

confidence: 99%

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

2020

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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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Section: Discussionmentioning

confidence: 99%

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

2020

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“…. σ n ) −1 and products of G ij , one can apply the algorithmic method of [76] to recover combinations of SL 2 -fixed Parke-Taylor factors in n+2 punctures, PT(a 1 , a 2 , a 3 , . .…”

Section: Recovering Parke-taylor Disk Integrandsmentioning

confidence: 99%

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

Mafra

Schlotterer

2020

J. High Energ. Phys.

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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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“…interplay of GEIs with the color-kinematics dual field-theory amplitudes obtained from the methods of [51,52]. The same questions will arise at higher genus [53,54].…”

Section: Discussionmentioning

confidence: 87%

Towards the n-point one-loop superstring amplitude. Part III. One-loop correlators and their double-copy structure

Mafra

Schlotterer

2019

J. High Energ. Phys.

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In this final part of a series of three papers, we will assemble supersymmetric expressions for one-loop correlators in pure-spinor superspace that are BRST invariant, local, and single valued. A key driving force in this construction is the generalization of a so far unnoticed property at tree level; the correlators have the symmetry structure akin to Lie polynomials. One-loop correlators up to seven points are presented in a variety of representations manifesting different subsets of their defining properties. These expressions are related via identities obeyed by the kinematic superfields and worldsheet functions spelled out in the first two parts of this series and reflecting a duality between the two kinds of ingredients.Interestingly, the expression for the eight-point correlator following from our method seems to capture correctly all the dependence on the worldsheet punctures but leaves undetermined the coefficient of the holomorphic Eisenstein series G 4 . By virtue of chiral splitting, closed-string correlators follow from the double copy of the open-string results.December 2018 †

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New BCJ representations for one-loop amplitudes in gauge theories and gravity

Cited by 92 publications

References 161 publications

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

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

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

Towards the n-point one-loop superstring amplitude. Part III. One-loop correlators and their double-copy structure

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