We revisit the conjectural method called Schubert analysis for generating the alphabet of symbol letters for Feynman integrals, which was based on geometries of intersecting lines associated with corresponding cut diagrams. We explain the effectiveness of this somewhat mysterious method by relating such geometries to the corresponding Landau singularities, which also amounts to “uplifting” Landau singularities of a Feynman integral to its symbol letters. We illustrate this $\textit{Landau-based Schubert analysis}$ using various multi-loop Feynman integrals in four dimensions and present an automated Mathematica notebook for it. We then apply the method to a simplified problem of studying alphabets of physical quantities such as scattering amplitudes and form factors in planar $$ \mathcal{N} $$ = 4 super-Yang-Mills. By focusing on a small set of Landau diagrams (as opposed to all relevant Feynman integrals), we show how this method nicely produces the two-loop alphabet of $\textit{n}$-point MHV amplitudes and that of the $\textit{n}$ = 4 MHV form factors. A byproduct of our analysis is an explicit representation of any symbol alphabet obtained this way as the union of various type-$\textit{A}$ cluster algebras.

China

Song He

Xuhang Jiang

Jiahao Liu

Germany

Qinglin Yang

Landau-based Schubert analysis

In this paper, we propose a construction of GLSM defects corresponding to Schubert cycles in Lagrangian Grassmannians, following recent work of Closset-Khlaif on Schubert cycles in ordinary Grassmannians. In the case of Lagrangian Grassmannians, there are superpotential terms in both the bulk GLSM as well as on the defect itself, enforcing isotropy constraints. We check our construction by comparing the locus on which the GLSM defect is supported to mathematical descriptions, checking dimensions, and perhaps most importantly, comparing defect indices to known and expected polynomial invariants of the Schubert cycles in quantum cohomology and quantum K theory.

Wei Gu

United States

Leonardo Mihalcea

Eric Sharpe

Weihong Xu

Hao Zhang

Hao Zou

Schubert defects in Lagrangian Grassmannians

Erratum to: On symbology and differential equations of Feynman integrals from Schubert analysis

While the QED photon amplitudes at full momentum so far have been calculated only up to the six-photon level, in the low-energy limit there are explicit formulas for all helicity components even at the N-photon level, obtained by Martin et al. in 2002. Here we use the worldline formalism to extend that result to the N-photon amplitudes in a generic constant field. For both scalar and spinor QED, we obtain compact representations for the low-energy limits of these amplitudes involving only simple algebra and a single global proper-time integral with trigonometric integrand.

Naser Ahmadiniaz

Misha A. Lopez-Lopez

Mexico

Christian Schubert

Low-energy limit of N-photon amplitudes in a constant field

A correction to this paper has been published: https://doi.org/10.1007/JHEP04(2023)072

Qu Cao

Yichao Tang

Erratum to: Cutting the traintracks: Cauchy, Schubert and Calabi-Yau

We take the first step in generalizing the so-called “Schubert analysis”, originally proposed in twistor space for four-dimensional kinematics, to the study of symbol letters and more detailed information on canonical differential equations for Feynman integral families in general dimensions with general masses. The basic idea is to work in embedding space and compute possible cross-ratios built from (Lorentz products of) maximal cut solutions for all integrals in the family. We demonstrate the power of the method using the most general one-loop integrals, as well as various two-loop planar integral families (such as sunrise, double-triangle and double-box) in general dimensions. Not only can we obtain all symbol letters as cross-ratios from maximal-cut solutions, but we also reproduce entries in the canonical differential equations satisfied by a basis of d log integrals.

On symbology and differential equations of Feynman integrals from Schubert analysis

The worldline formalism has previously been used for deriving compact master formulas for the QED N - photon amplitudes in vacuum, in a constant field and in a plane-wave field. Here we carry this program one step further by deriving master formulas for the scalar and spinor QED one-loop N-photon amplitudes in the background of the “parallel” special case of a combined constant and plane-wave field.

Czechia

Rashid Shaisultanov

Master formulas for photon amplitudes in a combined constant and plane-wave background field

In previous work on Schwinger pair creation in purely time-dependent fields, it was shown how to construct “solitonic” electric fields that do not create scalar pairs with an arbitrary fixed momentum. We show that this construction can be adapted to the fermionic case in two inequivalent ways, both closely related to supersymmetric quantum mechanics for reflectionless potentials, and both leading to the vanishing of the density of created pairs at certain values of the Pöschl-Teller like index <math><mi>p</mi></math> of the associated Schrödinger equation. For one of them, we are able to demonstrate that the pair noncreation can be interpreted as a quantum interference effect using the phase-integral formalism. Asymptotically for large <math><mi>p</mi></math>, here scalar particles are not created for integer <math><mi>p</mi></math> and fermions are not created for half-integer <math><mi>p</mi></math>. Thus, for any given momentum we can construct electric fields that create scalar particles but not spinor particles, and vice versa. In the scalar QED case, the solitonic fields had originally been found using the Gelfand-Dikii equation, which is related to the resolvent of the mode equation, and through it to the (generalized) KdV equation [I. M. Gelfand and L. A. Dikii, Asymptotic behavior of the resolvent of Sturm-Liouville equations and the Algebra of the Korteweg-De Vries equations, Russ. Math. Surv. 30, 5 (1975)]. This motivates us to develop for the spinor QED case, too, an evolution equation that can be considered as a fermionic generalization of the Gelfand-Dikii equation.

Alexander M. Fedotov

Evgeny G. Gelfer

South Korea

Sang Pyo Kim

Generalized Gelfand-Dikii equation and solitonic electric fields for fermionic Schwinger pair production

The symbol bootstrap has proven to be a powerful tool for calculating polylogarithmic Feynman integrals and scattering amplitudes. In this Letter, we initiate the symbol bootstrap for elliptic Feynman integrals. Concretely, we bootstrap the symbol of the twelve-point two-loop double-box integral in four dimensions, which depends on nine dual-conformal cross ratios. We obtain the symbol alphabet, which contains 100 logarithms as well as nine simple elliptic integrals, via a Schubert-type analysis, which we equally generalize to the elliptic case. In particular, we find a compact, one-line formula for the (2,2) coproduct of the result.

Denmark

Roger Morales

Anne Spiering

Matthias Wilhelm

Chi Zhang

Bootstrapping Elliptic Feynman Integrals Using Schubert Analysis

This is the second one of a series of four papers devoted to a first calculation of the scalar and spinor QED four-photon amplitudes completely off-shell. We use the worldline formalism which provides a gauge-invariant decomposition for these amplitudes as well as compact integral representations. It also makes it straightforward to integrate out any given photon leg in the low-energy limit, and in the present sequel we do this with two of the four photons. For the special case where the two unrestricted photon momenta are equal and opposite the information on these amplitudes is also contained in the constant-field vacuum polarisation tensors, which provides a check on our results. Although these amplitudes are finite, for possible use as higher-loop building blocks we evaluate all integrals in dimensional regularisation. As an example, we use them to construct the two-loop vacuum polarisation tensors in the low-energy approximation, rederive from those the two-loop β-function coefficients and analyse their anatomy with respect to the gauge-invariant decomposition. As an application to an external-field problem, we provide a streamlined calculation of the Delbrück scattering amplitudes in the low-energy limit. All calculations are done in parallel for scalar and spinor QED.

Colombia

Cristhiam Lopez-Arcos

The QED four-photon amplitudes off-shell: Part 2