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Zhenjie Li

Publications and source records attributed to Zhenjie Li.

27 records · Page 2Linked to original sources
arXiv2021

The Wilson-loop $d \log$ representation for Feynman integrals

We introduce and study the Wilson-loop ${\rm d}\log$ representation of certain Feynman integrals for scattering amplitudes in ${\cal N}=4$ SYM and beyond, which makes their evaluation completely straightforward. Such a representation was motivated by the dual Wilson loop picture, and it can also be derived by partial Feynman parametrization of loop integrals. We first introduce it for the simplest one-loop examples, the chiral pentagon in four dimensions and the three-mass-easy hexagon in six dimensions, which are represented by two- and three-fold ${\rm d}\log$ integrals that are nicely related to each other. For multi-loop examples, we write the $L$-loop generalized penta-ladders as $2(L{-}1)$-fold ${\rm d}\log$ integrals of some one-loop integral, so that once the latter is known, the integration can be performed in a systematic way. In particular, we write the eight-point penta-ladder as a $2L$-fold ${\rm d}\log$ integral whose symbol can be computed without performing the integration; we also obtain the last entries and the symbol alphabet of these integrals. Similarly we compute and study the symbol of the seven-point double-penta-ladder, which is represented by a $2(L{-}1)$-fold integral of a hexagon; the latter can be written as a two-fold ${\rm d}\log$ integral plus a boundary term. We comment on the relation of our representation to differential equations and resum the ladders by solving certain integral equations.

hep-th↗
arXiv2021

A Note on Letters of Yangian Invariants

Motivated by reformulating Yangian invariants in planar ${\cal N}=4$ SYM directly as $d\log$ forms on momentum-twistor space, we propose a purely algebraic problem of determining the arguments of the $d\log$'s, which we call "letters", for any Yangian invariant. These are functions of momentum twistors $Z$'s, given by the positive coordinates $α$'s of parametrizations of the matrix $C(α)$, evaluated on the support of polynomial equations $C(α) \cdot Z=0$. We provide evidence that the letters of Yangian invariants are related to the cluster algebra of Grassmannian $G(4,n)$, which is relevant for the symbol alphabet of $n$-point scattering amplitudes. For $n=6,7$, the collection of letters for all Yangian invariants contains the cluster ${\cal A}$ coordinates of $G(4,n)$. We determine algebraic letters of Yangian invariant associated with any "four-mass" box, which for $n=8$ reproduce the $18$ multiplicative-independent, algebraic symbol letters discovered recently for two-loop amplitudes.

hep-th↗
arXiv2021

The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations

We study the symbol and the alphabet for two-loop NMHV amplitudes in planar ${\cal N}=4$ super-Yang-Mills from the $\bar{Q}$ equations, which provide a first-principle method for computing multi-loop amplitudes. Starting from one-loop N${}^2$MHV ratio functions, we explain in detail how to use $\bar{Q}$ equations to obtain the total differential of two-loop $n$-point NMHV amplitudes, whose symbol contains letters that are algebraic functions of kinematics for $n\geq 8$. We present explicit formula with nice patterns for the part of the symbol involving algebraic letters for all multiplicities, and we find $17-2m$ multiplicative-independent letters for a given square root of Gram determinant, with $0\leq m\leq 4$ depending on the number of particles involved in the square root. We also observe that these algebraic letters can be found as poles of one-loop four-mass leading singularities with MHV or NMHV trees. As a byproduct of our algebraic results, we find a large class of components of two-loop NMHV, which can be written as differences of two double-pentagon integrals, particularly simple and absent of square roots. As an example, we present the complete symbol for $n=9$ whose alphabet contains $59\times 9$ rational letters, in addition to the $11 \times 9$ independent algebraic ones. We also give all-loop NMHV last-entry conditions for all multiplicities.

hep-th↗
arXiv2021

Feynman Integrals and Scattering Amplitudes from Wilson Loops

We study Feynman integrals and scattering amplitudes in ${\cal N}=4$ super-Yang-Mills by exploiting the duality with null polygonal Wilson loops. Certain Feynman integrals, including one-loop and two-loop chiral pentagons, are given by Feynman diagrams of a supersymmetric Wilson loop, where one can perform loop integrations and be left with simple integrals along edges. As the main application, we compute analytically for the first time, the symbol of the generic ($n\geq 12$) double pentagon, which gives two-loop MHV amplitudes and components of NMHV amplitudes to all multiplicities. We represent the double pentagon as a two-fold $\mathrm{d} \log$ integral of a one-loop hexagon, and the non-trivial part of the integration lies at rationalizing square roots contained in the latter. We obtain a remarkably compact "algebraic words" which contain $6$ algebraic letters for each of the $16$ square roots, and they all nicely cancel in combinations for MHV amplitudes and NMHV components which are free of square roots. In addition to $96$ algebraic letters, the alphabet consists of $152$ dual conformal invariant combinations of rational letters.

hep-th↗
arXiv2020

Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra

Stringy canonical forms are a class of integrals that provide $α'$-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebra, there exist completely rigid stringy integrals, whose configuration spaces are the so-called binary geometries, and for classical types are associated with (generalized) scattering of particles and strings. In this paper we propose a large class of rigid stringy canonical forms for another class of polytopes, generalized permutohedra, which also include associahedra and cyclohedra as special cases (type $A_n$ and $B_n$ generalized associahedra). Remarkably, we find that the configuration spaces of such integrals are also binary geometries, which were suspected to exist for generalized associahedra only. For any generalized permutohedron that can be written as Minkowski sum of coordinate simplices, we show that its rigid stringy integral factorizes into products of lower integrals for massless poles at finite $α'$, and the configuration space is binary although the $u$ equations take a more general form than those "perfect" ones for cluster cases. Moreover, we provide an infinite class of examples obtained by degenerations of type $A_n$ and $B_n$ integrals, which have perfect $u$ equations as well. Our results provide yet another family of generalizations of the usual string integral and moduli space, whose physical interpretations remain to be explored.

hep-th↗
arXiv2020

The study of calibration for the hybrid pixel detector with single photon counting in HEPS-BPIX

The calibration process for the hybrid array pixel detector designed for High Energy Photon Source in China, we called HEPS-BPIX, is presented in this paper. Based on the threshold scanning, the relationship between energy and threshold is quantified for the threshold calibration. For the threshold trimming, the precise algorithm basing on LDAC characteristic and fast algorithm basing on LDAC scanning are proposed in this paper to study the performance of the threshold DACs which will be applied to the pixel. The threshold dispersion has been reduced from 46.28 mV without algorithm to 6.78 mV with the precise algorithm, whereas it is 7.61 mV with fast algorithm. For the temperature from 5 to 60 , the threshold dispersion of precise algorithm varies in the range of about 5.69 mV, whereas it is about 33.21 mV with the fast algorithm which can be re-corrected to 1.49 mV. The measurement results show that the fast algorithm could get the applicable threshold dispersion for a silicon pixel module and take a shorter time, while the precise algorithm could get better threshold dispersion, but time consuming. The temperature dependence of the silicon pixel module noise is also studied to assess the detector working status. The minimum detection energy can be reduced about 0.83 keV at a 20 lower temperature.

physics.ins-det↗
arXiv2019

Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations

We compute the symbol of the first two-loop amplitudes in planar ${\cal N}=4$ SYM with algebraic letters, the eight-point NMHV amplitude (or the dual octagon Wilson loops). We show how to apply $\bar{Q}$ equations for computing the differential of two-loop $n$-point NMHV amplitudes and present the result for n=8 explicitly. The symbol alphabet for octagon consists of 180 independent rational letters and 18 algebraic ones involving Gram-determinant square roots. We comment on all-loop predictions for final entries and aspects of the result valid for all multiplicities.

hep-th↗
arXiv2018

Moduli Space of Paired Punctures, Cyclohedra and Particle Pairs on a Circle

In this paper, we study a new moduli space $\mathcal{M}_{n+1}^{\mathrm{c}}$, which is obtained from $\mathcal{M}_{0,2n+2}$ by identifying pairs of punctures. We find that this space is tiled by $2^{n-1}n!$ cyclohedra, and construct the canonical form for each chamber. We also find the corresponding Koba-Nielsen factor can be viewed as the potential of the system of $n{+}1$ pairs of particles on a circle, which is similar to the original case of $\mathcal{M}_{0,n}$ where the system is $n{-}3$ particles on a line. We investigate the intersection numbers of chambers equipped with Koba-Nielsen factors. Then we construct cyclohedra in kinematic space and show that the scattering equations serve as a map between the interior of worldsheet cyclohedron and kinematic cyclohedron. Finally, we briefly discuss string-like integrals over such moduli space.

hep-th↗