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.Comment: 23 pages, 7 figure

Crystal structure and electronic structure of quaternary semiconductors Cu2_2ZnTiSe4_4 and Cu2_2ZnTiS4_4 for solar cell absorber

We design two new I2-II-IV-VI4 quaternary semiconductors Cu$_2$ZnTiSe$_4$ and Cu$_2$ZnTiS$_4$, and systematically study the crystal and electronic structure by employing first-principles electronic structure calculations. Among the considered crystal structures, it is confirmed that the band gaps of Cu$_2$ZnTiSe$_4$ and Cu$_2$ZnTiS$_4$ originate from the full occupied Cu 3$d$ valence band and unoccupied Ti 3$d$ conducting band, and kesterite structure should be the ground state. Furthermore, our calculations indicate that Cu$_2$ZnTiSe$_4$ and Cu$_2$ZnTiS$_4$ have comparable band gaps with Cu$_2$ZnTSe$_4$ and Cu$_2$ZnTS$_4$, but almost twice larger absorption coefficient $\alpha(\omega)$. Thus, the materials are expected to be candidate materials for solar cell absorber.Comment: 4 pages, 4 figure

A nice two-loop next-to-next-to-MHV amplitude in N=4{\cal N}=4 super-Yang-Mills

We study a scalar component of the 8-point next-to-next-to-maximally-helicity-violating (N${}^2$MHV) amplitude at two-loop level in ${\cal N}=4$ super-Yang-Mills theory; it has a leading singularity proportional to the inverse of the four-mass-box square root and receives contributions from only two types of non-trivial integrals with one-loop infrared (IR) divergences. We compute such two-loop 8-point integrals by taking (double-)collinear limits of certain finite, dual-conformal-invariant integrals, and they nicely give the IR-safe ratio function after subtracting divergences. As the first genuine two-loop N${}^2$MHV amplitude computed explicitly, we find remarkable structures in its symbol and alphabet: similar to the next-to-MHV (NMHV) case, there are still 9 algebraic letters associated with the square root, and the latter also becomes a letter for the first time; unlike the NMHV case, such algebraic letters appear at either one or all of the second, third and last entry, and the part with three odd letters is particularly simple.Comment: 20 pages, 2 figures and an ancillary file containing symbols of two-loop integrals and the ratio functio

Notes on cluster algebras and some all-loop Feynman integrals

We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and (seven-point) double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is $D_2\simeq A_1^2$, we show that penta-box ladder has an alphabet of $D_3\simeq A_3$ and provide strong evidence that the alphabet of double-penta ladder can be identified with a $D_4$ cluster algebra. We relate the symbol letters to the ${\bf u}$ variables of cluster configuration space, which provide a gauge-invariant description of the cluster algebra, and we find various sub-algebras associated with limits of the integrals. We comment on constraints similar to extended-Steinmann relations or cluster adjacency conditions on cluster function spaces. Our study of the symbol and alphabet is based on the recently proposed Wilson-loop ${\rm d}\log$ representation, which allows us to predict higher-loop alphabet recursively; by applying such recursions to six-dimensional hexagon integrals, we also find $D_5$ and $D_6$ cluster functions for the two-mass-easy and three-mass-easy case, respectively.Comment: 28 pages, several figures; v2: typos corrected, functions of ladder integrals computed to higher loops; v3: more examples of double-penta-ladder integrals and discussions about their alphabet adde

The symbol and alphabet of two-loop NMHV amplitudes from Qˉ\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.Comment: 33 pages, 1 figure, revised version, typo corrected, accepted in JHE