We further investigate the 3d gauged linear sigma model (GLSM)/ quantum K-theory correspondence for partial flag manifolds $\textit{X}$ ≡ Fl( ; $\textit{n}$). This is a 3d uplift of the 2d GLSM/quantum cohomology correspondence with the 3d theory compactified on $$ {\mathbb{R}}^2\times {S}_{\beta}^1 $$ . Recently, a set of half-BPS line operators, called Schubert line defects, were constructed that correspond to the Schubert classes in the K-theory ring of $\textit{X}$. Utilizing algebro-geometric algorithms, we compute 2-point and 3-point correlation functions of these line operators in the 3d A-model regime of the theory. These are interpreted as genus-0 K-theoretic Gromov-Witten invariants, and they produce the K-theoretic Littlewood-Richardson coefficients of the quantum K-theory ring of $\textit{X}$. We show how this works explicitly in examples, going beyond the existing results in the literature. Taking the small $\textit{β}$ limit, we apply these techniques to the resulting 2d GLSM. We explicitly compute the quantum cohomology ring relations of $\textit{X}$ for some cases and match with existing results in the literature in examples.

Switzerland

Zhihao Duan

France

Osama Khlaif

China

Hao Zou

On the Schubert calculus of the quantum K-theory for partial flag manifolds: a 3d A-model perspective

In this work, we investigate three representative new-physics resonances that couple to Standard Model (SM) quarks through flavor-changing interactions involving the top quark. We identify the possible SMEFT operators at the electroweak scale and analyze their phenomenology.

Min Huang

Yandong Liu

Hao Zhang

Exploring new resonances with direct top flavor changing interactions * Min Huang and Hao Zhang is supported in part by the National Science Foundation of China (12575114, 12235001, 12075257)

We construct Schubert line defects in the 3d $$ \mathcal{N}=2 $$ supersymmetric gauged linear sigma model (GLSM) with target space a partial flag manifold $\textit{X}$ = Fl( ; $\textit{n}$), generalizing our construction for complete flag manifolds given in a companion paper (part I) [1]. In the context of the 3d GLSM/quantum K-theory correspondence, the Schubert line defects are constructed as 1d $$ \mathcal{N}=2 $$ supersymmetric gauge theories coupled to the 3d field theory, and they flow to objects supported on Schubert varieties $\textit{X}$ ⊆ $\textit{X}$ in the quantum K-theory. The flavored Witten index of the 1d defect is expected to compute the Chern character of [𝒪 ] — more precisely, it gives us a polynomial representative of the Schubert class in the quantum K-theory ring. We give strong evidence for this claim by showing in examples that the Witten indices of Schubert defects indeed reproduce a recently-defined set of polynomials that represent the Schubert classes in the Whitney presentation, which we call the parabolic Whitney polynomials. Moreover, upon using the quantum ring relations, we can convert these polynomials into seemingly new polynomials in the Toda presentation, which we call the parabolic quantum Grothendieck polynomials. These new polynomials specialize to known polynomials in various limits, including to the quantum Grothendieck polynomials in the case of the complete flag. In the 2d limit, our construction also realizes the Schubert classes [$\textit{X}$ ] in the quantum cohomology ring of the partial flag manifold, and the parabolic quantum Grothendieck polynomials then reduce to previously known parabolic quantum Schubert polynomials.

United Kingdom

Cyril Closset

Wei Gu

United States

Eric Sharpe

Schubert line defects in 3d GLSMs. Part II. Partial flag manifolds and parabolic quantum polynomials

We construct new half-BPS line defects in 3d $$ \mathcal{N}=2 $$ supersymmetric quiver gauge theories whose Higgs branches are complete flag manifolds $\textit{X}$ = Fl($\textit{n}$). Upon circle compactification, the bulk theory flows to a non-linear sigma model (NLSM) with target space $\textit{X}$ and the line defects flow to objects supported on Schubert varieties $\textit{X}$ ⊆ $\textit{X}$. These $\textit{Schubert line defects}$ form an important basis of the quantum K-theory of $\textit{X}$. They are realized as $$ \mathcal{N}=2 $$ supersymmetric quantum mechanics (SQM) quivers coupled to the 3d gauge theory. We show that the insertion of the Schubert line defect restricts the target space of the 3d gauged linear sigma model (GLSM) to the Schubert variety $\textit{X}$ , with the 1d degrees of freedom physically realizing a Bott-Samelson resolution of $\textit{X}$ . Moreover, we verify in examples that the 1d flavored Witten index of the quiver SQM reproduces the (equivariant) Chern character of the structure sheaf $$ {\mathcal{O}}_{X_w} $$ as a (double) quantum Grothendieck polynomial, generalizing previous results for $\textit{X}$ a Grassmannian manifold. Our construction thus provides a more direct realization of the 3d GLSM/quantum K-theory correspondence for complete flag manifolds. Finally, in the small-circle limit, we obtain a 0d-2d coupled system that realizes the Schubert classes [$\textit{X}$ ] in the quantum cohomology ring of $\textit{X}$.

Schubert line defects in 3d GLSMs. Part I. Complete flag manifolds and quantum Grothendieck polynomials

We study the inelastic charmonium ( $ J/\psi $ , $ \psi(2S) $ ) and bottomonium ( $ \Upsilon(nS) $ ) photoproduction and fragmentation processes in $\textit{p}$-$\textit{p}$ and $ \rm Pb$ - $\rm Pb$ collisions at LHC energies, where the ultra-incoherent photon emission is included. Using the NRQCD factorization approach, an exact treatment is developed. This approach recovers the Weizsäcker-Williams approximation (WWA) near the $ Q^{2}\sim0 $ region, where the Martin-Ryskin and BCCKL methods are used to avoid double counting. We calculated the $ Q^{2} $ , $\textit{y}$, $\textit{z}$, $ \sqrt{s} $ , and $ p_{T} $ dependent and total cross sections. Inelastic photoproduction and fragmentation were observed to contribute to heavy quarkonium production, particularly at large $ p_{T} $ . In addition, the contribution of the ultra-incoherent photon channel, which increases rapidly with quarkonium mass, is significant and begins to dominate the photoproduction processes for large $ p_{T} $ . We also obtained the complete WWA validity scopes of inelastic heavy quarkonium photoproduction in heavy-ion collisions. The WWA had high accuracy at high energies and for $\rm Pb $ - $\rm Pb $ collisions. However, current photon spectra are derived beyond the WWA scope, and double counting can occur when considering different channels simultaneously.

Zhi-Lei Ma

Zhun Lu

Hao Liu

Li Zhang

Inelastic heavy quarkonium photoproduction in $\textit{p}$-$\textit{p}$ and Pb-Pb collisions at LHC energies * Supported in part by the National Natural Science Foundation of China (12305092, 12233006, 12150013), the Xingdian Talent Support Program for Youth Project, and by the Yunnan Fundamental Research Projects (202401AU070206). Hao Liu is supported by the Scientific Research and Innovation Project of Postgraduate Students in the Academic Degree of YunNan University (KC-242410212)

In this study, we investigate a two-component scalar dark matter framework featuring two singlet scalar fields as dark matter candidates. To ensure vacuum stability, we employ copositive criteria for the scalar potential. We analyze four distinct copositive scenarios characterized by specific negative parameter configurations using direct detection constraints. A comprehensive parameter space scan is performed under joint constraints from the observed dark matter relic density and direct detection experiments. The different signs of couplings not only correspond to different copositive criteria but also contribute to different parameter spaces caused by interference. The allowed values of quartic couplings are different for the four different cases; however, they all require the new Higgs to play a dominant role in determining dark matter relic density within the viable parameter space.

Xin-Xin Qi

Hao Sun

Copositive criteria for a two-component dark matter model * Hao Sun is supported by the National Natural Science Foundation of China (12075043 and 12147205). XinXin Qi is supported by the National Natural Science Foundation of China (12447162)

Recent years have seen the development and growth of machine learning in high-energy physics. However, additional effort is required to continue exploring the use of machine learning to its full potential. To simplify the application of the existing algorithms and neural networks and to advance the reproducibility of the analysis, we developed HEP ML LAB ( $ \mathrm{hml}$ ), a Python-based, end-to-end framework for phenomenology studies. It covers the complete workflow from event generation to performance evaluation, and provides a consistent style of use for different approaches. We propose an observable naming convention to streamline the data extraction and conversion processes. In the KERAS style, we provide the traditional cut-and-count and boosted decision trees together with neural networks. We take the $W^+ $ tagging as an example and evaluate all built-in approaches with the metrics of significance and background rejection. With its modular design, HEP ML LAB is easy to extend and customize, and can be used as a tool for both beginners and experienced researchers.

Jing Li

HEP ML LAB: An end-to-end framework for applying machine learning to phenomenology studies * Hao Sun is supported by the National Natural Science Foundation of China (12075043)

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.

Germany

Leonardo Mihalcea

Weihong Xu

Schubert defects in Lagrangian Grassmannians

We present a counting formula that relates the number of physical Bethe states of integrable models with a twisted boundary condition to the number of states in the untwisted or partially twisted limit.

Hongfei Shu

Peng Zhao

Rui-Dong Zhu

Counting Bethe states in twisted spin chains

We study 2d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (2, 2) quiver gauge theories without flavor nodes. There is a special class of quivers whose gauge group ranks stay positive in any duality frame. We illustrate this with the Abelian Kronecker quiver and the Abelian Markov quiver as the simplest examples. In the geometric phase, they engineer an infinite sequence of projective spaces and hypersurfaces in Calabi-Yau spaces, respectively. We show that the Markov quiver provides an Abelianization of SU(3) SQCD. Turning on the FI parameters and the θ angles for the Abelian quiver effectively deform SQCD by such parameters. For an Abelian necklace quiver corresponding to SU(k) SQCD, we find evidence for singular loci supporting non-compact Coulomb branches in the Kähler moduli space.