Building upon the algebraic consistency construction of one-loop Bern-Carrasco-Johansson (BCJ) numerators for Yang-Mills (YM) and Yang-Mills-scalar (YMS) theories, we explore the expansion formula of one-loop Einstein-Yang-Mills (EYM) integrands (with a gluon loop) in terms of conventional one-loop YM integrands with quadratic propagators. We first express the EYM integrand by tree-level amplitudes according to the forward limit approach. Employing a two-step expansion strategy, the gluon-loop EYM integrand is decomposed into tree-level YM amplitudes under the forward limit, which are subsequently expanded into tree-level bi-adjoint scalar (BS) ones. We then prove that when the kinematic coefficients in both expansion steps satisfy the one-loop consistency conditions, the EYM integrand is finally expanded as a combination of YM integrands with quadratic propagators. The coefficients in this expansion formula coincide exactly with those in the expansion formula for YMS integrands. This correspondence highlights a shared kinematic structure, providing the proper foundation for constructing BCJ numerators in both YMS and EYM theories at one loop.

China

Yi-Jian Du

Chongsi Xie

Expansion formula of one-loop Einstein-Yang-Mills integrand

We employ the harmonic superspace methods to study a six-dimensional 𝒩 = (1$\textit{,}$ 0) supersymmetric gauge theory with higher derivatives coupled to a hypermultiplet in the adjoint representation. By introducing a certain non-minimal interaction between the gauge multiplet and the hypermultiplet, we demonstrate that the one-loop divergences in gauge superfield sector, which are present in the conventional formulation, are canceled. The resulting theory is off-shell one-loop finite in this sector, while preserving the gauge invariance and 𝒩 = (1$\textit{,}$ 0) supersymmetry. The cancelation mechanism is explicitly verified using both the background field method and the supergraph techniques. Thus, we present an example of the higher-derivative supersymmetric gauge theory in six dimensions which is finite in the vector multiplet sector.

Russia

I. Buchbinder

A. Budekhina

E. Ivanov

K. Stepanyantz

One-loop finiteness in higher-derivative 6$\textit{D}$, 𝒩 = (1, 0) super Yang-Mills — hypermultiplet system

We calculate the resummed perturbative free energy of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></math> supersymmetric Yang-Mills in four spacetime dimensions (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>SYM</mi></mrow><mrow><mn>44</mn></mrow></msub></mrow></math>) to order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>λ</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msup></math> in the ‘t Hooft coupling at finite temperature and zero chemical potential. All infrared divergences cancel when we include contributions from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>SYM</mi></mrow><mrow><mn>44</mn></mrow></msub></mrow></math> ring diagrams and the final result is both ultraviolet and infrared finite. Our result has special significance since order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>λ</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msup></math> is the highest order calculation that can be done with perturbation theory, because there are nonperturbative effects associated with the magnetic mass scale that come into play at order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>λ</mi><mn>3</mn></msup></math>. We compare results obtained with regularization by dimensional reduction (RDR), which preserves supersymmetry, and canonical dimensional regularization (DR). We also compare with a generalized Padé approximant constructed by matching the weak coupling result at order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>λ</mi><mn>2</mn></msup></math> and the large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> strong coupling result at order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>λ</mi><mrow><mo>−</mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></math>. Finally we make a comparison between our result and the QCD free energy and show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>SYM</mi></mrow><mrow><mn>44</mn></mrow></msub></mrow></math> has better convergence properties.

Canada

Margaret E. Carrington

Gabor Kunstatter

Saudi Arabia

Ubaid Tantary

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></mrow></math> supersymmetric Yang-Mills thermodynamics to order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>λ</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msup></math>

We construct symmetry topological field theories using the sandwich construction of Pulmann-Ševera-Valach that manifest the center symmetries of Chern-Simons theory and Yang-Mills theory as well as general relativity in the MacDowell-Mansouri formulation. The “filling” of the sandwich is an Alexandrov-Kontsevich-Schwarz-Zaboronsky sigma model whose target space is a Weil algebra, augmented with discrete degrees of freedom given by a choice of topological boundary condition.

United Kingdom

Leron Borsten

Dimitri Kanakaris

Hyungrok Kim

Sandwich construction of symmetry topological field theories for the center symmetries of Chern-Simons, Yang-Mills, and Einstein gravity

We show that, in the replica-broken sector of the Serreau–Tissier (ST) gauge fixing, the expansion of the replica determinant in the regulator $$\zeta $$ induces a nonlocal bilinear gluonic kernel with the same color and Lorentz structure as the quadratic part of the BRST-invariant Gribov horizon functional. This establishes an effective leading-order correspondence with the refined Gribov-Zwanziger (RGZ) horizon sector, rather than a reconstruction of the full nonlinear functional $$H(A^h)$$ . The induced scale satisfies $$\gamma _{\textrm{ind}}^4\propto \zeta $$ at leading order, up to scheme-dependent normalization and higher-order corrections. Depending on the replica phase, the ST sector yields either a local Curci–Ferrari (CF) screening mass or an induced RGZ-type horizon kernel, avoiding double counting of infrared scales.

Spain

Rodrigo Terin

Emergent Gribov horizon kernel from replica symmetry breaking in Yang–Mills theories

We employ the Transformer to learn patterns in two-dimensional lattice Yang-Mills theory. Specifically, we represent both Wilson loops and their expectation values as tokenized sequences. Taking the shape of Wilson loops as input, the model successfully predicts expectation values with high accuracy, demonstrating that the Transformer can effectively learn the supervised mapping from loop geometry to analytic results within the training distribution. We note that the 2D theory is exactly solvable and the target expressions belong to a restricted set of polynomial forms, which considerably simplifies the learning task. Our study differs from prior machine learning applications in lattice QCD by emphasizing analytical structures rather than numerical computations. We explore model performance under varying hyperparameters, training data sizes, and sequence lengths. This work serves as a proof-of-concept study toward extending such methods to higher dimensions and inspiring rigorous analytical derivations.

Zeyu Li

Guorui Zhu

Wenjie He

Bo Feng

Jiaqi Chen

Ming-xing Luo

Gang Yang

Application of transformer in 2D lattice Yang-Mills theory

We study two-body scattering as an SU(N)-equivariant map acting on tensor-product representation spaces and analyze the entanglement generated by the $\textit{S}$-matrix. This representation-theoretic perspective separates group structure from dynamics: the decomposition of $\textit{R}$⊗$\textit{R}$′ fixes the invariant operator algebra and therefore the qualitative entangling power of the process. For particles in the fundamental representation, End ($\textit{N}$⊗$\textit{N}$) = Span{𝕀, 𝕊}, so only the identity and swap directions preserve separability, whereas generic combinations generate entanglement. Adjoint-adjoint scattering involves a larger invariant algebra involving $\textit{d}$-tensors and is intrinsically entangling. In Yang-Mills theory one can use color-kinematics duality to show that the color kernel lies on a fixed ray of this operator space, yielding a universal maximum of the outgoing entanglement for scattering at right angles, $$ {E}_{\star}^{(2)}=\frac{3}{4} $$ for SU(2) and $$ {E}_{\star}^{(3)}\simeq 0.91 $$ , independent of kinematics. Dimension-six operators preserve this universality, while dimension-eight deformations populate new color sectors and shift $$ {E}_{\star}^{(N)} $$ , suggesting that entanglement in color space functions as a tomographic probe of effective operators. In helicity space, requiring maximally entangled inputs to scatter into maximally entangled outputs uniquely selects the Yang-Mills quartic coupling and enforces the color Jacobi identity, restating the on-shell Ward constraints as conditions on entanglement preservation. Our results suggest that the information-theoretic viewpoint unifies algebraic, geometric, and dynamical aspects of scattering.

United States

Kun-Feng Lyu

Rahul Muraleedharan

Kuver Sinha

Entanglement, Yang-Mills, and the scattering matrix as an SU(N)-equivariant kernel

In this work, we employ both theoretical and data-driven methods to derive the QCD vacuum energy, utilizing the GMOR relation, the low-energy theorem, and the equation of state from lattice QCD. The QCD vacuum energy is determined to be between around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><mn>163</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><msup><mo stretchy="false">)</mo><mn>4</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><mn>190</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><msup><mo stretchy="false">)</mo><mn>4</mn></msup></math>. With the assumptions of complete deconfinement, vanishing gluon condensate, full chiral-symmetry restoration, and the validity of perturbative QCD at baryon chemical potentials of order of the proton mass, a very specific kind of quark nugget is found to be less stable than ordinary nuclei.

Yang Bai

Ting-Kuo Chen

QCD vacuum energy and its implication for quark nugget stability

We present integrable deformations of Yang-Baxter (YB) sigma models based on the Courant-Hilbert (CH) construction. To this end, we employ the four-dimensional Chern-Simons theory, in which the CH construction is shown in arXiv:2509.22080. As a result, the CH construction works in an intricate way alongside the YB deformations. Remarkably, the resulting deformed action can also be expressed as the sum of the master formula Lagrangian and the trace of the energy-momentum tensor. This result indicates the universality of the correction term.

Japan

Osamu Fukushima

Takaki Matsumoto

Kentaroh Yoshida

Courant-Hilbert deformations of Yang-Baxter sigma models

We compute solutions to the Hermitian Yang-Mills equations on holomorphic vector bundles $\textit{V}$ via an alternating optimisation procedure founded on geometric machine learning. The proposed method is fully general with respect to the rank and structure group of $\textit{V}$, requiring only the ability to enumerate a basis of global sections for a given bundle. This enables us to compute the physically normalised Yukawa couplings in a broad class of heterotic string compactifications. Using this method, we carry out this computation in full for a heterotic compactification incorporating a gauge bundle with non-Abelian structure group.

Challenger Mishra

Justin Tan

Hermitian Yang-Mills connections on general vector bundles: geometry and physical Yukawa couplings