We study one-point functions of non-BPS single-trace operators on the Coulomb branch of planar $$\mathcal{N}=4$$ supersymmetric Yang-Mills theory. Holography relates them to overlaps between on-shell closed string states and a boundary state describing a probe D3-brane in $\textit{AdS}$ × $\textit{S}$ . Assuming that the D-brane preserves integrability, we formulate and solve integrable bootstrap equations satisfied by the boundary state at finite ’t Hooft coupling. This leads to a closed-form determinant expression for one-point functions at finite coupling, valid for sufficiently long operators. We test the result against direct field theory computations at tree level and one loop, finding perfect agreement.

Switzerland

Frank Coronado

United Kingdom

Shota Komatsu

Denmark

Sweden

Konstantin Zarembo

Coulomb branch and integrability

The angular distributions for the elastic scattering of the neutron-rich nucleus <math altimg='si7.svg'><msup><mrow /><mn>11</mn></msup></math>Be and proton-rich nucleus <math altimg='si6.svg'><msup><mrow /><mn>8</mn></msup></math>B on a <math altimg='si4.svg'><msup><mrow /><mn>58</mn></msup></math>Ni target have been measured at energies around six times the Coulomb barriers at the Radioactive Ion Beam Line in Lanzhou. The same measurements were made for <math altimg='si10.svg'><msup><mrow /><mn>7</mn></msup></math>Be and <math altimg='si3.svg'><msup><mrow /><mn>10</mn></msup></math>Be simultaneously. The data were analyzed with the optical model and continuum discretized coupled channels framework, and the results of both calculations could reasonably account for the experimental data. The coupling effects of breakup reaction on elastic scattering for <math altimg='si7.svg'><msup><mrow /><mn>11</mn></msup></math>Be projectile are much stronger than those of <math altimg='si10.svg'><msup><mrow /><mn>7</mn></msup></math>Be, <math altimg='si6.svg'><msup><mrow /><mn>8</mn></msup></math>B, and <math altimg='si3.svg'><msup><mrow /><mn>10</mn></msup></math>Be at this high incident energy region. Incorporating our previously published data with those from other laboratories, a systematic investigation into coupled channels effect via elastic scattering has been performed.

China

K. Wang

G. Yang

Y.Y. Yang

F.F. Duan

D.Y. Pang

Q. Liu

Y.M. Wei

S.W. Xu

J.B. Ma

P. Ma

Z. Bai

Z.Y. Sun

Elastic scattering of weakly-bound nuclei <math altimg='si7.svg'><msup><mrow /><mn>11</mn></msup></math>Be and <math altimg='si6.svg'><msup><mrow /><mn>8</mn></msup></math>B on <math altimg='si4.svg'><msup><mrow /><mn>58</mn></msup></math>Ni at energies above the Coulomb barriers

We study multiloop conformal integrals for four-point correlators of planar <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></math> super-Yang-Mills theory, and in particular those contributing to Coulomb branch amplitudes in the ten-dimensional lightlike limit, where linear combinations of such integrals are determined by the large R-charge octagons exactly known from integrability. Exploiting known results for integrands, we review those combinations of dual conformal invariant (DCI) integrals that must evaluate to determinants of ladders, generalizing the simplest cases of Basso-Dixon fishnet integrals; in this way, we summarize all-loop predictions for the integrands (which are extracted from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>f</mi></math>-graphs) contributing to components of Coulomb branch amplitudes, such as next-to-fishnet integrals. Moreover, this exercise produces new “magic identities,” i.e., certain combinations of DCI integrals equal zero, and we enumerate and simplify such identities up to six loops explicitly. On the other hand, most of these individual integrals have not been computed beyond three loops, and as a first step we consider a bootstrap program for DCI integrals based on their leading singularities and the space of pure functions. We bootstrap the 3 nontrivial DCI integrals for four-loop Coulomb branch amplitudes (providing an independent verification of the four-loop magic identity), which all take remarkably simple form as weight-8 single-valued harmonic polylogarithms. We also compute all leading singularities and a large portion of the pure functions for the 34 DCI integrals contributing to five-loop amplitudes, where not only some integrals evaluate to functions beyond harmonic polylogarithms, but they also contain lower-weight pieces individually.

Song He

Xuhang Jiang

Jiahao Liu

Yao-Qi Zhang

Notes on conformal integrals: Coulomb branch amplitudes, magic identities, and bootstrap

In recent years, the quantum computing method has been used to address the sign problem in traditional Monte Carlo lattice gauge theory (LGT) simulations. We propose that the Coulomb gauge (CG) should be used in quantum simulations of LGT. Since the redundant degrees of freedom of gauge fields can be eliminated in CG, the Hamiltonian in CG does not need to be gauge invariant, allowing the gauge field to be discretized naively. Then the discretized gauge fields and fermion fields should be placed on momentum and position lattices, respectively. Under this scheme, the CG condition and Gauss’s law can be conveniently preserved by solving for the polarization vectors from algebraic equations. Furthermore, we discuss the mapping of gauge fields to qubits and evaluate the associated qubit and gate cost of this framework. We point out that this formalism is efficient for simulating hadron scattering processes on future fault-tolerant quantum computers. Finally, we calculate the vacuum expectation value of the U(1) plaquette operator and the Wilson loop on a classical device to test the performance of our discretization scheme.

Japan

Tianyin Li

Quantum simulations of quantum electrodynamics in Coulomb gauge

The Coulomb branch of the $\textit{N}$ = 4 super-Yang-Mills theory is described by a D3-brane in the bulk of $\textit{AdS}$ × $\textit{S}$ . We show that the boundary conditions on the D-brane preserve integrability of the string sigma-model by constructing a dynamical and spectral parameter dependent reflection matrix that encodes an infinite number of conserved charges.

Ron Demjaha

String integrability on the Coulomb branch

In this study, we investigate the holographic Schwinger effect within the framework of $$\mathcal {N}=4$$ super Yang–Mills theory on the Coulomb branch (cSYM). The Coulomb branch is characterized by spontaneous conformal symmetry breaking via the Higgs mechanism, which dynamically generates a characteristic energy scale $$\Lambda $$ . We conduct a detailed analysis to examine how the dimensionless ratio of temperature to energy scale ( $$T/\Lambda $$ ) influences the quark–antiquark separation length, the potential barrier, and the critical electric field. Our findings indicate that the maximum quark–antiquark separation length exhibits a monotonic suppression as $$T/\Lambda $$ increases, suggesting that $$T/\Lambda $$ promotes the dissociation of quark-pair. Additionally, we observe that $$T/\Lambda $$ reduces both the height and width of the potential barrier. The critical electric field displays suppression with increasing $$T/\Lambda $$ . These results demonstrate that in strongly coupled cSYM, increasing $$T/\Lambda $$ enhances the Schwinger effect.

Rui Zhou

Zhou-Run Zhu

Holographic Schwinger effect in strongly coupled $$\mathcal {N}$$ = 4 super Yang–Mills plasma on the Coulomb branch

A basic datum of a rank-$\textit{r}$ $$ \mathcal{N} $$ =2 superconformal field theory (SCFT) is the $\textit{r}$-tuple of its Coulomb branch scaling dimensions, i.e., the scaling dimensions of a set of special protected scalar operators whose vevs generate the coordinate ring of the Coulomb branch of the theory. It is well known that when the coordinate ring is freely generated these scaling dimensions can only take values in a small set of rational numbers. But there are further constraints on which $\textit{r}$-tuples of these numbers can appear. The main aim of this work is to clarify what these are. Along the way we also compute explicitly the $\textit{r}$-tuples of allowed scaling dimensions for theories of ranks $\textit{r}$ = 2, 3, 4.

United States

Philip Argyres

Italy

Sergio Cecotti

Michele Zotto

Mario Martone

Robert Moscrop

Ben Smith

Allowed Coulomb branch scaling dimensions of four-dimensional $$ \mathcal{N} $$ = 2 SCFTs

The analysis of RHIC hydrogen gas jet target polarimeter measurements of transverse analyzing powers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>A</mi><mi mathvariant="normal">N</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> in proton-nucleus scattering requires accurate Coulomb corrections to both spin-flip and nonflip amplitudes. These corrections must cover a wide range of nuclear charges <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Z</mi></math> and form factor slopes, with flexibility to vary form factors during data fitting. To avoid technically challenging calculations involving a small but finite fictitious photon mass, the Coulomb correction to the nonflip electromagnetic amplitude with an exponential form factor was related to the corresponding correction for the spin-flip amplitude. This approach allows soft photon contributions to all amplitudes, including those with nonexponential form factors, to be calculated in the massless photon limit using only analytical expressions and numerically stable integrals with nonsingular integrands and finite integration limits. In addition, an absorptive correction to the spin-flip electromagnetic amplitude, which plays a critical role in spin effects in forward polarized proton-nucleus scattering, was accurately evaluated.

A.A. Poblaguev

Numerical calculation of Coulomb corrections in forward elastic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>p</mi><mo stretchy="false">↑</mo></msup><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>p</mi></mrow><mrow><mo stretchy="false">↑</mo></mrow></msup><mi>A</mi></mrow></math> scattering

Coulomb branches of vacua are the most universal moduli spaces that arise in local unitary interacting 4d $$ \mathcal{N} $$ = 2 superconformal field theories (SCFTs). In these theories, 1/2-BPS primaries parameterize the Coulomb branches and form (anti-)chiral rings. We define the notion of a Coulomb branch operator algebra, $$ {\mathcal{A}}_{\mathcal{C}} $$ , that contains these chiral and anti-chiral rings along with infinitely many more operators and products that are less protected by supersymmetry. Using a universal symmetry, $$ \mathcal{I}\cong {\mathbb{Z}}_2 $$ , that arises from studying the superconformal group, we give $$ \mathcal{I} $$ selection rules for $$ {\mathcal{A}}_{\mathcal{C}} $$ and, more generally, for arbitrary products in the local operator algebra of any 4d $$ \mathcal{N} $$ = 2 SCFT. Defining the notion of a “Coulombic” SCFT, we propose explanations for certain phenomena in a 4d/2d correspondence involving 4d $$ \mathcal{N} $$ = 2 theories and 2d vertex operator algebras. Finally, by considering deformations of $$ \mathcal{I} $$ , we explore the case of $$ \mathcal{N} $$ > 2 SCFTs.

Matthew Buican

Coulomb branch operator algebras and universal selection rules for $$ \mathcal{N} $$ = 2 SCFTs

The formalism developed in Guo and Gasparian [Toward extracting the scattering phase shift from integrated correlation functions, <a href="http://dx.doi.org/10.1103/PhysRevD.108.074504">Phys. Rev. D 108, 074504 (2023)</a>]; Guo [Toward extracting the scattering phase shift from integrated correlation functions. II. A relativistic lattice field theory model, <a href="http://dx.doi.org/10.1103/PhysRevD.110.014504">Phys. Rev. D 110, 014504 (2024)</a>]; and Guo and Lee [Toward extracting scattering phase shift from integrated correlation functions. III. Coupled channels, <a href="http://dx.doi.org/10.1103/PhysRevD.111.054506">Phys. Rev. D 111, 054506 (2025)</a>] that relates the integrated correlation functions for a trapped system to the infinite volume scattering phase shifts through a weighted integral is further extended to include Coulomb interaction between charged particles. The original formalism cannot be applied due to different divergent asymptotic behavior resulting from the long-range nature of the Coulomb force. We show that a modified formula in which the difference of integrated correlation functions between particles interacting with Coulomb plus short-range interaction and with Coulomb interaction alone is free of divergence, and has rapid approach to its infinite volume limit. Using an exactly solvable model, we demonstrate that the short-range potential scattering phase shifts can be reliably extracted from the formula in the presence of Coulomb interaction.

Peng Guo

Frank X. Lee

Toward extracting scattering phase shifts from integrated correlation functions. IV. Coulomb corrections