arXiv subjects

Khoi Le Nguyen Nguyen

Publications and source records attributed to Khoi Le Nguyen Nguyen.

8 recordsLinked to original sources

arXiv2026

Anomaly cancellation for two $U(1)$ factors

We show that solving the abelian part of the local anomaly cancellation conditions for a 4-d gauge theory whose gauge Lie algebra has an abelian summand with rank $K \geq 1$ is equivalent to the problem in algebraic/arithmetic geometry of finding $(K-1)$-dimensional projective linear subspaces of a cubic hypersurface over the rational numbers, where the cubic is determined by the data of the semisimple summand and the representation thereof carried by the Weyl fermions. We then use this reformulation to solve a variety of examples with rank 2. The simplest non-trivial example from physics, namely gauge Lie group $U(1)^2$ and six fermions, nevertheless has a rich (and well-studied) geometry: it corresponds to the Fano variety of lines in the Segre cubic primal threefold. This is a surface with 15 irreducible components that are planes (which correspond to lines lying in the 15 planes in the Segre cubic primal and which give rise to non-chiral fermions) and 6 components that are split del Pezzo surfaces of degree 5 (which give rise to chiral fermions). These components are all rational varieties, enabling all solutions to the anomaly cancellation conditions to be parametrized and their collective properties (e.g. their topology and asymptotic distribution) to be described.

hep-th↗

arXiv2025

Anomaly cancellation for a $U(1)$ factor

We use methods of arithmetic geometry to find solutions to the abelian local anomaly cancellation equations for a four-dimensional gauge theory whose Lie algebra has a single $\mathfrak{u}_1$ summand, assuming that a non-trivial solution exists. The resulting polynomial equations in the integer $\mathfrak{u}_1$ charges define a projective cubic hypersurface over the field of rational numbers. Generically, such a hypersurface is (by a theorem of Koll{á}r) unirational, making it possible to find a finitely-many-to-one parameterization of infinitely many solutions using secant and tangent constructions. As an example, for the Standard Model Lie algebra with its three generations of quarks and leptons (or even with just a single generation and two $\mathfrak{su}_3\oplus\mathfrak{su}_2$ singlet right-handed neutrinos), it follows that there are infinitely many anomaly-free possibilities for the $\mathfrak{u}_1$ hypercharges. We also discuss whether it is possible to find all solutions in this way.

hep-th↗

arXiv2025

The asymptotically-free gauge theories

We show how to classify the asymptotically-free gauge theories in four spacetime dimensions, focussing here on the case of purely fermionic matter. The classification depends on the fact (which we prove) that both the dimension and Dynkin index of irreducible representations of a simple Lie algebra are strictly increasing functions of each Dynkin label. This implies not only that the number of asymptotically-free representations of any one semisimple Lie algebra is finite, but also that they can be written down in a systematic fashion using tables for the asymptotically-free irreducible representations of simple Lie algebras, which we supply. These tables show that at most two out of a possible ten Dynkin labels can be non-zero and that no Dynkin label can exceed four. The extension to bosonic matter or supersymmetric theories is straightforward.

hep-th↗

arXiv2025

More varieties of 4-d gauge theories: product representations

Recently, we used methods of arithmetic geometry to study the anomaly-free irreducible representations of an arbitrary gauge Lie algebra. Here we generalize to the case of products of irreducible representations, where it is again possible to give a complete description. A key result is that the projective variety corresponding to $m$-fold product representations of the Lie algebra $\mathfrak{su}_n$ is a rational variety for every $m$ and $n$. We study the simplest case of $\mathfrak{su}_3$ (corresponding to the strong interaction) in detail. We also describe the implications of a number-theoretic conjecture of Manin (and related theorems) for the number of chiral representations of bounded size $B$ (measured roughly by the Dynkin labels) compared to non-chiral ones, giving a precise meaning to the sense in which the former (which are those most relevant for phenomenology) are rare compared to the latter. As examples, we show that, for both irreducible representations of $\mathfrak{su}_5$ and once-reducible product representations of $\mathfrak{su}_3$ that are non-anomalous, the number of chiral representations is asymptotically between $B (\log B)^5$ and $B^\frac{4}{3}$, while the number of non-chiral representations is asymptotically $B^2$. Despite this rarity of chiral, anomaly-free, product representations, we show that there are examples relevant for phenomenology, including one that gives an asymptotically-free gauge theory with Lie algebra $\mathfrak{su}_7$.

hep-th↗

arXiv2024

Impact of beam asymmetries at the Future Circular Collider e+e-

In this paper we present detailed simulations with asymmetric initial beam settings in the context of the proposed Future Circular Collider e+e- (FCC-ee) using the Xsuite framework. We compare simulated equilibrium bunch sizes and luminosities against an already existing analytical model, which shows remarkably good agreement for realistic small perturbations. We investigate the longitudinal top-up injection, the currently preferred injection scheme for the FCC-ee, using self-consistent simulations featuring beam-beam collisions with beamstrahlung and the injection process, for the first time. We present and assess the sensitivity and required precision of the nominal beam parameters in a potential real-life operation by providing first estimates of the tolerances in the initial asymmetry of several machine parameters, with respect to the 3D flip-flop mechanism, obtained from parameter scan simulations.

physics.acc-ph↗

arXiv2024

Varieties of four-dimensional gauge theories

We use algebraic geometry to study the anomaly-free representations of an arbitrary gauge Lie algebra for 4-dimensional spacetime fermions. For irreducible representations, the problem reduces to studying the Lie algebras $\mathfrak{su}_n$ for $n\geq 3$. We show that there exist equivalence classes of such representations that are in bijection with the rational points on a projective variety that are dense in a region of the underlying real variety diffeomorphic to $\mathbb{R}^{n-3}$. It follows that the chiral ones overwhelm the non-chiral ones for $n \geq 5$. We present an efficient algorithm to find explicit anomaly-free irreducible representations and discuss the generalization to reducible representations.

hep-th↗

arXiv2024

Two-dimensional gauge anomalies and $p$-adic numbers

We show how methods of number theory can be used to study anomalies in gauge quantum field theories in spacetime dimension two. To wit, the anomaly cancellation conditions for the abelian part of the local anomaly admit solutions if and only if they admit solutions in the reals and in the $p$-adics for every prime $p$ and we use this to build an algorithm to find all solutions.

hep-th↗

arXiv2024

Impact of bunch intensity asymmetry in colliders featuring strong beamstrahlung

An analytical investigation of beamstrahlung-induced blow-up in Gaussian beams with arbitrary dimensions is presented, using various approximations for the strength of the hourglass effect and crab waist scheme. The results, applied to the FCC-ee resonances, are compared with simulations and previous calculations, and relative luminosity values are also calculated. The stability of resultant conformations are analysed to rule out the existence of a flip-flop phenomenon in the longitudinal plane analogous to the well-known transverse counterpart. Implications for the top-up injection procedures are discussed, and a phenomenological model is proposed to study the transverse-longitudinal coupling in the blowup dynamics.

physics.acc-ph↗