Quaternions and Non-Commutative Geometry

In enlarging the field content of pure Yang-Mills theory to a cutoff dependent matrix valued complex scalar field, we construct a vectorial operator, which is by definition invariant with respect to the gauge transformation of the Yang-Mills field and with respect to a Stueckelberg type gauge transformation of the scalar field. This invariant operator converges to the original Yang-Mills field as the cutoff goes to infinity. With the help of cutoff functions, we construct with this invariant a regularized action for the pure Yang-Mills theory. In order to be able to define both the gauge and scalar fields kinetic terms, other invariant terms are added to the action. Since the scalar fields flat measure is invariant under the Stueckelberg type gauge transformation, we obtain a regularized gauge-invariant path integral for pure Yang-Mills theory that is mathematically well defined. Moreover, the regularized Ward-Takahashi identities describing the dynamics of the gauge fields are exactly the same as the formal Ward-Takahashi identities of the unregularized theory.

It is well known that there are no static non-Abelian monopole solutions in pure Yang-Mills theory on Minkowski space ${\mathbb{R}}^{3,1}$. I show that such solutions exist in $\mathrm{SU}(N)$ gauge theory on the spaces ${\mathbb{R}}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{2}$ and $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{S}^{1}\ifmmode\times\else\texttimes\fi{}{S}^{2}$ with Minkowski signature $(\ensuremath{-}+++)$. In the temporal gauge they are solutions of pure Yang-Mills theory on $\mathbb{T}\ifmmode\times\else\texttimes\fi{}{S}^{2}$, where $\mathbb{T}$ is $\mathbb{R}$ or ${S}^{1}$. Namely, imposing SO(3) invariance and some reality conditions, I consistently reduce the Yang-Mills model on the above spaces to a non-Abelian analog of the ${\ensuremath{\phi}}^{4}$ kink model whose static solutions give $\mathrm{SU}(N)$ monopole (-antimonopole) configurations on the space ${\mathbb{R}}^{1,1}\ifmmode\times\else\texttimes\fi{}{S}^{2}$ via the above-mentioned correspondence. These solutions can also be considered as instanton configurations of Yang-Mills theory in $2+1$ dimensions. The kink model on $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ admits also periodic sphaleron-type solutions describing chains of $n$ kink-antikink pairs spaced around the circle ${S}^{1}$ with arbitrary $n>0$. They correspond to chains of $n$ static monopole-antimonopole pairs on the space $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{S}^{1}\ifmmode\times\else\texttimes\fi{}{S}^{2}$ which can also be interpreted as instanton configurations in $2+1$ dimensional pure Yang-Mills theory at finite temperature (thermal time circle). I also describe similar solutions in Euclidean $\mathrm{SU}(N)$ gauge theory on ${S}^{1}\ifmmode\times\else\texttimes\fi{}{S}^{3}$ interpreted as chains of $n$ instanton--anti-instanton pairs.

In this article, we generalize the recursion methods based on cut equations [N. Arkani-Hamed, H. Frost, and G. Salvatori, .], originally developed for scalar theories, to gluons in pure Yang-Mills theory. In gauge theories, planar loop integrands are subtle to defined and obtained due to the existence of scaleless integrals. A critical challenge arises when constructing higher-loop integrands via recursion methods: integrands with or without all scaleless terms both are incompatible with the reconstruction formalism in Arkani-Hamed To address this, we introduce a integrand by systematically removing specific scaleless contributions, and develop an algorithmic implementation of the recursion to all-loop level. We explicitly demonstrate the framework by three steps and obtain the recursion formula in pure Yang-Mills theory. In the Supplemental Material, we provide the results up to the two-loop five-point integrand and the simplified result in the large-D limit for the three-loop four-point case.

β-Function in a non-covariant Yang-Mills theory

We study a well-defined formulation of pure Yang-Mills theories in the continuum which employs an action structured by gauge-invariant loop variables. At the classical level, we show how the aforementioned formulation comes arbitrarily close to the standard Yang-Mills action and also prove that it yields the correct equations of motion. We perform a path-integral quantization of the theory and establish, through direct calculation, strong-coupling confinement in the continuum. We claim that the advocated formulation constitutes a nonperturbative regularization of non-Abelian gauge theories in the continuum.

Asymptotic behavior in pure Yang-Mills theory with non-covariant gauges

The possibility of the confinement of external charge in a pure Yang-Mills theory is discussed. Only field configurations which almost obey the classical field equations are considered. The two- and three-dimensional models of Callan, Dashen, and Gross and of Polyakov are reviewed. If confinement occurs in a four-dimensional pure Yang-Mills theory it is concluded that field configurations which are far from obeying the Euclidean classical equations of motion must be predominant.

We have obtained the supersymmetric extension of a spectral triple that specifies a noncommutative geometry. We assume that the functional space |$\mathcal {H}$| consists of wave functions of matter fields and their superpartners included in the minimum supersymmetric standard model (MSSM). We introduce the internal fluctuations of the Dirac operator on the finite space as well as on the manifold by elements of the algebra |$\mathcal {A}$| in the triple. So, we obtain not only the vector supermultiplets that mediate |${\rm SU}(3)\otimes {\rm SU}(2)\otimes {\rm U}(1)_Y$| gauge degrees of freedom but also Higgs supermultiplets that appear in the MSSM from the same standpoint. According to the supersymmetric version of the spectral action principle, we calculate the square of the fluctuated total Dirac operator and verify that the Seeley–DeWitt coefficients give the correct action of the vector and Higgs supermultiplets. We also verify that the relation between the coupling constants of |${\rm SU}(3)$|⁠, |${\rm SU}(2)$|⁠, and |${\rm U}(1)_Y$| is same as that of SU(5) unification theory.

On the definition of spacetimes in noncommutative geometry: Part I

Gauge fields have a natural metric interpretation in terms of horizontal distance. The latest, also called Carnot-Caratheodory or subriemannian distance, is by definition the length of the shortest horizontal path between points, that is to say the shortest path whose tangent vector is everywhere horizontal with respect to the gauge connection. In noncommutative geometry all the metric information is encoded within the Dirac operator D. In the classical case, i.e. commutative, Connes's distance formula allows to extract from D the geodesic distance on a riemannian spin manifold. In the case of a gauge theory with a gauge field A, the geometry of the associated U(n)-vector bundle is described by the covariant Dirac operator D+A. What is the distance encoded within this operator ? It was expected that the noncommutative geometry distance d defined by a covariant Dirac operator was intimately linked to the Carnot-Caratheodory distance dh defined by A. In this paper we precise this link, showing that the equality of d and dh strongly depends on the holonomy of the connection. Quite interestingly we exhibit an elementary example, based on a 2 torus, in which the noncommutative distance has a very simple expression and simultaneously avoids the main drawbacks of the riemannian metric (no discontinuity of the derivative of the distance function at the cut-locus) and of the subriemannian one (memory of the structure of the fiber).

The Dirac operator in finite temperature QCD is equivalent to the Hamiltonian of an unconventional Anderson model, with on-site noise provided by the fluctuations of the Polyakov lines. The main features of its spectrum and eigenvectors, concerning the density of low modes and their localization properties, are qualitatively reproduced by a toy-model random Hamiltonian, based on an Ising-type spin model mimicking the dynamics of the Polyakov lines. Here we study the low modes of this toy model in the vicinity of the ordering transition of the spin model, and show that at the critical point the spectral density at the origin has a singularity, and the localization properties of the lowest modes change. This provides further evidence of the close relation between deconfinement, chiral transition and localization of the low modes.

Infrared properties of the glue propagator in non-abelian gauge theories

We study by numerical simulations on a lattice the behavior of the gauge-invariant two-point correlation functions of the gauge-field strengths across the deconfinement phase transition, both for the pure-gauge SU(3) theory and for full QCD with two flavors. Quenched data agree within errors with previous determinations, but have much higher statistics. A best-fit analysis of the data has been performed, both for the quenched and the full-QCD case, showing that the electric gluon condensate drops to zero at the deconfining\nphase transition.

Motivated by applications in thermal QCD and cosmology, we elaborate on a general method for computing next-to-leading order spectral functions for composite operators at vanishing spatial momentum, accounting for real, virtual as well as thermal corrections. As an example, we compute these functions (together with the corresponding imaginary-time correlators which can be compared with lattice simulations) for scalar and pseudoscalar densities in pure Yang-Mills theory. Our results may turn out to be helpful in non-perturbative estimates of the corresponding transport coefficients, which are the bulk viscosity in the scalar channel and the rate of anomalous chirality violation in the pseudoscalar channel. We also mention links to cosmology, although the most useful results in that context may come from a future generalization of our methods to other correlators.

Non-Abelian dark gauge forces that do not couple directly to ordinary matter may be realized in nature. The minimal form of such a dark force is a pure Yang-Mills theory. If the dark sector is reheated in the early universe, it will be realized as a set of dark gluons at high temperatures and as a collection of dark glueballs at lower temperatures, with a cosmological phase transition from one form to the other. Despite being dark, the gauge fields of the new force can connect indirectly to the Standard Model through non-renormalizable operators. These operators will transfer energy between the dark and visible sectors, and they allow some or all of the dark glueballs to decay. In this work we investigate the cosmological evolution and decays of dark glueballs in the presence of connector operators to the Standard Model. Dark glueball decays can modify cosmological and astrophysical observables, and we use these considerations to put very strong limits on the existence of pure non-Abelian dark forces. On the other hand, if one or more of the dark glueballs are stable, we find that they can potentially make up the dark matter of the universe.