Yang-Mills Lagrangian Research Articles

BRST symmetry of non-Lorentzian Yang-Mills theory

We explore the realization of BRST symmetry in the non-Lorentzian Yang-Mills Lagrangian within the context of Galilean and Carrollian Yang-Mills theory. Firstly, we demonstrate the nilpotent property of classical BRST transformations and construct corresponding conserve charges for both cases. Then, we analyze the algebra of these charges and observe the nilpotent properties at the algebraic level. The findings of this study contribute to an understanding of BRST symmetry in non-Lorentzian Yang-Mills Lagrangians and provide insights into the algebraic properties of related conserve charges.

Renormalization of gluonic leading-twist operators in covariant gauges

We provide the all-loop structure of gauge-variant operators required for the renormalisation of Green’s functions with insertions of twist-two operators in Yang-Mills theory. Using this structure we work out an explicit basis valid up to 4-loop order for an arbitrary compact simple gauge group. To achieve this we employ a generalised gauge symmetry, originally proposed by Dixon and Taylor, which arises after adding to the Yang-Mills Lagrangian also operators proportional to its equation of motion. Promoting this symmetry to a generalised BRST symmetry allows to generate the ghost operator from a single exact operator in the BRST-generalised sense. We show that our construction complies with the theorems by Joglekar and Lee. We further establish the existence of a generalised anti-BRST symmetry which we employ to derive non-trivial relations among the anomalous dimension matrices of ghost and equation-of-motion operators. For the purpose of demonstration we employ the formalism to compute the N = 2, 4 Mellin moments of the gluonic splitting function up to 4 loops and its N = 6 Mellin moment up to 3 loops, where we also take advantage of additional simplifications of the background field formalism.

Double copy for Lagrangians at trilinear order

We present a novel double-copy prescription for gauge fields at the Lagrangian level and apply it to the original double copy, couplings to matter and the soft theorem. The Yang-Mills Lagrangian in light-cone gauge is mapped directly to the mathcal{N} = 0 supergravity Lagrangian in light-cone gauge to trilinear order, and we show that the obtained result is manifestly equivalent to Einstein gravity at tree level up to this order. The application of the double-copy prescription to couplings to matter is exemplified by scalar and fermionic QCD and finally the soft-collinear effective QCD Lagrangian. The mapping of the latter yields an effective description of an energetic Dirac fermion coupled to the graviton, Kalb-Ramond, and dilaton fields, from which the fermionic gravitational soft and next-to-soft theorems follow.

Five-dimensional non-Lorentzian conformal field theories and their relation to six-dimensions

We study correlation functions in five-dimensional non-Lorentzian theories with an SU(1, 3) conformal symmetry. Examples of such theories have recently been obtained as Ω-deformed Yang-Mills Lagrangians arising from a null reduction of six-dimensional superconformal field theories on a conformally compactified Minkowski space. The correlators exhibit a rich structure with many novel properties compared to conventional correlators in Lorentzian conformal field theories. Moreover, identifying the instanton number with the Fourier mode number of the dimensional reduction offers a hope to formulate six-dimensional conformal field theories in terms of five-dimensional Lagrangian theories. To this end we show that the Fourier decompositions of six-dimensional correlation functions solve the Ward identities of the SU(1, 3) symmetry, although more general solutions are possible. Conversely we illustrate how one can reconstruct six-dimensional correlation functions from those of a five-dimensional theory, and do so explicitly at 2- and 3-points. We also show that, in a suitable decompactification limit Ω → 0, the correlation functions become those of the DLCQ description.

On the Lagrangian formulation of the double copy to cubic order

We investigate the Lagrangian formulation of the double-copy correspondence between gauge theories and gravity, up to the cubic order. Building on the definition of the double-copy field as a convolution of two vectors, we obtain free gravitational Lagrangians as products of two Yang-Mills Lagrangians, in a form amenable to be easily extended to the massive case. We discuss the origin of these results from tensionless strings and show the existence of gauge fixings that mix the two spin-one sectors and lead to an alternative, especially simple, version of the free Lagrangian. We then construct cubic vertices for the full double-copy multiplet, comprising a graviton, a two-form and a scalar particle, by means of the Noether procedure. Both at the free and at the cubic level the result gets uniquely fixed only upon imposing, on top of gauge invariance, a left-right Lorentz symmetry ruling contraction of indices among double-copy fields. Whereas the outcome nicely matches the cubic interactions of mathcal{N} = 0 supergravity, including the gauge-invariant coupling between the scalar particle and the two-form, such a twofold Lorentz symmetry seems to conflict with the perturbative reconstruction of spacetime geometry.

\u03ba-Minkowski-deformation of U(1) gauge theory

We construct a noncommutative kappa-Minkowski deformation of U(1) gauge theory, following a general approach, recently proposed in JHEP 08 (2020) 041. We obtain an exact (all orders in the non-commutativity parameter) expression for both the deformed gauge transformations and the deformed field strength, which is covariant under these transformations. The corresponding Yang-Mills Lagrangian is gauge covariant and reproduces the Maxwell Lagrangian in the commutative limit. Gauge invariance of the action functional requires a non-trivial integration measure which, in the commutative limit, does not reduce to the trivial one. We discuss the physical meaning of such a nontrivial commutative limit, relating it to a nontrivial space-time curvature of the undeformed theory. Moreover, we propose a rescaled kappa-Minkowski noncommutative structure, which exhibits a standard flat commutative limit.

Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5

In the quest for the mathematical formulation of M-theory, we consider three major open problems: a first-principles construction of the single (abelian) M5-brane Lagrangian density, the origin of the gauge field in heterotic M-theory, and the supersymmetric enhancement of exceptional M-geometry. By combining techniques from homotopy theory and from supergeometry to what we call super-exceptional geometry within super-homotopy theory, we present an elegant joint solution to all three problems. This leads to a unified description of the Nambu-Goto, Perry-Schwarz, and topological Yang-Mills Lagrangians in the topologically nontrivial setting. After explaining how charge quantization of the C-field in Cohomotopy reveals D’Auria-Fré’s “hidden supergroup” of 11d supergravity as the super-exceptional target space, in the sense of Bandos, for M5-brane sigma-models, we prove, in exceptional generalization of the doubly-supersymmetric super-embedding formalism, that a Perry-Schwarz-type Lagrangian for single (abelian) mathcal{N} = (1, 0) M5-branes emerges as the super-exceptional trivialization of the M5-brane cocycle along the super-exceptional embedding of the “half ” M5-brane locus, super-exceptionally compactified on the Hořava-Witten circle fiber. From inspection of the resulting 5d super Yang-Mills Lagrangian we find that the extra fermion field appearing in super-exceptional M-geometry, whose physical interpretation had remained open, is the M-theoretic avatar of the gaugino field.

Higgs fields induced by Yang–Mills type Lagrangians on gauge-natural prolongations of principal bundles

We address some new issues concerning spontaneous symmetry breaking. We define classical Higgs fields for gauge-natural invariant Yang–Mills type Lagrangian field theories through the requirement of the existence of canonical covariant gauge-natural conserved quantities. As an illustrative example, we consider the ‘gluon Lagrangian’, i.e. a Yang–Mills Lagrangian on the [Formula: see text]-order gauge-natural bundle of [Formula: see text]-principal connections, and canonically define a ‘gluon’ classical Higgs field through the split reductive structure induced by the kernel of the associated gauge-natural Jacobi morphism.

Mathematical quantum Yang–Mills theory revisited

A mathematically rigorous relativistic quantum Yang-Mills theory with an arbitrary semisimple compact gauge Lie group is set up in the Hamiltonian canonical formalism. The theory is non-perturbative, without cut-offs, and agrees with the causality and stability principles. This paper presents a fully revised, simplified, and corrected version of the corresponding material in the previous papers DYNIN[11] and [12]. The principal result is established anew: due to the quartic self-interaction term in the Yang-Mills Lagrangian along with the semisimplicity of the gauge group, the quantum Yang-Mills energy spectrum has a positive mass gap. Furthermore, the quantum Yang-Mills Hamiltonian has a countable orthogonal eigenbasis in a Fock space, so that the quantum Yang-Mills spectrum is point and countable. In addition a fine structure of the spectrum is elucidated. KEYS: Millennium Yang-Mills problem; Finite propagation speed; Sobolev inequalities; Nuclear vector spaces; Infinite-dimensional holomorphy; Friedrichs operator extensions; Variational spectral principle; Symbols and spectral theory of pseudo-differential operators.

A non-abelian quasi-particle model for gluon plasma

We propose a quasi-particle model for the thermodynamic description of the gluon plasma which takes into account non-abelian characteristics of the gluonic field. This is accomplished utilizing massive non-linear plane wave solutions of the classical equations of motion with a variable mass parameter, reflecting the scale invariance of the Yang-Mills Lagrangian. For the statistical description of the gluon plasma we interpret these non-linear waves as quasi-particles with a temperature dependent mass distribution. Quasi-Gaussian distributions with a common variance but different temperature dependent mean masses for the longitudinal and transverse modes are employed. We use recent Lattice results to fix the mean transverse and longitudinal masses while the variance is fitted to the equation of state of pure $SU(3)$ on the Lattice. Thus, our model succeeds to obtain both a consistent description of the gluon plasma energy density as well as a correct behaviour of the mass parameters near the critical point.

Computing the Lagrangians of the Standard Model II. The Ghost Term

We follow up an earlier attempt to compute the Yang–Mills Lagrangian density from first principles. In that work, the Lagrangian density emerged replete with a Feynman–’t Hooft gauge fixing term. In this note we find that similar methods may be applied to produce the concomitant ghost term. Our methods are elementary and entirely and straightforwardly algebraic. Insofar as one of our first principles in the earlier computation was the Schwinger Action Principle, which is a differential version of the Feynman path integral, our computation here may be viewed as a differential version of the Faddeev–Popov functional integral approach to generating the ghost Lagrangian. As such, it avoids all measure theoretic difficulties and ambiguities, though at the price of generality.

Spin connection as Lorentz gauge field in Fairchild’s action

We propose a modified gravitational action containing besides the Einstein–Cartan term some quadratic contributions resembling the Yang–Mills Lagrangian for the Lorentz spin connections. We outline how a propagating torsion arises and we solve explicitly the linearized equations of motion on a Minkowski background. We identify among torsion components six degrees of freedom: one is carried by a pseudo-scalar particle, five by a tachyon field. By adding spinor fields and neglecting backreaction on the geometry, we point out how only the pseudo-scalar particle couples directly with fermions, but the resulting coupling constant is suppressed by the ratio between fermion and Planck masses. Including backreaction, we demonstrate how the tachyon field provides causality violation in the matter sector, via an interaction mediated by gravitational waves.

Non-Abelian clouds around Reissner-Nordström black holes: The existence line

A known feature of electrically charged Reissner-Nordstr\"om-anti-de Sitter planar black holes is that they can become unstable when considered as solutions of Einstein--Yang-Mills theory. The mechanism for this is that the linearized Yang-Mills equations in the background of the Reissner-Nordstr\"om (RN) black holes possess a normalizable zero mode, resulting in non-Abelian (nA) magnetic clouds near the horizon. In this work we show that the same pattern may occur also for asymptotically flat RN black holes. Different from the anti-de Sitter case, in the Minkowskian background the prerequisite for the existence of the nA clouds is $i)$ a large enough gauge group and $ii)$ the presence of some extra interaction terms in the matter Lagrangian. To illustrate this mechanism we present two specific examples, one in four and the other in five dimensional asymptotically flat spacetime. In the first case, we augment the usual $SU(3)$ Yang-Mills Lagrangian with higher order (quartic) curvature term, while for the second one we add the Chern-Simons density to the $SO(6)$ Yang-Mills system. In both cases, an Abelian gauge symmetry is spontaneously broken near a RN black hole horizon with the appearance of a condensate of nA gauge fields. In addition to these two examples, we review the corresponding picture for anti-de Sitter black holes. All these solutions are studied both analytically and numerically, existence proofs being provided for nA clouds in the background of RN black holes. The proofs use shooting techniques which are suggested by and in turn offer insights for our numerical methods. They indicate that, for a black hole of given mass, appropriate electric charge values are required to ensure the existence of solutions interpolating desired boundary behavior at the horizons and spatial infinity.

Yang-Mills condensate as dark energy: A nonperturbative approach

Models based on Yang-Mills condensate (YMC) have been advocated in the literature and claimed to be successful candidates to explain dark energy. Several instantiations of this simple idea have been considered, the most promising of which are reviewed here. Nevertheless, results previously attained heavily relied on the perturbative approach to the analysis of the effective Yang-Mills action, which is only adequate in the asymptotically-free limit, and were extended into a regime, the infrared limit, in which confinement is expected. We show that if a minimum of the effective Lagrangian in $\theta \!=\! - F_{\, \, \mu \nu}^a \, F^{a \mu \nu}/2$ exists, a YMC forms that drives the Universe toward an accelerated de Sitter phase. The details of the models depend weakly on the specific form of the effective Yang-Mills Lagrangian. Using non-perturbative techniques mutated from the functional renormalization group procedure, we finally show that the minimum in $\theta$ of the effective Lagrangian exists, thus YMC can actually take place. The non-perturbative model has properties similar to the ones of the perturbative model. In the early stage of the universe, the YMC equation of state has an evolution that resembles the radiation component, i.e. $w_y \rightarrow 1/3$. However, in the late stage, $w_y$ naturally runs to the critical state with $w_y =-1$, and the universe transits from matter-dominated into dark energy dominated stage only recently, at a redshift the value of which depends on the initial conditions that are chosen while solving the dynamical system.

Classical Higgs fields on gauge gluon bundles

Classical Higgs fields and related canonical conserved quantities are defined by invariant variational problems on suitably defined gauge gluon bundles. We consider Lagrangian field theories which are assumed to be invariant with respect to the action of a gauge-natural group. As an illustrative example we exploit the ‘gluon Lagrangian’, i.e. a Yang-Mills Lagrangian on the (1, 1)-order gauge-natural bundle of SU (3)-principal connections. The kernel of the gauge-natural Jacobi morphism for such a Lagrangian, by inducing a reductive split structure, canonically defines a ‘gluon classical Higgs field’.

Interaction of the vector-meson octet with the baryon octet in effective field theory

We analyze the constraint structure of the interaction of vector mesons with baryons using the classical Dirac constraint analysis. We show that the standard interaction in terms of two independent SU(3) structures is consistent at the classical level. We then require the self-consistency condition of the interacting system in terms of perturbative renormalizability to obtain relations for the renormalized coupling constants at the one-loop level. As a result we find a universal interaction with one coupling constant which is the same as in the massive Yang-Mills Lagrangian of the vector-meson sector.

Computing the Lagrangians of the standard model

Ordinary quantum logic has well known pathologies rendering it useless for the purposes of computation. However, loosely related logics, based upon variants of Girard's Linear Logic, have been found useful in the context of quantum computation. In one sense, the use of such computational schemes affords a meta level view of the possible provenance of certain expressions not otherwise apparent. Since such logics are presumed to encapsulate the essential behavior of quantum “resources” we may entertain the question as to whether this logical or computational approach could have any bearing upon quantum physics itself. In this article we address the question of the genesis of certain fundamental Lagrangians, namely those occurring in the standard model. If a certain set of sentences in a logic are added to the set of axioms of the logic the resulting structure is generally called a theory by logicians. In this paper we shall introduce a version of such a logic and deduce some of its physical ramifications. Namely, we will show that there is a single type of sequent that, when added to the logical calculus at hand as an axiom, generates in the theory so defined, series whose leading terms match exactly the Yang–Mills Lagrangian density (including a gauge fixing term) and also the Einstein–Hilbert Lagrangian density, most of the remaining terms being negligible at low intensities in both cases. By expanding the logic somewhat, in the manner of second quantization, we are able also to give an account of interaction terms in the Yang–Mills case. This shows that there is a common form ancestral to all the Lagrangians of the standard model in the ensemble of “evolutionary” trees provided by deductions in a certain clearly specified logic, and reveals the differences between the Yang–Mills and gravitational kinetic terms. Thus we acquire a new paradigm for “unification” of the fundamental forces at the level of the underlying logic.

A nonperturbative method for the Yang–Mills Lagrangian

Using the properties of the partition function for a Yang–Mills theory we compute simple relations among the renormalization constants. In the particular case of the background gauge field method we obtain that the all orders beta function for the gauge coupling constant contains only the first two orders coefficients different than zero and thus corresponds to the 't Hooft scheme.

Supersymmetric gauge theory on a class of cocalibrated G2-structures

Given a seven-dimensional cocalibrated G2-structure that satisfies an additional trace condition, we construct a supersymmetric Yang–Mills–Lagrangian on it, with the supersymmetry parameters being a distinguished generalized Killing spinor of the G2-structure. The construction is based on a reinterpretation of certain generalized Killing spinors as parallel spinors in an ambient space. The results generalize previous attempts for flat G2-structures and those admitting geometric Killing spinors.

Action, quantum computation and the structure of the Yang–Mills Lagrangian

We seek to interpret the notions of action and Lagrangian in the quantum domain in terms of a quantum computational paradigm. This effort was inspired by work of Toffoli in the classical case. We claim that in this paradigm an action integral may itself be regarded as a quantum “macroprogram” with the generating Lagrangian density playing the role of a quantum “microprogram” describing an underlying quantum process, in rough analogy with the way a \(\lambda \)-expression (in simply typed \(\lambda \)-calculus) may be unfolded to reveal an underlying natural deduction. We carry out such an unfolding for the Yang–Mills Lagrangian (including a gauge fixing term) to reveal a simple underlying quantum microprogram which in this case effects the variation of a fermion tuple consequent upon a quantized, generically deformed, gauge rotation of the tuple.