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22 pages, 328 KB

Open AccessArticle

Spontaneous BRST Symmetry Breaking in Infrared QCD

by Angelo Raffaele Fazio and Adam Smetana

Universe 2026, 12(5), 138; https://doi.org/10.3390/universe12050138 - 7 May 2026

Viewed by 275

We present a novel proposal for the effective Lagrangian of the low-energy Yang–Mills quantum field theory. The proposed effective Lagrangian exhibits the spontaneous BRST symmetry breaking. We build on the Fujikawa model that we couple to the Yang–Mills elementary field sector, motivated by the analogy with Chiral Quark Model. We interpret the Fujikawa fields as effective fields, composites of the elementary gluon and ghost fields. In order to justify the existence of two massless Nambu–Goldstone modes among the Fujikawa fields, we require not only the BRST but also the anti-BRST invariance of the effective Lagrangian, with both being spontaneously broken. The most striking consequence of that is the emergence of the effective gluon and ghost masses. We reproduce the Curci–Ferrari model as a special case of our effective model upon the spontaneous BRST symmetry breaking. In order to reproduce also the non-nilpotent modified-BRST symmetry, characteristic for the Curci–Ferrari model, we modify our effective Lagrangian to be invariant with respect to the extended-BRST symmetry, which mixes the elementary and Fujikawa field sectors, and which is nilpotent. The Curci–Ferrari model is reproduced by the elementary field sector of the resulting Lagrangian. The remaining Fujikawa-field-dependent terms guarantee the underlying nilpotent extended-BRST symmetry, which is now hidden in the sense of the spontaneous symmetry breaking. Full article

(This article belongs to the Section Field Theory)

17 pages, 336 KB

Open AccessArticle

On the Geometric Structure of Hyperbolic Clifford Bundles and Associated Spin Groups

by Eduardo Notte-Cuello

Axioms 2026, 15(4), 286; https://doi.org/10.3390/axioms15040286 - 14 Apr 2026

Viewed by 387

Abstract

In this paper, we present the spinor structure associated with the Hyperbolic Clifford algebra of a real n-dimensional vector space V, which is denoted by

C l (H_{V})

. Unlike the standard Clifford algebra, the Hyperbolic Clifford algebra [...] Read more.

In this paper, we present the spinor structure associated with the Hyperbolic Clifford algebra of a real n-dimensional vector space V, which is denoted by

C l (H_{V})

. Unlike the standard Clifford algebra, the Hyperbolic Clifford algebra

C l (H_{V})

simultaneously accommodates both multiforms and multivectors in a single algebraic structure, making it the natural framework—known as the “mother algebra”—for the study of superfields in theoretical physics and for generalizing the Clifford bundle formalism to hyperbolic structures arising in gravitational theories. The orthogonal groups and orthogonal transformations associated to the hyperbolic space

H_{V}

are presented. The Clifford–Lipschitz group and the Pin and Spin groups associated with

C l (H_{V})

are defined. Then, the frame bundle and spinor structure associated to Hyperbolic Clifford algebra is derived. Full article

(This article belongs to the Special Issue Complex Variables in Quantum Gravity)

16 pages, 368 KB

Open AccessArticle

Finiteness of

N = 4

Super-Yang–Mills Effective Action in Terms of Dressed

N = 1

Superfields

by Igor Kondrashuk and Ivan Schmidt

Particles 2023, 6(4), 993-1008; https://doi.org/10.3390/particles6040063 - 8 Nov 2023

Cited by 11 | Viewed by 4793

Abstract

We argue in favor of the independence on any scale, ultraviolet or infrared, in kernels of the effective action expressed in terms of dressed

N = 1

superfields for the case of

N = 4

super-Yang–Mills theory. Under “scale independence” of the effective [...] Read more.

We argue in favor of the independence on any scale, ultraviolet or infrared, in kernels of the effective action expressed in terms of dressed

N = 1

superfields for the case of

N = 4

super-Yang–Mills theory. Under “scale independence” of the effective action of dressed mean superfields, we mean its “finiteness in the off-shell limit of removing all the regularizations”. This off-shell limit is scale independent because no scale remains inside these kernels after removing the regularizations. We use two types of regularization: regularization by dimensional reduction and regularization by higher derivatives in its supersymmetric form. Based on the Slavnov–Taylor identity, we show that dressed fields of matter and of vector multiplets can be introduced to express the effective action in terms of them. Kernels of the effective action expressed in terms of such dressed effective fields do not depend on the ultraviolet scale. In the case of dimensional reduction, by using the developed technique, we show how the problem of inconsistency of the dimensional reduction can be solved. Using Piguet and Sibold formalism, we indicate that the dependence on the infrared scale disappears off shell in both the regularizations. Full article

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51 pages, 565 KB

Open AccessReview

BRST and Superfield Formalism—A Review

by Loriano Bonora and Rudra Prakash Malik

Universe 2021, 7(8), 280; https://doi.org/10.3390/universe7080280 - 1 Aug 2021

Cited by 13 | Viewed by 3247

Abstract

This article, which is a review with substantial original material, is meant to offer a comprehensive description of the superfield representations of BRST and anti-BRST algebras and their applications to some field-theoretic topics. After a review of the superfield formalism for gauge theories, we present the same formalism for gerbes and diffeomorphism invariant theories. The application to diffeomorphisms leads, in particular, to a horizontal Riemannian geometry in the superspace. We then illustrate the application to the description of consistent gauge anomalies and Wess–Zumino terms for which the formalism seems to be particularly tailor-made. The next subject covered is the higher spin YM-like theories and their anomalies. Finally, we show that the BRST superfield formalism applies as well to the

N = 1

super-YM theories formulated in the supersymmetric superspace, for the two formalisms go along with each other very well. Full article

(This article belongs to the Special Issue Gauge Theory, Strings and Supergravity)

32 pages, 360 KB

Open AccessArticle

Superfields, Nilpotent Superfields and Superschemes

by María Antonia Lledó

Symmetry 2020, 12(6), 1024; https://doi.org/10.3390/sym12061024 - 17 Jun 2020

Cited by 1 | Viewed by 2598

Abstract

We interpret superfields in a functorial formalism that explains the properties that are assumed for them in the physical applications. We study the non-trivial relation of scalar superfields with the defining sheaf of the supermanifold of super spacetime. We also investigate in the present work some constraints that are imposed on the superfields, which allow for non-trivial solutions. They give rise to superschemes that, generically, are not regular, that is they do not define a standard supermanifold. Full article

28 pages, 642 KB

Open AccessReview

Quantum Cosmology in the Light of Quantum Mechanics

by Salvador J. Robles-Pérez

Galaxies 2019, 7(2), 50; https://doi.org/10.3390/galaxies7020050 - 24 Apr 2019

Cited by 6 | Viewed by 6782

Abstract

There is a formal analogy between the evolution of the universe, when it is seen as a trajectory in the minisuperspace, and the worldline followed by a test particle in a curved spacetime. The analogy can be extended to the quantum realm, where the trajectories are transformed into wave packets that give us the probability of finding the universe or the particle in a given point of their respective spaces: the spacetime in the case of the particle and the minisuperspace in the case of the universe. The wave function of the spacetime and the matter fields, all together, can then be seen as a super-field that propagates in the minisuperspace and the so-called third quantisation procedure can be applied in a parallel way as the second quantisation procedure is performed with a matter field that propagates in the spacetime. The super-field can thus be interpreted as made up of universes propagating, i.e., evolving, in the minisuperspace. The analogy can also be used in the opposite direction. The way in which the semiclassical state of the universe is obtained in quantum cosmology allows us to obtain, from the quantum state of a field that propagates in the spacetime, the geodesics of the underlying spacetime as well as their quantum uncertainties or dispersions. This might settle a new starting point for a different quantisation of the spacetime. Full article

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11 pages, 11001 KB

Open AccessArticle

Soliton and Similarity Solutions of Ν = 2, 4 Supersymmetric Equations

by Laurent Delisle and Véronique Hussin

Symmetry 2012, 4(3), 441-451; https://doi.org/10.3390/sym4030441 - 8 Aug 2012

Cited by 4 | Viewed by 6596

Abstract

We produce soliton and similarity solutions of supersymmetric extensions of Burgers, Korteweg–de Vries and modified KdV equations. We give new representations of the τ -functions in Hirota bilinear formalism. Chiral superfields are used to obtain such solutions. We also introduce new solitons called [...] Read more.

(This article belongs to the Special Issue Supersymmetry)

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