^{1}

^{,*}

^{1,2}

^{,†}

^{1}

^{,‡}

^{3}.

In this work we present an algebraic proof of the renormazibility of the super-Yang-Mills action quantized in a generalized supersymmetric version of the maximal Abelian gauge. The main point stated here is that the generalized gauge depends on a set of infinity gauge parameters in order to take into account all possible composite operators emerging from the dimensionless character of the vector superfield. At the end, after the removal of all ultraviolet divergences, it is possible to specify values to the gauge parameters in order to return to the original supersymmetric maximal Abelian gauge, first presented in

In the understanding of the quark confinement mechanism, some formulations of the Yang-Mills theory in specific gauge conditions, like the Landau gauge, the maximal Abelian gauge, the Curci-Ferrari gauge, etc., are explored. In particular, the maximal Abelian gauge permits us to approach the notion of Abelian projection, one of the main ideas regarding quark confinement

In fact, supersymmetric theories at finite temperature can reveal fundamental properties, similar to those of weak interactions in a plasma of quarks and gluons. Thus, it may be possible to study the transition from the deconfinement phase in an analogous way to the nonsupersymmetric case, i.e., to analyze phase transitions due to the emergence of singularities in the Abelian sector of a super-Yang-Mills theory.In this work, our focus is on the proof of the renormazibility of the super-Yang-Mills theory in the supersymmetric maximal Abelian gauge, as proposed in

See Table

The paper is organized as follows. Section

References

In this section we perform a brief review on the supersymmetric formulation of the maximal Abelian gauge, first presented in

See also the Appendix for a review of the maximal Abelian gauge for the ordinary Yang-Mills theory in the

The algebra of the generators and some useful identities can be found in the Appendix.

In order to make clear our notations and conventions, let us display here the vector superfield in terms of its components:

We also have in Eq.

It is well known that in the quantization of a gauge field theory (being supersymmetric or not) an additional condition (or a constraint, or a gauge-fixing condition) for the gauge field needs to be implemented. Following

Since we have defined the gauge conditions of the SSMAG, Eqs.

Actually, the correct quantization of a gauge theory is an open problem until now. The Faddeev-Popov method is considered to be correct only at the perturbative level, but, at the nonperturbative level, other effects, such as the Gribov ambiguity problem, show up and have to be taken into account. In this work we will restrict ourselves to the Faddeev-Popov quantization method.

needs the implementation of a gauge-fixing condition. The Faddeev-Popov method corresponds to a way of introducing a constraint in the functional integral for a gauge theory. In this method the SYM action needs to be supplemented, in the Feynman path integrals, by a term including such a constraint. Then, according to the Faddeev-Popov method, the SYM action is replaced byThe off-diagonal chiral ghosts

The off-diagonal antichiral ghosts

The diagonal chiral ghosts

The diagonal antichiral ghosts

Transformations of the components of the vector superfield

Transformations of the components of chiral superfields

Transformations of the components of the antichiral superfields

In this section we would like to study the symmetry content of the action

In general, if the combination

Now we are able to present the set of Ward identities enjoyed by the action

Mass dimension

The BRST symmetry can be written as a functional identity as follows:

The classical equations of motion of the diagonal Lagrange multipliers

In contrast with the diagonal equations of motion of the Lagrange multipliers

From the diagonal antighost equations the following identities can be obtained

Some authors call these equations the ghost equations. In our nomenclature, however, we have decided to name the equations obtained from the functional derivatives of the antighost fields (chiral or antichiral) as “antighost equations” and the identities obtained from the functional derivatives of the ghost fields as “ghost equations.”

:Actually they can be recovered but they are completely innocuous.

^{,}

A detailed discussion on the Ward identities in the ordinary maximal Abelian gauge can be found in

Another important symmetry for the renormalization procedure is the diagonal ghost equation, given by

Noticing that the vector superfield

Then, due to the above-mentioned ambiguity, it is necessary to redefine the gauge-fixing conditions

Remember here that

We display here a full set of Ward identities enjoyed by the action

The new Slavnov-Taylor identity now includes the BRST doublet of sources

The new diagonal gauge-fixing equations now assume the forms:

The off-diagonal gauge-fixing equations are now very similar to the diagonal ones (we will turn to this point later):

The new diagonal antighost equations are modified by the presence of the sources

The new off-diagonal antighost equations assume a simpler form in this generalized formulation:

The new

The new diagonal rigid symmetry is also generalized as follows:

Finally, the new diagonal ghost equation is generalized in the new formulation in order to accommodate the new BRST sources. We can also notice that classical breaking term remains the same:

Furthermore, a deeper look at the diagonal and off-diagonal gauge-fixing equations, Eqs.

As a final comment, we would like to emphasize that the identities displayed above are, in principle, valid only at the classical level. At the quantum level it is first necessary to prove the absence of anomalies. This is in fact one of the main steps of the algebraic proof of the renormalization. The study of anomalies for such identities in superspace was exhaustively discussed in the literature; see, e.g.,

Our next step will be to determine the most general invariant counterterm which can be freely added to all order in perturbation theory, allowing us to remove all divergences of the theory. Such a counterterm is generically written as

Then, in order to find an explicit expression for the counterterm,

Through the action of the linearized operator

In its simplest form,

A final comment emerges from a comparison between the supersymmetric and ordinary cases. In the study of the renormalization of the Yang-Mills action quantized in the maximal Abelian gauge, the absence of the off-diagonal gauge-fixing and antighost equations as genuine Ward identities gives rise to extra interaction terms among the ghost fields. In fact, quartic interaction ghost terms naturally emerge, as a diagrammatic analysis reveals, and the original gauge can only be defined modulo an extra gauge parameter

In this work, we have concluded the algebraic proof of the renormalizability of a

The proof presented here is very similar to the one presented in

Also, in

Another problem that can be investigated in the GSMAG is the Gribov problem

The Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil), the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), and the SR2-UERJ are gratefully acknowledged. M. A. L. Capri is a level PQ-2 researcher under the program Produtividade em Pesquisa-CNPq, Grant No. 307783/2014-6 and is a Procientista under SR2-UERJ. R. C. T. is supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) under the program Doutorado Sanduíche no Exterior (PDSE), Grant No. 88881.188419/2018-01.

In this section, we make a brief review on the ordinary maximal Abelian gauge. This gauge arises from the breaking of color symmetry, which generates a separation of the structure of the group

To impose the minimum condition it is necessary to take into account also

The Landau gauge can also be defined as an extreme condition of an auxiliary functional. In this case, the suitable functional is