# Non-perturbative to perturbative QCD via the FFBRST

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## Abstract

Recently a new type of quadratic gauge was introduced in QCD in which the degrees of freedom are suggestive of a phase of abelian dominance. In its simplest form it is also free of Gribov ambiguity. However this gauge is not suitable for usual perturbation theory. The finite field dependent BRST (FFBRST) transformation is a method established to interrelate generating functionals for different effective versions of gauge fixed field theories. In this paper we propose a FFBRST transformation suitable for transforming the theory in the new quadratic gauge into the standard Lorenz gauge Faddeev–Popov version of the effective lagrangian. The task is made interesting by the fact that the effective lagrangian is invariant under two different BRST transformations which leads to suitable extension of the previous procedures to accomplish the required result. We are thus able to identify a field redefinition to go from a non-perturbative phase of QCD to perturbative QCD.

## 1 Introduction

*x*. Several proposals to establish abelian dominance in the infrared (IR) use what is called Abelian Projection [10]. Such and other algebraic gauges are usually non-covariant. But as introduced in [7, 8] the above gauge is in fact covariant . The gauge prima facie is an ambiguity free gauge as it is algebraic in nature. Thus, the quadratic gauge shares the same property of being free of Gribov copies as the axial gauges \(n_\mu A^{\mu a} = f^a(x)\) and the flow gauge \(\alpha A_0^a = \nabla \cdot \vec {A^a}\) [11] despite being non linear. The Faddeev–Popov determinant in this gauge is given by

*a*will be understood when it appears repeatedly, including when repeated

*thrice*as in the ghost terms above. In particular,

*a*,

*b*and

*c*each runs independently over 1 to \(N^2-1\). We should note that ghost Lagrangian does not have kinetic terms and hence the ghosts do not propagate in this theory and make no loop contributions. They act like auxiliary fields, but playing an important role in the IR. With this understanding, we write the full effective Lagrangian density in this quadratic gauge as

*S*-matrix. These issues were studied in Refs. [7, 8].

The form of the second term of the expression (4) appearing in the ghost lagrangian contains ghost bilinears multiplying terms quadratic in gauge fields. Hence if the non-propagating ghosts are assumed to be frozen they amount to a non-zero mass matrix for the gluons. To strengthen this connection it is necessary to assume that the vacuum corresponds to ghost condensation. This was achieved through introducing a Lorenz gauge fixing term for one of the diagonal gluons, in addition to the purely quadratic terms of Eq. (1). This gauge fixing gives the propagator to the corresponding ghost field. Using this ghost propagator, one can give nontrivial vacuum values to bilinears \( \overline{c^a} c^c \) within the framework Coleman-Weinberg mechanism as described in [7, 8]. We shall revisit the point in the next section also.

The resulting mass matrix for the gluons has \(N(N - 1)\) non-zero eigenvalues only and thus has nullity \( N -1\). Thus, the \(N(N - 1)\) off-diagonal gluons acquire masses and the rest \(N-1\) diagonal gluons remain massless. The massive off-diagonal gluons are presumed to provide evidence of Abelian dominance, which is a signature of quark confinement. This and other phenomena that emerge in this gauge, such as the avoidance of Gribov ambiguity were studied explicitly in [7, 8, 9]. Quark confinement and Gribov ambiguity are important non-perturbative issues. And this gauge therefore proves to be important in studying non-perturbative regime of QCD.

The finite field dependent BRST (FFBRST) transformation was introduced for first time in Ref. [6] by integrating infinitesimal usual BRST transformations. Such FFBRST transformations have exactly the same form as the infinitesimal ones, with the difference that the infinitesimal global anti-commuting parameter is replaced by an anti-commuting but finite parameter dependent on space time fields, but with no explicit dependence on space time coordinates. The meaning of “finite anti-commuting parameter” is that if we calculate the Green’s functions for such parameters between vacuum and a state with gauge and ghost fields we get finite values as opposed to infinitesimal values. Being finite in nature FFBRST transformation does not leave the path integral measure invariant even though other properties of usual infinitesimal BRST transformation are intact. Thus the generating functional to a BRST invariant theory is not invariant under FFBRST. Jacobian of such finite transformation provides a non-trivial factor which depends on FFBRST parameter.

Due to this non-trivial Jacobian FFBRST transformations are simultaneously field redefinitions as well as BRST transformations on the fields being redefined. They are thus capable of connecting generating functionals of two different BRST invariant theories and have been used to study different gauge field theoretic models with various effective actions [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. In this paper we construct an appropriate FFBRST transformation to establish the connection at the level of generating functionals between the recently introduced quadratic gauge with substantial implications in the non-perturbative QCD [7, 8, 9] and the familiar Lorenz gauge which is suitable to describe the perturbative QCD. This is novel connection since previous connections were either between two gauges suitable for only perturbative sector e.g., connection between Lorenz and axial gauges or they had no such unique field theoretic meaning attached to them. We should here mention that the same FFBRST however does not explicitly connect the vacuum in the quadratic gauge with which non-Perturbative phase of QCD is associated to the vacuum in Lorenz gauge to which perturbative phase of QCD corresponds. To understand this, we first discuss the vacua of both the theories.

*SU*(

*N*) symmetric ghost condensation of bilinears [7, 8] which is non perturbative in nature since the non perturbative confining phase corresponds to this vacuum [7, 8] and it arises at Lagrangian level only as is clear from Eq. (4). Such a vacuum does not exist in the theory of Lorenz gauge. The vacuum in the Lorenz gauge is provided by the mix condensate of gluon and ghost, \(\langle F^{\mu \nu }F_{\mu \nu } + \overline{c}c \rangle \) arising from the following sorts of loops (Fig. 1) [29].

## 2 Connecting two different regimes

*SU*(3) singlets.

*x*, it is not the differential equation in the \(\delta \overline{d^3 }\) and the \(\Box \) is an invertible operator (similar concept has appeared in the case of Lagrangian in the literature, see for example Ref. [30]). However, these transformations are not useful for FFBRST technique since the transformations of \(G^3\) and \(\Box d^3\) are trivial. Therefore, we need to introduce a new set of BRST transformations under which the action (6) is also invariant. This clearly shows that the passage from BRST to FFBRST transformations is non trivial. This conclusion has never been obvious from earlier works. The transformations are as follows

*J*. This

*J*can further be cast as a local functional of fields, \( e^{iS_J}\) (where the \(S_J\) is the action representing the Jacobian factor

*J*) if the following condition is met [6]

*a*is summed over. This FFBRST transformation is particularly different among others [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] due to the unique form of transformations (11) and by the fact that the field dependent parameter in Eq. (18) contains two ghosts \(\overline{c}, d\) with two different transformation properties unlike others where there is only one ghost. We now calculate the change in the Jacobian \(\frac{1}{J}\frac{dJ}{d\kappa } \) due to the FFBRST with the parameter in Eq. (18), under which the measure changes \(\mathcal {D}\phi (\kappa ) \rightarrow J(\kappa )\mathcal {D}\phi (\kappa )\) as

## 3 Conclusion

The spirit of BRST invariance was to establish the unitarity of the S-matrix in gauge theories whose gauge fixed versions contain ghost degrees of freedom. This technique was substantially extended in the FFBRST approach to permit field redefinitions transforming the effective action with one possible gauge fixing to that of another. In some of the recent earlier work the interesting features of a purely quadratic gauge condition without the usual Lorenz condition have been studied and shown to lead to several interesting properties of the non-perturbative QCD vacuum in the IR limit. At first site the effective degrees of freedom entering here, the off diagonal gluons with masses, appear unrelated to those entering the perturbation theory calculations and which are compatible with the elegant UV properties of Yang–Mills theories. In this paper we have resorted to the FFBRST technique to establish a direct formal connection between the two varieties of the QCD effective lagrangians. Several technical difficulties are encountered in this process and it has required us to make suitable extensions to the FFBRST method. In particular a new auxiliary field is required to ensure nilpotency of the modified BRST transformations. The resulting field redefinitions which connect the degrees of freedom capturing the IR behaviour of QCD vacuum with those of the UV version suitable to perturbative computations need to be studied further. Also, the extensions of the FFBRST technique proposed here can be put to use for other similar problems.

## Notes

### Acknowledgements

We acknowledge the fruitful discussion with Prof. Urjit A. Yajnik. This work is partially supported by Department of Science and Technology, Govt. of India under National Postdoctoral Fellowship scheme with File No. ‘PDF/2017/000066’. One of us (BPM) acknowledges the support from Physics Department, IIT-Bombay for a visitation during which the work was initiated.

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