Background field method and nonlinear gauges

TL;DR

This paper introduces a reformulation of the background field method for Yang-Mills theories using a superalgebra framework, allowing consistent implementation of nonlinear gauges through invariance principles.

Contribution

It presents a novel superalgebra-based reformulation that facilitates the use of nonlinear gauges in background field methods for Yang-Mills theories.

Findings

Enables consistent use of nonlinear gauges in background field method.

Provides a superalgebra framework for BRST and background transformations.

Ensures invariance of gauge-fixing functional under background transformations.

Abstract

We present a reformulation of the background field method for Yang-Mills type theories, based on using a superalgebra of generators of BRST and background field transformations. The new approach enables one to implement and consistently use non-linear gauges in a natural way, by using the requirement of invariance of the fermion gauge-fixing functional under the background field transformations.

Equations

$${\delta }_{\xi }{A}^i=R_{\alpha }^i(A){\xi }^{\alpha }.$$

$$R_{\alpha }^i(A)z^{\alpha }=0⟹z^{\alpha }=0,$$

$$[{\delta }_{{\xi }_1},{\delta }_{{\xi }_2}]A^i={\delta }_{{\xi }_3}{A}^i\text{with}{\xi }_3^{\alpha }=F_{\beta \gamma }^{\alpha }{\xi }_2^{\gamma }{\xi }_1^{\beta }.$$

$$R_{\alpha ,j}^i(A)R_{\beta }^j(A)-(-1{)}^{{\epsilon }_{\alpha }{\epsilon }_{\beta }}{R}_{\beta ,j}^i(A)R_{\alpha }^j(A)=-R_{\gamma }^i(A)F_{\alpha \beta }^{\gamma },$$

$${\delta }_{\omega }{S}_0(A+\mathcal{B})=0$$

$${\delta }_{\omega }^{(q)}{A}^i=\left[R_{\alpha }^i(A+\mathcal{B})-R_{\alpha }^i(\mathcal{B})\right]{\omega }^{\alpha },{\delta }_{\omega }^{(c)}{\mathcal{B}}^i=R_{\alpha }^i(\mathcal{B}){\omega }^{\alpha },$$

$${\delta }_{\omega }^{(c)}{\mathcal{B}}^i+{\delta }_{\omega }^{(q)}{A}^i=R_{\alpha }^i(A+\mathcal{B}){\omega }^{\alpha }.$$

$$\displaystyle\phi^{A}=\big{(}A^{i},B^{\alpha},C^{\alpha},\bar{C}^{\alpha}\big{)}.$$

$$S_{\text{FP}}(\varphi ,\mathcal{B})=S_0(A+\mathcal{B})+S_{\text{gh}}(\varphi ,\mathcal{B})+S_{\text{gf}}(\varphi ,\mathcal{B}),$$

$$S_{\text{gh}}(\varphi ,\mathcal{B})={\bar{C}}^{\alpha }\frac{{\delta }_r{\chi }_{\alpha }(A,\mathcal{B})}{\delta A^i}{R}_{\beta }^i(A+\mathcal{B})C^{\beta }$$

$$\displaystyle\quad S_{\text{gf}}(\phi,\mathcal{B})\,=\,B^{\alpha}\Big{(}\chi_{\alpha}(A,\mathcal{B})+\frac{\xi}{2}g_{\alpha\beta}B^{\beta}\Big{)}.$$

$${\chi }_{\alpha }(A,\mathcal{B})=F_{\alpha i}(\mathcal{B})A^i,$$

$$\displaystyle\delta_{\text{BRST}}\,\phi^{A}\,=\,\big{(}\hat{R}_{\text{BRST}}\,\phi^{A}\big{)}\lambda,$$

$${\hat{R}}_{\text{BRST}}(\varphi ,\mathcal{B})=\frac{{\delta }_r}{\delta {\varphi }^A}{R}_{\text{BRST}}^A(\varphi ,\mathcal{B}),$$

$$\displaystyle R_{\text{BRST}}^{A}(\phi,\mathcal{B})\,=\,\Big{(}R^{i}_{\alpha}(A+\mathcal{B})C^{\alpha},\,\,0,\,\,-\frac{1}{2}F^{\alpha}_{\beta\gamma}C^{\gamma}C^{\beta}(-1)^{\varepsilon_{\beta}},\,\,(-1)^{\varepsilon_{\alpha}}B^{\alpha}\Big{)}.$$

$${\hat{R}}_{\text{BRST}}^2(\varphi ,\mathcal{B})=0.$$

$$\Psi (\varphi ,\mathcal{B})={\bar{C}}^{\alpha }{\chi }_{\alpha }(A,B,\mathcal{B}),\text{mbox}with\epsilon (\Psi )=-\text{gh}(\Psi )=1.$$

$$S_{\text{FP}}(\varphi ,\mathcal{B})=S_0(A+\mathcal{B})+{\hat{R}}_{\text{BRST}}(\varphi ,\mathcal{B})\Psi (\varphi ,\mathcal{B}).$$

$${\delta }_{\omega }^{(q)}{B}^{\alpha }=-F_{\gamma \beta }^{\alpha }{B}^{\beta }{\omega }^{\gamma },{\delta }_{\omega }^{(q)}{C}^{\alpha }=-F_{\gamma \beta }^{\alpha }{C}^{\beta }{\omega }^{\gamma }(-1{)}^{{\epsilon }_{\gamma }},{\delta }_{\omega }^{(q)}{\bar{C}}^{\alpha }=-F_{\gamma \beta }^{\alpha }{\bar{C}}^{\beta }{\omega }^{\gamma }(-1{)}^{{\epsilon }_{\gamma }}.\text{mbox}$$

$${\hat{R}}_{\omega }(\varphi ,\mathcal{B})=\frac{{\delta }_r}{\delta {\mathcal{B}}^i}{\delta }_{\omega }^{(c)}{\mathcal{B}}^i+\frac{{\delta }_r}{\delta {\varphi }^A}{\delta }_{\omega }^{(q)}{\varphi }^A,\epsilon ({\hat{R}}_{\omega })=0.$$

$$[{\hat{R}}_{\text{BRST}},{\hat{R}}_{\omega }]=0.$$

$${\delta }_{\omega }{S}_{\text{FP}}(\varphi ,\mathcal{B})={\hat{R}}_{\omega }(\varphi ,\mathcal{B})S_{\text{FP}}(\varphi ,\mathcal{B})=0⟺{\hat{R}}_{\omega }(\varphi ,\mathcal{B})\Psi (\varphi ,\mathcal{B})=0.$$

$${\hat{R}}_{\omega }(\varphi ,\mathcal{B})\Psi (\varphi ,\mathcal{B})={\bar{C}}^{\alpha }{\delta }_{\omega }{\chi }_{\alpha }(\varphi ,\mathcal{B})-F_{\gamma \beta }^{\alpha }{\bar{C}}^{\beta }{\omega }^{\gamma }(-1{)}^{{\epsilon }_{\gamma }}{\chi }_{\alpha }(\varphi ,\mathcal{B})=0$$

$${\delta }_{\omega }{\chi }_{\alpha }(\varphi ,\mathcal{B})=-{\chi }_{\beta }(\varphi ,\mathcal{B})F_{\alpha \gamma }^{\beta }{\omega }^{\gamma }.$$

$${\hat{R}}_{\omega }(\varphi ,\mathcal{B})={\hat{\mathcal{R}}}_{\alpha }(\varphi ,\mathcal{B}){\omega }^{\alpha },\epsilon ({\hat{\mathcal{R}}}_{\alpha })={\epsilon }_{\alpha }$$

$${\hat{R}}_{\text{BRST}}^2=0,[{\hat{R}}_{\text{BRST}},{\hat{\mathcal{R}}}_{\alpha }]=0,[{\hat{\mathcal{R}}}_{\alpha },{\hat{\mathcal{R}}}_{\beta }]=-{\hat{\mathcal{R}}}_{\gamma }{F}_{\beta \alpha }^{\gamma },$$

$${\hat{R}}_{\text{BRST}}(\varphi ,\mathcal{B})S_{\text{FP}}(\varphi ,\mathcal{B})=0\text{and}{\hat{\mathcal{R}}}_{\alpha }(\varphi ,\mathcal{B})\Psi (\varphi ,\mathcal{B})=0$$

$$S_{\text{ext}}(\varphi ,\mathcal{B},{\varphi }^{\ast })=S_{\text{FP}}(\varphi ,\mathcal{B})+{\varphi }_A^{\ast }{\hat{R}}_{\text{BRST}}(\varphi ,\mathcal{B}){\varphi }^A,$$

$$Z(\mathcal{J},\mathcal{B},{\varphi }^{\ast })=\int \mathcal{D}\varphi \exp \left\{i\left[S_{\text{FP}}(\varphi ,\mathcal{B})+{\mathcal{J}}_A{\varphi }^A+{\varphi }_A^{\ast }{\hat{R}}_{\text{BRST}}(\varphi ,\mathcal{B}){\varphi }^A\right]\right\},$$

$$(-1{)}^{{\epsilon }_i}\frac{{\delta }_{\ell }{R}_{\alpha }^i(A+\mathcal{B})}{\delta A^i}+(-1{)}^{{\epsilon }_{\beta }+1}{F}_{\beta \alpha }^{\beta }=0,$$

Full text

Keywords:  background field method, nonlinear gauge, Yang-Mills theories, quantum gravity theories

1 Introduction

The standard approach to the Lagrangian quantization of gauge theories [1, 2, 3] assumes the violation of the gauge invariance of the action. In some cases, e.g., in the semiclassical gravity theory [4], this makes all considerations and also practical calculations quite complicated. The same certainly concerns quantum gravity. Fortunately, there is a useful approach to the quantization of gauge theories, known as the background field method (BFM) [5, 6, 7]. Within the BFM one can explicitly preserve the gauge invariance of an effective action in all stages, and thus all physical results are reproduced using the effective action of background fields (background effective action). In many cases, the calculations in gauge theories in flat and curved space-time are performed by means of the BFM, while the general theorems concerning gauge invariant renormalizability and gauge fixing dependence rely on the standard version of quantization. This situation makes it at least highly desirable to construct a completely consistent formulation of the BFM for gauge theories and quantum gravity, such that the mentioned general theorems could be formulated directly within the formalism that is used for calculations and, e.g., for the analysis of power counting in new non-conventional models of quantum field theory (see, for instance, [8, 9]).

The BFM has been the object of extensive investigation from different viewpoints (see e.g. [10, 11, 12, 13, 14, 15, 16, 17, 18, 19]), and in the last years the interest in this method has grown rapidly, since it is supposed to solve many important problems of gauge theories [20, 21, 22, 23, 24, 25, 26, 9]. One of the standard assumptions of the BFM is related to the special choice of gauge generators and gauge-fixing conditions, that are typically linear in the quantum fields. Indeed, even though the gauge fixing may admit a lot of arbitrariness, linearity is always assumed in the framework of the BFM. At the same time, the consistent formulation of the quantization method should certainly be free of such a restriction, as the choice of the gauge fixing is arbitrary and is not related to the physical contents of the initial gauge theory. In the present paper we propose a generalization of the standard formulation of the BFM, in the sense that is can be used for a wide class of non-linear gauge-fixing conditions and together with some types of non-linear gauge generators. This new point of view on the BFM is based on the superalgebra of generators of the fundamental symmetries of this formalism (namely, the BRST [27, 28] and the background field symmetries [5, 6, 7]).

It is well known that a judicious choice of gauge condition can yield a considerable simplification of calculations in quantum field theory, and non-linear gauges are often used with this purpose (see, for instance, [31, 32, 29, 30, 15, 33]). On the other hand, such gauges emerge naturally in the framework of the effective field theory approach, after the massive degrees of freedom are integrated out and we are left with the gauge theory of relatively light quantum fields [34]. Thus, the consistent implementation of non-linear gauge fixing in all approaches to quantization looks relevant from many viewpoints.

The general analysis of renormalization of gauge theories under non-linear gauge conditions has been presented in many papers, including [35, 36, 37, 38, 39, 40]. Nonetheless, non-linear gauges are not frequently used in the background field formalism. A remarkable example of simultaneous use of the BFM and a non-linear gauge is the two-loop calculation in quantum gravity [15], which confirmed the previous calculation of [41]. This correspondence is certainly a positive signal, making even more clear the need for a general discussion on this subject, which is not present in the literature.

The idea that the gauge-fixing function should transform in a covariant way under background field transformations (a condition that is trivially satisfied for the usual linear gauge), as a form of preserving the background field invariance of the Faddeev-Popov action, is present in many discussions on Yang-Mills theories [42, 21]. Nevertheless, the applicability of the BFM with non-linear gauges is not a consensus. For example, in [43] it is mentioned that the Gervais-Neveu gauge [31] could be used within the BFM for Yang-Mills theory, while in [21] the linearity of the gauge-fixing condition and gauge generators is regarded as an important factor for the consistency of the quantum formalism.

As it was already mentioned above, in the present work we introduce a new geometric point of view on the BFM based on the operator superalgebra which underlies the method. As the main application of this formalism, we obtain necessary conditions for the consistent application of the BFM [5, 6, 7] for Yang-Mills type theories with non-linear gauge-fixing conditions and non-linear gauge generators. The work is organised as follows. In Sec. 2 we briefly review the BFM applied to gauge theories. The main original results are presented in Sec. 3, including the discussion of non-linear gauges for generalised Yang-Mills theories in the BFM. In Sec. 4 some aspects of the Yang-Mills theory and quantum gravity are explicitly considered as examples of the general result. Finally, in Sec. 5 we draw our conclusions. The DeWitt’s condensate notations [44] are used throughout the paper.

2 Background field formalism for gauge theories

As a starting point, consider a gauge theory of fields $A^{i}$, with Grassmann parity $\varepsilon_{i}=\varepsilon(A^{i})$. A complete set of fields $\,A^{i}=(A^{\alpha k},A^{m})\,$ includes fields $A^{\alpha k}$ of the gauge sector and also fields $A^{m}$ of the matter sector of a given theory.

The classical action $S_{0}(A)$ is invariant under gauge transformations

[TABLE]

Here $\xi^{\alpha}(x)$ is the transformation parameter with parity $\varepsilon(\xi^{\alpha})=\varepsilon_{\alpha}$, while $R^{i}_{\alpha}(A)$ (with $\varepsilon(R^{i}_{\alpha})=\varepsilon_{i}+\varepsilon_{\alpha}$) is the generator of gauge transformation. In our notation $i$ is the collection of internal, Lorentz and continuous (spacetime coordinates) indices. We assume that the generators are linearly independent, i.e.,

[TABLE]

and satisfy a closed algebra with structure coefficients $F^{\gamma}_{\alpha\beta}$ that do not depend on the fields,

[TABLE]

From (3) it follows that $\,F^{\gamma}_{\alpha\beta}=-(-1)^{\varepsilon_{\alpha}\varepsilon_{\beta}}F^{\gamma}_{\beta\alpha}$ and

[TABLE]

where we denote the right functional derivative by $\delta_{r}X/\delta A^{i}=X_{,i}$. In principle, the generators $R^{i}_{\alpha}(A)$ may be non-linear in the fields. Further restrictions on the generators will be introduced in the next section, motivated by quantum aspects of the theory.

Let us formulate the theory within the BFM by splitting the original fields $A^{i}$ into two types of fields, through the substitution $A^{i}\longmapsto A^{i}+\mathcal{B}^{i}$ in the initial action $S_{0}(A)$. It is assumed that the fields $\mathcal{B}^{i}$ are not equal to zero only in the gauge sector 111 In gauge theories without spontaneous symmetry breaking one can introduce background fields in the sector of gauge fields without loss of generality [22, 26].. These fields form a classical background, while $A^{i}$ are quantum fields, that means being subject of quantization, e.g., these fields are integration variables in functional integrals. It is clear that the total action satisfies

[TABLE]

under the transformation $\,A^{i}\longmapsto A^{\prime i}=A^{i}+R^{i}_{\alpha}(A+\mathcal{B})\omega^{\alpha}$. On the other hand, the new field $\mathcal{B}^{i}$ introduces extra new degrees of freedom and, thence, there is an ambiguity in the transformation rule for each of the fields $A^{i}$ and $\mathcal{B}^{i}$. This ambiguity can be fixed in different ways, and in the BFM it is done by choosing the transformation laws

[TABLE]

defining the background field transformations for the fields $A^{i}$ and $\mathcal{B}^{i}$, respectively. The superscript ($q$) indicates the transformation of the quantum fields, while that of the classical fields is labelled by ($c$). Thus, in Eq. (5) one has $\delta_{\omega}=\delta^{(q)}_{\omega}+\delta^{(c)}_{\omega}$. Indeed, the background field transformation rule for the field $A^{i}$ was chosen so that

[TABLE]

In order to apply the Faddeev-Popov quantization scheme [1] in the background field formalism, one has to introduce a gauge-fixing condition for the quantum fields $A^{i}$, and extra fields $C^{\alpha}$, $\bar{C}^{\alpha}$ and $B^{\alpha}$. To simplify notation we denote by $\phi=\left\{\phi^{A}\right\}$ the set of all quantum fields

[TABLE]

The Grassmann parity of the fields is $\,\varepsilon(\phi^{A})=\varepsilon_{A}=(\varepsilon_{i},\varepsilon_{\alpha},\varepsilon_{\alpha}+1,\varepsilon_{\alpha}+1)$, while their ghost numbers are $\,\text{gh}(C^{\alpha})=-\text{gh}(\bar{C}^{\alpha})=1\,$ and $\,\text{gh}(A^{i})=\text{gh}(B^{\alpha})=0$. The corresponding Faddeev-Popov action in the BFM reads

[TABLE]

where the ghost and gauge-fixing actions are defined as

[TABLE]

and

[TABLE]

In this expression $\,\xi\,$ is a gauge parameter that has to be introduced in the case of a non-singular gauge condition, and $g_{\alpha\beta}$ is an arbitrary invertible constant matrix such that $g_{\beta\alpha}=g_{\alpha\beta}(-1)^{\varepsilon_{\alpha}\varepsilon_{\beta}}$. We recall that in the BFM only the gauge of the quantum field is fixed by $\chi_{\alpha}(A,\mathcal{B})$, while the symmetry for the background fields may be preserved. The standard choice of $\,\chi_{\alpha}(A,\mathcal{B})\,$ in the BFM is of the type

[TABLE]

which is a gauge fixing condition linear in the quantum fields.

In the framework of Faddeev-Popov quantization, the gauge symmetry of the initial action is replaced by the global supersymmetry (BRST symmetry) of the Faddeev-Popov action (9), defined by the transformation [27, 28]

[TABLE]

where $\lambda$ is a constant anticommuting parameter. The generator reads

[TABLE]

where

[TABLE]

Regarding the Grassmann parity, one has $\varepsilon(R^{A}_{\text{BRST}}(\phi,\mathcal{B}))=\varepsilon_{A}+1$. The BRST transformations are applied only on quantum fields, thus, $\delta_{\text{BRST}}\,\mathcal{B}^{i}=0$. Moreover, it is possible to show that the BRST operator is nilpotent, i.e.,

[TABLE]

Let us note that in the existing literature there is a formulation of the BFM in terms of extended BRST differentials [17, 18, 19, 20], that coincides with $\hat{R}_{\text{BRST}}(\phi,\mathcal{B})$ in the sector of fields $\phi$. At the same time, the BRST variation of the background fields is equal to new ghost fields, so that in the extended field space the usual BRST algebra takes place. In contrast to this approach, in the next Section we propose a new superalgebra, underlying the BFM which enables us to consider non-linear gauges and non-linear gauge generators.

For the subsequent discussion, it is useful to introduce the fermion gauge-fixing functional

[TABLE]

The Faddeev-Popov action can then be cast in the form

[TABLE]

The BRST symmetry of $S_{\text{FP}}(\phi,\mathcal{B})$ can be easily verified in this representation by applying the nilpotency property of the operator $\hat{R}_{\text{BRST}}$.

Let us point out that in order to achieve economic notations, in Eq. (17) and hereafter we let $\,\chi_{\alpha}\,$ to depend also on the auxiliary field $B^{\alpha}$. This is a practically useful way of taking into account the possibility of non-singular gauge conditions (see Eq. (11)). Nonetheless, further discussion and consequent results do not require any kind of a priori specific dependence of the gauge-fixing functional on $A^{i},B^{\alpha}$ and $\mathcal{B}^{i}$.

3 BFM compatible gauge functionals

In this section we propose a new point of view and a generalization of the standard BFM for gauge theories, that is based on using a superalgebra of generators of all the symmetries underlying the method and works for a wide class of gauge fixing conditions.

Apart from the global supersymmetry, a consistent formulation of the BFM requires that the Faddeev-Popov action be invariant under background field transformations. The former symmetry is ensured in the representation (18) of the Faddeev-Popov action, for any choice of gauge-fixing functional $\Psi$. Therefore, it is possible to extend considerations to a more general case in which $\Psi(\phi,\mathcal{B})=\bar{C}^{\alpha}\chi_{\alpha}(\phi,\mathcal{B})$, where the gauge-fixing functions $\chi_{\alpha}(\phi,\mathcal{B})$ depend on all the fields under consideration and satisfy the conditions $\varepsilon(\chi_{\alpha})=\varepsilon_{\alpha}$ and ${\rm gh}(\chi_{\alpha})=0$. On the other hand, the presence of the background field symmetry is not immediate — especially in the case of non-linear gauges — as the gauge-fixing functionals depend on the background fields. In what follows we derive necessary conditions that the fermion gauge-fixing functional should satisfy to achieve the consistent application of the BFM. As it has been already mentioned above, during this process we shall also find restrictions on the form of the generators $R^{i}_{\alpha}$ of gauge symmetry.

Let us extend the transformation rule (6) to the whole set of quantum fields, as

[TABLE]

Following the procedure used for the BRST symmetry, one can define the operator of background field transformations,

[TABLE]

The gauge invariance of the initial classical action implies that $\hat{{R}}_{\omega}(\phi,\mathcal{B})\,S_{0}(A+\mathcal{B})=0$. Furthermore, it is not difficult to verify that the background gauge operator commutes with the generator of BRST transformations, i.e.,

[TABLE]

Combining this result with the representation (18) of the Faddeev-Popov action, we get

[TABLE]

In other words, the Faddeev-Popov action is invariant under background field transformations if and only if the fermion gauge-fixing functional is a scalar with respect to this transformation.

The condition (22) constrains the possible forms of the (extended) gauge-fixing function $\chi_{\alpha}(\phi,\mathcal{B})$, as the relation

[TABLE]

fixes the transformation law for $\chi_{\alpha}(\phi,\mathcal{B})$,

[TABLE]

Therefore, in order to have the invariance of the Faddeev-Popov action under background field transformations it is necessary that the gauge function $\chi_{\alpha}$ transforms as a tensor with respect to the gauge group. This requirement can be fulfilled provided that $\chi_{\alpha}(\phi,\mathcal{B})$ is constructed only by using tensor quantities. Thus, Eq. (24) may impose a restriction on the form of gauge-fixing functions which are non-linear on the fields $A^{i}$. Let us anticipate that an example of this kind will be presented in Sec. 4.

In order to complete the geometric point of view on the BFM for gauge theories, let us introduce the generators $\hat{\mathcal{R}}_{\alpha}=\hat{\mathcal{R}}_{\alpha}(\phi,\mathcal{B})$ of the background field transformation, defined by the rule

[TABLE]

It is possible to verify that the generators of BRST and background field transformation satisfy the relations

[TABLE]

that define a superalgebra of the symmetries underlying the BFM. It is important to note that the last relation in (26) reproduces the gauge algebra (4), when restricted to the sector of fields $\mathcal{B}^{i}$, and provides its generalization to the whole set of fields $\phi^{A}$.

At this point we can conclude that

[TABLE]

represent necessary conditions for the consistent application of the BFM. The first relation is associated to the validity of the Ward identity (Slavnov-Taylor identities in the case of Yang-Mills fields) and the gauge independence of the vacuum functional 222 Let us mention that the gauge independence of the vacuum functional is needed for the gauge independent $S$-matrix and hence is a very important element for the consistent quantum formulation of a gauge theory [2, 45]., while the second relation is called to provide the invariance of the effective action in the BFM with respect to deformed (in the general case) background field transformations. In what follows we shall consider these statements explicitly. To this end, it is convenient to introduce the extended action

[TABLE]

where $\phi^{\ast}=\{\phi^{\ast}_{A}\}$ denote the set of sources (antifields) to the BRST transformations, with the parities $\varepsilon(\phi^{\ast}_{A})=\varepsilon_{A}+1$. The corresponding (extended) generating functional of Green functions reads

[TABLE]

where \mathcal{J}_{A}=\big{(}J_{i},\,\,J^{(B)}_{\alpha},\,\,\bar{J}_{\alpha},\,\,J_{\alpha}\big{)} (with the parities $\varepsilon(\mathcal{J}_{A})=\varepsilon_{A}$) are the external sources for the fields $\phi^{A}$. The BRST symmetry, together with the requirement that the generators $R^{i}_{\alpha}$ of gauge transformation satisfy

[TABLE]

implies the Ward identity

[TABLE]

The relation (30) plays an important role in the derivation of the Ward identity insomuch as it ensures the triviality of the Berezenian related to the change of integration variables in the form of BRST transformations. In Yang-Mills theories, for instance, the relations (30) are satisfied due to antisymmetry properties of the structure constants.

The generating functional $W(\mathcal{J},\mathcal{B},\phi^{\ast})$ of connected Green functions is defined in a usual way,

[TABLE]

and the identity (31) can be cast into the form

[TABLE]

In order to construct the effective action (generating functional of vertex functions), let us introduce the set of mean fields $\Phi=\{\Phi^{A}\}$ with

[TABLE]

respectively for the fields $A^{i}$, $B^{\alpha}$, $C^{\alpha}$ and $\bar{C}^{\alpha}$, such that

[TABLE]

The (extended) effective action is defined as

[TABLE]

and it satisfies the Ward identity

[TABLE]

with the anti-bracket of two functionals defined by the rule [2, 3]

[TABLE]

Now, in order to explore the background field symmetry it is useful to switch off the antifields and deal with the traditional generating functions, e.g.,

[TABLE]

Then, let us perform the change of the functional integration variables $\phi^{A}\mapsto\varphi^{A}(\phi)=\phi^{A}+\delta_{\omega}^{(q)}\phi^{A}$ in (29). If the Berezenian associated to this change of variables is $\text{Ber}(\varphi)=1$, it follows

[TABLE]

It is clear that if the generators $R^{i}_{\alpha}$ of gauge transformations are linear, then $\text{Ber}(\varphi)=1$ trivially when using the dimensional regularisation, as the change of variables is linear. On the other hand, in the more general scenario with non-linear generators we meet a situation similar to that of BRST transformations, where it was necessary to impose the condition (30) to ensure the triviality of the Berezenian. At the first sight it seems that the background field symmetry would introduce further requirements on the generators of gauge transformations. Nonetheless, when computing $\text{Ber}(\varphi)$ one finds out that the contribution from the field $B^{\alpha}$ compensates the one from $C^{\alpha}$ (or $\bar{C}^{\alpha}$, which transforms under the same rule), leading to

[TABLE]

Therefore, the condition (30) also ensures the triviality of the Berezenian for the change of variables $\phi^{A}\mapsto\phi^{A}+\delta_{\omega}^{(q)}\phi^{A}$. This remarkable result means that, as it will be clear from what follows, for the gauge theories considered here the consistent use of the BFM does not impose an additional requirement on the generators besides those that are already needed in the framework of traditional approach (i.e., without using the BFM).

After this important digression on the Berezenian, let us rewrite Eq. (40) in the form

[TABLE]

where we introduced the notation

[TABLE]

If the condition $\hat{{R}}_{\omega}(\phi,\mathcal{B})\Psi(\phi,\mathcal{B})=0$ (see Eq. (22)) for the invariance of the Faddeev-Popov action with respect to the background field transformations is satisfied, Eq. (42) can be written in the closed form

[TABLE]

or, in terms of the generating functional of connected Green functions,

[TABLE]

Here $1$ is the unite vector in the space of the fields.

Finally, for the effective action, the invariance of the Faddeev-Popov action with respect to background field transformations implies that

[TABLE]

where $\hat{{R}}_{\omega}(\Phi,\mathcal{B})$ is defined by the substitution $\phi^{A}\mapsto\Phi^{A}$ in Eq. (20) with the (deformed) transformations of the mean fields (cf. (6) and (19))

[TABLE]

In the sector of fields $\mathfrak{A}^{i}$, the generator is given by

[TABLE]

where $\Gamma^{\prime\prime-1}$ is the inverse matrix of second derivatives of the effective action,

[TABLE]

Let us stress that the relation (46) follows from (44), which only holds if the condition $\hat{{R}}_{\omega}(\phi,\mathcal{B})\,\Psi(\phi,\mathcal{B})=0$ is satisfied. The property (46) is crucial to the BFM inasmuch as, when the mean fields $\Phi$ are switched off, it ensures the gauge invariance of the functional $\Gamma(\mathcal{B})=\Gamma(\Phi,\mathcal{B})|_{\Phi^{A}=0}$, namely,

[TABLE]

It is worth mentioning that the previous expressions can be simplified if the generators $R^{i}_{\alpha}$ of gauge transformations are linear in fields $A^{i}$. In this case one can define the background field transformation of all the external sources as

[TABLE]

Then, Eqs. (44) and (45) boil down to

[TABLE]

and

[TABLE]

In other words, the invariance of the Faddeev-Popov action with respect to the background field transformations implies the invariance of the generating functionals of Green functions with respect to the joint transformation $\delta_{\omega}^{(q)}\mathcal{J}_{i}$ of the external sources and $\delta_{\omega}^{(c)}\mathcal{B}$ of the background field. Moreover, in what concerns the effective action, under this circumstance the transformation (47) of the mean fields $\mathfrak{A}^{i}$ are not deformed, i.e.,

[TABLE]

For the sake of completeness, let us compare the generating functionals in the background field formalism and in the traditional one — and, ultimately, their relations with $\Gamma(\mathcal{B})$. Consider the generating functional of Green functions which corresponds to the standard quantum field theory approach, but in a very special gauge fixing,

[TABLE]

In the last expression all the dependence of the quantity $Z_{2}(\mathcal{J})$ on the external field is only through the gauge-fixing functional. Thus, this functional depends the external field $\mathcal{B}^{i}$, but since this dependence is not of the BFM type, $Z_{2}(\mathcal{J})$ is nothing else but the conventional generating functional of Green functions of the theory, defined by $S_{0}$ in a specific $\mathcal{B}^{i}$-dependent gauge. One of the consequences is that any kind of physical results does not depend on $\mathcal{B}^{i}$. Furthermore, in Eq. (55) $\hat{R}_{\text{BRST}}(\phi)$ is defined by setting $\mathcal{B}^{i}=0$ in Eq. (15). The arguments of $\Psi$ are written explicitly, showing that we assume that $A^{i}$ only occurs in a specific combination with $\mathcal{B}^{i}$. We stress that, being formulated in the traditional way (i.e., not in the BFM), $\,Z_{2}(\mathcal{J})\,$ per se does not impose any constraint on the linearity of the gauge-fixing fermion $\Psi$ with respect to the quantum field $A^{i}$.

Making some change of variables in the functional integral, it is easy to verify that (39) is related to (55) in the following way:

[TABLE]

Accordingly, for the generating functional of connected Green functions one has

[TABLE]

where $Z_{2}(\mathcal{J})=e^{iW_{2}(\mathcal{J})}$. Recall that, according to (35),

[TABLE]

Similarly,

[TABLE]

Following the same line, let us define the effective action associated to $Z_{2}(\mathcal{J})$, as

[TABLE]

A moment’s reflection shows that

[TABLE]

In other words, the effective action $\Gamma(\Phi,\mathcal{B})$ in the background field formalism is equal to the initial effective action in a particular gauge with mean field $\,\mathfrak{A}_{2}^{i}=\mathfrak{A}^{i}+\mathcal{B}^{i}$ — or, switching off the mean fields, $\Gamma(\mathcal{B})=\Gamma_{2}(\mathfrak{A}_{2})|_{\mathfrak{A}_{2}^{i}=\mathcal{B}^{i}}$. We point out that the aforementioned particularity of the gauge is not associated to its linearity with respect to the quantum fields, but to its dependence on $\mathcal{B}$ (see Eq. (55)).

4 Two particular cases

In this section we present the applications of the formalism described above, to the Yang-Mills and quantum gravity theories.

4.1 Yang-Mills theory

As an example of the results presented in the previous section, let us consider the case of the pure Yang-Mills theory, defined by the action

[TABLE]

where $\,F^{a}_{\mu\nu}(A)=\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu}+f^{abc}A^{b}_{\mu}A^{c}_{\nu}\,$ is the field strength for the non-Abelian vector field $A_{\mu}$, taking values in the adjoint representation of a compact semi-simple Lie group. Being a particular case of the more general theory described above, it is instructive to present the correspondence with the notations used in Sec. 3, namely

[TABLE]

Here the structure coefficients $f^{abc}$ of the gauge group are constant. The action (62) is invariant under the gauge transformations defined by the generator $D^{ab}_{\mu}(A)$ with an arbitrary gauge function $\omega^{a}$ with $\varepsilon(\omega^{a})=0$. In the Faddeev-Popov quantization, the Grassmann parity of the fields $\,\phi^{A}=(A^{a}_{\mu},B^{a},C^{a},\bar{C}^{a})$ is, respectively, $\varepsilon_{A}=(0,0,1,1)$.

The background field formalism for Yang-Mills theory comprises the definition of the background field transformation (see Eqs. (6) and (19))

[TABLE]

Note that, in agreement to (6), the generator of the transformation in the sector of fields $A^{a}_{\mu}$ reads

[TABLE]

and thus all the quantum fields transform according the same rule. Also, the condition (22) for the background field invariance of the Faddeev-Popov action reads

[TABLE]

This requirement can be fulfilled provided that $\,\chi^{a}(A,B,\mathcal{B})\,$ is constructed only by using vectors (in the group index) such as $B^{a}$, $A^{a}_{\mu}$, $F^{a}_{\mu\nu}(\mathcal{B})$, $D^{ab}_{\mu}(\mathcal{B})A^{b}_{\nu}$, $D^{ab}_{\mu}(\mathcal{B})D^{bc}_{\nu}(\mathcal{B})A^{c}_{\alpha}$ and so on. The most simple gauge-fixing function compatible with this condition is

[TABLE]

that turns out to be the most popular gauge-fixing function for Yang-Mills theory. In principle, nonetheless, non-linear gauges which satisfy (66) can also be used. One simple example is

[TABLE]

which is quadratic on the quantum field. Note, however, that the gauge resultant from the substitution of $F^{a}_{\mu\nu}(\mathcal{B})$ by $F^{a}_{\mu\nu}(A)$ in (68) is not admissible, as it violates the transformation law (66). We also point out that the specific dependence of the gauge condition on the auxiliary field $B^{a}$ (and also on the ghost fields, in the generalisation proposed in Sec. 3) is not critical, as it already satisfies the vector transformation law.

We conclude that, in principle, it is possible to use non-linear gauge fixing conditions for the quantum fields in Yang-Mills theories formulated in the BFM, provided that the relation (66) is satisfied. Such procedure may introduce parameters with negative mass dimension, which could affect renormalization; however, a general analysis on this issue is beyond the scope of the present work.

4.2 Quantum gravity

As a second example, consider the case of quantum gravity theories, defined by the action $S_{0}(g)$ of a Riemann metric $g=\{g_{\mu\nu}(x)\}$ with $\varepsilon(g)=0$, and which is invariant under general coordinate transformations. The generator of such transformation is linear and reads

[TABLE]

Therefore, for an arbitrary gauge function $\omega^{\alpha}$ with $\varepsilon(\omega^{\alpha})=0$ one has $\delta_{\omega}g_{\mu\nu}=R_{\mu\nu\sigma}(g)\omega^{\sigma}$, or, writing all the arguments explicitly,

[TABLE]

In this case, the structure functions are given by

[TABLE]

which satisfy $F^{\alpha}_{\beta\gamma}(x,y,z)=-F^{\alpha}_{\gamma\beta}(x,z,y)$, as usual.

In terms of the notation used in Sec. 3 one has the correspondence

[TABLE]

According to (6) and (19), in the BFM one defines the background field transformation

[TABLE]

where $V^{\alpha}$ denotes any of the vector fields $C^{\alpha}$, $\bar{C}^{\alpha}$ and $B^{\alpha}$. Thence, the condition (22) for the background field invariance of the Faddeev-Popov action now reads $\delta_{\omega}\chi_{\alpha}=-\omega^{\sigma}\partial_{\sigma}\chi_{\alpha}-\chi_{\sigma}\partial_{\alpha}\omega^{\sigma}-\chi_{\alpha}\partial_{\sigma}\omega^{\sigma}$, i.e., the gauge-fixing function must transform as a vector density. Writing the density part explicitly, as it is standard for gravity theories, in the BFM one has $\chi_{\alpha}(\phi,\bar{g})=\sqrt{-\,\mbox{det}\,\bar{g}}\,\tilde{\chi}_{\alpha}(\phi,\bar{g})$, with $\tilde{\chi}_{\alpha}$ transforming as a vector field,

[TABLE]

Differently from the case of Yang-Mills theory (see (4.1)), in gravity theories all the fields under consideration transform as tensor ones with respect to the group algebra. Therefore, the condition (76) is automatically satisfied for any gauge-fixing condition, including non-linear ones. Furthermore, since in gravity the coupling constant has negative dimension, it is natural that non-linear gauge and field parametrizations have fruitful applications in these models (see, e.g., [15]). Indeed, the gauge that is linear in one parametrization becomes non-linear in another one, as one can observe, e.g., in the recent paper [46].

5 Conclusions

We introduced a new framework for formulating the BFM in a general type of Yang-Mills theories. The new approach is based on identifying the superalgebra (26) composed by the generators of BRST and background field transformations. It is shown that the condition for consistent use of the BFM is that the gauge-fixing fermion functional $\Psi$ must be a scalar with respect to the background field transformation. This condition produces a restriction on the admissible forms of the non-linear gauge-fixing condition used within the BFM. At the same time, these restrictions open the way for using a wide class of gauge-fixing conditions, including the gauges which are non-linear with respect to quantum fields. The considerations presented above can be directly extended to the case when the gauge-fixing function $\chi_{\alpha}$ depends on all quantum fields under consideration, including the ghost fields, as discussed in Sec. 3. As an application of general formulation, we discussed the non-linear gauge conditions for the Yang-Mills and quantum gravity theories in the background field formalism.

Realistic models of fundamental interactions [47] use the mechanism of spontaneous symmetry breaking to generate the masses of the physical particles [48]. Up to now there is no consistent formulation of gauge theories with spontaneous symmetry breaking within the BFM. All quantum studies of a such kind of gauge theories are restricted to the one-loop approximation (see, e.g., early papers [49, 50] and recent ones [51, 52]). For example, an attempt to consider the spontaneous symmetry breaking in the presence of external gravity may lead to serious complications [53]. It is clear that the consistent analysis of invariant renormalization in this case is a very challenging problem.

Acknowledgements

B.L.G. and P.M.L. are grateful to the Department of Physics of the Federal University of Juiz de Fora (MG, Brazil) for warm hospitality during their long-term visits. The work of P.M.L. is supported partially by the Ministry of Science and Higher Education of the Russian Federation, grant 3.1386.2017 and by the RFBR grant 18-02-00153. This work of I.L.Sh. was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq under the grant 303635/2018-5 and Fundação de Amparo à Pesquisa de Minas Gerais - FAPEMIG under the project APQ-01205-16.