# Perturbative quantization of Yang–Mills theory with classical double as gauge algebra

## Abstract

Perturbative quantization of Yang–Mills theory with a gauge algebra given by the classical double of a semisimple Lie algebra is considered. The classical double of a real Lie algebra is a nonsemisimple real Lie algebra that admits a nonpositive definite invariant metric, the indefiniteness of the metric suggesting an apparent lack of unitarity. It is shown that the theory is UV divergent at one loop and that there are no radiative corrections at higher loops. One-loop UV divergences are removed through renormalization of the coupling constant, thus introducing a renormalization scale. The terms in the classical action that would spoil unitarity are proved to be cohomologically trivial with respect to the Slavnov–Taylor operator that controls gauge invariance for the quantum theory. Hence they do not contribute gauge invariant radiative corrections to the quantum effective action and the theory is unitary.

## 1 Introduction

Nonreductive metric Lie algebras are Lie algebras that (i) cannot be written as a direct product of semisimple and Abelian Lie algebras but (ii) admit a metric, where by a metric is meant a nondegenerate symmetric bilinear form that is invariant under the adjoint action. Here we will be concerned with perturbative quantization of Yang–Mills theory for a particular class of such algebras, known as classical doubles. These algebras describe the gauge symmetries in a variety of problems, including three-dimensional gravity [1, 2], asymptotically flat solutions to the Einstein equations in three and four dimensions [3, 4, 5], string actions in doubled space [6], or \(1/N_{\mathrm{color}}\) expansions for baryons in QCD [7].

Many WZW models [8, 9, 10, 11, 12, 13, 14] based on nonreductive Lie algebras, though of a different type, have a simpler structure than the WZW models based on semisimple Lie algebras. This suggests considering Yang–Mills theories in four dimensions and investigate if the simplifications introduced in two dimensions by going nonreductive carry through to four dimensions. The problem was addressed in Ref. [15] for a class of nonreductive algebras called double extensions. One-loop radiative corrections for certain models were computed and it was argued that, if renormalizability is assumed, there would not be higher-loop corrections. An apparent lack of unitarity was found.

The classical double, we denote it as \(\mathfrak {g}_\ltimes \), of any real Lie algebra \(\mathfrak {g}\) is a Lie algebra of dimension twice the dimension of \(\mathfrak {g}\) that admits a metric. This metric determines a Yang–Mills Lagrangian for the field that results from gauging the algebra. For \(\mathfrak {g}\) simple, the self-antiself dual instantons of the \(\mathfrak {g}_\ltimes \) Yang–Mills theory in four-dimensional Euclidean space have been studied elsewhere [16]. Every \(\mathfrak {g}_\ltimes \) instanton has an embedded \(\mathfrak {g}\) instanton with the same instanton number and twice the number of collective coordinates. This doubling of degrees of freedom and the simpler structure of classical doubles as compared to double extensions suggest considering perturbative quantization of Yang–Mills theory with gauge algebra \(\mathfrak {g}_\ltimes \).

The theory is shown to have UV divergences at one loop but no radiative corrections at higher loops. As in the classical case, the physical degrees of freedom of the quantum theory are doubled with respect to the \(\mathfrak {g}\) theory. In particular, the first and only coefficient (since there are no radiative corrections beyond one loop) of the beta function is twice that of the \(\mathfrak {g}\) theory. To disentangle truly gauge invariant one-loop corrections from those due to gauge fixing, the Slavnov–Taylor operator for the quantum theory is used. The term in the classical action that would spoil unitarity is cohomologically trivial with respect to the Slavnov–Taylor operator, so in the quantum effective action it can be put to zero through a field redefinition and poses no problem for unitarity.

The manuscript is organized as follows. Section 2 contains a brief reminder of classical doubles \(\mathfrak {g}_\ltimes \) and their Lie groups \(\mathrm{G}_\ltimes \). Classical Yang–Mills theory with gauge group \(\mathrm{G}_\ltimes \) is formulated in Sect. 3, and the path integral generating the theory’s Green functions is derived. With the mind set in avoiding miscounting the theory’s degrees of freedom, special consideration is given to gauge fixing, and three derivations for the gauge fixing terms in Landau gauge are presented. Section 3 also discusses the emergence of the classical theory as a limit of Yang–Mills theory with gauge algebra the direct product \(\mathfrak {g}\times \mathfrak {g}\). Section 4 contains our perturbative analysis, with the calculation of one-loop 1PI radiative corrections and the proof that there are no higher-loop corrections. The one-loop divergences are removed in Sect. 5 by adding a gauge invariant counterterm consistent with unitarity, so the one-loop contribution to the quantum effective action is positive definite. We conclude in Sect. 6.

## 2 Semidirect products of Lie algebras and their groups

*n*with basis \(\{T_a\}\) and commutation relations \([T_a,T_b]=f_{ab}{}^c\,T_c\). As a vector space, \(\mathfrak {g}\) has a dual vector space \(\mathfrak {g}^*\). Take on \(\mathfrak {g}^*\) the canonical dual basis \(\{Z^a\}\), given by \(Z^a(T_b)={\delta }^a{}_b\). The classical double of \(\mathfrak {g}\), denoted by \(\mathfrak {g}_\ltimes \), is the Lie algebra of dimension 2

*n*with basis \(\{T_a,Z^b\}\) and commutators

Denote by \(\mathrm{G}_{\ltimes },\,\mathrm{G}\), and \(\mathrm{N}\) the Lie groups obtained by exponentiation of the algebras \(\mathfrak {g}_{\ltimes },\,\mathfrak {g}\), and \(\mathfrak {g}^*\). Every element *h* of \(\mathrm{G}\) and every *n* of \(\mathrm{N}\) can be uniquely written as \(h=e^T\) and \(n=e^Z\), for some *T* in \(\mathfrak {g}\) and some *Z* in \(\mathfrak {g}^*\). In turn, every element *g* of \(\mathrm{G}_{\ltimes }\) can be uniquely written as \(g=hn\). The product \(g_3=g_1g_2\) of two elements \(g_1=h_1n_1\) and \(g_2=h_2n_2\) of \(\mathrm{G}_{\ltimes }\) is given by \(g_3=h_3n_3\), with \(h_3=h_1h_2\) and \(n_3=(h_2^{-1}n_1h_2)n_2\). It is clear that \(g_3\) is in \(\mathrm{G}_{\ltimes }\) since the Campbell–Hausdorff formula and the commutators (2.1) imply that \(h_3\) is in \(\mathrm{G}\) and \(n_3\) is in \(\mathrm{N}\). It is very easy to see that the group \(\mathrm{G}_{\ltimes }\) is the semidirect product of \(\mathrm{G}\) with the normal Abelian subgroup \(\mathrm{N}\). Since \(\mathrm{N}\) is isomorphic to \(\mathbf {R}^n\), the group \(\mathrm{G}_{\ltimes }\) is noncompact.

*t*is an arbitrary real parameter. It is trivial to show that they satisfy the Jacobi identity for all

*t*. Hence they define a Lie algebra, call it \(\mathfrak {g}_{\ltimes }^{\,t}\), that reduces to \(\mathfrak {g}_{\ltimes }\) in the limit \(t\rightarrow 0\). Furthermore, \(\mathfrak {g}_{\ltimes }^{\,t}\) admits the invariant metric

## 3 The gauge fixed classical action

*g*an arbitrary group element in \(\mathrm{G}_{\ltimes }\),

*g*is a coupling constant, is gauge invariant. We are interested in perturbatively quantizing Yang–Mills theory with Lagrangian \(\mathcal{L}_{ ym }\). To obtain a path integral that generates the theory’s Green functions, we next fix the gauge. We do this in three different ways.

*Gauge fixing I*. Introduce a ghost field \(\varvec{c}\), an antighost field \(\varvec{\bar{c}}\) and a Lagrange multiplier field \(\varvec{b}\),

*s*by its action on the fields,

*s*commutes with \({\partial }_{\mu },\,\varvec{A}_{\mu }\) and \(\varvec{b}\), and anticommutes with \(\varvec{c}\) and \(\varvec{\bar{c}}\). The BRS transformations for the t and z components of the fields can be obtained either from the definition of

*s*in Eq. (3.8) and the commutations relations (2.5), or directly from Eqs. (3.3) and (3.4). They read

*Gauge fixing II*. Equivalently, the path integral (3.13) can be derived as follows. In the naive path integral

- (i)Average over \(\varvec{f}\) with Gaussian type weight. That is, introduce in the measure$$\begin{aligned} \quad \int [\mathrm{d}\varvec{f}] \exp \Big [ - \frac{1}{2{\alpha }g^2} \int \!\mathrm{d}^4 x\; {\Omega }(\varvec{f},\varvec{f}) \,\Big ]\,. \end{aligned}$$
- (ii)Exponentiate \({\delta }\big ({\partial }\varvec{A}- \varvec{f}\big )\) by means of an auxiliary field \(\varvec{b}\),$$\begin{aligned} \quad {\delta }\big ({\partial }\varvec{A}- \varvec{f}\big ) = \int [\mathrm{d}\varvec{b}]~ \exp \Big [ \frac{i}{g^2} \!\int \! {\Omega }(\varvec{b}, {\partial }\varvec{A}-\varvec{f})\,\Big ] \,. \end{aligned}$$
- (iii)Write the determinantas a path integral over Grassmann fields \(c_1^a,\,c_2^a\), and \(\bar{c}_{1a},\,\bar{c}_{2a}\),$$\begin{aligned} \quad \Delta _{{\partial }\varvec{A}} \!=\! det ~{\delta }^{(4)}(x-y) \begin{pmatrix} {\partial }^{\,{\mu }} D_c{\,}^a_{\mu }(x) &{} 0 \\ f_{bc}{}^a\,{\partial }^{\,{\mu }} A_{ Z {\mu }}^b(x) &{} {\partial }^{\,{\mu }} D_c{\,}^a_{\mu }(x) \end{pmatrix} \end{aligned}$$$$\begin{aligned} \quad \Delta _{{\partial }\varvec{A}\!}= & {} \int [\mathrm{d}\bar{c}_1]\, [\mathrm{d}\bar{c}_2]\, [\mathrm{d}c_1]\, [\mathrm{d}c_2] ~ \exp \! \\&\times \Big [ - \>\frac{1}{g^2}\! \int \! \mathrm{d}^4 x \> \big (\, \bar{c}_{1a}\,{\partial }D c_1^a + \bar{c}_{2a} \,f^a{\!}_{cb}\,\\&\times {\partial }_{\mu }(A^{c{\mu }}_{ T } c^b_1) + \bar{c}_{2a}\,{\partial }D c_2^a \big ) \Big ] . \end{aligned}$$

^{1}

*Gauge fixing III*. The observation at the end of Sect. 2 concerning deformations of \(\mathfrak {g}_{\ltimes }\) suggests that classical \(\mathrm{G}_{\ltimes }\) Yang–Mills theory can be regarded as a limit of \(\mathrm{G}\times \mathrm{G}\) Yang–Mills theory. To see this, consider two copies of a Yang–Mills theory, both with gauge group the semisimple Lie group \(\mathrm{G}\). Label the copies with the subscripts \(+\) and \(-\), so that \(A^a_{\pm \mu },\, F^a_{\pm \mu \nu }\), and \(g_\pm \) denote their gauge fields, field strengths and coupling constants. Fix in both copies a Lorenz gauge by introducing ghost, antighost, and auxiliary fields \(c^a_\pm ,\,\bar{c}^a_\pm \), and \(b^a_\pm \), and a BRS operators

*s*given by

*g*and a parameter

*t*as

## 4 Radiative corrections

To explicitly calculate radiative corrections, we take in this section \(\mathfrak {g}\) to be \(\mathfrak {su} (N) \). The group \(\mathrm{G}\) is then \( SU(N) \) and for \(\mathrm{G}_{\ltimes }\) we use the notation \( SU(N) _{\ltimes }\).

*h*of \( SU(N) \) is written as \(e^T\), the elements

*T*of \(\mathfrak {su}(N)\) in the defining (fundamental) representation are traceless antihermitean matrices. We normalize the structure constants \(f_{ab}{}^c\) of \(\mathfrak {su}(N)\) by requiring \(f_{ca}{\,}^df_{db}{\,}^c=\mathrm{N}{\delta }_{ab}\). This gives for the Killing form \(k_{ab}=\mathrm{N}{\delta }_{ab}\) and amounts to taking \(\mathrm{tr} [ T_{(R)a} T_{(R)b} ]=C_2 {\delta }_{ab}\) in a representation

*R*, with \(C_2=\mathrm{N}\) in the adjoint representation and \(C_2=-1/2\) in the defining representation. For \({\omega }_{ab}\) in Eq. (2.6), we take

*Feynman rules in Lorenz gauge*. Introduce external sources \(\varvec{K}^{{\mu }\!}\) and \(\varvec{H}\) for the nonlinear BRS transforms \(s \varvec{A}_{\mu }\) and \(s\varvec{c}\),

Consider for comparison conventional \( SU(N) \) Yang–Mills theory in Lorenz gauge. Its gauge fixed Lagrangian is recovered from \(\mathcal{L}_\ltimes \) by setting all the z components equal to zero. To avoid confusion, we reserve the subscripts t and z for the field components of the \( SU(N) _\ltimes \) theory, and use \(A^a_{\mu },\,b^a,\,\bar{c}^a,\,c^a\), and \(K^a_{\mu },\,H^a\) without subscripts for the fields and the nonlinear BRS sources of the \( SU(N) \) theory. The gauge field and ghost free propagators of the \( SU(N) \) theory are given by \(D^{ab}_{{\mu }{\nu }}(p)\) and \(\Delta ^{ab}(p)\) in Eqs. (4.4) and (4.5), which are equal to the tz and zt free propagators of the \( SU(N) _\ltimes \) theory. The Feynman rules for the vertices \(A^3,\,A^4,\,\bar{c}Ac,\,KAc\), and *Hcc* of the \( SU(N) \) theory are as in Figs. 2 and 3.

We now proceed to compute radiative corrections. To regulate whatever UV divergences may occur, we will use dimensional regularization with \(D=4-2\epsilon \), so from now on all diagrams and Green functions should be understood as dimensionally regularized. Since dimensional regularization manifestly preserves BRS invariance, the dimensionally regularized Green functions will solve the functional identities associated to BRS invariance.

*One-loop radiative corrections*. The only 1PI one-loop diagrams that occur in perturbation theory have all their external legs of type t. To prove this, note first that the vertices of the theory, see Figs. 2 and 3, have either no or one leg of type z. Assume now that there is a 1PI one-loop diagram with an external z leg, and call \(U_1\) to the vertex to which the leg is attached. All the other legs of \(U_1\) will be of type t. To close a loop, two of these t legs must be internal. Since there are no tt propagators, each internal t leg must propagate into type z. Each one of the resulting z legs will in turn be attached to a different vertex. Call these vertices \(U_2\) and \(U_3\). From \(U_2\) and \(U_3\) only t legs will come out. One may go on and introduce new vertices, but the loop will never close since there are no tt propagators to join two t legs. Hence 1PI one-loop diagrams have all their external legs of type t. The only nonzero 1PI Green functions at one loop are then \(\langle \Psi _{1 t }(p_1) \ldots \Psi _{n t }(p_n)\rangle _{ SU (N)_\ltimes }\), where \(\Psi _{i t }\) stands for any of the fields \(A^a_{ T \mu },\, \bar{c}^a_{ T },\, c^a_{ T }\) or the sources \(K^a_{ T \mu },\, H^c_{ T }\).

UV divergent 1PI Green functions in \( SU(N) _\ltimes \) theory and their counterterms

1PI UV divergent Green function | Contribution from \(\bar{\Gamma }_1^{{\epsilon }}\) | Contribution from \(\bar{\Gamma }_1^{ ct }\) |
---|---|---|

\(\langle A^a_{ T \mu }(-p) A^b_{ T \nu }(p) \rangle \) | \(\big ( \frac{13}{3} - {\alpha }\big ) \,C_{\epsilon }\,\Pi _{\mu \nu }^{ab}(p)\) | \( -\,\big (\,c_1 + 2c_2\big )\, \Pi ^{ab}_{{\mu }{\nu }}(p) \) |

\(\langle A^a_{ T \mu }(p) A^b_{ T \nu }(q) A^c_{ T \rho }(k) \rangle \) | \( \big ( \frac{17}{6}-\frac{3}{2}\,{\alpha }\big ) \, C_{\epsilon }\, V^{abc}_{\mu \nu \rho }(p,q,k) \) | \( -\,(c_1+3c_2)\,V^{abc}_{\mu \nu \rho }(p,q,k)\) |

\(\langle A^a_{ T \mu }(p) A^b_{ T \nu }(q) A^c_{ T \rho }(k) A^d_{ T \sigma }(r) \rangle \) | \(\big (\frac{4}{3}-2{\alpha }\big )\, C_{\epsilon }\,W^{abcd}_{\mu \nu \rho \sigma }\) | \( -\,\big (c_1 + 4c_2\big )\, W^{abcd}_{\mu \nu \rho \sigma }\) |

\(\langle \bar{c}^a_{ T }(-p) c^b_{ T }(p) \rangle \) | \(\frac{1}{2}\, (3-{\alpha })\,C_{\epsilon }{\delta }^{ab} p^2\) | \((c_2-c_3)\, {\delta }^{ab} p^2\) |

\(\langle \bar{c}^a_{ T }(p) A^b_{ T \mu }(q) c^c_{ T }(k) \rangle \) | \(i{\alpha }C_{\epsilon }\, f^{abc}\,p_{\mu }\) | \(ic_3 f^{abc}\,p_{\mu }\) |

\(\langle K^a_{ T {\mu }}(-p) \, c^b_{ T }(p) \rangle \) | \(\frac{1}{2}\,(3-{\alpha })\,C_{\epsilon }\, {\delta }^{ab}\,i p_\mu \) | \((c_2-c_3)\,{\delta }^{ab}\,i p_\mu \) |

\(\langle K^a_{ T {\mu }}(p) \, A^b_{ T {\nu }}(q)\, c^c_{ T }(k) \rangle \) | \(-{\alpha }C_{\epsilon }\,f^{abc} {\delta }_{{\mu }{\nu }} \) | \(-c_3 f^{abc} {\delta }_{{\mu }{\nu }} \) |

\(\langle H^a_{ T }(p) \, c^b_{ T }(q)\, c^c_{ T }(k) \rangle \) | \(-\,{\alpha }C_{\epsilon }\,f^{abc}\) | \(-\,c_3\,f^{abc} \) |

*Vanishing of 1PI radiative corrections beyond one loop*. Any 1PI *n*-loop diagram can be obtained by joining two external legs in a 1PI (*n*–1)-loop diagram. In our case, since 1PI one-loop diagrams have all their external legs of type t and there are no tt propagators, it is impossible to have two- and higher-loop 1PI diagrams.

We end this section by noting that, again because 1PI Green functions have all their external legs of type t and to these it is only possible to attach free tz propagators, the only on-shell Green functions that receive radiative corrections are those having all their external legs of type z.

## 5 The BRS identity, renormalization, and unitarity

*X*. Cohomologically trivial corrections are of the form \(\Delta X\), originate in gauge fixing and do not contribute to on-shell amplitudes.

^{2}The most general solution over this space has the form

*X*is any local integrated functional of mass dimension three and ghost number \(-1\). Note that the cohomologically nontrivial part of the solution (5.6) does not have a term

*Y*given by

*s*and the Slavnov–Taylor operator \(\Delta \) over the space of local integrated functionals of mass dimension four and ghost number zero. While \(S_{ tt }\) and \(S_{ tz }\) are both nontrivial with respect to

*s*, only \(S_{ tt }\) is nontrivial with respect to \(\Delta \).

Recall now that the classical action is the sum of the terms \(S_{ tz }\) and \(S_{ tz }\). The first one of them is positive definite and the second one is not. This would seem to point to a loss of unitarity. This, however, is only apparent since, being cohomologically trivial with respect to \(\Delta \), \(S_{ tz }\) does not carry gauge invariant radiative corrections in the quantum effective action.

*X*given by

*Multiplicative renormalization*. The subtraction performed by the counterterm \(\bar{\Gamma }_1^\mathrm{ct}\) in Eqs. (5.6) and (5.10) is equivalent to multiplicative renormalization. To see this, recall that in multiplicative renormalization, the fields and the coupling constant in the tree-level action \(\bar{\Gamma }_0\) in Eq. (5.4) are regarded as bare fields \(\{\varvec{A}_{0\mu }, \varvec{c}_0, \varvec{G}_{0\mu }, \varvec{H}_0\}\) and bare coupling constant \(g_0\). Renormalized quantities are then introduced through the equationsand

*X*is given by Eq. (5.10) and

*U*has the form

All in all, the only gauge invariant radiative corrections are those in \(c_1\), which account for a renormalization of the coupling constant. This introduces a renormalization scale in the quantum effective action and the quantum theory is asymptotically free.

## 6 Discussion

The pattern observed for the gauge invariant degrees of freedom in the quantum theory resembles very much that for the self-antiself dual instantons of the classical theory [16]. In the classical case, the number of collective coordinates of the \( G _\ltimes \) instantons is twice that of the embedded \( G \) instantons, yet \({\omega }_{ab}F^a_{ T \mu \nu }F^{b\mu \nu }_{ Z }\) does not contribute to the instanton number. Now the gauge invariant radiative corrections are doubled and \({\omega }_{ab}F^a_{ T \mu \nu }F^{b\mu \nu }_{ Z }\) is cohomologically trivial with respect to the Slavnov–Taylor operator.

Our discussion may have some implications for Yang–Mills theories with more general nonreductive real metric Lie algebras. There is a structure theorem [17] that states that all real metric Lie algebras are direct products of Abelian algebras, simple real Lie algebras, and double extensions \(\mathfrak {d}(\mathfrak {h},\mathfrak {g})\) of a real metric Lie algebra \(\mathfrak {h}\) by an algebra \(\mathfrak {g}\).^{3} The double extension \(\mathfrak {d}(\mathfrak {h},\mathfrak {g})\) is obtained [14, 17] by forming the classical double \(\mathfrak {g}_\ltimes \) and then by acting with \(\mathfrak {g}\) on \(\mathfrak {h}\) via antisymmetric derivations. Incidentally we mention that the classical double \(\mathfrak {g}_\ltimes \) can be viewed as the double extension of the trivial algebra by \(\mathfrak {g}\).

According to the theorem, since \(\mathfrak {h}\) must be metric, three possibilities must be considered for \(\mathfrak {h}\) in forming double extensions \(\mathfrak {d}(\mathfrak {h},\mathfrak {g})\). The first one is that \(\mathfrak {h}\) is a simple real Lie algebra. In this case [14], the algebra of antisymmetric derivations of \(\mathfrak {h}\) is \(\mathfrak {h}\) itself and the double extension \(\mathfrak {d}(\mathfrak {h},\mathfrak {h})\) is isomorphic to the direct product \(\mathfrak {h}\times \mathfrak {h}_\ltimes \). The resulting Yang–Mills theory then separates into two Yang–Mills theories, not interacting with each other, one with gauge group \( H \) and one with group \( H _\ltimes \). The second possibility is that \(\mathfrak {h}\) is Abelian, of dimension *m*. Being Abelian, any nondegenerate, symmetric bilinear form on \(\mathfrak {h}\) is a metric, and it can always be brought to a diagonal form with all the entries in the diagonal equal to \(+1\) and \(-1\). If the number of occurrences of \(+1\) is *p*, and the number of occurrences of \(-1\) is *q*, the algebra of antisymmetric derivations of \(\mathfrak {h}\) is any subalgebra of \(\mathfrak {s}\mathfrak {o}(p,q)\) [14]. Many of the nonsemisimple WZW models considered in the literature [8, 9, 10, 11, 12, 13] and their four-dimensional Yang–Mills analogs [15] fall into this class. In this instance unitarity remains an open problem. We think that a thorough analysis of the corresponding Slavnov–Taylor operator should shed some light on the problem. The third possibility for \(\mathfrak {h}\) is that it is a double extension, which takes us back to the starting point.

## Footnotes

- 1.A more precise notation for the t and z components of a \(\mathfrak {g}_\ltimes \)-valued field is \(\varvec{\Phi } = \Phi ^{T_a}T_a + \Phi ^{Z_a}Z_a\). Not to load the writing, we have used instead \(\Phi ^a_{ T \!}:=\Phi ^{T_a}\) and \(\Phi ^a_{ Z \!}:=\Phi ^{Z_a\!}\). Using \({\Omega }\) and \({\Omega }^{-1}\) to lower and raise indices, one hasHence \((\bar{c}_{1a} , \bar{c}_{2a})\) in Eq. (3.17) is nothing but \((\bar{c}_{T_a}, \bar{c}_{Z_a})\).$$\begin{aligned} \Phi _{T_a}&= {\omega }_{ab}\,( \Phi ^b_{ T } + \Phi ^b_{ Z }) \,,&\qquad \Phi _{Z_a}&= {\omega }_{ab}\,\Phi ^b_{ T } \,\\ \Phi ^a_{ T }&= {\omega }^{ab}\, \phi _{Z_b} \,,&\Phi ^a_{ Z }&= {\omega }^{ab}\,(\Phi _{T_b} - \Phi _{Z_b}) \,. \end{aligned}$$
- 2.
Both the BRS operator and the Slavnov–Taylor operator have mass dimension one and ghost number 1.

- 3.
The theorem goes further and specifies the nature of \(\mathfrak {g}\) in the double extension.

## Notes

### Acknowledgments

This work was partially funded by the Spanish Ministry of Economy and Competitiveness through Grant FPA2014-54154-P and by the European Union Cost Program through Grant MP 1405.

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