Composite operator and condensate in the $SU(N)$ Yang-Mills theory with $U(N-1)$ stability group

Recently, some reformulations of the Yang-Mills theory inspired by the Cho-Faddeev-Niemi decomposition have been developed in order to understand confinement from the viewpoint of the dual superconductivity. In this paper we focus on the reformulated $SU(N)$ Yang-Mills theory in the minimal option with $U(N-1)$ stability group. Despite existing numerical simulations on the lattice we perform the perturbative analysis to one-loop level as a first step towards the non-perturbative analytical treatment. First, we give the Feynman rules and calculate all renormalization factors to obtain the standard renormalization group functions to one-loop level in light of the renormalizability of this theory. Then we introduce a mixed gluon ghost composite operator of mass dimension two and show the BRST invariance and the multiplicative renormalizability. Armed with these results, we argue the existence of the mixed gluon-ghost condensate by means of the so-called local composite operator formalism, which leads to various interesting implications for confinement as shown in preceding works.


I. INTRODUCTION
The dual superconductivity picture [1][2][3][4] represents one of the most popular attempts to explain color confinement. In order for this mechanism to work, the existence of magnetic monopoles is crucial, raising the question of how to extract them from the underlying theory. Famous examples of such monopole configurations are the Dirac-monopole in the Abelian Maxwell theory [5], represented by a singular gauge field, or the 't Hooft-Polyakov monopole in non-Abelian gauge theories, which, however, relies on the presence of an adjoint scalar field [6,7]. Dealing with pure Yang-Mills theory, one has to find a way to define the monopole in the absence of any scalar field. For SU (2), even in this case, a possibility to obtain the monopoles is given by performing a gauge-covariant decomposition of the gauge field, the Cho-Duan-Ge-Faddeev-Niemi-Shabanov decomposition [8][9][10][11][12][13][14][15][16][17][18][19], which also has been extended to the general SU (N ) case by Cho [11,12,20,21] and Faddeev-Niemi [15][16][17]. It relies on the introduction of the so-called adjoint color-field n(x), which is used to define the decomposition of the gauge field A µ = V µ +X µ into the residual (or restricted) field V µ and the coset (or remaining) field X µ .
Recently, this idea has been readdressed under a different viewpoint by regarding this decomposition merely as a non-linear change of variables. The resulting reformulation of the Yang-Mills theory has been first performed in the SU (2) case [22] and later extended to the general SU (N ) case [23]. It turned out, however, that for N ≥ 3 the decomposition is no longer unique, as the gauge field is decomposed into the part lying in the stability group H and its remainder SU (N )/H. But already in the N = 3 case, for example, there are two options for the stability group, (I.1) The first case is referred to as maximal option and involves the definition of two color-fields. In the second case, the minimal option, we only need one color-field to define the decomposition. Only for SU (2) the choice is unique and the maximal and minimal options are equivalent.
In this paper we will consider the decomposition in the minimal option, that is decomposing G = SU (N ) into the stability group H = U (N − 1) and the coset G/H = SU (N )/U (N − 1). However, counting the degrees of freedom in the color-field extended Yang-Mills theory one finds that they exceed those of the original Yang-Mills theory. This is taken care of by means of the so-called reduction condition, Solving this differential equation gives n as a functional of A µ , eliminating the superfluous degrees of freedom. Even though this procedure is reminiscent of a "gauge fixing" from the extended Yang-Mills theory to a theory equipollent to the original SU (N ) Yang-Mills theory, it should be remarked that the idea behind it is conceptually different from the usual gauge fixing.
Another key aspect within this reformulation is that one could introduce a gauge invariant mass term for the homogeneously transforming coset field [22], This leads to the idea of a non-vanishing covariant coset gluon condensate that could lead to a dynamically generated mass term coming from the quartic self-interaction term. In fact, at least in the SU (2) and SU (3) case the "abelian" dominance implied by such a condensate has already been observed in lattice simulations in terms of an exponential falloff of the covariant coset field twopoint function in the infrared, suggesting the dynamical generation of the gluon mass [24,25]. Moreover, in preceding works the assumption of a non-zero condensate lead to many more implications such as removal of the Nielsen-Olesen instability in the Savvidy vacuum [26,27], the Faddeev-Niemi model (describing glueballs as knotsolitons) as a low-energy effective theory of this reformulated Yang-Mills theory [28,29], or quark confinement at low temperatures [30]. For a more detailed introduction of the reformulated Yang-Mills theory and its main features please see the review [31].
The main goal of this paper is to complement the lattice simulations by an analytical study of the coset field mass generation. In order to do so we consider a slightly modified dimension-2 operator, as suggested in [32]. Here, ξ is the "gauge fixing" parameter corresponding to the reduction condition and the index a runs over the coset space SU (N )/U (N − 1). The pure gluon condensate is recovered for ξ = 0. It should be remarked that this operator has already been analyzed in different gauges, such as for a non-decomposed gauge group (and thus with the index running over the whole gauge group) and in the usual covariant gauge fixing [33], within the Curci-Ferrari gauge [34,35], or within the maximal Abelian gauge (MAG) [4,36,37]. In order to construct a well-defined effective action, a new method called local composite operator (LCO) formalism has been developed [38,39] and used to show not only the existence of the "full" gluon condensate A A µ A µA in linear covariant gauges [40,41] but also the existence of the gluon-ghost condensate in both aforementioned gauges [42][43][44]. We would like to readdress this issue within our novel decomposition.
The paper is organized as follows. In the second section we set up the Lagrangian, explaining the decomposition of the gauge field and the incorporation of the "gauge fixing" related to the reduction condition. We then briefly discuss the one-loop renormalization and calculate all renormalization group (RG) functions. The third section is dedicated to the proper introduction of the composite operator O = 1 2 X a µ X µa −iξC aC a . We prove its (on-shell) BRST invariance and the multiplicative renormalizability to one-loop level. In the fourth section we use the LCO formalism in order to deal with divergences quadratic in the source, coming from the composite operator source term JO. In the last section, the one-loop effective potential for the composite operator is calculated and the existence of the condensate is discussed.

II. LAGRANGIAN IN THE MINIMAL OPTION
Before discussing the decomposition we restrict ourselves to the case of a space-time independent colorfield, thus discarding the monopole degrees of freedom. The analytical treatment of the dynamical color-field is a highly complicated task and plays only a minor role when investigating the coset field condensate. Therefore, considering a fixed color-field is sufficient for our purposes. In particular, we choose the color-field to be the last Cartan generator, where the generators are normalized as 2Tr(T A T B ) = δ AB . Even after the choice (II.1), the theory still has the local (and global) U (N − 1) symmetry, because the constant form of the color-field is maintained under the U (N − 1) gauge transformations, as it is supposed to transform in the adjoint way, The gauge field is then decomposed into A µ = V µ + X µ , where the residual field V µ takes value in U (N − 1) and the covariant coset field X µ takes value in the coset space SU (N )/U (N − 1). It is convenient to . In a suitable basis we write where the generators obey the following commutator relations, and the last relation is further decomposed according to In the SU (3) case for example the different indices take the values a ∈ {4, 5, 6, 7}, j ∈ {1, 2, 3} and γ = 8. The SU (2) case is special in the sense that the decomposition reads SU (2) → SU (2)/U (1) × U (1) and therefore the residual field does not possess the SU (N − 1) part. We simply find a ∈ {1, 2} and γ = 3. Using the fact that in this decomposition the only non-vanishing structure constants are f abJ and f JKL (note however that f γKL = 0), the Yang-Mills Lagrangian is decomposed as, and the covariant derivative is defined with respect to the residual field, Finally, we want to remark that due to the fixing of the color-field n, the originally gauge invariant mass term m 2 X Tr (X µ X µ ) loses its gauge invariance. However, one is at least able to construct a BRST invariant composite operator containing the coset gluon condensate, cf. section III.

A. BRST invariance and gauge fixing
In the following, we want to incorporate the reduction condition (I.2) by means of a "gauge fixing". First, let us recall the BRST transformation δ B , and the anti-BRST transformationδ B , where we have introduced the Nakanishi-Lautrup field N = (N a , N j .N γ ), the ghosts C = (ω a , C j , C γ ) and anti-ghostsC = (ω a ,C j ,C γ ) according to the three parts SU (N )/U (N − 1), SU (N − 1) and U (1), respectively. The quantityN finally is defined asN = g[C,C] − N . Both transformations are nilpotent, δ 2 B =δ 2 B = 0 and satisfy {δ B ,δ B } = 0. It is shown [31] that the reduction condition (I.2) can be cast into the form D ab µ X µb = 0. (II.10) Then we can introduce the "gauge fixing term" as (II.11) Beside its different interpretation this also differs from the standard gauge fixing procedure by the last term, generating the four-ghost interaction after performing the BRST transformation. This is necessary since we deal with a non-linear gauge fixing, in which case the four-ghost interaction preserves the renormalizability of the theory [45]. Indeed, we find where the four-ghost interactions are obtained.
We are left with fixing the residual U (N −1) symmetry. We choose the simple Lorenz gauge, where according to the decomposition U (N − 1) = SU (N − 1) × U (1) two different gauge fixing parameters are introduced, Even though two different gauge fixing parameters are introduced, it can be shown that both the SU (N − 1) and the U (1) part of the residual gauge fixing Lagrangian are independently invariant under the global U (1), the global SU (N − 1) and the combined global U (N − 1) transformations, irrespective of the choice of λ and α. In particular, it is shown later that λ and α receive different one-loop corrections, such that this distinction is actually necessary from the viewpoint of renormalization.
Finally, the Nakanishi-Lautrup field is integrated out, which casts the "gauge fixing" Lagrangian into the form (II.14) 15) and the equations of motion for the Nakanishi-Lautrup field read This completes the "gauge fixing" and leaves us with a BRST invariant Lagrangian We proceed with the one-loop analysis of our theory, (II.18) The induced propagators are shown in Fig. 1,

FIG. 1. Propagators
and imply the Feynman rules  with the Feynman rules where we have defined δq := (2π) D δ(q).
Moreover, we find six four-field vertices, see Fig. 3, and with defining Iµν,ρσ = gµρgνσ−gµσgνρ we obtain the Feynman rules Let us point out the differences to some other common gauges. As mentioned before, our "gauge condition" D ab µ X µb = 0 is non-linear unlike for example the standard Lorenz gauge. This requires the four-ghost vertex (II.34) in order to render our theory renormalizable. Furthermore, the non-linearity gives rise to the ξ-dependent corrections in the four-gluon interaction (II.30) as well as in the two-gluon-twoghost interactions (II.31) and (II.33). These features are also observed in the MAG. However, in the MAG the coset field takes value in SU (N )/U (1) N −1 , where the quotient is Abelian and thus f JKL = 0. In our decomposition however, the coset field takes value in In fact, only the U (1) generator T γ commutes with all other generators of the quotient, f γKL = 0 while f jkl = 0. On the other hand, in the MAG one has f abc = 0 while in our decomposition f abc = 0. Once more we emphasize that in the case of SU (2) our decomposition and the MAG are equivalent. Finally, we state some color-algebra relations that are required in the upcoming one-loop calculations. Starting from the SU (N ) and SU (N − 1) identities and using the fact that only f abj , f abγ and f jkl are nonvanishing structure constants one derives using the Jacobi identity

B. One-loop analysis
We start our one-loop analysis with the introduction of the renormalization factors Note that we took the same renormalization factor for the coset ghost and anti-ghost, while for the residual ghost they must be chosen independently [46]. Furthermore, one has to distinguish between the SU (N − 1) and U (1) gauge field and ghosts. Hereafter, the subscript R is dropped again. The corresponding counterterm Lagrangian is given in appendix A together with the relation between the counterterms and the renormalization factors. The renormalization is done within dimensional regularization.
We begin with the coset gluon self-energy. The one-loop correction reads 1 (II. 38) In dimensional regularization only the last diagram will contribute. The divergent part is calculated as The renormalization factors are expanded and consequently, We proceed with the self-energy of the residual field, starting with the SU (N − 1) part, as V j + V γ means one has to calculate the sum of two diagrams, one with the propagator of the SU (N − 1) field V j and one with the propagator of the U (1) field V γ .
Here, only the last four diagrams contribute in dimensional regularization. Their divergent parts read as follows, The sum is transverse and we therefore find The case of the U (1) part is more simple, since f γKL = 0. The one-loop correction is given by Again, only the last two diagrams have to be calculated and contain the divergent parts yielding a purely transverse correction and thus Next, we turn to the ghost self-energy, starting with the coset ghosts. The one-loop correction is given by (II.57) Only the last diagram contributes in dimensional regularization and the divergent part reads For the residual part we find that the U (1) ghost C γ does not enter any interaction vertex. Therefore, the self-energy correction is zero and we immediately obtain We complete the self-energy analysis by considering the SU (N − 1) ghosts. Their one-loop correction is given by only one diagram Thus, we find the relation The next step is to obtain the renormalization factor of the Yang-Mills coupling by renormalizing the V j C kC l vertex. Its one-loop correction reads The divergent parts of the diagrams read as follows, Therefore, we obtain the relation Using the equations (II.48) and (II.63) we find As expected, this is the standard result for the Yang-Mills coupling in pure Yang-Mills theory. Finally, we consider the X a ω bC J vertex. The one-loop correction reads (II.70) Note that the second and third diagram only contribute to the vertex with an external SU (N − 1) antighost, while the first and last diagram contribute in both cases. The first diagram's divergent part is found to be while the second diagram has the divergent part The two diagrams that only contribute in the J = j case have the divergent parts Adding up all contributions we find after some color algebra (II.75) Together with (II.63) this implies which implies using (II.61) This completes the one-loop analysis. We want to finish this section by summarizing all the corresponding RG functions. We define them by for the fields A, the parameters B and the Yang-Mills coupling, respectively. Then we obtain: We find that the running of α and λ according to µ ∂α ∂µ = αγα and µ ∂λ ∂µ = λγ λ implies the existence of both the "symmetric" as well as the "asymmetric" fixed point, Even though the latter one implies an "asymmetric" gauge fixing of the U (1) and SU (N − 1) part of the residual field, no problem occurs as the invariance of the residual gauge fixing Lagrangian under global U (N − 1) color transformations is completely independent of the parameters α and λ. Moreover, as mentioned before, for N = 2 our decomposition coincides with the MAG. In that case, our results are in full agreement with the existing literature, see for example [46][47][48][49][50]. In particular, note that the λ dependent terms in γX and γω coming from the SU (N − 1) part of U (N − 1) vanish in this case, which reflects the fact that for N = 2 we have the decomposition SU (2)/U (1) × U (1), i.e. the SU (N − 1) part of the residual field is absent.

III. BRST INVARIANCE AND MULTIPLICATIVE RENORMALIZABILITY OF THE COMPOSITE OPERATOR
A. BRST invariance of the composite operator The first step in the proper introduction of the composite operator O = 1 2 X a µ X µa − iξω aωa is to add a source term to the action, and to show that the BRST invariance of the action is preserved. The source shall satisfy δBJ = 0. Using (II.8) the composite operator transforms as Replacing N a by its equation of motion (II.16) we find and therefore the (on-shell) BRST invariance is maintained after introducing the source term for the composite operator. Yet two problems remain to be solved. The first is the proof of multiplicative renormalizability of the composite operator, at least to one-loop level. This will be given below. The second and more involved problem are the divergences proportional to J 2 which are generated by the source term. This will be postponed to the next section.

B. Multiplicative renormalizability of the composite operator
The composite operator OR = 1 2 X a µ X µa R − ξ [iω aωa ] R can in principle mix with any condensate that has the same mass dimension and quantum number. We therefore have to set up the renormalization matrix The matrix elements are calculated by inserting OR into the various two-point functions and requiring the cancellation of the resulting divergences [33], using the Feynman rules for the operator insertions as shown in Fig. 4.

FIG. 4. Feynman rules for operator insertions.
At this point we only state the result, since the calculations are quite lengthy. The details are presented in appendix B. The renormalization matrix is shown to have the form where the one-loop part Z (1) contains 10 non-vanishing elements given by . (III.6) Using the fact that to one-loop level the inverse of the renormalization matrix reads we can invert equation (III.4), obtaining The composite operator is thus renormalized as This yields the condition −Z 3 . (III.10) Indeed, we find −Z (III.12) Thus, the composite operator is one-loop multiplicatively Again, this result is in agreement with the existing N = 2 MAG results, e.g. [44]. According to equation (III.8) the existence of the coset gluon condensate seems to induce a residual field condensate V j µ V µj (V γ µ V µγ ) due to a non-vanishing of the matrix entries Z ). However, no BRST invariant combination of mass dimension two operators including the residual field condensate can be constructed. This renders such a condensate non-physical and thus we continue to discuss the composite operator O only.
Finally, for later use we furthermore introduce the composite operator anomalous dimension, which reads to one-loop level

IV. LCO FORMALISM
As mentioned before, the introduction of the composite operator source term leads to new divergences quadratic in the source. To treat these divergences the so-called LCO formalism has been developed in [38,39] and also has been applied to similar gluon-ghost composite operators for example in the usual Lorenz gauge and in the MAG [40,41,44]. In order to make this paper self-contained, we briefly introduce the LCO formalism, thereby mainly following the lines of [44].
In order to cure the aforementioned divergences we extend the Lagrangian by adding where κ is an a priori arbitrary parameter and the second term is understood to be a pure counterterm. Since we already proved the multiplicative renormalizability of the composite operator we define J0 = Z −1/2 O J such that J0O0 = JO. The running of the generating functional then becomes where η = µ ∂ ∂µ κ. Its running behaviour will allow us to determine κ if we assume that it only runs implicitly through its dependence on g and ξ, as shown below. By noting that κ and δκ have mass dimension [κ] = [δκ] = D − 4 = −2 we find that starting from 0 = µ ∂ ∂µ the RG function of κ can be written as with the inhomogeneity Next, we use the assumption that the auxiliary parameter κ = κ(g 2 , ξ, µ) depends on µ only implicitly via g 2 (µ) and ξ(µ). The equation (IV.4) then becomes Expanding in g 2 this implies that the solution can be written as κ(g 2 , ξ) = κ0 g 2 + κ1 + 2 κ2g 2 + . . . , (IV.7) where we temporarily introduced . At this stage it becomes obvious that we unfortunately need to perform (n + 1)-loop calculations in order to determine κ to n-loop. For example, assuming all quantities have been determined to two-loop level we have the expansions β g 2 = −2 g 2 + β1 g 4 + β2 g 6 , δκ = δκ0 + δκ1,1 + δκ1,2 2 g 2 , γO = γO,0 g 2 + γO,1 g 4 , γ ξ = γ ξ,0 g 2 + γ ξ,1 g 4 , (IV.8) and equation (IV.6) implies 1 g 2 : 2 κ0 − 2 κ0 = 0, (IV.9) g 0 : 2γO,0 κ0 + κ0β1 − ξγ ξ,0 ∂ ξ κ0 + 2δκ0 = 0, (IV.10) where δ1 = −ξγ ξ,1 ∂ ξ κ0 + 2γO,1 κ0 + β2κ0 + 4δκ1,1 The first equation is satisfied identically while the second equation implies the ODE for κ0, Therefore, knowledge of the one-loop quantities γ ξ,0 , γO,0, β1 and δκ0 is necessary to obtain the tree-level part κ0. The solution of this ODE is plugged into equation (IV.11) to obtain κ1. However, one comment needs to be made about the inhomogeneity δ1. When taking the limit → 0 the last term in equation (IV.12) can only be finite if the bracket vanishes identically. This is guaranteed from the fact that if the theory is renormalizable, the finiteness of equations (IV.2)-(IV.4) implies the finiteness of δ and therefore there is no need to consider the terms proportional to 1/ in δ, as they must vanish by construction [51]. In fact, based on the results in [41] for the case of the "full" gluon composite operator A A µ A µA and in Lorenz gauge with arbitrary gauge parameter this condition can be explicitly checked and is found to be satisfied. It should be remarked on the other hand that if one is interested in the mere existence of the condensate, knowledge of κ0 is sufficient, thus avoiding this subtlety in determining κ1.
Before we turn to the calculation of the last ingredient for the ODE (IV.13), that is the one-loop part of δκ, let us note that there actually exist two ways of calculating this quantity and also γO, depending on the interpretation of the composite operator source J. One possibility is to regard J as a constant parameter and therefore treat γO as a mass renormalization. Hence, all calculations are performed using a massive gluon propagator, which is quite cumbersome especially in higherloop calculations. This has been adopted in the original version of the LCO formalism. Alternatively, in [52] it has been suggested to treat J as a non-dynamical field that interacts with the gluon. In this case, the calculations can be performed using massless propagators and the renormalization is done by inserting the composite operator into two-point functions in order to obtain γO, while δκ is obtained by inserting the composite operator into the vacuum bubbles, requiring the quantity O(x)O(y) to be finite. It actually was the second viewpoint that we used to prove the one-loop multiplicative renormalizability of our composite operator in section III B. Both approaches seem to be equivalent as for example the results derived in [52] agree with those in [40].
To obtain δκ0 it is convenient to return to the viewpoint of J being a mass, then the one-loop correction to the generating functional is given by where the second line is obtained using the orthonormality of the transverse and longitudinal gluon propagator. Adopting dimensional regularization and taking the derivative with respect to J twice we find the -divergent part proportional to J 2 , and therefore (IV. 16) We are now ready to solve the differential equation (IV.13) for κ0. A particular solution is given by The homogeneous part is solved as and therefore which implies the general solution As discussed in [44] the minimum of the effective potential should be independent of the gauge fixing parameter, allowing us to choose the integration constant C arbitrarily. In practice, the result for the vacuum energy may explicitly depend on ξ due to the mixing between different orders of perturbation theory. This could only be avoided if one knew the potential up to infinite order. Nevertheless, in the next section we will motivate a reasonable choice for C.

V. EFFECTIVE POTENTIAL AND EXISTENCE OF THE CONDENSATE
Before we calculate the one-loop effective potential there is still one problem left. Because of the introduction of the terms proportional to J 2 the generating functional lost its usual interpretation as an energy density. However, following [44] this can easily be circumvented by performing a Hubbard-Stratonovich transformation, introducing the auxiliary field σ as Here, a normalization constant was absorbed into the path integral measure. The parameters A and B are chosen such that the J 2 term and the JO term of the original Lagrangian are cancelled, for example A = −g and B = −gκ. The modified Lagrangian then reads From equation (V.1) we also find that the vacuum expectation values of O and the auxiliary field σ at J = 0 are related as Provided the auxiliary field has a non-zero vacuum expectation value and using we note that Lσ contains the mass term for the coset gluon and ghosts, with the tree-level masses Thus, to answer whether the condensate exists or not, we need to calculate the effective potential for the auxiliary field. Decomposing the potential into V = V0 + V1 with the tree part V0 and the one-loop part V1 we immediately find the tree-level part For the one-loop correction we have Within dimensional regularization, the calculation of the logarithms can be done analogously to section IV. Adopting the M S-scheme we find whereμ 2 = 4πµ 2 e −γ . Next we are looking for the stationary points, Besides the solution σ = 0 we find another stationary point σ * , providing the squared mass m 2 X given by Based on these results, we want to discuss the open issue of fixing the integration constant in the solution for κ0(ξ), equation (IV.20). First of all, we need to recover the correct UV limit, σ * → 0 as g 2 → 0,which implies that H1 must be negative and thus ξ 2 < 3. This is consistent with the fact that the "physical" region for ξ is the close vicinity of ξ = 0. 2 Then, from the tree potential (V.7) we learn that κ0 should be positive in order to have a bounded-from-below tree part. The choice guarantees that κ0 is positive for all ξ within the close vicinity of ξ = 0. Moreover, for this choice we find that for ξ = 0 the function H2 becomes an irrelevant constant, while for H1 we obtain, Introducing the experimentally accessible and RG invariant QCD scale ΛQCD as usual, , (V. 16) we find that at ξ = 0 and to one-loop order the coset gluon mass becomes proportional to Λ 2 QCD , Therefore, assuming that ξ only changes marginally withμ around ξ = 0 we obtain an RG invariant coset gluon mass.
More explicitly, to one-loop order Consequently, the vacuum energy is calculated as The first term in equation (V.19) is replaced using dV dσ σ * = 0, which yields (V.20) Plugging this into the equation (V.19) we obtain where in the last line we used that the gluon mass is given by m 2 X = gσ * κ 0 and the result is in full agreement with the N = 2 MAG case [44]. Together with the condition ξ 2 < 3 we indeed find that the energy for this vacuum is negative and therefore the condensate is energetically favoured. At first sight, the dependence of the vacuum energy on the parameter ξ is problematic, as one should obtain the gauge independent result. However, our "gauge" is different from the usual treatment in the sense that it removes superfluous degrees of freedom from the extended Yang-Mills theory, in order to recover the theory equipollent to the SU (N ) Yang-Mills theory. This suggests we set ξ = 0 and thus our result hints at the existence of the non-zero coset field condensate, at least to one-loop level.

VI. CONCLUSION
In this paper we investigated the decomposition of the G = SU (N ) Yang-Mills theory with respect to the stability group H = U (N − 1). We proved the one-loop renormalizability of this theory and explicitly obtained all the involved RG functions.
An important feature of this theory is the fact that one can introduce a gauge invariant mass term for the cosetfield Xµ ∈ Lie(G/H). This is interesting from the viewpoint that the existence of a non-zero condensate XµX µ = 0 directly leads to many implications such as quark confinement at low temperature. While it is true that the mass term for the coset gluon is gauge invariant within the original version of the reformulated Yang-Mills theory, it loses its gauge invariance because the color-field is considered to be fixed within this paper. However, we showed that one can at least introduce the on-shell BRST invariant composite operator O = Tr G/H XµX µ − 2iξCC to investigate the possibility of a coset gluon condensate. As an intermediate step, by taking into account the mixing with condensates of the same quantum number, we obtained the one-loop renormalizability of this composite operator.
In the second part of the paper we used these results to discuss the existence of the condensate by means of the lo-cal composite operator formalism. Consequently, after performing a Hubbard-Stratonovich transformation, we obtained the one-loop effective potential V (σ) for the auxiliary field σ, where the vacuum expectation values of σ and O are related as σ = g O . Indeed, we found a non-zero stationary point σ * away from the origin. However, the corresponding vacuum energy V (σ * ) explicitly depends on the parameter ξ. This would be a problem in the usual gauge fixing framework, but our reduction condition has a different meaning as it reduces the enlarged color-field extended gauge symmetry back to the theory equipollent to the SU (N ) Yang-Mills theory. In other words, even though the reduction condition is imposed, the full SU (N ) gauge symmetry is preserved. But again, due to the fixing of the color-field, the situation changes. The reduction condition appears as a gauge fixing-like term for the coset gluon. Nevertheless, we take the standpoint that according to the previous argument, we should adopt the "physical" choice ξ = 0 in order to incorporate the reduction condition in an δ-function like manner. In this case, the value V (σ * ) is negative and a non-zero coset gluon condensate is energetically favoured. Certainly, these considerations need to be improved, for example by discussing the existence of the condensate within a non-perturbative approach such as the functional renormalization group.

Appendix A: Counterterm Lagrangian
In this appendix we set up the counterterm Lagrangian corresponding to Then the counterterm Lagrangian is written as where the coefficients ∆i are expressed in terms of the renormalization factors as In the main text we determine ∆1, ∆2, ∆3, ∆4, ∆5, ∆6, ∆7, ∆8 and ∆9 by considering one-loop self-energy corrections. Next, we consider corrections to the V j C kC l -vertex and the XωC J -vertex, yielding ∆25, ∆26 and ∆27, respectively. This is sufficient to determine all the renormalization factors to one-loop level.
Appendix B: Determining the renormalization Matrix of the composite operator In this appendix we briefly explain how to determine the renormalization matrix of the composite operator renormal-ization, deriving the diagrammatic equations for the renormalization matrix elements. This is done by inserting the composite operator into the propagators of the fields.

a. Insertion into XX
(B.1) From this it follows that Z1 has the tree part.
(B.2) From this it follows that Z7 does not have the tree part.
3) From this it follows that Z13 does not have the tree part.
(B.4) From this it follows that Z19 does not have the tree part.
(B.5) From this it follows that Z25 does not have the tree part.
From this it follows that Z31 does not have the tree part.

b. Insertion into
From this it follows that Z2 does not have the tree part.
From this it follows that Z8 does have the tree part.
From this it follows that Z14 does not have the tree part.
From this it follows that Z20 does not have the tree part.
From this it follows that Z26 does not have the tree part.
From this it follows that Z32 does not have the tree part.
c. Insertion into ωω (B.13) From this it follows that Z3 does not have the tree part.
(B.14) From this it follows that Z9 does not have the tree part.
(B.15) From this it follows that Z15 does have the tree part.
(B.16) From this it follows that Z21 does not have the tree part.
(B.17) From this it follows that Z27 does not have the tree part.
(B.18) From this it follows that Z33 does not have the tree part.
From this it follows that Z4 does not have the tree part.
From this it follows that Z10 does not have the tree part.
From this it follows that Z16 does not have the tree part.
From this it follows that Z22 does have the tree part.
From this it follows that Z28 does not have the tree part.
From this it follows that Z34 does not have the tree part.
From this it follows that Z5 does not have the tree part.
From this it follows that Z11 does not have the tree part.
From this it follows that Z17 does not have the tree part.
From this it follows that Z 23 does not have the tree part.
From this it follows that Z 29 does have the tree part.
From this it follows that Z 35 does not have the tree part. From this it follows that Z 36 does have the tree part. At this stage we have proven the following form of the renormalization matrix, We therefore can reconsider the equations (B.1) -(II.36) to obtain the diagrammatic equations for the elements of Z (1) . From equations (B.1)-(B.6) we find Z (II.41) (II.42) (II.51) (II.52) Finally, from the equations (II.31) -(II.36) we find Z 3(λ 2 + 6)ξ + λ 2 + 3ξ 2 + 3 4ξ .