Building models of quarks and gluons with an arbitrary number of colors using Cartan-Polyakov loops

In this work we introduce the concept of Cartan-Polyakov loops, a special subset of Polyakov loops in the fundamental representation of the SU( N c ) group, with charges k = 1 , . . . , ( N c − 1) / 2. They constitute a suﬃcient set of independent degrees of freedom to parametrize the thermal Wilson line. Using properties of the characteristic polynomial of the thermal Wilson line, we write a non-Cartan-Polyakov loop charge decomposing formula. This formalism allows one to readily build eﬀective models of quarks and gluons with an arbitrary number of colors. We apply it to the Polyakov − Nambu − Jona-Lasinio model and to an eﬀective glue model, in the mean ﬁeld approximation, showing how to directly extend these models to higher values of N c .

From the phenomenological point of view, several effective models of QCD have been built in order to study the properties of the conjectured quarkyonic phase. Some of these models were based on the Polyakov−Nambu−Jona-Lasinio (PNJL) model, in which quark-quark interactions are point-like, with a static gluon field in the temporal direction acting as a background field. These models possess all the global symmetries of QCD, including the confinement/deconfinement transition in the sense of the breaking of the Z(N c ) symmetry and which can be measured by the expectation value of the Polyakov loop. This quantity is an exact order parameter in the pure glue theory but only approximate when quarks are introduced in the Lagrangian. The Polyakov loop is extremely used as an order parameter to study the deconfinement transition in both ab-initio calculations like lattice QCD, in Dyson-Schwinger equations, and in phenomenological models like the Polyakov-quark-meson model or the PNJL model.
In this work we show how to build models of quarks and gluons with an arbitrary number of colors, in a systematic way. More precisely, we will build a version of the Polyakov−Nambu−Jona-Lasinio model, in the mean field approximation and consider a glue effective potential, both for any number of colors. Being able to study such models systematically can be quite useful in order to better understand the deconfinement phase transition, the conjectured quarkyonic phase and the properties of quarks and gluons.

II. CENTER SYMMETRY AND THE POLYAKOV LOOP
In the imaginary time formalism, often used to study quantum field theory at finite temperature, boson and fermion fields must respect periodic and antiperiodic boundary conditions, respectively. Using this prescription, it can be shown that, at the Lagrangian level, the boson fields of the Yang-Mills theory have symmetric boundary conditions with respect to transformations of the discrete group Z(N c ), the center of the SU(N c ). The introduction of fermion fields, on the other hand, explicitly breaks such invariance. Hence, Yang-Mills theory at finite temperature have boundary conditions that are invariant under Z(N c ) transformations while QCD does not share the same property, due to the presence of the quark fields. However, even in the pure glue theory, such symmetry can be spontaneously broken.
The Polyakov loop, Φ, is a gauge invariant quantity, that can be used as an exact order parameter for the spontaneous symmetry breaking of the Z(N c ) symmetry in the Yang-Mills theory [31]: if the Polyakov loop is zero, the Z(N c ) symmetry is preserved while, if it is non-zero, the symmetry is broken.
One can check the gauge invariance of the Polyakov loop by considering the gauge transformation of the thermal Wilson loop, L(x), from which one can define the Polyakov loop. The Wilson line can be defined along a path in spacetime, from an initial position (x i ) to a final position (x f ), as follows: Here, P is the path-ordering operator, g is the strong interaction coupling and A µ is the gauge field 1 . The Wilson line of the gauge transformed field is given by: By considering a closed path (x i = x f = x), one gets the so-called Wilson loop, which is a gauge invariant quantity due to the cyclic property of the trace: Considering the Wilson loop wrapped in imaginary time (i.e., at finite temperature with periodic boundary conditions) one defines the thermal Wilson line, 1 Under a gauge transformation, the gauge field Aµ(x) transforms as: Here, A 4 = igA a µ λa 2 δ µ 0 , is the gluon field in the temporal direction. This quantity is also gauge invariant. Finally, one can define the Polyakov loop on a spatial coordinate x, in the fundamental representation of the SU(N c ) group, Φ(x), as the trace of the thermal Wilson loop with the field A 4 written in the fundamental representation [31,32]: Here, the trace is made over color space and the index F indicates that the matrix A 4 is written in the fundamental representation of the SU(N c ) group. The Polyakov loop transforms akin to the fermion fields and only the identity element of the Z(N c ) group maintains the symmetry indeed, under a global Z(N c ) transformation, it transforms as a field with charge one [31,33].
The physical interpretation behind the two previously discussed phases, corresponding to zero and non-zero Polyakov loop, can be understood by evaluating the Helmholtz free energy of a single quark in a background of gluons. To this end, one can solve the static Dirac equation coupled to a static background field and use it to calculate the Helmholtz free energy of a single quark in a background of gluons [31,34,35]. It yields that, if the Polyakov loop is finite, a single quark can be created in the gluonic background with a finite amount of energy while, if the Polyakov loop is zero, an infinite amount of energy is required [31,35]. Hence, a finite Polyakov loop would correspond to a deconfined phase of matter while, a zero Polyakov loop to a confined phase.
As already pointed out, the Polyakov loop is often incorporated in phenomenological models of QCD, like the PNJL or the PQM models, in order to study the properties of the phase diagram of strongly interacting matter and explore the relation between the restoration of chiral symmetry and the deconfinement transition. In the following sections we will show how to build both a PNJL model and and effective glue potential, with an arbitrary number of colors, which can be used to explore the properties of strongly interacting matter.

III. CARTAN-POLYAKOV LOOPS
In this section, we will introduce the concept of Cartan-Polyakov loops which will allow to readily derive quark and gluon models with an arbitrary number of colors.
Consider the thermal Wilson line defined in Eq. (4), L. Allying the unity determinant property of this quantity (it is an element of the SU(N c ) group) with the fact that, at every spacetime point, this matrix can be diagonalized with a gauge transformation [31], one can conclude that this quantity has a total of N c − 1 degrees of freedom, the same number as the rank of the SU(N c ) group [36]. Knowing the value of N c − 1 traces of different powers of this matrix, allows one to build a system of N c − 1 equations to the same number of unknown matrix entries. Hence, the traces can be used to parametrize the thermal Wilson line [37].
As introduced earlier, the Wilson line in the fundamental representation transforms as a field with charge +1, under a Z(N c ) symmetry transformation. The antifundamental representation transforms as a field with charge −1. As pointed out in [38], the charges are defined modulo N c , meaning that the antifundamental charge, −1, is equivalent to the charge N c − 1.
One can build Polyakov loops in higher dimensional representations, using combinations of fundamental and antifundamental Wilson lines. Examples include Wilson lines in the adjoint representation of SU(N c ), twoindex symmetric, antisymmetric and other representations, with higher index counts. These different possible representations transform under Z(N c ) as fields with different charges. For more information on different representations of the Wilson line, see [38][39][40]. In particular, in Ref. [38] it was shown how the classification procedure can be made in the large N c limit.
In SU(3), the fundamental and antifundamental Polyakov loops constitute a sufficient set from which one can build any other higher representation Polyakov loop. For larger values of N c however this is not the case [41] and a higher number of degrees of freedom must be considered. In this work we use Polyakov loops with different charges, in the fundamental representation, in order to easily build effective models containing both gluon and quark degrees of freedom, at an arbitrary N c . More specifically, we will show how to build the Polyakov−Nambu−Jona-Lasinio model, in the mean field approximation and an effective gluon potential, for increasing values of N c . Such constructions can be very helpful in better understanding the thermodynamics of QCD at large N c , as well as, studying the properties of the conjectured quarkyonic phase in the QCD phase diagram or determining the behavior of the critical endpoint for increasing N c [28].
We define the Polyakov loop with charge k, as the trace of the k−power of the thermal Wilson line, in a particular representation, R, of the SU(N c ) group. Polyakov loops with negative charge, are defined analogously, as the trace of the k−power of the adjoint thermal Wilson line [33]. Formally, we write: Here, N R is the dimension of the representation. In the literature, the Polyakov loop variable in the fundamental representation is denoted by Φ and Φ, see Eq. (5). Thus, in our notation, the following equivalence holds: Φ = ℓ F,1 and Φ = ℓ F,1 . The zero charge Polyakov loop is trivially defined by the normalized trace of the N R ×N R identity matrix, ℓ R,0 = ℓ R,0 = 1. In other words, the Polyakov loop with charge k is the normalized trace of a thermal Wilson line wrapped k times around the compact imaginary time dimension [19]: Interestingly, for a given number of colors, these objects are not independent of each other. Indeed, a given Polyakov loop with charge k can be written as a function of other Polyakov loops, with different charges. This is connected to a previously mentioned property of the thermal Wilson line: it has a total of N c − 1 independent degrees of freedom. As discussed earlier, one can parametrize the thermal Wilson line by calculating the trace of N c − 1 powers of this matrix. Since a Polyakov loop with charge k is simply the trace of the k−power of the Wilson thermal line, there is a total of N c − 1 independent charges. This immediately poses the question about which charges should be chosen as degrees of freedom to parametrize a phenomenological model.
This motivates the introduction of a special subset of Polyakov loops, which we denote by Cartan-Polyakov loops. For a specific number of colors, N c , we denote the subset of N c − 1 Polyakov loops in the fundamental representation, with charges chosen to be independent of each other, by Cartan-Polyakov loops and the remaining ones, by non-Cartan-Polyakov loops. In this work we choose the Cartan-Polyakov loops to be those with charges k = 1, . . . , (N c − 1)/2. Hence, the Cartan-Polyakov loop set, C Nc , is given by: Although this choice is made a priori and may seem arbitrary, it is guided by simplicity: although other choices exist 2 , this particular one produces straightforward one-to-one mappings from the Cartan loops to non-Cartan loops while, for other choices, this may not be the case. For N c = 3, for example, the Cartan loops, are given by the set C 3 = ℓ F,1 , ℓ F,1 . For the cases in which N c is an even number, we opted to remove the element ℓ F,(Nc−1)/2 from the set defined in Eq. (9). The reason for this is based on the fact that, for an even number of colors, there is freedom to choose between ℓ F,(Nc−1)/2 and ℓ F,(Nc−1)/2 . In other words, picking one of them to include in the Cartan set allows one to write the other in terms of the other loops in the Cartan set. In this work, for simplicity, we mainly focus on results for odd N c : for such case there is a symmetry between the number of ℓ F,k and ℓ F,k contained in the Cartan set.
In the following, we will write recurrence relations, which will allow us to, within a fixed representation of the SU(N c ) group, decompose a non-Cartan-Polyakov loop with charge k, in terms of Polyakov loops with smaller charges, k − 1, k − 2, . . .. The idea is to use such recurrence relation in order to build effective models of quarks and gluons whose only degrees of freedom are the Cartan-Polyakov loops. For example, when building a model with N c = 5, the Cartan loops, are given by the set C 5 = ℓ F,1 , ℓ F,2 , ℓ F,1 , ℓ F,2 . However, in the construction process, non-Cartan-Polyakov loops such as, ℓ F,3 , ℓ F,4 , ℓ F,3 , ℓ F,4 , will arise. Using the recurrence relation that we will derive in the next section, enables one to write these non-Cartan-Polyakov loops as functions of loops with smaller charge, more specifically the ones present in the set C 5 .
We note that it may not be always possible to write effective models in terms of Cartan-Polyakov loops. Indeed in some models arises the necessity to involve elements of the Wilson thermal line matrix [40]. The mapping between the matrix elements of L R and the Cartan-Polyakov loops is not straightforward.

A. Decomposition of non-Cartan-Polyakov loops
In order to derive the non-Cartan-Polyakov loop charge decomposing formula, we will make use of an identity between the coefficients of the characteristic polynomial of an unitary matrix. Consider the characteristic polynomial of the thermal Wilson line in a particular representation, L R . It can be written as (see Appendix A): Here, c i (L R ) are coefficients calculated using (see Eq. (A4)): The matrix J R,k will be essential in the derivation of the reducing formula. It is a k × k, square matrix defined by (see Eq. (A4)): ℓ R,k was defined in Eq. (6). One can also write it in terms of matrix elements, (J R,k ) i,j as, Here, H(n) is the discrete version of the Heaviside step function, H(n) = 0 if n < 0 or H(n) = 1 if n ≥ 0.
We will also need the matrix J R,k , which has an identical structure to the one defined above but with the replacement, ℓ R,k → ℓ R,k . Its definition comes from the characteristic polynomial of the adjoint thermal Wilson line, p(L † R , λ), with coefficients given by, Excluding the upper triangular section of the matrix J R,k , all its entries are Polyakov loops in the representation R, with different charges, ranging from 1 to k. Also, the higher charge Polyakov loop, ℓ R,k , is only present in the entry (k, 1). All the other entries in the matrix are numbers or Polyakov loops of charge smaller than k.
To this end, consider the Laplace expansion of the determinant on the first column of the matrix J R,k . It can be written as: Here, J R,k is the square (k − 1) × (k − 1) matrix, obtained by removing the i−th line and j−th column of matrix J R,k . The determinant in the first term, det J (k,1) R,k , can be readily calculated by the product of the diagonal entries since the matrix J (k,1) R,k is of the lower triangular type. It yields: The determinant of the matrix J R,k can then be written as: Using the coefficients of the characteristic polynomial of the thermal Wilson line, defined in Eq. (11), we can write the determinant in the left-hand side of the above equation as: At this point we still cannot solve the equation for ℓ R,k since the coefficients, c k (L R ), still strongly depend on ℓ R,k . So, we make use of a special property of these coefficients. If the N −dimensional square matrix M is unitary, then the following holds: with j = 0, 1, 2, 3, . . . , N − 1, N . Here, c j (M ) and c j (M † ) are the j−coefficients of the characteristic polynomial of matrices M and M † , respectively. This property is shown in Appendix A. For our purposes, it will allows us to write c k (L R ) in terms of c NR−k (L † R ). This might not sound as much progress however, the highest charge Polyakov loop present in the coefficient c NR−k (L † R ) is ℓ R,NR−k . As long as N R − k > 0, all the Polyakov loops present in c NR−k (L † R ) will have a smaller charge than ℓ R,k . Hence, applying the identity written in Eq. (18) to the thermal Wilson line, L R , we can write Eq. (17) as: We have used the fact that the thermal Wilson line is an special unitary matrix, det [L R ] = 1. Finally, we can solve this equation for ℓ R,k to yield the generating formula to decompose a non-Cartan-Polyakov loop into combinations of loops with smaller charges: In here, we have used Eq. (11) to write, c NR−k (L † R ) = (−1) N R −k /(NR−k)! det J R,NR−k and applied the following identity, (−1) 2NR−3k−1 = (−1) k+1 . The most important feature in this equation is that the right-hand side does not depend on the non-Cartan-Polyakov loop of charge k. Indeed, both det J R,NR−k and det J (i,1) R,k , depend only Polyakov loops with charge smaller than k.
One can also derive a very similar formula using the characteristic polynomial of −L R , instead of L R i.e., p(−L R , λ). Such polynomial is can be written as: Similar to the previous characteristic polynomial, the coefficients are given by: The k × k matrix R R,k used above, is defined by: The matrix elements are given by: Following the same steps as before, one can arrive at the following decomposing formula: Once again, the matrix R R,k , has the same structure as the matrix defined in Eq. (23) with the replacement: We highlight that both formulas, Eqs. (20) and (25), are only valid for k < N R . In order to decompose non-Cartan loops with charge k ≥ N R , one can use the Cayley-Hamilton theorem. Using this theorem, we can write: Multiplying by p−powers of the thermal Wilson line, dividing by the dimension of the representation and taking the trace, yields the following equality: Hence, recognizing the different Polyakov loops, for k ≥ N R + p, we can write: This non-Cartan-Polyakov loop decomposing equation is valid for k ≥ N R . One can also use the identity given in Eq. (18) in order to introduce the adjoint Polyakov loops.
In order to show some examples in which Eqs. (20) and (28) are useful, lets consider Polyakov loops in the fundamental representation of SU(N c ), with increasing number of colors. Consider first N c = 3. For this particular case, the Cartan loops are given by the set C 3 = ℓ F,1 , ℓ F,1 . Polyakov loops with higher charge (so-called non-Cartan-Polyakov loops) can be written as functions of the loops in this set. The charge 2 loop, ℓ F,2 , can be calculated using Eq. (20). It yields: For the higher charges, we use Eq. (28): . . . Naturally, the non-Cartan-Polyakov loops decomposition formula maintains the symmetry charge of the loop. For each decomposed non-Cartan-Polyakov loop, each term in its decomposition, has the same charge as the original non-Cartan-Polyakov loop. In the case of ℓ F,2 for example, both ℓ 2 F,1 and ℓ F,1 transform as fields of charge +2. One can readily obtain the respective adjoint Polyakov loops by replacing, ℓ R,k → ℓ R,k and ℓ R,k → ℓ R,k .
Increasing the number of colors to N c = 5 changes the set that constitute the Cartan loops. In this new case the set is: Once again, using Eq. (20), we can decompose the non-Cartan loops, with charge k < 5, in terms of the Cartan set as follows: Again, the higher charge Polyakov loops can be decomposed using Eq. (28): . . .
The formalism developed earlier allows one to go on decomposing non-Cartan-Polyakov loops for even higher values of N c . Naturally, as N c increases the number of terms in each decomposition gets larger and larger. For N c = 9, the set is 4 , for N c = 11, one must also join ℓ F,5 and ℓ F,5 and so on. The number of terms that appear in the decomposition of non-Cartan-Polyakov loops into Cartan-Polyakov loops naturally increases with the number of colors. For example, for N c = 11, the first non-Cartan loop, ℓ F,6 , has 17 terms while, for N c = 21, the first non-Cartan is ℓ F,11 , which contains 97 different terms.
These relations can be checked by hand if one considers a particular gauge choice. In the so-called Polyakov loop gauge, the A 4 field in the fundamental representation is diagonal and static [42] and it can be expressed by the N c eigenphases q i , applies. Thus, in this particular gauge, Polyakov loops of different charges can be written as: Using these gauge fixed Polyakov loops in the fundamental representation, one can check the results coming from the non-Cartan-Polyakov decomposing formulas.

IV. APPLICATIONS
In this section we show two examples of how to construct models of quarks and gluons with an increasing number of colors. First, we consider the PNJL model in the mean field approximation. Secondly, we consider a glue effective potential.

A. The mean-field Polyakov−Nambu−Jona-Lasinio model at arbitrary Nc
The NJL model is widely used as an effective model of QCD due to its simplicity, symmetry properties and ability to display dynamical chiral symmetry breaking in the vacuum and restoration at high temperatures/densities. Some applications of this model include the study of in-medium meson behavior, transport coefficients of quark matter, the equation of state of hybrid neutron stars and the phase diagram of strongly interacting matter with great emphasis on the location of the conjectured critical endpoint. This model however, lacks the ability to describe the confinement-deconfinement transition. In the Polyakov version of the model, a static gluonic background field is coupled to the quarks sector, allowing for the description of both the chiral and deconfinement transitions [32,[44][45][46].
The Polyakov loop can be included in the model by minimally coupling a background gluon field in the time direction, A 0 = −iA 4 , to the Lagrangian density via the covariant derivative. It can be defined as: Here, the covariant derivative is defined as . , m N f is the quark current mass matrix (with N f the number of quark flavors). The different quark-quark interactions are contained in the term L int ψ, ψ and, for our purposes, its exact definite form is not important. This interaction term not only includes the dynamical chiral symmetry breaking 4-quark scalar-pseudoscalar interaction, L ⊃ (ψλ a ψ) 2 + (ψiγ 5 λ a ψ) 2 , but it also includes other multi-quark interactions, like the 't Hooft determinant, eight quark-quark interactions, explicit chiral symmetry breaking interactions, vector interactions, etc [47][48][49][50].
The effective potential, U eff (A 4 , T ) incorporates the spontaneous breaking of the Z(N c ) symmetry at some finite temperature. This potential will be discussed in much detail in later sections where we will define a version of it for any number of colors.
The Euclidean generating functional for the PNJL model, can be written as: Here, S E is the Euclidean action of the PNJL model. The path integral over the temporal gluon field A 4 is to be made over all of its N c − 1 components and j 4 is a source for the temporal gluon field.
To deal with the path integral over the quark fields, we apply the mean field approximation. In this scheme, the multi-quark interactions present in L int , can be written as the product of quark bilinear operators with the introduction of mean fields, φ i . After the linearization of the product between these operators, a quadratic Lagrangian in the quark fields is obtained and the quark fields can be integrated out from the generating function [48]. The generating functional becomes: The only thing left to do is to solve the path integral over the temporal gluon field. Different approaches to deal with the path integral over the A 4 field, can be found in the literature, ranging from using the Functional Renormalization Group, a Gaussian model or using the Weiss mean field approximation, see [40,51,52]. One can parametrize the SU(N c ) group volume in terms of the diagonal elements of the SU(N c ) algebra, i.e., the Cartan elements of the algebra [53][54][55][56]. Hence, we can separate the integration over A 4 in Eq. (41) in two contributions, one made over the diagonal generators of the SU(N c ) algebra and another over the non-diagonal generators. One can integrate over the non-diagonal components to yield the so-called Haar measure of the SU(N c ) group, H [54,57,58]: Here, A diag 4 is the set of diagonal components of A 4 . In the already introduced Polyakov gauge, A 4 is diagonal and traceless, meaning it can be parametrized in terms of the Cartan elements, A 4 = i∈diag A (i) 4 t i with t i the diagonal generators of the SU(N c ) algebra i.e., the Cartan subalgebra. For N c = 3, for instance, we can write 4 λ 8 , with λ 3 and λ 8 the diagonal Gell-Mann matrices. Thus, in Eq. (41) one can integrate out the non-diagonal elements of A 4 leaving only the path integral over the diagonal components, which contains all the physics of the model [54,57]: Here, κ is some proportionality constant and ] is the Haar measure, which results from integrating the non-diagonal components of A 4 . The Haar measure can be absorbed on the definition of the Polyakov effective potential, U eff . As a matter of fact, the Haar measure has been understood to be a crucial part of the potential by favoring the confined phase [33].
At this point one needs to solve the path integral over the diagonal components of the field A 4 . The standard approach is to consider that the functional integration is dominated by the stationary point: quantum fluctuations are neglected and only the classical configuration contributes to the path integral [49,59]. Thus, the functional integration can be dropped and the diagonal components of the A 4 field can be treated as mean fields. The values of these fields are calculated by requiring the effective action to be stationary, dS E dA diag 4 = 0.
Usually, for N c = 3, the requirements of stationarity of the effective action with respect to the diagonal components of the A 4 field are replaced by stationarity with respect to the values of the charge one Polyakov loop and its adjoint in the fundamental representation, Φ and Φ [59].
, changing variables is allowed as long as the Jacobian of the transformation is non-singular [53]. For other values of N c , ℓ F,1 = Φ and ℓ F,1 = Φ are not enough degrees of freedom to express the Haar measure. Thus, for other values of N c , we will consider that these change of variables are allowed and replace the variables of the model: instead of using the set of Cartan elements of the A 4 field, we will use the Cartan-Polyakov loops set defined in Eq. (9).
Hence, within the mean field approximation, the Lagrangian density of the PNJL model can be written as the one of fermionic quasiparticles with an effective mass, M , effective chemical potential,μ, and effective mean field potential, U . Using the Matsubara formalism, one can calculate the thermodynamic potential of the model, Ω(T, µ), to be given by: Here, β = 1 /T , E = p 2 + M 2 and the constant Ω 0 is the value of the thermodynamic potential in the vacuum, Ω 0 = Ω 0, 0, φ i , A diag 4 . The potential U ( φ i ) contains the contribution coming from the mean fields, φ i . The trace must be calculated over flavor indexes while the determinant is calculated in color space. The mean fields are fixed by requiring for thermodynamic consistency, dΩ/d φ i = 0 [60]. The subscript "reg" in the integration, serves only has an acknowledgment that the PNJL model has divergent integrals and some regularization scheme must be employed to render these integrals finite. Some techniques include the 3-momentum or the Pauli-Villars regularization schemes.
Until now, we have not yet specified the number of colors. In order to obtain the thermodynamic potential for a particular number of colors, one must write the effective Polyakov loop potential and calculate the determinant inside the logarithms, for the chosen N c .

The fermionic determinant at arbitrary Nc
Lets start by evaluating the fermionic contribution for a given number of colors. The fermionic determinants are given by: Here, we have defined the non-zero quantities: The goal is to calculate the determinants for an arbitrary N c using only the corresponding set of Cartan-Polyakov loops. To this end, we will write the determinants in Eqs. (45) and (46) We To solve this issue, we will use a trick in order to simplify the practical calculations while also automatically re-writing the sum as a function of Cartan-Polyakov loops only. For simplicity, in the following discussion, we will focus on the case in which the number of colors is odd. The even number of colors case is a straightforward extension.
Consider the identity given in Eq. (18), which states that we can relate the coefficient c k (−L F ), with the coefficient c Nc−k (−L † F ). If we employ this identity, we can break the original sum in Eq. (48) in the middle, in two distinct sums and write, where we have used c j (−L F ) = c N −j (−L † F ). One can check that both the first and second sums on the righthand side of this equation, contains coefficients that will only produce Polyakov loops with charges in the interval [1, . . . , (N c − 1)/2]. This range of charges is the same as the one which defines the Cartan-Polyakov loop set. Hence, the determinant can be written only as a function of Cartan-Polyakov loops.
From an analytically point of view, another nice feature of this trick is that by employing it, one only needs to calculate half of the coefficients. At first glance, one has to calculate N c − 1 coefficients, starting with the coefficient with k = 1 up to k = N c − 1. However, one can calculate the coefficients, c k (−L F ), up to k = (N c − 1)/2 and the remaining coefficients, related to c k (−L † F ), can be obtained from the substitution, ℓ F,k → ℓ F,k .
Thus, for an odd number of colors, we can re-write the sum using Eq. (49), and write the determinant in Eq. (45) as: For the case in which N c is an even number, the idea is the same with the small caveat that one of the sums will have an additional term.
The calculation of the determinant, defined in Eq. (46), follows the same steps. One can gets the same result as the one shown in Eq. (50) with the following replacements: − → +, L F → L † F and L † F → L F . The explicit calculation of the coefficients follows from Eq. (22). Hence, one can readily calculate the determinants defined in Eqs. (45) and (46), for an arbitrary N c . The problem of writing the fermionic part of the PNJL model for an arbitrary number of colors is reduced to calculating the c k (−L F ) coefficients up to (N c − 1)/2. The first coefficients are: . . .
As an example, for N c = 9, the fermionic determinant containing L F , is given by: The determinant containing L † F , can be obtained from the above expression with the appropriate substitutions: ℓ F,k → ℓ F,k , ℓ F,k → ℓ F,k and µ → −µ.

The Polyakov loop effective potential at arbitrary Nc
We now turn our attention to the effective Polyakov loop potential, U eff , which can be written using the Ginzburg−Landau theory of phase transitions. Within this approach, the potential must respect the symmetry properties of the original theory.
For N c = 3, one can build this potential using the charge one Polyakov loop in the fundamental representation (and its conjugate), ℓ F,1 = Φ and ℓ F,1 = Φ. Different phenomenological potentials were introduced in the literature including the logarithmic, polynomial and polynomial-logarithmic. They all share a kinetic term, proportional to ΦΦ and higher order terms that are essential to build a bounded potential, with the correct symmetries (the kinetic term introduces an additional U(1) symmetry which must be removed by other terms [53,55,61]), which reproduces a first-order phase transition at some finite temperature.
In the logarithm potential, the non-quadratic term of the potential is given by the logarithm of the Haar measure. As already seen, this contribution naturally appears when one integrate out the non-diagonal components of the A 4 field [32,57]. This potential, for N c = 3, is widely defined as [32,59,62]: The quantity inside the logarithm is the Haar measure for N c = 3. The temperature dependent parameters [59,62] are: The parameters T 0 , a 0 , a 1 , a 2 and a 3 are fixed by reproducing lattice QCD results [63,64] with T 0 fixing the temperature of the phase transition [45]. The parameter a 0 is fixed by requiring the Steffan-Boltzmann limit when T → ∞ while the parameter b 3 is fixed by requiring the second-order phase transition to occur at T = T 0 [45,59]. The gluon contribution to the Steffan-Boltzmann pressure is PSB /T 4 = 2(N 2 c −1)π 2 /90. Hence, the a 0 parameter can be readily defined at diverging temperatures, T → ∞, a 0 = 4(N 2 c −1)π 2 /90. Thus, one can conclude that this parameter is N c dependent. Since b 3 can be calculated from the remaining parameters, including a 0 , it is also N c dependent. When building an effective Polyakov loop potential for any number of colors, this has to be taken into account.
For N c = 3, following lattice QCD calculations at zero chemical potential, the T 0 parameter is usually fixed to 270 MeV [65,66]. Different approaches have been used in the literature to fix the value of T 0 , specially when studying systems at finite chemical potential. As a matter or fact, in [67], the parameter T 0 was given an explicit dependence on the the number of flavors of quarks considered, due to renormalization group arguments.
The polynomial potential is very similar with the major difference that higher powers of ℓ F,1 = Φ and ℓ F,1 = Φ are incorporated in the potential [45,68]. The polynomial-logarithmic potential, as stated in the name, is a mixture between the logarithmic and polynomial potentials [69]. Another important property of the logarithmic potential is its diverging nature in the limit ℓ F,1 = Φ → 1 and ℓ F,1 = Φ → 1. This ensures that the Polyakov loops do not exceed unity [59].
The question that now arises is the following: how to define an effective Polyakov loop potential for any given value of the number of colors? The studies performed in Refs. [25,27], built effective Polyakov loop potentials for any number of colors based on the polynomial potential in order to study the phase diagram at large N c . However, such potentials were written only as a function of ℓ F,1 = Φ and ℓ F,1 = Φ and Polyakov loops with larger charges were neglected. At large N c , the fundamental loop with charge one may indeed be the most important degree of freedom however, in this work, we want to write a general expression for any N c , without any a priori approximations (apart from the mean field one). After building such model, then one may perform a large N c approximation. Now, we propose to simply extend the logarithmic potential to any value of N c . Using the polynomial potential, would mean having to consider different interaction terms between different Cartan-Polyakov loops in such a way to build Z(N c ) invariant and U(1) breaking interactions. The Haar measure automatically gives interaction terms which respect the symmetries of the model, yielding an appropriate effective potential at any number of N c . To this end we consider that the potential can be written solely in terms of the previously introduced Cartan-Polyakov loops. Hence, for the kinetic part of the potential we simply consider a sum over quadratic terms of the Cartan-Polyakov loop set: This term is invariant under Z(N c ) and under an unwanted U(1) symmetry which must be removed by the remaining contributions. As expected, the non-quadratic part of the potential is given by the Haar measure, for a given N c , written in terms of Cartan-Polyakov loops. How to accomplish this task will be discussed later. So, the effective Polyakov loop potential, at arbitrary N c , is given by: Here, we define the temperature and color dependent parameters A(N c , T ) and B, in the same way as the temperature parameters a(T ) and b(T ) in Eqs. (57) and (58): As before, the A 0 (N c ) parameter have to be fixed in such a way to recover the Steffan-Boltzmann limit for a gas of gluons with SU(N c ) symmetry. The remaining parameters must ensure that a second-order phase transition occurs at T 0 at least for N c = 3. At higher values of N c other behaviors might be expected [8]. We did not introduce a color dependence in the T 0 parameter. Some lattice results, seem to indicate that T 0 very weakly depends on the number of colors [8,[70][71][72]. Indeed, in such studies, the deconfinement temperature T 0 , as a function of the number of colors was found to be, where, σ is the N c −independent string tension. In [70,71], the values obtained were α 1 = 0.582(15) and α 2 = 0.43 (13), in Ref. [72], α 1 = 0.9026(23) and α 2 = 0.880 (43) while, in Ref. [73], α 1 = 0.5949(17) and α 2 = 0.458 (18).
The exact functional dependence of the parameters in effective Polyakov loop parameters, on the number of colors, is beyond the scope of the present work. In order to find such dependencies, one can perform phenomenological studies involving results coming from lattice QCD with N c > 3. The current status of the studies seems to agree that the transition of pure Yang-Mills is of second order for N c = 2 and N c = 3, and of first-order for N c ≥ 4 [8]. The most straightforward way to give some N c dependence to the parameters would be to simply consider the Steffan-Boltzmann requirement enforced on A 0 and fix the remaining parameters at N c = 3.
We now turn our attention to evaluating the argument inside the logarithm of Eq. (60), H(C Nc ), for a given N c .
Since we wrote the fermionic determinant and the kinetic part of the potential, using Cartan-Polyakov loops as degrees of freedom, we also want to write the non-quadratic part of the potential in terms of such variables.
In the following, we show how one can write the argument of the logarithm in Eq. (60), the Haar measure, in terms of Cartan-Polyakov loops. The Haar measure is given by [33,43,58,74,75]: Here, q i are N c eigenphases [43]. This measure can be written using the Vandermonde determinant. The Vandermonde matrix, V , of a set of variables z 1 , . . . , z m , is defined as [75]: The matrix elements of this quantity can readily be written: The determinant of this matrix, known as the Vandermonde determinant, can be calculated to yield: The relation between the Vandermonde determinant and the Haar measure is 3 : Where, W (q 1 , . . . , q Nc ) = V † (e iq1 , . . . , e iqN c )V (e iq1 , . . . , e iqN c ). The matrix elements of W (q 1 , . . . , q Nc ) are given by: Here, z l = e iq l . Recognizing that the diagonal entries are N c and that the non-diagonal entries of this matrix are the fundamental Polyakov loops with different charges (see Eqs. (37) and (38)), one can write the matrix W as [75]: Thus, the Haar measure is exclusively written in terms of Polyakov loops of different charges (and its adjoints), in the fundamental representation. It is given by: Using the recursion formula in Eq. (20), it is possible to write the Haar measure above using only the set of Cartan-Polyakov loops, C Nc . Hence, H(ℓ F,1 , . . . , ℓ F,Nc−1 , ℓ F,1 , . . . , ℓ F,Nc−1 ) = H(C Nc ). For N c = 3, we can write: Using Eq. (29), we have ℓ F,2 = ℓ F,2 (ℓ F,1 , ℓ F,1 ). This yields the usual result used inside the logarithm in Eq.  (43). From this definition one can expect the complexity of the calculation to largely increase for increasing values of N c , leading to polynomials with several interaction terms. For instance, for N c = 5, the Haar measure is given by: Here, ℓ F,3 , ℓ F,4 , ℓ F,3 and ℓ F,4 , are non-Cartan-Polyakov loops that must be decomposed in terms of Cartan loops using the previously derived decomposing formulas, in particular, Eqs. (33) and (34). Written in terms of the Cartan-Polyakov loop set, the Haar measure for N c = 5 has 106 different symmetry conserving interaction terms. For N c = 7 and N c = 9 this number increases to 3589 and 164856, respectively. The reason for this rapid increase in the number of terms can be traced back to combinatorics: increasing N c increases the number of elements inside the Cartan set, which in turn, boosts the number of possible Z(N c ) invariant combinations of Cartan-Polyakov loops.

The thermodynamic potential at arbitrary Nc
Using the results obtained so far, we are finally able to write a thermodynamic potential for the PNJL model, in the mean field approximation, for an arbitrary number of colors: As already stated, C Nc , is the set of Cartan-Polyakov loops, the coefficients c k (−L F ) and c k (−L † F ) can be calculated with Eq. (22) and the effective Polyakov loop potential is given by Eq. (60). The mean field potential, U ( φ i ) can also be N c dependent. For instance, in the NJL model the four scalar-pseudoscalar quark-quark interaction can be considered as being inversely proportional to N c due to the QCD counting rules [76]. As already discussed, the values of the mean fields, φ i , and of the Cartan-Polyakov loops, C Nc , can be obtained by deriving the gap equations of the model through: dΩ/d φ i = dΩ dC i Nc = 0. It is important to note that the fermionic part of the thermodynamic potential written above, by construction, only depends on Cartan-Polyakov loops. However, the effective Polyakov loops potential, U eff (C Nc , T ), contains the Haar measure which depends explicitly on non-Cartan-Polyakov loops, in the fundamental representation, with charges up to N c − 1 (see Eq. (71)). Of course, as already discussed, one can make use of the decomposing formula given in Eq. (20) (or Eq. (25)) in order to write the Haar measure in terms Cartan-Polyakov loops only. However, when dealing with the mean field approximation, the values of the Cartan-Polyakov loops are fixed by solving the gap equations, dΩ dC i Nc = 0. The derivative of the Haar measure with respect to an element of the Cartan-Polyakov loop set, ℓ F,k (the same can be applied to derivatives with respect to ℓ F,k ), is: Here, we used Eq. (71) without the N Nc c coefficient and dW /dℓ F,k is a tangent matrix, whose matrix elements are derivatives of Polyakov loops, in the fundamental representation, with charges 1, . . . , N c − 1, with respect to ℓ F,k (or ℓ F,k ). Explicitly, it is given by: When differentiating a Cartan-Polyakov loop, the result is a delta function. However, this matrix contains non-Cartan-Polyakov loops with charges in the interval (N c + 1)/2, . . . , N c − 1. For these cases one can calculate these derivatives using:  74) can be a good starting point to write down a thermodynamic potential for the large N c limit. Such potential could then be used to study the phase diagram of strongly interacting matter at large N c , and the quarkyonic phase of matter. We leave both aforementioned studies, the numerical analysis and the large−N c model, as future research endeavors.
We would like to highlight that the formalism employed here, to define the PNJL for different values of N c , can also be applied to other fermionic models as long as the thermodynamic potential can be written in a similar way to the one given in Eq. (44).

B. Glue effective potential at arbitrary Nc
In this section we apply the concept of Cartan-Polyakov loops to extend the one-loop gluon effective potential to arbitrary values of N c . The starting point is the usual one-loop contribution to an effective potential of gluons with a spatially uniform Polyakov loop [56,[77][78][79]: Here, E g = p 2 + M 2 g , with M g a possible gluon effective mass [56,79], and L A the thermal Wilson line in the adjoint representation of the SU(N c ) group (see Eq. (4)). This contribution to the effective glue potential drives the system to a state of spontaneously broken Z(N c ) symmetry [43]. In order to study the transition from a Z(N c ) symmetric state to the broken state, one can consider a confining contribution to the potential in the form of the Haar measure [43] which, as previously discussed, can be written in terms of the Cartan-Polyakov loop set, C Nc . Hence, the total effective glue potential, Ω, is given by: Here, k(N c , T ) is to be considered a phenomenological parameter, similar to the one introduced in the PNJL model at arbitrary N c , B(N c , T ) (see the discussion in section IV A 2). This result was also derived for the N c = 3 case in Ref. [56] where the the background field approximation was applied to the generating functional for the SU(3) Yang-Mills theory, in the presence of a uniform gluon field. This model was also considered in Ref. [43], in order to study fluctuations of the order parameter in an SU(N c ) effective model.
The strategy to extend this model is the same used for the PNJL model. Here however, the confining part of the potential written in Eq. (82) was already defined for an arbitrary number of colors. Indeed in section IV A 2 we discussed how to write the Haar measure using only the elements present in the Cartan-Polyakov loop set.
The final step is to evaluate the gluonic determinant, present in Eq. (81), for an arbitrary number of colors. The gluonic determinant is defined by: Here, N A = N 2 c − 1, is the dimension of the adjoint representation and we have defined the non-zero quantity: As performed for the fermionic determinant, the goal is to calculate the above determinant for an arbitrary N c using only the corresponding Cartan-Polyakov loop set. To this end, consider the characteristic polynomial p L A , h −1 g (see Eq. (10)). We can write the gluonic determinant as: Where we considered that det [L A ] = 1. The coefficients, c k (L A ), which arise in the sum, are calculated using Eq. (11) with the appropriate choice of representation. Observing the structure of the matrix J A,k , defined in Eq. (12), one can conclude that the k coefficient will depend on the non-Cartan-Polyakov loops ℓ A,k , with charges 1, . . . , k. Since we want to write the model in terms of Cartan-Polyakov loops, we must find a way to write the non-Cartan-Polyakov loop of charges k, in the adjoint representation, ℓ A,k , in terms of Cartan-Polyakov loops.
In order to accomplish this one can use some results from group theory.
A known result from representation theory allows for the decomposition of the product between the fundamental and the conjugate representations of the SU(N c ) group (F and F, respectively) with its adjoint and the singlet representations [80]: This result can be proven using Young tableaux or by sandwiching a fundamental thermal Wilson line the antifundamental between two SU(N ) generators in order to get an object with adjoint indexes [38]. Since powers of the thermal Wilson line, L k , are elements of the SU(N c ) group, we can write the following identities using character theory [79]: tr L k ρ⊗σ = tr L k ρ tr L k σ .
Where ρ and σ are different representations of the SU(N c ) group. Using these properties alongside the decomposition made in Eq. (86), one can write: Thus, one can write the Polyakov loop with charge k, in the adjoint representation, in terms of Polyakov loops with charge k, in the fundamental and antifundamental representations. It is given by [33,38]: As mentioned for the non-Cartan loops in the fundamental representation, this relation can also be checked by hand by considering a particular gauge choice, e.g., using the already mentioned Polyakov loop gauge [56].
Here, we have written the coefficients first in terms of charged Polyakov loops in the adjoint representation and then used Eq. (90) alongside the non-Cartan decomposing formulas, in order to write the coefficients in terms of Cartan-Polyakov loops. These results agree with the ones present in the literature, see [56]. For N c = 5, one has to calculate twelve coefficients, the first three are given by: One can immediately observe that the complexity of the coefficients written in terms of Cartan-Polyakov loops rapidly rises for high order coefficients and with the number of colors. Indeed, the c 12 (L A ) coefficient, for example, has 104 different terms when written in terms of the Cartan-Polyakov loop set. In practice, for numerical calculations, the overall size of the analytical expressions for a given coefficient at fixed N c is not important. In such approach, using as variables the Cartan-Polyakov loops, one can build all non-Cartan-Polyakov loops using the decomposing formulas and then use Eq. (90) to directly obtain different charged Polyakov loops in the adjoint representation and then get the coefficients.
Furthermore, since we are dealing with a gluonic theory, the coefficients only have terms which preserve Z(N c ) symmetry. Naturally, the opposite was found for the fermionic coefficients where the terms explicitly broke the Z(N c ) symmetry.
Finally, one can write the glue effective potential defined in Eq. (81), for any value of N c using only Cartan-Polyakov loops. It is given by: It is known that the mean field approximation is not suitable to study gluon effective potentials written in terms of fundamental and anti-fundamental Polyakov loops [74]. The reason for such can be understood in the basis of the relation between the adjoint and the fundamental loops, written in Eq. (90). In the naive mean field approximation, considered in this work, one can see that in the confined phase, where the expectation values of charged Polyakov loops in the fundamental representation are zero, the expectation value of charged adjoint Polyakov loops are negative [74]. As pointed out in [33], this implies that charged Polyakov loops in the adjoint representation, ℓ A,k , cannot be used as an order parameters for gluon confinement. Also, the thermodynamic potential obtained in Eq. (98) leads to a negative pressure, negative entropy density and thus, to thermodynamic instability [56,74]. This was discussed in [56], using the glue model of Eq. (81) for N c = 3, by considering the low temperature expansion of the effective action. In such limit, the last term inside the logarithm, (−1) (N 2 c −1) exp −(N 2 c − 1)βE g , prevails and the incorrect sign for Bose statistics in front of the exponential is obtained for N c = 3 [56]. To overcome this feature of the glue effective potential for N c = 3, in the same study, the authors proposed an hybrid model of glueballs in the confined phase and Polyakov loops in the deconfined phase, with the transition between models occurring at the transition temperature, for more details see [56]. We point out an interesting feature of the arbitrary N c effective potential written in Eq. (98): the incorrect Bose statistics sign is obtained for odd number of colors, for an even number of colors the sign would be the one expected for a system of bosons.
Although one can use the approach used here to extended glue effective potentials to arbitrary values of N c , as discussed, the simple mean field approach might not be sufficient and corrections coming from connected diagrams in Eq. (90), might be necessary [33]. However, at large N c , these correction become negligible since expectation values factorize [38]. Thus, in this limit, the relation between the adjoint loops of charge k and the fundamental and anti-fundamental loops of charge k become much simpler: In fact, at large N c , the expectation value of any Polyakov loop can be written as powers of the expectation value of the fundamental and anti-fundamental Polyakov loops [38]. Hence, at large N c this model is a proper tool to study the behavior of the deconfining transition of strongly interacting matter.
As a final note, we point out that this model could be used to extend the PNJL model at arbitrary N c . For instance, instead of using the Haar measure and the kinetic term proposed in Eq. (59) for the effective Polyakov loop potential in the PNJL model, one could use the above effective glue potential. Indeed, one can show that for N c = 3 under certain assumptions, it is possible to recover from Eq. (98) the −ΦΦ used in the usual N c = 3 PNJL model, which is essential in order to get a first-order phase transition [56]. Not only that, using this effective glue potential for the gluonic sector of the arbitrary N c PNJL would remove the necessity of defining so many N c −dependent phenomenological parameters that are present in the effective Polyakov loop potential defined in Eq. (60). In such case, the only parameter coming from the gluonic sector would be the one in front of the Haar measure [56]. Of course, the problem of the negative contribution to the pressure in the confined phase, for ℓ F,k = 0, would have to resolved [43,56].

V. CONCLUSIONS
In this work we defined Cartan-Polyakov loops, a special subset of Polyakov loops in the fundamental representation of the SU(N c ) group which, under Z(N c ) transformations, behave like fields with charges k = 1, . . . , (N c − 1)/2. This set can be used as independent degrees of freedom to parametrize effective models of quarks and gluons with increasing number of colors. When building such phenomenological models, non-Cartan-Polyakov loops, Polyakov loops with charges higher than (N c − 1)/2, naturally arise. In order for these models to be written as functions of the loops contained in the Cartan set, we had to derive a decomposing formula for non-Cartan-Polyakov loops. Such formula was derived using properties of the characteristic polynomial of unitary matrices and the Cayley-Hamilton theorem. Fixing some values of N c , we consistently decompose some non-Cartan loops until they were only functions of Cartan-Polyakov loops. Finally, we showed how to build two distinct effective models for increasing N c , including a version of the PNJL model and an effective glue potential. These models were built using the naive mean field approximation commonly used when dealing with the PNJL model: the Cartan-Polyakov loops were directly substituted by their respective expectation values and the expectation values of products between them factorized.
The formalism developed in this work, allows for a consistent study of increasing the number of colors in effective models of quarks and gluons. Numerically, one can build tools that, for a given N c , automatically builds the Cartan-Polyakov loop and calculate all the necessary non-Cartan-Polyakov loops. In the case of the PNJL model, for example, one can build numerical tools in order to solve the model, within the mean field approximation, for a fixed value of N c . Having the solution of such model, one can calculate the phase diagram of the model for that particular value of N c , especially the behavior of the critical endpoint and the interplay between the chiral transition and the deconfinement transition. One can then repeat the calculation to increasing values of N c and study the behavior of the critical endpoint for large values of N c , as well as study the conjectured quarkyonic phase of matter at high chemical potentials and moderate temperatures. Indeed, in Ref. [28], the fate of the critical endpoint at large N c was studied using a Polyakov loop quark-meson model by varying the number of colors. Within this work it was also confirmed the existence of the quarkyonic phase, in which chiral symmetry is restored but quarks are still confined [28]. It would be very interesting to see if the similar features were obtained within the model proposed in this work. The glue effective model defined at arbitrary N c can also be used to study fluctuations of the confined-deconfined order parameters, given by different susceptibilities of the Polyakov loops and the curvature masses associated with the Cartan-Polyakov loop set, for any value of N c . As discussed in [43], these observables can provide useful information about the link between the vacuum structure and gluon properties.
As future work, we plan to use the formalism developed in this paper in order to build numerical tools which are prepared to build the PNJL model at any given N c . As pointed out, this can be accomplished recursively with the goal to study the behavior of the critical endpoint with increasing N c , without further approximations, except the ones introduced in this work. At the same time, we will build an effective PNJL model at large N c . In this limit, we plan to study the phase diagram of the model and the properties of the expected quarkyonic phase at large baryon densities. Having a numerical calculation together with a large N c approximation will allow us to compare both approaches and better understand the behavior of the strongly interacting matter phase diagram at large N c .
Using this definition and the properties of the unitary matrix M , one can write the polynomial p(M, λ) as follows: Here, we assumed that λ is non-zero and defined λ = λ −1 . The determinant on the right hand side of this equation, det M † − λI , can also be written in terms of a characteristic polynomial, p M † , λ . It yields: Comparing this polynomial, with the polynomial in Eq. (A5), one can establish the following identities: Where we used Eq. (A3). We highlight that N is the dimension of the unitary matrix M and c j (M ) is calculated from Eq. (A4).
The total derivative of this non-Cartan loop, ℓ R,a , with respect to the Cartan-Polyakov loop ℓ R,b , is given by: As expected, this formula also works recursively: in order to get dℓ R,a /dℓ R,b one needs to first calculate all the total derivatives of non-Cartan loops with smaller charges, dℓ R,i /dℓ R,b , with i ∈ { (Nc+1) /2, . . . , a − 1}. Likewise, one can write the total derivative of a non-Cartan loop with respect to some adjoint Cartan loop, dℓ R,a dℓ R,b or dℓ R,a dℓ R,b and dℓ R,a dℓ R,b . They are given by: The different partial derivatives necessary to evaluate the total derivatives, can be obtained by partially differentiating the decomposing formulas derived in this work. If the charge a is smaller than N R , the partial derivative can be calculated using the decomposing formula of Eq. (20) (a similar calculation can be done using Eq. (25) as a starting point) while, for charges a ≥ N R , one can calculate the partial derivative using Eq. (28). Lets first deal with the case a < N R . Calculating the partial derivative of Eq. (20), with respect to the Cartan-Polyakov loop ℓ R,b , yields: We also used the fact that ∂ ∂ℓ R,b det J R,NR−a = 0, since the matrix J R,NR−a only depends on adjoint Cartan-Polyakov loops. We recall that in the notation used in throughout this work, the subscript (i,1) in the matrix J (i,1) R,a is equivalent to the the matrix J R,a with the i−th line and first column removed. The matrix ∂JR,a ∂ℓ R,b can be found very easily by taking into account its definition given in Eq. (12). It is given by: With these tools one is able to evaluate the total derivatives of quantities which depend explicitly on non-Cartan-Polyakov loops, for some fixed number of colors, as is the case with the Haar measure (see Eq. (71)). Hence, using these tools, one can readily evaluate the derivative of the Haar measure with respect to some element of the Cartan-Polyakov loop set.