Schrödinger functional boundary conditions and improvement for N >3

The standard method to calculate non-perturbatively the evolution of the running coupling of a SU(N ) gauge theory is based on the Schrödinger functional (SF). In this paper we construct a family of boundary fields for general values of N which enter the standard definition of the SF coupling. We provide spatial boundary conditions for fermions in several representations which reduce the condition number of the squared Dirac operator. In addition, we calculate the Oa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O}(a) $$\end{document} improvement coefficients for N >3 needed to remove boundary cutoff effects from the gauge action. After this, residual cutoff effects on the step scaling function are shown to be very small even when considering non-fundamental representations. We also calculate the ratio of Λ parameters between the MS¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\mathrm{MS}} $$\end{document} and SF schemes.


Introduction
Asymptotically free theories, such as gauge theories coupled to fermionic matter fields, 1 are characterized by having a coupling which becomes small at short distances. This property enables reliable perturbative calculations of physical quantities at large energies. A dimensionful scale is dynamically generated through the process of dimensional transmutation. Typically, this scale is associated in perturbation theory with the Λ parameter, i.e. a multiplicative constant of the integrated beta function.
The non-perturbative evolution of the running coupling in different gauge field theories from the low energy sector to the high energy regime has been the central goal of many studies. The standard approach is the use of a finite size scaling technique based on the Schrödinger functional (SF), in which the size of the system is associated to the renormalization scale [1]. This method was successfully used to calculate the scale evolution of the coupling in the SU(2) [2] and SU(3) [3] Yang-Mills theories and in QCD [4]. Motivated by ideas of physics beyond the standard model (BSM), in the last decade this method has JHEP11(2014)074 also been applied to study the SU(4) pure gauge theory [5] and several theories containing matter transforming under higher dimensional representations of the gauge group or a large number of fermions in the fundamental representation [6][7][8][9][10][11][12][13][14].
However, for lattices accessible in typical numerical simulations SF schemes are affected by lattice artifacts arising from the bulk and from the boundaries of the lattice. These can be removed, following Symanzik's improvement program, by adding the corresponding counterterms to the action at the bulk and the boundaries. Symanzik's program was successfully carried out in [1,[15][16][17][18], where the improvement coefficients necessary to remove O(a) effects from the coupling were calculated in perturbation theory.
For theories beyond QCD the situation is still inconclusive. A program for the nonperturbative study of SU(N ) gauge theories in the large N limit [19] started in the last decade driven by interest from string theory. As part of that program, in [5] the ratio Λ MS / √ σ between the lambda parameter in the MS scheme and the string tension σ was calculated for the SU(4) theory aiming to obtain extrapolations of the N dependence of the Λ MS / √ σ in the large N limit. There, the dominant systematic errors are due to the lattice artifacts present by using an unimproved action. For the case of theories with non-fundamental fermions, although the O(a) improvement coefficients are known, the remaining higher order cutoff effects have been reported to be very large if the standard setups, which work fine for QCD, are naively exported.
In the last few years a new renormalized coupling based on the gradient flow (GF) has been proposed for step scaling studies [20][21][22][23][24]. Compared to the original SF coupling based on a background field (see bellow), the gradient flow coupling has the advantage that considerably smaller statistics are required for obtaining a similar accuracy.
However, there are some situations where the original SF coupling is superior compared to the gradient flow. First of all, it has been observed that while the GF coupling works better at large physical volumes, at small volumes the SF coupling fares better than the gradient flow [25]. Also, in the pure gauge theory, relevant for the large N limit, the generation of configurations is so fast compared to the measurement of the gradient flow that the reduced accuracy can be overcome with increased statistics. In addition, in BSM lattice studies one is often interested in the existence of a nontrivial infra-red fixed point. The value of the coupling constant g at the fixed point is a renormalization scheme dependent quantity and it differs between Schrödinger functional and gradient flow schemes. Therefore, it is possible that in a specific scheme the coupling is too strong at the fixed point or it is on the wrong side of a bulk phase transition. This is true even if the fixed point is visible in other schemes. The only study, we are aware of, that compares these two methods with the same action found the gradient flow coupling to be about twice the Schrödinger functional coupling [26]. Moreover, due to the property of continuum reduction [27], at large N it is possible to do simulations at small lattice volumes where the SF coupling is known to perform well.
This work completes the Schrödiger functional framework to study the phase diagram of strongly interacting gauge theories [28] with any N or representation. In the paper we generalize the boundary conditions for the gauge fields in the SF to obtain a family of schemes useful for arbitrary N with a good signal to noise ratio in lattice simulations.

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Moreover, the O(a) improvement coefficients are obtained to one loop order in perturbation theory. For this, we calculate the one loop running coupling in our family of SF schemes following closely the discussions in [1,15] and adapting them to arbitrary N . The values obtained for the boundary improvement coefficients are valid for any choice of Dirichlet boundary conditions at the temporal boundaries. With this knowledge we relate the Λ parameters between our SF schemes and the more widely used MS scheme. Another appealing property of the present family of schemes is that, together with an appropriate choice of spatial boundary conditions for the fermions, they lead to a setup for which higher order cutoff effects due to fermions are very small even for non-fundamental representations. Preliminary results of this work have been published in [29].
The paper is organized as follows: in section 2 we recall some concepts concerning the Schrödinger functional and collect a set of formulas useful for the remaining discussion. In section 3 the generalized boundary conditions are provided. The calculation of the improvement coefficients is presented in section 4, where we also discuss the effect that the fermionic spatial boundary conditions have on the residual higher order cutoff effects. The matching of the Λ parameters to the MS scheme is done in section 5. We conclude in section 6.

Schrödinger functional
In this section we briefly recall the ideas introduced in [1,15,30] and collect the expressions necessary for the subsequent discussion. We refer the interested reader to the original articles for further detail.
The Schrödinger functional is the euclidean propagation amplitude between a field configuration C at time 0 and another field configuration C at time T , which has a path integral representation given by with Dirichlet boundary conditions specified for the gauge fields U and the fermion fields ψ andψ.
In the present work, we are interested in the O(a) improved Wilson action The pure gauge part is the standard SU(N ) Wilson gauge action The spatial components of the gauge fields at the temporal boundaries (t = 0 and t = T ) satisfy nonhomogeneous Dirichlet boundary conditions

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The boundary gauge fields W k and W k can be parametrized as W k (x) = exp(aC k (η)), W k (x) = exp(aC k (η)), (2.5) where C k (η) and C k (η) are taken to be homogeneous, abelian and spatially constant [1], and they depend on a dimensionless parameter η. A specific form for these boundary matrices is derived in section 3 for gauge group SU(N ) with arbitrary N . In the spatial directions the gauge fields are taken to be periodic U µ (t, x) = U µ (t, x + Lk).
The weight w(P ) = 1 except for the spatial plaquettes at the boundaries for which w(P ) = 1 2 . Due to the particular choice of boundary conditions for the gauge fields, the spatial boundary plaquettes give only a constant contribution to the action and can be ignored. It is well known that within SF schemes, the mere presence of temporal boundaries constitutes an extra source of lattice artifacts. Removal of these effects has first been studied in [1,16,17], where it was shown that the O(a) lattice artifacts coming from the boundaries can be canceled by tuning the weight w(p) = c t (g 0 ) for the temporal plaquettes attached to the boundaries, where c t is the coefficient of a dimension 4 counterterm localized at the boundaries [1]. The perturbative expansion of c t is where c where D WD is the improved Wilson-Dirac operator The operator F µν (x) is the symmetrized lattice field strength tensor, σ µν = i 2 [γ µ , γ ν ] and the operators D µ and D * µ are the covariant forward and backward derivatives which are defined in eq. (A.8). The improvement coefficient c sw can be determined perturbatively [16,31] and non-perturbatively [32,33]. To the lowest order in perturbation theory c sw = 1 [18]. The removal of O(a) effects arising from the interplay between fermions and the SF boundaries requires the addition of another dimension 4 counterterm at the boundaries. Since this does not contribute to the observables studied further in this work at the present order in perturbation theory, we ignore it from now on and refer the reader to the original literature [17] for further details.
The fermionic fields satisfy the following boundary conditions where P ± = 1 2 (1 ± γ 0 ). The boundary conditions in the spatial directions are periodic up to a phase [30]: (2.11)

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The phase is usually chosen so that the smallest eigenvalue of the squared Dirac operator is large [30]. In this situation, the condition number (i.e. the ratio between the highest an lowest eigenvalues) is small, which improves the speed of the known inversion algorithms. However, the value of θ also has an effect on the convergence of the 1-loop perturbative coupling to its continuum limit. This is discussed in subsection 4.3. The boundary conditions for the gauge fields in eqs. (2.4) and (2.5) induce a constant chromo-electric background field V µ (x) in the space-time. The variable η in the boundary fields eq. (2.5) parametrizes a curve of background fields. A renormalized coupling can be defined [1] as a response of the system to a deformation of the background field with the effective action Γ = − ln Z. The normalization constant is defined so that g 2 = g 2 0 to the lowest order of perturbation theory. One of the central quantities in numerical simulations is the step scaling function (2.14) This is required for reconstructing non-perturbatively the scale evolution of the running coupling. In presence of a lattice regulator, the deviations of the lattice counterpart of the step scaling function Σ(u, L/a) from the continuum σ(u) can be used to monitor the size of cutoff effects (see subsection 4.3).

1-loop expansion
The renormalized coupling eq. (2.12) is suitable for both perturbative and non-perturbative evaluation. The 1-loop calculation of eq. (2.12) was done in [1,3] for the pure gauge theory in SU (2), and extended to accommodate fermions in [30]. 2 Non fundamental fermions have been considered in [34][35][36]. In the present work we extend the previous calculations to arbitrary N . Although the main strategy of the calculation follows closely previous works, some care has to be taken to generalize those ideas without complicating the calculation.
In the present subsection we collect some formulas necessary for the subsequent discussion and leave all technical details on the calculation to appendices A and B. After going through the gauge fixing procedure [1], the effective action is expanded to 1-loop as Here Γ 0 is the classical action. The 1-loop term Γ 1 in the effective action can be written as

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where ∆ 0 , ∆ 1 and ∆ 2 are the quadratic ghost, gluonic and fermionic operators respectively. The explicit forms of the operators ∆ i are given in appendix A. The renormalized coupling eq. (2.12) is also expanded in perturbation theory According to eq. (2.15), the 1-loop coefficient receives an independent contribution from ghost, gauge, and fermionic fields The gauge and fermionic contributions to eq. (2.19) can be calculated independently. The gauge part is given by The calculation of h 0 (L/a) and h 1 (L/a) has been described in great detail for SU(2) in [1] and the calculation has been done for N = 3 in [3]. In appendix A we give the generalization of the calculations to N ≥ 3. The calculation of the fermionic part p 1,1 (L/a) is straight forward to generalize to any boundary fields and to any representation of the gauge group. One just needs to replace the link variables in the Wilson Dirac operator eq. (2.8) with their counterparts in the desired representation. Thus we will refer the interested reader to the original paper [15].
The continuum and lattice step scaling functions are given to first order in perturbation theory by with σ 1 = 2b 0 ln(2). The 1-loop coefficient b 0 of the beta function is given in an arbitrary representation by where the color group invariants are defined as in the representation R of SU(N ).

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Similarly as in eq. (2.19), the step scaling functions can be separated into a gauge and a fermionic part, This allows us to study separately cutoff effects due to gauge and fermion fields independently.

Boundary fields for N > 2
In this section we present a generalization of the boundary fields for N > 2. The selection of the boundary fields is only limited by the requirement that there is a unique and stable classical solution to system. In practice, this limits us to Abelian boundary fields W k and W k which can be written as in eq. (2.5), where Now the classical solution, i.e. the background field, can be written as It is shown in [1] that the solution V µ (x) is absolutely stable if the vectors φ and φ satisfy eq. (3.2) and These conditions define a fundamental domain, which is an irregular (N − 1)-simplex and has vertices at points . . . (3.11) To define a renormalized coupling we can choose any two different points inside the fundamental domain to set up the boundary fields. A different choice leads to a different renormalization scheme, which can be matched to each other using perturbation theory (see section 5). However, there are practical considerations in selecting the boundary fields, namely the signal to noise ratio in the Monte Carlo simulations and the size of higher order lattice artifacts. Our choice is based on the attempt to maximize the signal to noise ratio as in practice the minimization of the higher order lattice artifacts often leads to a low signal, which neglects the gains of a better continuum extrapolation.
To obtain a maximal signal strength we have two competing requirements. We need to twist the gauge fields as much as possible while staying away from the boundaries of the fundamental domain. This is because the coupling is proportional to the twist and because closeness of the instability of the classical solution increases noise. According to these considerations we choose φ to be in the middle of a line connecting X 1 and the centeroid of the fundamental domain To determine φ we find a transformation which is a map from the fundamental domain to itself and mirrors the vertices. First we define a simple map R i,j (φ) that reflects the points in the fundamental domain with respect to a (N − 2) dimensional hyperplane. The hyperplane R i,j (φ) goes through vertices X k , k = i, j and intersects the line connecting X i and X j at the middle. For N > 3 the function R i,j (φ) is not in general a mapping from the fundamental domain to itself, but we can define a composite mapping is a mapping from the fundamental domain to itself and written in components it has a simple form To define the coupling we choose a one parameter curve of background fields φ + t(η). We select it in a way that the results are equivalent to those of the SU(3) theory given in [3], 4 i.e. we select t(η) so that it changes sign under the mapping M(φ) and points JHEP11(2014)074  In the lattice computations it is advisable not to set t(η) beforehand, but to measure a complete N − 1 dimensional basis which can be used to construct a generic curve. Each curve corresponds to a different renormalization scheme.

Fermionic spatial boundary conditions
Recalling the spatial boundary conditions for the fermion fields eq. (2.11), we still have to choose a particular value for the angles θ k . For simplicity, we consider the same angle in all spatial directions θ = θ k , k = 1, 2, 3.
We then fix θ, following the criteria introduced in [15], so that the minimum eigenvalue λ min of the fermion operator ∆ 2 is as large as possible. This leads to a small condition number which optimizes the speed of the numerical inversion of the operator.
The values of θ leading to a maximum λ min depend on the background field and also on the fermion representation being considered. For the fundamental representation, the profile of smallest eigenvalues λ min as a function of θ is shown in figure 2 for the different gauge groups considered in this work excluding the case of SU (2).
Although the maximum of λ min is achieved at different values of θ for every gauge group considered, the choice θ = π/2 is always close to the maximum and hence leads to JHEP11(2014)074   a small condition number. For homogeneity in the definition of a renormalization scheme in the subsequent calculations, we will fix θ = π/2 for all values of N . As we will show in subsection 4.3, this choice of θ together with the family of background fields defined in this work will lead to a setup for which higher order cutoff effects are highly suppressed even for non-fundamental representations. Although the choice of θ = π/2 is taken considering the fundamental representation, this value also leads to reasonably small condition numbers for the symmetric and antisymmetric representations. For the adjoint representation the smallest condition number is obtained for θ = 0. However, we decide to stick to the choice θ = π/2 also in this case since it leads to a situation were higher order lattice artifacts are highly reduced. 5 For the case of SU(2), we choose θ = 0 for the fundamental representation but leave θ = π/2 for the symmetric/adjoint. In order to extract the coefficients r n,i as accurately as possible we first evaluate p 1,0 (L/a) and p 1,1 (L/a) adapting the strategies in [1,15] to general N (see appendices A and B for details on the calculation). Once the series of data for p 1,i (L/a) is produced, the coefficients r n,i can be extracted using a suitable fitting procedure.
In the pure gauge case, we calculated p 1,0 (L/a) for values of L/a ∈ {6, 8, . . . , 100} and then used the "Blocking" method described in [39] to obtain the values of the asymptotic coefficients. The calculation was done using floating point precision with 50 decimal places for 2 ≤ N ≤ 8 and with quadruple precision for N > 8. To control the error we compared the results and errors obtained with different level of accuracy. Since the asymptotic form eq. (4.1) is expected to be valid as a/L → 0, we consider only values of L/a ⊂ [28,100] when extracting the coefficients r 0,0 and r 0,1 . This choice produced the most reliable values for the coefficients and their relative errors. As a check we also reproduced the known value of s 0,0 = 2b 1,0 to a similar degree of accuracy.
Concerning the fermionic part, values for p 1,1 (L/a) were produced at quadruple precision in the range L/a ⊂ [4, 64] (for even and odd values) for all gauge groups and representations considered in this work. This was enough to obtain the asymptotic coefficients in eq. (4.1) to very high precision (see tables 2 and 3).

Results
In table 2 we give the values for the coefficients r 0,0 and r 1,0 . From r 1,0 we can extract the gauge contribution to the boundary improvement coefficient c where C 2 (R) is the quadratic Casimir operator in the representation R. For completeness, we also include here the value of the fermionic part c This was calculated for the fundamental representation in [15] and later extended to other represesentations in [35]. In the present work we have been able to reproduce the value of c (1,1) t with similar accuracy, which is a further check on the correctness of the whole calculation.

Residual cutoff effects
The determination of the gauge and fermion contributions to c       This is true for the family of background fields defined in section 3 and for the value of θ chosen in subsection 4.1. A different choice of parameters, however, can lead to very high residual cutoff effects even after boundary O(a) improvement is implemented. 6 In order to check this, we study the dependence of δ 1,1 on the parameter θ for different values of N in a range θ ⊂ [0.45π, 0.57π]. The cases of N = 3 and 6 are displayed in figure 6. Other gauge groups show a very similar behavior. The residual cutoff effects δ 1,1 depend strongly on θ. Clearly, a poor choice of θ might lead to situations with very large higher order cutoff effects. 7 It is remarkable that the value θ = π/2, established in subsection 4.1 to obtain a condition number as small as possible, also leads to a situation where higher order cutoff effects are highly suppressed.
A very similar picture is observed when considering any of the 2-index representations.     large if particular care is not taken in the choice of BF [34][35][36]. The magnitude of δ 1,1 for the 2-index representations strongly depends on the angle θ in a very similar way as it is shown in figure 6 for the fundamental representation. It is then possible to tune θ to minimize cutoff effects without the need of modifying the BF [35,36]. What is remarkable of the family of background fields proposed in this work is that for the fundamental, symmetric and antisymmetric representations, values of θ which lead to small condition numbers also lead to small higher order lattice artifacts in the step scaling function. This is not true for the adjoint representation since, as discussed in subsection 4.1, the condition number is minimized for θ = 0. It is also remarkable that cutoff effects for all the representations considered are minimized for the same value θ = π/2.

Matching the Λ parameter to MS
In this section we calculate the relation Λ SF /Λ MS of Λ parameters in our family of SF schemes and the MS scheme. This relation is essential for obtaining the ratio Λ MS / √ σ from SF simulations. We provide numerical values of Λ SF /Λ MS for the pure gauge theories and for the theories with 2 fundamental fermions. For completeness, we derive an expression (see eq. (5.12)) for the ratio Λ SF /Λ MS as a function of N , N f and the representation R, which might be useful also for future BSM studies using the SF.
The Λ parameter is a renormalization group invariant and scheme dependent quantity given by (in a generic scheme X) It is a dimensionfull scale dynamically generated by the theory. In subsection 4.2.2 we have performed the computation of the SF coupling 8 g SF in the Schrödinger Functional scheme to one loop order in perturbation theory, i.e. we have calculated the renormalized coupling as an expansion in terms of the bare coupling g 0 g 2 SF (L) = g 2 0 + p 1 (L/a)g 4 0 + O(g 6 0 ), (5.2) where, after doing a continuum extrapolation, To be able to compare the results at different values of N , we are interested in a relation between α SF = g 2 SF /4π and some scheme where N is only a parameter. For that we choose the usual MS scheme, defined at infinite volume and at high energies. The relation between the running coupling in the two schemes can be written as an expansion where s is a scale parameter and c i (s) are the coefficients relating the couplings in the two schemes at each order in perturbation theory.

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The relation between the Λ parameter in the SF and MS scheme is given by where c 1 (1) is the coefficient of the 1-loop relation (5.4). Note that eq. (5.5) is an exact relation even though it depends on the 1-loop coefficient relating the couplings in two different schemes. For determining the coefficient c 1 (s) in eq. (5.4), we first use the known relation between α MS in the MS and the bare coupling α 0 [17,30,33,40], which at 1-loop is given by The 1-loop coefficient d 1 (s) is given for generic N and fermionic representation R by where The coefficient k 1 of the gauge part is taken from [17,30,33] and reads The coefficient K 1 is a representation independent function of the tree level coefficient c  with r 0,i being the continuum coefficients in the series (4.1). Knowing this, the relation between Λ parameters in eq. (5.5) can be given as a function of the parameters N , N f and T R and of the coefficients r 0,i Finally, in table 5 we collect the values of the ratio of Λ parameters for the schemes studied in this work and for the pure gauge theory (N f = 0) and for 2 flavors of fundamental fermions. 9 Ratios of lambda parameters for 2 index representations can be recovered using eq. (5.12) and the corresponding coefficients from table 3.  Table 5. Ratios between Λ parameters in the SF and MS schemes, for the pure gauge theory and for 2 flavors of fundamental fermions.

Conclusions
We have studied the Schrödinger functional boundary conditions and the perturbative O(a) improvement for SU(N ) gauge theories with general N . The improvement coefficient c is obtained also for all values of N . Additionally we provide the matching between the SF and MS schemes for a wide range of theories including fermions in various representations. This enables a precision study of the coupling and the determination of Λ MS in the large N limit. The fermionic twisting angle θ is also studied and we found out that the value θ = π/2 is a good compromise between the simulation speed and the minimization of the O(a 2 ) lattice artifacts in the perturbative 1-loop lattice step scaling function.

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of different variables with our choice of background field and basis for SU(N ) generators are shown in appendix B. In the following we will work in lattice units i.e. the lattice spacing a = 1. Additionally repeated Latin indices a, b, c, . . . are not summed over and repeated Greek indices α, β, γ, . . . are always summed over unless otherwise stated in the formula. Latin indices run from 1, 2, 3 and Greek ones from 0, 1, 2, 3.
The operators we are interested in are defined as There is no summation over µ in the r.h.s. of the eq. (A.2). The star product in eq. (A.2) which maps an N ×N matrix M and an SU(N ) matrix X to an SU(N ) matrix is defined as In eq. (A.3) the operator D WD is the same as in eq. (2.8) with c sw = 1. The first step is to find a suitable basis for the SU(N ) generators. This is a basis that is invariant under the star product defined in eq. (A.4). In practice we want to find generators X a that satisfy cosh G 0k X a = χ c a X a , sinh G 0k X a = χ s a X a , with arbitrary coefficients χ c a and χ s a . The hyperbolic sine and cosine of the non-zero elements of the field strength tensor are A basis that satisfies eq. (A.5) for the non-diagonal generators are the ladder operators defined as where n and m are the matrix indices and a(j, k) is the color index. The properties of a(j, k) are given in table 6. The generators X a are normalized as Tr X a X b = − 1 2 δ a,b . The diagonal generators can be chosen in any way that satisfies eq. (A.5).
The boundary conditions generate a background field V µ (x) defined in eqs.   Table 6. The values of the color index a(j, k) as a function of the dummy indices j and k. When 1 ≤ a ≤ (N 2 − N )/2 the generators X a have a non-zero element in the upper and for (N 2 − N )/2 < a ≤ N 2 − N in the lower triangle.
and through them to the operators ∆ s . The next step is to calculate the covariant derivatives with the background field V µ (x) when q(x) = q a (x)X a is proportional to a generator X a . The covariant derivatives can then be written in a general form is the three momentum. Next we will show how one can calculate the determinant of operators ∆ s . In [1] it has been shown that for an operator ∆ that satisfies for matrices A, B and C and an eigenvalue equation We will next use these properties of the ∆ s operators.
Since the operator ∆ 0 is invariant under spatial translations and constant diagonal gauge transformations its eigenfunctions are of the form Operating with ∆ 0 on ω a (x) we get Clearly the operator ∆ 0 is similar to the operator in eq. (A.14) and thus the strategy shown can be used. Using the eq. (A.15) with ξ = 0 i.e. We will then move on to the more challenging case of ∆ 1 . The eigenfunctions of the operator ∆ 1 have the general form where normalization 10 matrix R a µν (t) is a diagonal 4 × 4 matrix with R a 00 (t) = −i, R a kk (t) = e i(p k +fa(t))/2 , k = 1, 2, 3. (A.26) Again we can operate with ∆ 1 on the eigenfunction eq. (A.25) which yields 10 Adding R a µν ensures that the matrices A a µν (t), B a µν (t) and C a µν (t) in the recursion relation are real.

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where the matrices A a µν (t), B a µν (t) and C a µν (t) are We have used the following short handed notation The operator ∆ 1 is also similar to the case in eq. (A.14) and the same strategy can again be exploited. Additionally the boundary conditions of ψ a µ (t) in eq. (A.27) are With this we can first calculate the determinant of ∆ 1 in the more general case where the boundary conditions are given by eq. (A.33). Setting ξ = 0 in eq. (A.15) we get With the help of these equations we find F a µν (t) which has the property , (A.37)

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are the first nonzero components of ψ a µ (t). The matrix F a µν (t) is where the projection operator P µν is With F a µν (t) we will be able to construct a matrix M a µν that couples v a µ from eq. (A.37) and the boundary condition eq. We can then move on to the case of ∆ 1 where a > N 2 − N i.e. for diagonal generators X a and when p = 0. In this case the boundary conditions are given by eq. (A.32) and ψ a 0 (t) and ψ a k (t) components decouple since the matrices A a µν (t), B a µν (t) and C a µν (t) are diagonal

B Chosen basis for the diagonal generators and the values of the coefficients which depend on the background field
In appendix A we showed how the 1-loop coupling can be calculated for a generic background field and basis of generators. In here we will specify the basis that we have selected as well as the values of the coefficients χ c a , χ s a and f a (t). We have chosen a basis given by