Center-symmetric Landau gauge: Further signatures of confinement

In a recent article [1], we have identified new signatures for the Yang-Mills deconfinement transition, based on the finite-temperature longitudinal or (chromo-)electric gluon propagator as computed in the center-symmetric Landau gauge. Here, we generalize these considerations into a systematic study of the center symmetry identities obeyed by the correlation functions in this gauge. Any violation of these constraints signals the breaking of center symmetry and can thus serve as a probe for the deconfinement transition.


I. INTRODUCTION
Functional methods are by now a well developed corpus of approaches in the framework of non-abelian gauge theories [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] that can bring valuable complementary information in situations where Monte-Carlo lattice simulations are the least efficient.One limitation of functional methods, as compared to the lattice, is, however, that the primary quantities they give access to are gaugedependent correlation functions.Although observables can be reconstructed from the correlation functions in principle, this strongly rests on the accuracy at which the latter are computed and on their particular relation to the observables under consideration.
A natural question that emerges is then whether it could be possible to extract relevant physical information directly from the correlation functions themselves, an idea that can be further elaborated in at least two distinct directions.The most obvious one is to try to identify gauge-independent features of the correlation functions [21], which are then more prone to encapsulate observable information.A different strategy is based on the idea that certain physical questions could be addressed directly from gauge-dependent features of the correlation functions, in certain, well chosen gauges.A paradigmatic example is the question of symmetries and their breaking, which usually underlies the phase structure of the system under consideration.
In particular, in recent years, both lattice simulations [22][23][24][25][26][27] as well as various analytical studies [29][30][31][32][33] have searched for signatures of the Yang-Mills deconfinement phase transition within the Landau gauge gluon propagator.However, even though some change of behavior is seen in the vicinity of the transition and some quantities extracted from the propagator seem to behave as order parameters, no clear connection has been established with the breaking of center symmetry. 1Moreover, in the particular case of the SU(2) gauge group, where the transition is known to be second order, the Landau gauge gluon propagator does not seem to become critical.
In order to explain these observations, in a series of works [1,35,36], we have put forward the idea that the Landau gauge might not be the most appropriate gauge to analyze the deconfinement transition, in particular when it comes to identifying signatures of symmetry breaking at the level the correlation functions.One can actually understand this on very general grounds since, as we argue below, the gauge-fixed action associated to a chosen gauge fixing does not necessarily reflect the symmetries of the problem.Even though this feature has no influence on the way the symmetry constrains the observables,2 it can have a strong impact on whether and how the symmetries manifest themselves at the level of the correlation functions.
To be more specific, consider a gauge theory defined by some non-gauge-fixed action S[A] and consider a physical symmetry A → A ′ such that S[A ′ ] = S [A].When switching to a gauge-fixed formulation, that same physical transformation does not need to be a symmetry of the gauge-fixed action S gf [A].In fact, one could more generally have 3 with gf ′ a gauge fixing that can differ from the original one gf.Indeed, although not representing a symmetry of the action in a given gauge, the identity (1) is actually sufficient for the physical symmetry to be manifest at the level of the observables ⟨O[A]⟩ gf .For instance, assuming a linear transformation and because the observables do not depend on the gauge fixing, we can write the following chain of identities: 4 This gives a symmetry constraint for the considered observable in any chosen gauge and can then be used as a probe for the breaking of the symmetry under consideration.A well known example of such type of observables is the Polyakov loop that probes the center-symmetry in pure Yang-Mills theories.
If we consider instead a correlation function ⟨C[A]⟩ gf , and assuming again a linear transformation C[A ′ ] = LC[A], the previous chain of identities stops one step earlier because correlation functions depend explicitly on the gauge fixing: In this case, one obtains a relation between the correlation functions in the two different gauges gf and gf ′ but certainly not a constraint on the correlation functions of a given gauge.For this reason, the correlation functions cannot in general be used as probes for the breaking of physical symmetries.
There is one important exception, however, corresponding to the case where the chosen gauge fixing is invariant under the considered physical transformation, that is gf ′ = gf.In this case, the chain of identities ( 4) can be continued one step further, just as for the case of observables (but not for the same reason), and one obtains symmetry constraints for the correlation functions themselves, that can be used as order parameters for the symmetry at hand.
In this work, we consider one particular example of such symmetry invariant gauge fixings, the recently introduced center-symmetric Landau gauges [35] which are invariant under the center symmetry of pure SU(N) Yang-Mills theories at finite temperature [37][38][39][40][41][42] and which are thus adapted to the study of the deconfinement transition from the correlation functions.Some of these aspects were analyzed in Ref. [1] using specific Lorentz/color projections of the gluon two-point function.The present work makes the discussion more general by extending it to any correlation function and any Lorentz/color projection.
The article is organized as follows.In the next section, we define the notion of center-symmetric gauge field backgrounds and the associated center-symmetric Landau gauges.We also particularize to backgrounds (and thus, gauges) that are, in addition, charge conjugation invariant, and also invariant under particular color rotations.Section III analyzes the constraints on correlation functions associated to color invariance.The discussion of charge conjugation and center symmetry is more subtle because one needs to pay attention to the fact that physical symmetries act on gauge fields modulo genuine gauge transformations.One convenient way to handle this aspect of the theory is through the notion of Weyl transformations and Weyl chambers which we discuss in Sec.IV.Sections.V, VI and VII are then devoted to a systematic analysis of the constraints on the correlation functions that derive from charge conjugation and center symmetry for N = 2 and N = 3, as well as the identi-fication of new order parameters for center symmetry.Additional details are gathered in the Appendixes.

II. CENTER-SYMMETRIC LANDAU GAUGES
In what follows, we consider SU(N) Euclidean Yang-Mills theories within the framework of background Landau gauges [43][44][45][46].The latter actually refers to a family of gauges parametrized by a background gauge field configuration Ā that, in a sense, plays the role of an infinite collection of gauge-fixing parameters.The gauge condition is where stands for the adjoint covariant derivative in the presence of the background.
A given background defines a particular choice of gauge within the class of background Landau gauges.For instance, when the background is taken equal to zero, one retrieves the standard Landau gauge ∂ µ A a µ = 0. Here, we are interested in the subclass of center-symmetric Landau gauges [35] obtained by choosing, instead, centersymmetric backgrounds which we now define in more detail.

A. Center-symmetric backgrounds
A center-symmetric background Āc is defined by the condition Here, G denotes the group of gauged SU(N) matrices U (τ, ⃗ x) obeying the particular boundary conditions5 Any U ∈ G acts on the gauge field as where we have defined A µ ≡ A a µ t a .It should be stressed, however, that only those U 0 ∈ G 0 correspond to genuine gauge transformations, that is transformations that do not alter the state of the system.In contrast, any U ∈ G with k ̸ = 0 transforms at least one observable, the Polyakov loop [47], and should therefore be considered as a physical transformation.
Actually, because U and U 0 U act on the Polyakov loop in the same way, these physical transformations are defined only modulo multiplication by elements of G 0 .This is of course in correspondence with the fact that two gauge field configurations A and A U0 connected by an element of G 0 should be interpreted as two equivalent representations of the same physical state.In turn, this explains the particular definition of center-invariant configurations given in Eq. (7). 7This definition can actually be replaced by a simpler condition, namely where U 1 denotes the set (not a group) of gauged SU(N) matrices that fulfil Eq. ( 8) with k = 1.We shall stick to this simpler formulation in what follows.It should also be mentioned that a given centersymmetric background can obey additional symmetries.It can happen for instance that it is invariant under charge conjugation in the following sense: where X t denotes the transposed of X.Finally, the background could be invariant under certain elements of G 0 (in general global transformations): The center-symmetric backgrounds that we consider below obey such additional symmetries which we also exploit.

B. Symmetry constraints
The main interest of center-symmetric backgrounds and center-symmetric Landau gauges is that the gaugefixed action is invariant under center transformations.By this, we mean that This is to be contrasted to what happens for a gauge choice corresponding to an arbitrary background Ā.
In this case, one has instead S Ā[A] = S ĀU [A U ], with ĀU ̸ = Ā for any U ∈ U 1 .The latter identity connects the gauge-fixed actions in two different gauges, corresponding respectively to Ā and ĀU , and is thus quite different from Eq. ( 14) which is a symmetry identity within a single gauge, corresponding to the choice Ā = Āc .The symmetry identity (14) implies constraints on the correlation functions as computed in the centersymmetric Landau gauge.Take first the one-point function ⟨A⟩ Āc .If the symmetry is not broken, then we must have This means that the one-point function should also correspond to a center-symmetric configuration.Any departure from this expectation signals the breaking of center symmetry.
Next, if we consider a connected correlation function with δA ≡ A − ⟨A⟩ Āc transforming as we have, when the symmetry is not broken, It is easily seen that the vertex functions obey similar identities. 8These constraints can serve as probes of deconfinement since the violation of any of these identities signals the breaking of center symmetry.We stress that the converse is not true as some of these identities could be further protected by other (unbroken) symmetries even when center symmetry breaks.This in particular the case when (12) or (13) apply.From these equations, one can indeed derive similar constraints as (18), provided one replaces U c with the appropriate U C or U 0 .We will see below that some of the constraints derived from charge conjugation coincide with some of the constraints derived from center symmetry.Since charge conjugation is not expected to break spontaneously, these particular constraints cannot be used as probes of the breaking of center symmetry.
A remark on notation is now in order.From now on, we shall omit the label "connected" when writing correlation functions, and neither should we use the label Āc .It is implicitly assumed that, with the exception of the onepoint function, the notation ⟨A a1 µ1 (x 1 ) • • • A an µn (x n )⟩ refers either to a connected correlation function or to a vertex function, computed in the center-symmetric Landau gauge.Moreover, we shall use a condensed notation such that, unless specifically stated, both Lorentz indices and position arguments are combined into one single index.

C. Constant, temporal and diagonal backgrounds
In practice, one does not need to determine all possible backgrounds complying with Eq. ( 11) but it is enough to find just one.Particularly simple examples are obtained by first restricting to backgrounds of the form where the t j 's provide a maximal set of commuting generators of the algebra.One then looks for specific values rc of r that correspond to center-symmetric backgrounds.A convenient way to find these particular values is through the use of Weyl transformations and Weyl chambers, see below for further details, as well as the discussion in Ref. [34].
In the SU(2) case, one possible choice is with t j = σ 3 /2, whereas in the SU(3) case, one can take with t j ∈ {λ 3 /2, λ 8 /2}.We shall restrict to these choices in what follows.We will see below that not only do they fulfil Eq. ( 11), but they also comply with Eqs. ( 12) and (13).

III. COLOR CONSTRAINTS
The presence of a background makes the color structure of the various correlation functions more intricate than in the Landau gauge.For backgrounds of the form (19), the color structure remains simple, however, due to the fact that the background, and thus the gauge-fixed action, is invariant under global color rotations of the form U θ = e iθ j t j .This is just a particular example of Eq. ( 13), with similar consequences on the correlation functions as the constraints (18) provided one replaces U c with the corresponding U θ .To make the most of this symmetry, and in fact of the other symmetries as well, it will be convenient to work within a Cartan-Weyl basis {t κ } whose definition we now recall.

A. Cartan-Weyl bases
By construction, the generators t κ of a Cartan-Weyl basis simultaneously diagonalize the adjoint action of the t j 's: It can be helpful to recall these relations using a quantum mechanical language: the labels κ are real-valued, (N − 1)-dimensional vectors that collect the "quantum (eigen)numbers" κ j that a given "(eigen)state" t κ acquires under the action of the various charges [t j , ].
In more technical terms, the vectors κ are the adjoint weights of the algebra.
The adjoint weights can be of two types.If they are non-zero, they are called roots and are represented using the first letters of the Greek alphabet, κ = α, β, . . .The roots are non-degenerate, meaning that there is only one eigenstate t α associated to a given root α.It is also generally true that, if α is a root, then −α is a root as well.Aside from the roots, there is also a vanishing adjoint weight.It is degenerate9 because any t j is an eigenstate with vanishing charges.One can again write these states as t κ provided one sets κ = 0 (j) .This notation should be understood as representing multiple copies of the null vector, needed to distinguish the various degenerate zerocharge states t 0 (j) = t j .Of course, this label should be interpreted as nothing else but the null vector when appearing in algebraic expressions (that is anytime it is not used as a label).

B. Constraints
From Eq. ( 22), it is easily deduced that the adjoint action of U θ on the generators t κ of a Cartan-Weyl basis reads with X • Y ≡ X j Y j .It follows that, within a Cartan-Weyl basis, U κλ θ = e iθ•κ δ κλ and the correlation functions are invariant under the transformation FIG. 1. SU(3) Weyl chambers in the (r3, r8)-plane and their relation to the Weyl transformations and the roots.The red vectors represent the roots multiplied by 4π.The elements of G0 are generated by translations along these vectors and reflections with respect to lines orthogonal to these vectors that go through the origin.Equivalently, they are generated by all possible reflections with respect to lines orthogonal to the roots and displaced by any multiple of 2π times the corresponding root.The corresponding symmetry axes define a paving of the (r3, r8)-plane into physically equivalent regions, known as Weyl chambers.
The constraints on the correlation/vertex functions take then the form This implies that the non-vanishing correlators are necessarily such that color is conserved in the sense In particular, the non-vanishing components of the two-point function G κλ µν ≡ ⟨A κ µ A λ ν ⟩ are necessarily such that κ + λ = 0.In the SU(2) case, it immediately follows that G κλ µν = G λ µν δ κ(−λ) .In the SU(3) case, we cannot conclude this as yet, however, because there are two neutral modes and we could have κ + λ = 0 with κ = 0 (3)  and λ = 0 (8) .What can be said without any further assumption is that and but the structure of G 0 (j) 0 (j ′ ) µν needs still to be investigated.
Let us now study, in a similar way, the constraints deriving from charge conjugation and center invariance.
Their study is slightly more delicate because the corresponding symmetries, ( 7) and ( 12), involve an element U 0 ∈ G 0 which we still need to characterize.This can be done with the help of Weyl transformations whose definition we recall in the next section.

C. Defining weights
Before doing so, it is useful to generalize Eq. ( 22) beyond the adjoint representation.In particular, when diagonalizing the defining action 10 all the t j 's: one obtains the defining weights ρ, which, just as the roots, are real-valued, (N − 1)-dimensional vectors.For instance, in the SU(2) case, there are two defining weights ρ 1 = 1/2 and ρ 2 = −1/2, while in the SU(3) case, there are three weights )) and ρ 3 = (0, −1/ √ 3).The defining weights are closely connected to the roots since the latter arise as all possible differences of two distinct weights α kl = ρ k − ρ l , thus explaining the notation that we introduced above.In the SU(N) case at least, the pair of weights that decompose a given root is unique.Sometimes, given a root α, we might want to access the corresponding weights which we denote 11 ρ α and ρα such that α = ρ α − ρα .
Let us finally mention that the SU(N) weights are all such that whereas for two distinct weights ρ and ρ ′ , see App. A. If follows in particular that the scalar product ρ • α between a defining weight and a root can only take a certain number of values: Similarly, given two roots α and β, one has FIG. 2. Transformation of a Weyl chamber under charge conjugation.The colored chamber represents the various locations of the Weyl chamber along the transformation process.We have chosen a point and a particular axis of the Weyl chamber to ease orientation as the Weyl chamber is transformed.In the first two figures, the blue items represent the transformations that will be applied to the Weyl chamber, Ā → − Āt and Wα 12 respectively, while in the third figure, the blue item represents the combined effect of these two transformations, which corresponds to a transformation of the original Weyl chamber into itself, more specifically a reflection with respect to its horizontal symmetry axis.

IV. WEYL TRANSFORMATIONS
A Weyl transformation is a particular element of G 0 .It is a global color transformation associated to a given root α as Details on the choice of the two factors that enter this definition are given in App.C together with a number of properties.In particular we will need to know the adjoint action of W α on the color algebra.

A. Action on the algebra
It is shown in App.C that In order to alleviate the notation, we have explicitely used the fact that the SU(N) roots are unit vectors.Otherwise, α in the RHS of Eqs. ( 34) and ( 35) needs to be replaced by α/ √ α 2 .We also note that Eq. ( 34) would remain unchanged were we not to include the second factor in Eq. ( 33).However, this factor is crucial in order to avoid uninteresting but annoying extra factors in Eq. ( 35) as we could show in the SU(N) case, see App. C. Let us finally mention that the combination β − 2(β • α)α appearing in the RHS of this equation takes different forms depending on the relation between α and β: This combination corresponds to the reflection of β with respect to an hyperplane orthogonal to α.

B. Weyl chambers
The Weyl transformations play an important role in identifying center-symmetric backgrounds among the backgrounds of the form (19).As explained in Ref. [34], one first restricts to the subgroup G0 of transformations of G 0 that keep the background of the form (19).These transformations are seen to be of the form [34] where W is a global color rotation that leaves the diagonal part of the algebra globally invariant, and with s such that Interestingly, the Weyl transformations W α provide examples of global color rotation that leave the diagonal part of the algebra globally invariant, as follows from Eq. (34).Similarly, given a root α, one always has see App. C.
FIG. 3. Transformation of a Weyl chamber under a center transformation.The colored chamber represents the various locations of the Weyl chamber along the transformation process.We have chosen a point and a particular axis of the Weyl chamber to ease orientation as the Weyl chamber is transformed.In the first three figures, the blue items represent the transformations that will be applied to the Weyl chamber, V−ρ 1 (τ ), Wα 12 and Wα 31 respectively, while in the fourth figure, the blue item represents the combined effect of these three transformations, which corresponds to a transformation of the original Weyl chamber into itself, more specifically a rotation by an angle 2π/3 around its center.
Therefore, one can generate all transformations of G0 from the elementary transformations W α and V α (τ ).In the space of backgrounds of the form (19), the latter correspond to simple geometrical transformations: reflections with respect to hyperplanes orthogonal to α and translations of r by 4πα.By combining those transformations, one obtains a more interesting generating set, namely the reflections with respect to hyperplanes orthogonal to α, displaced by any multiple of 2πα.The benefit of this generating set is that it subdivides the space of backgrounds of the form (19) into regions, known as Weyl chambers, that are connected to each other by elements of G0 and are thus physically equivalent.
In the SU(2) case, from the roots given above, one obtains that the Weyl chambers are the intervals r ∈ [2πk, 2π(k + 1)].In the case of SU(3), the Weyl chambers are equilateral triangles, see Fig. 1.Once the Weyl chambers have been identified, one can easily construct the particular transformations U 0 that appear in Eqs.(7) and (12).The idea is the same in both cases.A physical transformation such as charge conjugation or an element of G typically displaces a given Weyl chamber.By using elements of G0 , one can bring the Weyl chamber back to its original location.In doing so, one generates a transformation of a given Weyl chamber into itself.The fixed points of this transformation correspond to backgrounds obeying (7) or (12) and the so-constructed combinations of elements of G 0 provide the transformations U 0 appearing in these equations.
In the SU(2) case, the transformation Āµ → − Āt µ is itself an element iσ 2 ∈ G 0 .Therefore, Eq. ( 12) is fulfilled for any choice of background since one can choose U 0 to be the inverse of this element.It follows that charge conjugation imposes no constraint in this case.
In the SU(3) case, in contrast, the transformation Āµ → − Āt µ is not an element of G 0 .For backgrounds of the form (19), it corresponds to the transformation r → −r.Under this transformation, the Weyl chamber that is highlighted in the first plot of Fig. 2 is transformed as shown in the second plot of that same figure.To bring the Weyl chamber back to its original location, one can use the reflection (r 3 , r8 ) → (−r 3 , r8 ) which corresponds to W α12 .From this, not only do we deduce that all backgrounds of the form (19) with r = (r 3 , 0) comply with Eq. ( 12), 12 but we also identify U 0 in this equation with W α12 .

D. Center transformations
We can proceed similarly to construct the transformation U 0 that appears in Eq. ( 7) or, more directly, the transformation U c that appears in Eq. (11).First of all, it can be shown that, modulo elements of G0 , the transformations of U 1 that leave the Cartan subalgebra globally invariant are winding transformations of the form V −ρ (τ ), with ρ one of the defining weights of the algebra. 13The winding transformations act on the algebra as where we have used similar considerations as in Eq. ( 23).
In the space of backgrounds of the form (19), they correspond to translations by −4πρ.
In the SU(2) case for instance, if we choose to work on the Weyl chamber [0, 2π], we see that V −ρ1 (τ ) transforms it into [−2π, 0].This Weyl chamber can be brought back to its original location by applying W α12 .Eventually, this produces the transformation r → 2π − r that leaves the original Weyl chamber globally invariant and identifies r = π as a fixed point.It follows that the transformation U c that appears in (11) is the transformation and that the center-symmetric background is indeed (20).
As for the SU(3) case, a similar argumentation leads to corresponding to the center-symmetric background (21).
For a graphical representation of this construction, see Fig. 3.We mention that simple rules to permute the order between the various W α or between the W α and V ρ are provided in App. C.
We are now fully equipped to investigate the constraints that charge conjugation and center symmetry impose on correlation functions.

V. CHARGE CONJUGATION CONSTRAINTS
As already mentioned above, charge conjugation imposes no constraints in the SU(2) case.
In the SU(3) case, we have seen that, for backgrounds of the form (19) with r = (r 3 , 0), the transformation Āµ → − Āt µ combined with W α12 is a symmetry.Using Eqs. ( 34)- (35), we find that this symmetry acts on the generators of the algebra as and We can further use color rotation invariance and redefine the above transformation such that it appears as 14 t ±α12 → −t ±α12 , 14 To this purpose, we consider the color rotation e i2πρ j 3 t j and exploit the fact that It follows that the correlation/vertex functions are invariant under the transformation and corresponding to a change of sign of the components in the color 0 (8) and ±α 12 directions and a permutation of the components in the ±α 23 and ±α 31 directions.
Let us now investigate the consequences of this symmetry on the correlation functions, first using some examples and then in full generality.

Neutral sector
Consider a correlation function whose external legs are all in the neutral sector: The constraints from charge conjugation invariance read As a consequence, all correlators with an odd number of components along the color 8 direction need to vanish: We expect these constraints to always be valid since charge conjugation should not be spontaneously broken.

Two-point function
In particular, for the two-point function, it follows that Combined with Eq. ( 26) and (27), this implies as in the SU(2) case.We mention, however, that we do not know at this point whether and how the neutral diagonal elements G 0 (3) 0 (3)   µν and G 0 (8) 0 (8)   µν are connected to each other.We will come back to this question below as it is linked to center symmetry.
In the charged sector, we have that as follows from (49) but we cannot tell at this point whether and how they are connected to the components along the ±α 12 directions.Again, we will come back to this question below.

Three-point function
For the three-point functions, we have Similarly as well as and Finally

B. General case
To derive the general constraints from charge conjugation invariance, it is convenient to introduce the field A σ µ ≡ A 3 µ + σiA 8 µ (with σ = ±1) which transforms as A general correlation function takes then the form which we write formally as We note that we have the constraint as follows from color conservation, see the discussion below Eq. ( 25).Now, because α 12 + α 23 + α 31 = 0, this rewrites as (p + j − q − i)α 12 + (k + j − ℓ − i)α 23 = 0. Since α 12 and α 23 are linearly independent, this eventually leads to p − q = i − j = k − ℓ.Moreover, correlation functions involving A 3 µ and A 8 µ rather than A ± µ can be obtained through appropriate linear combinations of (62).
With this compact notation, the constraints due to charge conjugation invariance read ⟨(+) mn (12) pq (23) kℓ (31) One easily checks that this identity contains the constraints already derived above and allows one to generate all other possible constraints related to charge conjugation invariance.
VI. Z2-SYMMETRY CONSTRAINTS Using Eqs. ( 34)-( 36) and Eqs. ( 41)-( 42), we find that the action of U c on the SU(2) algebra reads as well as where we have used that ρ 1 • α 12 = 1/2.From this, one reads the corresponding U c and deduces that the correlation functions are invariant under the transformation where δA was defined below Eq. ( 16) and τ µ stands for the Euclidean time argument associated to the index µ.
Let us now analyze the consequences of this symmetry on the correlation functions, first using some examples, and then in full generality.

Neutral sector
If we consider correlation functions that are purely in the neutral sector, the constraint (18) takes the form Thus, functions with an even number of external legs are unconstrained, while those with an odd number of external legs should vanish as long as center symmetry is not broken.

Two-point function
In particular there is no constraint on the two-point function in the neutral sector. 15 On the contrary, from Eq. (69), we find the following constraint on the two-point function in the charged sector where τ µ and τ ν are the Euclidean time arguments associated with the fields carrying the indices µ and ν respectively.In other words (we now make the position arguments explicit) In Fourier space, this means that G with N = (1, ⃗ 0).Any violation of this identity signals a breaking of center symmetry.

Three-point function
For the three-point function, either all color directions are neutral and then or, only one is neutral and then In Fourier space, this leads to 15 In Refs.[1,35], we have seen that the two-point function in the neutral sector develops a zero-mode at the deconfinement transition.Although this is a combined consequence of center symmetry and of the second order nature of the transition in the SU(2) case, it is not of the same type than the symmetry constraints that we are presently discussing.The latter apply indeed over the whole confining phase and not just at the transition.

B. General case
A general SU(2) correlation function takes the form which we denote more simply as ⟨(3) n (12) p ⟩.The constraints due to center symmetry read We of course retrieve the previously obtained constraints as particular cases of this general identity.

VII. Z3-SYMMETRY CONSTRAINTS
Using Eqs. ( 34)-( 36) and Eqs. ( 41)-( 42), we find again that the neutral and charged sectors decouple under the adjoint action of U c .In the neutral sector, the transformation corresponds to a rotation by an angle 2π/3, whereas in the charged sector, we find In terms of the gauge field, the rotation by an angle 2π/3 in the neutral sector reads while in the charged sector, we have If we consider correlation functions that are purely in the neutral sector, the constraint (18) takes the form Thus, correlation functions such that σ 1 + • • • + σ n / ∈ 3Z need to vanish if the center symmetry is not broken.
For the two-point function, without loss of generality, 16  we can consider ⟨A + µ A + ν ⟩ and then Taking the real and imaginary parts of this identity, we find that both and if center symmetry is not broken.
In fact, the second identity is always fulfilled due to the constraints from charge conjugation invariance, see Sec.V. On the other hand, the first combination has no reason to remain 0 if center symmetry is broken.One could invoke color rotation invariance but the fact that the gauge fixing introduces a preferred color direction along λ 3 /2 prevents us from doing so.We deduce that this second combination can be used as an (infinite collection of) order parameter(s) for center-symmetry.We have tested this hypothesis in Ref. [1] for the case of the chromo-electric component of the propagator in the zerofrequency limit.We now see that this should apply to the chromo-magnetic component as well and for any value of the external momentum.This will be studied in a future work.
In the charged sector, we find that is In Fourier space, this means that G α12(−α12) µν with N = (1, ⃗ 0).Any violation of these identities signals a breaking of center symmetry.On the other hand, the fact that G α23(−α23) µν and G α31(−α31) µν agree with each other is a consequence of charge conjugation invariance, as we have seen in Sec.V.

Three-point function
Consider first the case where all the external legs are in the neutral sector.Without loss of generality, we can consider ⟨A + µ A + ν A − ρ ⟩ as the other cases are obtained from permutations or complex conjugation.We then find Taking the real and imaginary parts this gives and Writing similar formulas for permutations of (x, y, z) and (µ, ν, ρ), we find that these relations are equivalent to and A priori each of these identities could be used as a probe for the deconfinement transition.However, the second set is always (trivially) fulfilled for our particular choice of background due to charge conjugation invariance.

Consider now a three-point function involving charged modes. Let us first consider the correlators ⟨A
which can interpret as two independent equations giving two correlation functions in terms of the third one.Equivalently, this rewrites and Upon using charge conjugation invariance, the only nontrivial information that arises from center symmetry is and since the other correlators are fixed through Eqs. ( 57)-( 59).
Finally we find the constraint The argument in the previous section is in fact more general.

VIII. SU(N) CASE
Let us now see how the previous considerations extend to the SU(N) case.

A. Weyl chambers
Recall that the Weyl chambers appear as the result of the tiling of the Cartan subalgebra by the network of hyperplanes orthogonal to the roots and displaced from the origin by any multiple of 2π times the corresponding root, see Sec.IV B. To construct the Weyl chambers more explicitly, one can proceed as follows.
First, one selects N − 1 out of the N defining weights.We denote them as ρ 1 , . . ., ρ N −1 . 17It is easily seen that they form a basis [48].Next, denoting the remaining weight as which allow one to rewrite any other root as a linear combination with integer coefficients: with j, k ̸ = N .The reason for particularizing the roots α j is that it is convenient to first determine the regions delimited by the hyperplanes orthogonal to the α j and then to determine how these regions are further subdivided by the hyperplanes orthogonal to the other roots.
Let us now take a point r = r j t j in the Cartan subalgebra.It lies in the hyperplane orthogonal to α j displaced from the origin by 2πα j n if and only if r • α j = 2πn (recall that the roots are unith length vectors).To make the most of this condition, it is convenient to decompose r along the basis formed by the vectors 4πρ 1 , . . ., 4πρ N −1 : Then, and, thus, the considered hyperplane corresponds to the equation x j = n.This, in turn, shows that the regions 17 This labelling does not need to be the particular one chosen in Eq. (A2).
delimited by the hyperplanes associated to the α j are the regions delimited by the lattice generated by vectors 4πρ 1 , . . ., 4πρ N −1 .In what follows, it will be sufficient to consider the parallelepiped defined by these vectors, corresponding to x k ∈ [0, 1] in Eq. ( 108).
Let us now see how the remaining hyperplanes further subdivide this parallelepiped.Consider an hyperplane orthogonal to α jk and displaced by a multiple n of 2π times this root.That r belongs to that hyperplane means again that r • α jk = 2πn.However, we now have and then, the equation of the hyperplane is , the only hyperplanes that split the parallelepiped into non-trivial regions correspond to n = 0, that is x j = x k .We deduce that one possible Weyl chamber in the considered parallelepiped is the one defined by the equations 0 The other Weyl chambers in that same parallelepiped correspond to 0 where σ is any permutation of 1, . . ., N − 1.This also implies that the parallelepiped is subdivided into (N −1)! Weyl chambers.These Weyl chambers can be given yet another useful characterization in terms of the defining weights.Let us consider for instance the Weyl chamber 0

and let us define the variables
It is easily seen that when the x j span the considered Weyl chamber, the only constraints on the y k are y k ∈ [0, 1] and N k=1 y k = 1.Moreover, one can easily retrieve the x j from the y j as We can then write It follows that the considered Weyl chamber is the convex hull of the points Since we could have labelled the weights as wanted, the general rule is that, for any choice ρ 1 , . . ., ρ N −1 of N − 1 weights, the convex hull of one Weyl chamber attached to the origin.The other Weyl chambers attached to the origin correspond to the convex hull of and 0, with σ is any permutation of 1, . . ., N − 1.

B. Confining configurations
The previous characterization of the Weyl chambers attached to the origin is quite useful.In particular, it can be used to find symmetry invariant points within the Weyl chamber.Let us first decompose an arbitrary element of the Cartan subalgebra along the basis 4πη 1 , . . ., 4πη N −1 : Since η N = 0, we can extend this decomposition into a decomposition along an affine basis with The considered Weyl chamber corresponds to the extra constraints z k ∈ [0, 1].Its vertices 4πη k correspond to those points with one of the coordinates z k equal to 1 and the rest equal to 0.
Let us now associate to each value of r, the "classical" Polyakov loop Its value at the vertices of the Weyl chamber is Now and thus where we have used that the term with the Θ-function in Eq. ( 116) does not contribute to the exponential in Eq. ( 114).We have thus found that the value of the classical Polyakov loop at the vertex 4πη k is nothing but the k th center element of SU(N).Consider now a center transformation with associated center element e i2π/N .Since the Polyakov loop is multiplied by this center element, we deduce that the vertices of the Weyl chamber are transformed as 4πη 1 → 4πη 2 , 4πη 2 → 4πη 3 , . . ., 4πη N −1 → 4πη N = 0 and 4πη N = 0 → 4πη 1 .From the point of view of a passive transformation, this means that the coordinates z k in Eq. ( 113) are transformed as Then, the only invariant points are those such that this common coordinate needs to be 1/N .It follows that the center-symmetric point in the considered Weyl chamber is where we have used that We can similarly consider the case of charge conjugation which transforms the Polyakov loop ℓ associated to a particle into the Polyakov loop ℓ * of the corresponding anti-particle.We deduce that, under charge conjugation, a vertex 4πη k of associated Polyakov loop e −i2πk/N is transformed into the vertex of associated Polyakov loop e i2πk/N = e −i2π(N −k)/N , that is 4πη N −k .From the point of view of a passive transformation, this means that the coordinates z k in Eq. ( 113) are transformed as z k → z N −k .We then need to distinguish two cases depending on whether or not there exists a k such that k = N − k, that is depending on whether N is even or odd.
If N is even, all coordinates are transformed into different ones, expect for z N/2 which is mapped into itself: , the value of z N/2 being unconstrained.This represents an affine space of dimension N − 1 − (N/2 − 1) = N/2.In the SU(2) case, this is again a line, corresponding to the whole Weyl chamber as we have already seen above.In the SU(4) case, this would correspond to a plane, see for instance [34].

C. Symmetry constraints
The transformations of the Weyl chamber into itself associated to center transformations can be seen as resulting from the application of a winding transformation V −ρ (τ ) that translates the Weyl chamber into a different one by a vector −4πρ, followed by a sequence of Weyl transformations which correspond to reflections with respect to the facets of the Weyl chamber, in order to bring the Weyl chamber back to its original position. 18 In order to find the appropriate sequence of reflections, let us consider the Weyl chamber defined by the vertices Under the winding transformation V ρ1 (τ ) it becomes the Weyl chamber of vertices To continue, let us first remark that the action of the reflection w.r.t. the hyperplane orthogonal to a given root α jk on the collections of weights ρ h is only to flip ρ j and ρ k .This can be easily checked using the property (A5). 19It follows that, by successively applying W α12 , W α23 , . . .W α (N −1)N , one transforms the Weyl chamber (119) into the Weyl chamber (120).We have thus found that This can be rewritten in an alternative form using the crossing rules given in App.C.3 and by noticing that the action of the reflection w.r.t. the hyperplane orthogonal to a given root α jk on the collections of roots α jℓ is to flip α jk and α kj , α jℓ and α kℓ , as well as α hk and α hj .This can again be easily checked using the property (A5).We then find In this form this is a generalization of Eqs. ( 43) and (44).
In the SU(3) case, the comparison actually requires exchanging the labels 2 and 3 because, with the partialcular labelling (A2), the Weyl chamber that we considered in the main text is ρ 1 , ρ 1 + ρ 3 , ρ 1 + ρ 3 + ρ 2 .There are many other forms of U c obtained by exchanging some of the W 's. In what follows, we denote the general form as The combination of the above Weyl transformations results in an isometry of the Weyl chamber into itself, centered around the confining configuration of the Weyl chamber.We note that since V −ρ (τ ) acts like a translation for backgrounds of the form (19), the combined action of the Weyl transformations only corresponds to the same isometry centered about the origin of the algebra.We denote this isometry by (123) 18 The same considerations apply to charge conjugation upon replacing the winding transformation by Aµ → −A t µ and adapting the sequence of Weyl transformations. 19Using this remark, one can also easily deduced that the facets of the considered Weyl chamber lie either within the hyperplanes orthogonal to α k(k+1) = ρ k − ρ k+1 , with k = 1, . . ., N − 1, that go through the origin, or within the hyperplane orthogonal to α N 1 displaced by 2πα N 1 with respect to the origin.
We now would like to analyze the constraints of the symmetry U c on the correlation functions.
Before doing so, it will be convenient to rewrite the action (34)-( 35) of a Weyl transformation on the various generators of the algebra in a more compact form.To this purpose, recall that another notation for t j is t 0 (j) where the use of 0 as a label emphasizes the fact that the generators t j are vanishing-charge states, while the label (j) is used to distinguish these various degenerate states.Now, 0 (j) should be understood as the zero vector associated with the direction j in the commuting subalgebra.We can more generally associate a zero to an arbitrary direction u j .To this purpose, we define t 0 (u) ≡ u j t j .Then, we notice that We can now combine Eqs. ( 34) and ( 35) into the single formula where R α • κ ≡ κ − 2(κ • α)α denotes the geometrical reflection of the vector κ with respect to an hyperplane orthogonal to α.In particular, under this reflection, a zero 0 (u) is mapped onto another zero 0 (v) .The nuance, however, is that it is not the same zero since v = R α • u.
Similarly, we can define the action of the isometry (123) on any type of label κ, denoted I • κ in what follows.It is obtained by repeated action of the R αj and we note in particular that I • 0 (u) ≡ 0 (I•u) .Returning to the symmetry constraints associated to the transformation U c , let us evaluate U c t κ U † c .Upon repeated use of Eq. (125), it is found that In terms of the gauge field, this corresponds to the transformation Since ( 127) is a symmetry within the center-symmetric Landau gauge, we obtain the following constraint on the (connected) correlation functions in this gauge: This formula compares well with those obtained in the SU(2) and SU(3) case.We stress that and thus, while the charged labels transform according to I −1 , the neutral components of the field transform according to I.This is also what we observe in the above examples, see for instance Eqs. ( 95)-(97).

IX. CONCLUSIONS
We have performed a systematic study of the centersymmetry constraints on the correlation functions computed within the center-symmetric Landau gauge, a class of background Landau gauges where the background is chosen to be center-symmetric in a sense that we have precisely defined.We have specified to backgrounds that comply with other symmetries as well such as charge conjugation and invariance under particular color rotations, whose consequences we have also thoroughly investigated.As a result of our analysis, we have identified new signatures for the deconfinement transition from the correlation functions in those gauges, extending the results obtained in Ref. [1].
The analysis made in that reference was restricted to the (color) neutral, chromo-electric sector because this is where the transition usually occurs.In the SU(2) case, we found a sharp signature of the transition signalled by a divergence of the zero-momentum propagator.However, this does not really qualify as an order parameter since there exists no phase over which this quantity is constant.The present analysis shows that a more standard SU(2) order parameter can be constructed from the (color) charged components of the propagator, both in the chromo-electric and chromo-magnetic sectors, see Eq. ( 74).Other order parameters can be constructed from the three-point function, both in the purely neutral sector (75), or in a sector mixing neutral and charged components (77).This can also be generalized to higher order correlators (79).
Similar conclusions hold for the SU(3) gauge group.In that case, we had already identified an order parameter from the chromo-electric propagator in the neutral sector.The present analysis extends this conclusion to the chromo-magnetic sector, see Eq. (85), while revealing various other order parameters from the charged sector, see Eq. (89).The three-point function leads to four different order parameters, Eqs.(93), (102), ( 103) and (104).Finally, we have extended our analysis to the SU(N) case.
All these results confirm that the center-symmetric Landau gauge put forward in Ref. [35] is a good gauge for the study of the confinement-deconfinement transition within functional approaches.Indeed, in this gauge, the transition is encoded directly within the building blocks that sustain these approaches, that is the correlation functions for the primary fields and does not require the computation of more involved order parameter such as the Polyakov loop.
In a work in preparation, we shall confront these expectations to one-loop calculations within the Curci-Ferrari model [49], a model accounting for some of the low energy aspects of the gauge fixing in (background) Landau gauges [50].Similar calculations for the gluon three-point function are also in progress.Finally, it would be interesting to see whether similar ideas extend to the lattice implementation of center-symmetric gauge fixings.Work in this direction is also in progress.

Action outside the Cartan subalgebra
We now would like to study the action of U α (z) outside the Cartan subalgebra, that is compute We need to consider various cases.
c. Finally, when β is equal neither to α nor to −α but α + β is a root, we have we have where we have used that σ (−α)(α+β) = σ αβ .More generally reconstruct W α modulo an element of the center.But the action on the algebra does not depend on θ and neither can the center-element since the center group is discrete.

Crossing rules
Now that we know that the W α are a convenient way to define the Weyl transformations let us determine some useful "crossing rules" for the latter.To this purpose, let us evaluate W α W β W † α .We have but we still need to show that the two expressions between brackets in the last line correspond to defining weights.This is actually a well-known result: the set of weights is invariant under Weyl transformations.This is seen by evaluating W † α t j W α |ρ⟩.One has W † α t j W α |ρ⟩ = (t j − 2(α k t k )α j )|ρ⟩ = (ρ j − 2(ρ • α)α j )|ρ⟩ (C24) and thus Since W α is invertible, W α |ρ⟩ is non-zero and therefore ρ − 2(ρ • α)α is one of the defining weights.
We have thus shown that Another way of recalling this rule is in the form of crossing rules In words, W α can cross W β but in doing so, it replaces β by β − 2(β • α)α in its wake.

Appendix D: Winding transformations
Given a fundamental weight ρ, we define V ρ (τ ) = e i τ β 4πρj t j , (D1) known as a winding transformation.These transformations correspond to non-trivial center transformations.They act on the algebra as It is useful to derive the relation between these transformations and the previously defined Weyl transformations.For instance  The first factor is nothing but U α (θ = 4πρ • ατ /β).Using Eq. (C14), we then arrive at The identities (D5) and (D7) can again be conveniently recast in the form of crossing rules: and Once more, W α can cross V ρ but, in doing so, it replaces ρ by ρ − 2(ρ • α)α in its wake.
)α12 µν (Q) are related by a frequency shift of 2πT .With our convention ∂ µ → −iQ µ for the Fourier transform, this reads

,
that is a permutation 12 → 31 → 23 → 12 with appropriate phase factors.Let us now analyze the consequences of this symmetry on the correlation functions.
by a mere shift of the external frequency by 2πT .With our convention ∂ µ → −iQ µ for the Fourier transform, this reads ) b. Next, we consider the case where β = α or β = −α.Consider for instance β = α.Then