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Search for the QCD Ground State

Within the Euclidean effective action approach we propose criteria for the ground state of QCD. Despite a nonvanishing field strength the ground state should be invariant with respect to modified Poincar\'e transformations consisting of a combination of translations and rotations with suitable gauge transformations. We have found candidate states for QCD with four or more colours. The formation of gluon condensates shows similarities with the Higgs phenomenon.

1. It has been known for a long time that the perturbative vacuum cannot be the true ground state of QCD [1]. If one considers the Euclidean effective action Γ as a function of the colour-magnetic and colour-electric fields B z i , E z i , one finds states with B, E different from zero for which the value of Γ is lower than for B = E = 0. This observation constitutes the basis for many models of condensates of composite operators as F µν z F z µν [2]. Since the existence of states with lower energy than the perturbative vacuum is clearly visible in the effective action for the gluon field A µ , one may wonder what is the true QCD ground state in this language. 1 The Euclidean effective action for a pure Yang-Mills theory should be bounded below and we know that the state A µ = 0 (or a gauge-equivalent state) is not the state of lowest action. There must therefore exist an absolute minimum of Γ with F z µν = 0. An immediate worry is then the apparent breakdown of Euclidean rotation symmetry (corresponding to Lorentz-symmetry) for any nonvanishing value of the antisymmetric tensor field F µν . This seems to be in contradiction with the observed Poincaré symmetry (global d-dimensional rotations and translations in the Euclidean language) of our world which requires a ground state invariant with respect to these symmetries. Furthermore, no parity P , time reversal T or charge conjugation C violation is observed in strong interactions and the QCD ground state must respect these discrete symmetries as well.
The situation is less dramatic if one realizes that the standard implementation of rotations and translations is not the only way to realize the Poincaré symmetry.
One may define modified Poincaré transformations by combining the standard space transformations with appropriate gauge transformations. An example for the SO(3) rotation group is well known for instantons [3] where space rotations are combined with SU(2) gauge rotations to form a new rotation group. With respect to the combined symmetry transformations the instanton is invariant. From the observational 1 In the present context "ground state" means the classical field configuration which constitutes the absolute minimum of the Euclidean effective action. It should not be confused with the quantum mechanical vacuum state. point of view there is no way of distinction between the new "combined" rotations and the standard rotations. Furthermore, the combined rotations act as standard rotations on any gauge-invariant state. Another example for a ground state with a modified translation group are spin waves [4]. 2 The aim of the present letter is to search for a ground state for QCD with F z µν = 0 which nevertheless preserves a new version of Poincaré symmetry as well as P, T and C symmetry.
The four-dimensional rotation symmetry SO(4) is locally equivalent to the direct It is obvious that this group structure remains unchanged if we combine SU(2) L or SU(2) R or both with appropriate SU(2) subgroups of the gauge group SU(N) C .
(We discuss here general N and will later discuss special properties for N = 3.) Denoting by τ 1 , τ 2 the generators of two groups SU(2) 1 , SU(2) 2 commuting with SU(2) L , SU(2) R one obtains new rotation symmetries with generators We will also consider the possibility that only SU(2) L (or SU(2) R ) is modified and therefore admit ǫ L , ǫ R = 0 or 1. The new generators τ ′ L and τ ′ R fulfil the same commutation relations (1) as τ L and τ R provided the subgroups SU(2) 1 and SU (2) (There are two inequivalent embeddings I, II of SO(4) in SU(4).) For the embedding II one may put all gauge fields A z µ to zero except for those in one (2,2) representation which we denote in the standard SO(4) vector notation by A α µ , α = 1...4. The state with constant nonvanishing a is manifestly invariant under standard translations as well as under the combined SO(4) rotation group (2) (with ǫ L = ǫ R = 1). An arbitrary space rotation acting on the index µ of A α µ can be compensated by an appropriate SO(4) C gauge rotation acting on α. (In a more group-theoretical language A α µ belongs to the representation (2,2,2,2) with respect to SU(2) L ×SU(2) R ×SU(2) C1 × SU(2) C2 and this representation contains a singlet (proportional to (6)) with respect .) It is easy to verify that the commutation relations between modified rotations ( τ ′ L , τ ′ R ) and translations P µ are the same as for the usual rotations since the gauge transformations of SO(4) C are coordinate independent ([ τ 1,2 , P µ ] = 0). The state (6) is therefore invariant under a modified version of Poincaré symmetry. (The discrete symmetries P, T, C will be discussed later.) Hence it is a possible candidate for the ground state of a pure SU(4) C gauge theory. The field strength for the gauge field (6) is easily computed with T α the generators of

SO(4) C and
One finds a nonvanishing value with g the gauge coupling and f γ αβ the structure constants of SO(4). We emphasize, nevertheless, that (6) is not the only possible ground state candidate. Ground state candidates corresponding to the embedding (I) can also be found. They can be described in a way similar to the discussion for N = 2, 3 to which we will turn next.
Let us now address the realistic case of SU(3) C . There are two inequivalent embeddings of SU(2) C according to which the gluon octet transforms as For the first embedding the fundamental three-dimensional representation (the quarks) transform as a triplet with respect to the SO(3) subgroup of SU (3). The second embedding corresponds to the SU(2)-"isospin" symmetry and the results based on this embedding can be applied to the case N = 2 as well. In four dimensions there is obviously no possible choice of a constant gauge field A z µ as in (6) since the representation (2,2) with respect to SO(4) space rotations cannot be matched with any of the representations (9). A constant field A µ does not contain a singlet with respect to SU(2) L × SU(2) R . There is a mismatch between a large number of dimensions (d = 4 in the present case) and a small number of colours (N C = 2, 3).
The possibility of a constant gauge field exists, nevertheless, for the threedimensional theory, which is relevant as an effective theory at high temperature.
Here the rotation symmetry is reduced to SO(3) and it is now easy to find a rotation and translation invariant ground state In four dimensions the state (10) is not an acceptable ground state for the zero temperature theory since it is not invariant with respect to "boosts" (or the full SO(4) rotations). It is, however, a candidate for the ground state at high temperatures where the SO(4) symmetry is not respected for "Euclidean time" compactified on a torus. We observe that the configuration (10) gives rise to non-vanishing constant colour magnetic fields ∼ a 2 whereas the colour electric fields vanish for s = 0.
If there is a transition from a zero temperature ground state with a symmetry between electric and magnetic fields to the "asymmetric state" (10) at high temperature the difference between the electric and magnetic condensate could be an interesting signal.  With respect to SU(2) L × SU(2) R the field strength F z µν decomposes into two irreducible representations G z µν and H z µν transforming as (3,1) and (1,3): with ε µνρσ the totally antisymmetric tensor ε 1234 ≡ ε 1230 = 1. 3 In terms of colourmagnetic and colour-electric fields 3 We use often the index 0 instead of 4 to be close to a Minkowski notation. Greek indices always run from 1 to 4 (or 0) whereas latin indices run from 1 to 3.
the self-dual and anti-self-dual fields can be written in the form Now G is invariant under SU(2) R and transforms as a triplet with respect to SU(2) L .
We may therefore form a singlet with respect to SU(2) ′ L if G z µν belongs to a triplet representation G α µν (α = 1...3) in the decomposition (9). Similar to (6) the singlet with respect to SU(2) ′ L × SU(2) R reads now   (5)). In addition, the SU(3) C group is here completely broken without any residual U(1) C symmetry commuting with SU(2) ′ L × SU(2) R . With respect to the three dimensional rotation group diag (SU(2) ′ L × SU(2) R ) the fields in G µν and H µν have the same spectrum, i.e. singlets, vectors, spin 2 and spin 3 tensors for the embedding I and additional half integer spin states for the embedding II. With respect to the modified boosts, however, the fields in G µν and H µν transform quite differently. This is not a worry of principle since colour neutral bound states will always have standard Lorentz-transformation properties. It points, nevertheless, to a strong violation of left-right symmetry and one wonders how such a spectrum can be consistent with parity. 4. In addition to the Poincaré symmetry we will now also require parity conservation for the ground state. Again, we envisage the possibility that parity is realized as a modified transformation in combination with a suitable discrete gauge transformation. The standard parity transformation (x → x ′ ) reverses the sign of the electric fields It therefore maps and is obviously violated by the state (15). Let us ask if there could be a modified parity transformation with the property (19) and leaving the state (15) invariant.
(Such a transformation would belong to the classP of generalized parity transformations discussed in [5].  We should mention at this place that the parity problem is absent for the ground state candidate (6) for SU(4) C (or other gauge groups containing O(4)). In fact, the configuration (6) violates the standard parity transformation P . We may nevertheless combine P with an automorphism of the SO(4) gauge group which reverses the sign of three components of A α µ , i.e. A α µ → −A α µ for α = 1...3. The state (6) is left invariant by the combined reflection. The global SU(4) C gauge transformations contain the required automorphism.
For SU(4) C one may also construct a ground state candidate based on the embedding (I) (5) by having a nonvanishing G µν for the (3,1) representation according to (15) and similarly nonvanishing H µν for the (1,3) representation. The modified parity reflection is again combined from the standard parity P and a suitable discrete gauge transformation. We remember that such a state would have constant F µν , but not constant A µ . 5. We may summarize the preceding observations by the statement that neither the gauge field nor the field strength can be constant for a realistic ground state of four-dimensional QCD with three colours. This implies that also translation symmetry cannot be realized in the standard way. Standard translations have to be combined with suitable gauge transformations 4 . We require that any infinitesimal translation of the ground state gauge field A z µ (x) can be compensated by a corresponding infinitesimal gauge transformation with gauge parameter θ z µ (x) The combined transformations form the modified translation group with generators (The modified "momentum operators" act in a standard way on tensors, whereas for gauge fields the inhomogeneous part of the transformation (22) has to be included.) Invariance of the ground state gauge field A z µ (x) under generalized Lorentz rotations requires similarly The corresponding generalized (Lorentz) rotation operators are Here the "spin operators" S µν correspond to rotations acting on the vector index of A µ and commute with angular momentum L µν and the "gauge part" G µν Both S µν and L µν obey separately the SO(4) commutation relations (The spin generators S µν are linear combinations of τ L , τ R in eq. (1).) In consequence, we associate to every element l i of the standard Poincaré group an element g i of the gauge group G such that the combination g i l i leaves A µ invariant.
This defines a map With respect to the group multiplication the invariance of A µ implies for this map and it is easy to see that h 12 must be a gauge transformation which leaves A µ invariant. (We denote by H the subgroup of gauge transformations leaving A µ invariant (h 12 ∈ H).) Let us discuss the case that H is trivial (only the identity element) or that the g i can be chosen from a subgroupG ⊂ G commuting with H.
Then we can put h 12 = 1 in eq. (29) and the mapping (28) is a group homomorphism.
The elements g i are uniquely determined in this case by the invariance condition The image of f is then isomorphic to the Poincaré group P 4 .
By virtue of this isomorphism the generalized translation generators P µ (23) in the different directions must commute Similarly, the modified "angular momenta" M µν must obey the same SO(4) commutation relations as L µν . In general L µν and G µν do not commute for η depending on x, and we obtain similar to (30) the consistency condition The operators M µν and P ρ must generate the generalized Poincaré group. From we get the additional consistency relation if h 12 = 1 in eq. (29) (see above).
It is amazing to see how the ground state problem for QCD in the framework of the Euclidean effective action turns into an interesting but difficult group-theoretical problem. It is not yet clear to us what is the best way for its solution. The observation may be helpful that all gauge singlets contracted from the ground state field A µ must be invariant under the standard Poincaré and parity transformations 5 , i.e. This gives the masses of (some of) the glueballs. In this context it is interesting to observe that the excitations of the gauge field have indeed different spins. The field A µ can describe a rather rich glueball spectrum which is not restricted to spin one states. On the other hand the gauge group may be completely broken. (For certain ground state candidates a residual global gauge symmetry could also persist.) The phenomenon of "gluon condensates" can be associated in this language with the "spontaneous symmetry-breaking of the gauge symmetry" by non-perturbative 5 One may check that the configuration (16) does not obey (34).
effects! In this respect there are analogies with the Higgs phenomenon: The role of the scalar field is now played by gauge fields proportional to the ground state field A µ . This field corresponds to a scalar with respect to the modified Poincaré transformations. The ground state field is equivalent to the vacuum expectation value of this "scalar" excitation [7]. The (generalized) scalar excitation corresponding to the Higgs boson is always present within the spectrum of glueball states since the gauge field must contain a singlet with respect to the generalized (Lorentz) rotations.
We have seen that the "embedding problem" of finding a generalized larger than an infrared cutoff, q 2 > k 2 , are included. The effective action obtains then in the limit k → 0. The dependence of Γ k on the scale k is governed by an exact evolution equation [9]. It is encouraging to observe in this context that the lowest order invariant changes its sign for small k [10], i.e.
Here the vanishing of Z F,k is directly related to a diverging renormalized gauge coupling and occurs at the confinement scale Λ conf . The negative value of Z F,k for small k is a clear sign of the instability of the perturbative vacuum and the onset of "gluon condensation" or "nonperturbative spontaneous symmetry-breaking of the gauge group". A calculation of the k-dependence of the coefficients of higher invariants (e.g. ∼ (F z µν F µν z ) 2 ) is in progress [11] and should shed some light on the properties of the ground state in terms of the values of various gauge invariants formed from the ground state field A µ . In particular, it would be interesting to know if a convariantly constant field strength