New Z(3) strings

We consider a Yang-Mills-Higgs theory with the gauge group SU(3) broken to its center Z(3) by two scalar fields in the adjoint representation and obtain new Z(3) strings asymptotic configurations with the gauge field and magnetic field in the direction of the step operators.


Introduction
String or vortex solutions may appear naturally in non-Abelian theories if the vacuum manifold has a nontrivial fundamental group, i.e., π 1 (G/G 0 ) = 1, where G and G 0 are, respectively, the gauge and the unbroken gauge groups.These configuration are topological and may be relevant in many areas such as condensed matter physics [1], astrophysics and cosmology [2] and high energy physics, in special in Grand Unified Theories [3] and the quark confinement problem [4,5].In order to have finite energy per unit length, or finite tension, the string solutions are asymptotically constructed by gauge transformations of the vacuum configuration and the associated group elements can be related to the fundamental group elements.This relation classify the solutions into topological classes, the strings belonging to the same equivalence class being homotopic to each other.These features can be seen in the Abelian Higgs Model [6] which couples a U (1) gauge field to a complex scalar.This symmetry is spontaneously broken by a non vanishing vacuum configuration φ vac , which vanishes the potential.Hence the asymptotic scalar field can be written as φ(ϕ) = exp(ia(ϕ))φ vac and therefore provides a mapping from the circle S 1 at spatial infinity to the circle S 1 in the internal space.These mappings can be classified into homotopy classes and this prevents non trivial solutions to be continuously deformed into the vacuum configuration.These string solutions may also appear in theories with non-Abelian groups and they were initially studied in [7,8,9].
It is believed that confinement in the strong coupling regime of QCD is due to chromoelectric strings called QCD strings.Many properties of QCD strings have been studied using lattice QCD.On the other hand, it is believed that QCD string in the strong coupling may be dual to chromomagnetic strings in weak coupling [4,5].Since QCD strings appear in gauge theories with non-Abelian groups (without U (1) factors) and are believed to be associated to center elements, in the last years we have analyzed some properties of chromomagnetic Z N strings which appear in Yang-Mills-Higgs theories with arbitrary simple gauge group G broken to its center Z(G) by two scalar fields [10,11,12].These Z N strings are associated to coweights of G and their topological sectors associated to center elements as was analyzed in detail in [11].Similarly to QCD strings, the tensions of the BPS Z N strings can satisfy the Casimir law [10,11].We showed also that the magnetic charges of the monopoles, which appear in the first symmetry breaking, are always integer linear combinations of the magnetic charges of the Z N strings, which allows the monopole confinement by the Z N strings.
In all our previous works we consider Z N strings with gauge field and magnetic field in the direction of the Cartan subalgebra (CSA).However it is possible to have string or vortex solutions with gauge fields as combinations of step operators as has been done for strings in theories with gauge group SO (10) broken to SU (5) × Z 2 [13], SU (2) broken to Z 2 [14] and SU (2) × U (1) [15].In this paper we consider a Yang-Mills-Higgs theory with gauge group SU (3) broken to its center Z 3 and construct the asymptotic configurations with gauge fields and magnetic fields as a combination of the su(3) step operators which we shall call Estrings.
The paper is organized as follows: in this paper, in section 2 we define our conventions adopted through the paper and define a new basis for su (3).In section 3 we choose a vacuum configuration that spontaneously breaks the symmetry and allows for the existence of Z 3 strings.In section 4 we define group elements associated to generators which are not in the Cartan subalgebra, show them to satisfy the condition (4.2) and then obtain the asymptotic form of the vector and the scalar fields.

Conventions
Let us consider a Yang-Mills theory with the Lagrangian where φ s , s = 1, 2, are complex scalar fields in the adjoint representation of the gauge group G whose Lie algebra is g.The covariant derivative is defined as and the field strength tensor is The so-called Cartan-Weyl basis decomposes the generators of a simple Lie algebra into the Cartan elements H i and the step operators E α satisfying where α is said to be a root of the algebra.Given an algebra of rank r, the roots belong to an r-dimensional vector space, Φ(g), called root space.This space is dual to the space containing the weights λ which satisfy The basis of these spaces are the simple roots α i and the fundamental weights λ i , respectively.The fundamental co-weights and simple co-roots are defined as

respectively, and they satisfy λ
The step operators have the following commutation relations among themselves, where N αβ are antisymmetric coefficients.
We define the generators where c i = ∑ r j=1 (K −1 ) i j , with K i j = 2α i • α ∨ j being the elements of the Cartan matrix associated to g.These generators form an su(2) algebra which is called the principal su(2) subalgebra of g.
From the so-called principal element, where h is the Coxeter number of g, one can show that the generators of the algebra satisfy which provides g with a Z h grading, The elements of the subspace g k has degree k and only g 0 forms a subalgebra, which corresponds to the Cartan subalgebra formed by the generators H i .Let ψ denote the highest root.We can expand ψ in the basis of simple roots α i as ψ = ∑ r i=1 n i α i , where n i are positive integers.Then, the Coxeter number can be written as h = 1 + ∑ r i=1 n i and we can conclude that A Lie algebra can have more than one different Cartan subalgebra.We can obtain a new Cartan subalgebra in following way: we can define the degree one generator E = ∑ r i=0 √ m i E α i , where α 0 = −ψ, m 0 ≡ 1 and the other real constants are m i = n i α 2 i /2 (no sum in the repeated indices).This generator appears naturally in Affine Toda field theories [16,17,18] Remembering that for arbitrary integers p i the group elements exp where we used the fact that since exp (2πip i λ ∨ i • H) lies in the center of the group, it commutes with P. In particular for the algebra su(3) the grading is The generators E and E † are given by or in terms of h i , It is convenient to define a new basis where the other six su(3) generators are given by where X i M has degree M and satisfies , with [M] denoting M modulo 3.This set of generators has the normalization in the adjoint representation and the commutation relations From (2.7) and the fact that X i † 0 = X i 0 , the set X i M have only seven non vanishing independent commutation relations, which read Note that by calculating ) one can write the above commutation relations in a more compact form where C i jk MN = 0, ±1, according to the above commutation relations.With this new basis it turned out to be easier to compute the commutators needed to obtain the asymptotic fields.

Vacuum configuration
with v = v i λ ∨ i having all v i different from zero, commute only with the generators in the Cartan subalgebra and therefore it spontaneously breaks the gauge symmetry to the maximal torus U (1) r .The elements belonging to this unbroken group are then written as exp ic j λ ∨ j • H with c j being real parameters.We can further consider another scalar field vacuum with all b j = 0, and by Baker-Campbell-Hausdorff (BCH) formula one can show that the only elements with ω belonging to the co-weight lattice of G. Since these are just the center elements of G, we see that this vacuum configuration produces a spontaneous symmetry breaking pattern Let us consider the same potential discussed in [10,11], where m is a real mass parameter, which accept vacuum solutions of the form (3.1) and (3.2).In the case of G = SU (3) these vacuum solutions can be written as where ψ = α 1 + α 2 = λ 1 + λ 2 is the highest root of SU (3), and where a 1 and a 2 are real constants.This vacuum configuration spontaneously breaks the gauge symmetry in the pattern giving rise to a multiply connected vacuum manifold which allows the existence of Z 3 strings.

Asymptotic Z 3 string solutions
For a theory with gauge group G broken to its center Z(G), the energy per unit length or string tension of a static topological non-Abelian string, considering W 0 = 0 = W 3 , is given by where i = 1, 2 denotes directions perpendicular to the string.In order to the string tension be finite, the asymptotic form of the fields must be related to the vacuum configuration by a gauge transformation, In order to be single valued configurations the group element g(ϕ) satisfies Notice that by assuming g(0) as the identity, then the above condition leads to g(2π) ∈ Z(G).We can consider that g(ϕ) = exp(iϕM).( Then exp(2πiM) ∈ Z(G) and M must be diagonalizable which implies that M must be a normal generator, that is, M, M † = 0.In order to fulfill this condition, one can consider that M = ω.H is a linear combination of Cartan generators.Then, the vector ω is in the co-weight lattice of the gauge group to guarantee that exp(2πiω • H) ∈ Z(G), as was considered in [10,11].As it can be seen from equations (4.1) and (4.3), the asymptotic gauge field is in the Cartan subalgebra of g.However, this is not the only possible choice for M. From Eq. (2.2) we can see that we can consider M = p i λ ∨ i • h, p i ∈ Z.For simplicity we shall consider only the theory with gauge group G = SU (3).In this case we shall have recalling that the algebra su(3) is simply laced so that λ ∨ i = λ i .Solving (2.4) for λ i • h we find It can be checked explicitly that g(2π) belongs indeed to the center of SU (3).In effect, in the 3 dimension representation and a direct calculation shows that the powers of M 1 are given by where are the Jacobsthal Numbers.Then it is easy to compute exp (2πinM 1 ) = exp 4πin 3 1 ∈ Z(SU( 3)).
In a similar way, Therefore the group elements given by (4.4) are in the center of SU (3).
For simplicity we shall adopt the notation where λ = 1 or λ = exp(−iπ/3).Let us consider the group element g(ϕ) = exp(iϕnM λ ), n ∈ Z. Then the asymptotic gauge field reads or in polar coordinates Let us define the generators which satisfy Using these relations and the BCH formula we obtain the asymptotic form of φ 1 as Similarly we define such that these generators obey and then we obtain Therefore, for these strings, the gauge field and the magnetic field take value in the direction of the step operators and we will call them E-strings.Since we are considering the same potential as in our previous works, this theory also have Z 3 string in the direction of the Cartan subalgebra, which we shall call H-strings.Due to relation (2.2), E-strings belong to the same three topological sectors as the H-string.Which one is stable will depend on the form of the ansatz and energetic considerations which we will analyse in another work.It is interesting to note that in the symmetry breaking pattern (3.3), there appear monopoles in the first breaking, which get confined in the second breaking [10].However, since the monopoles have magnetic flux in the Cartan direction, the will get confined only by H-strings.Then, in order to have finite energy the E-strings should not end in monopoles but close into itself.
. It is a normal generator, i.e., [E, E † ] = 0, which means it can be diagonalized.Therefore the generators E and E † belong to a new Cartan subalgebra h ′ [19] generated by orthogonal generators h 1 , h 2 , . . ., h r , which are related by a similarity transformation to the generators H 1 , H 2 , ..., H r of the Cartan subalgebra h, that is,