Flux dualization in broken SU(2)
aa r X i v : . [ h e p - t h ] D ec Preprint typeset in JHEP style - HYPER VERSION
Flux dualization in broken SU(2)
Chandrasekhar Chatterjee, Amitabha Lahiri [email protected] , [email protected] Department of Theoretical SciencesS. N. Bose National Centre for Basic SciencesBlock JD, Sector III, Salt Lake, Kolkata 700 098, W.B. India.
Abstract:
An SU(2) gauge theory is broken to U(1) by an adjoint scalar to producemagnetic monopoles. At a lower scale, this U(1) is further broken by a fundamental scalarto produce tubes of magnetic flux. We dualize the resulting theory to write an effectivetheory in terms of the macroscopic string variables. The monopoles are attached to theends of the strings, and the flux is confined in the tubes. . Introduction:
It is widely believed that color confinement in the strong coupling regime should be aphenomenon dual to monopole confinement in a color superconductor at weak coupling. Inthis picture, the QCD vacuum behaves like a dual superconductor, created by condensationof magnetic monopoles, in which confinement is analogous to a dual Meissner effect. Quarksare then bound to the ends of a flux string [1, 2, 3] analogous to the Abrikosov-Nielsen-Olesen vortex string of Abelian gauge theory [4, 5].A construction of flux strings in the Weinberg-Salam theory was suggested by Nambu [6],in which a pair of magnetic monopoles are bound by a flux string of Z condensate. Themagnetic monopoles are introduced by hand. If we demand that the magnetic monopolesshould appear from the underlying gauge theory, we need an additional adjoint scalar field.Such a construction of flux string, involving two adjoint scalar fields in an SU(2) gaugetheory, has been discussed in [5, 7]. Recently there has been a resurgence of interest in suchconstructions [8, 9, 10, 11]. We have previously shown explicitly that [12] an SU(2) gaugetheory broken by two adjoint scalar fields at different energy scales has configurations ofmagnetic monopoles bound by flux strings.In this paper we consider an SU(2) gauge theory coupled to an adjoint scalar fieldas well as a fundamental scalar field. The two fields break the symmetry at two scales.At the higher scale the adjoint scalar breaks the symmetry down to U(1) and produces’t Hooft-Polyakov magnetic monopoles [13, 14, 15]. The fundamental scalar breaks theremaining U(1) symmetry at a lower scale and produces a flux string.Our starting point is the Lagrangian L = −
12 Tr ( G µν G µν ) + Tr ( D µ φD µ φ ) + 12 ( D µ ψ † )( D µ ψ ) + V ( φ, ψ ) . (1.1)Here φ is in the adjoint representation of SU (2) , φ = φ i τ i with real φ i and ψ is a fundamen-tal doublet of SU (2), with V ( φ, ψ ) some interaction potential for the scalars. The SU (2)generators τ i satisfy Tr ( τ i τ j ) = δ ij . The covariant derivative D µ and the Yang-Mills fieldstrength tensor G µν are defined as( D µ φ ) i = ∂ µ φ i + gǫ ijk A jµ φ k , (1.2) G iµν = ∂ µ A iν − ∂ ν A iµ + gǫ ijk A jµ A kν , (1.3)( D µ ψ ) α = ∂ µ ψ α − igA iµ τ iαβ ψ β . (1.4)We will sometimes employ vector notation, in which D µ ~φ = ∂ µ ~φ + g ~A × ~φ , (1.5) D µ ψ = ∂ µ ψ − igA µ ψ , (1.6) ~G µν = ∂ µ ~A ν − ∂ ν ~A µ + g ~A µ × ~A ν , etc . (1.7)Obviously, ~φ and φ represent the same object. The simplest form of the potential V ( φ, ψ )that will serve our purpose is, V ( φ, ψ ) = − λ | φ | − v ) − λ ψ † ψ − v ) − V mix ( φ, ψ ) . (1.8)– 1 –ere v , v are the parameters of dimension of mass and λ , λ are dimensionless couplingconstants. The last term V mix ( φ, ψ ) includes all mixing terms in the potential, whichinvolve products of the two scalar fields in some way. We will take V mix ( φ , ψ ) = 0 for now,so v and v are the local minima of the potential, and we will refer to them as the vacuumexpectation values of φ and ψ .The adjoint scalar φ acquires a vacuum expectation value (vev) ~v which is a vectorin internal space, and breaks the symmetry group down to U(1). The ’t Hooft-Polyakovmonopoles are associated with this breaking. The other scalar field ψ also has a non-vanishing vev v which is a vector in the fundamental representation. This vector can beassociated uniquely with a vector in the adjoint space which is free to wind around ~v . Acircle in space is mapped to this winding, giving rise to the vortex string. We then dualizethe fields as in [16, 17, 18, 19, 20] to write the action in terms of string variables.The idea of two-scale symmetry breaking in SU(2), the first to produce monopolesand the second to produce strings, has appeared earlier [21]. Later this idea was usedin a supersymmetric setting in [22, 23, 24], where the idea of flux matching, followingNambu [6] was also included. The model we discuss in this paper, with one adjoint andone fundamental scalar, has been considered previously in [10]. Here we construct the fluxstrings explicitly in non-supersymmetric SU(2) theory with ’t Hooft-Polyakov monopolesof the same theory attached to the ends. The internal direction of symmetry breaking isleft arbitrary, so that the magnetic flux may be chosen to be along any direction in theinternal space. We also dualize the variables to write the effective theory of macroscopicstring variables coupled to an antisymmetric tensor, and thus show explicitly that the fluxat each end of the string is saturated by the magnetic monopoles, indicating confinementof magnetic flux.
2. Magnetic monopoles
We assume that v , the vacuum expectation value of φ , is large compared to the energyscale we are interested in. Below the scale v , we find the φ vacuum, defined by theequations D µ ~φ = 0 , (2.1) | φ | = v . Below v , the original SU(2) symmetry of the theory is broken down to U(1). At lowenergies the theory is essentially Abelian, with the component of A along φ remainingmassless. We can now write the gauge field below the scale v as ~A µ = B µ ˆ φ − g ˆ φ × ∂ µ ˆ φ , (2.2)where B µ = ~A µ · ˆ φ and ˆ φ = ~φ /v [25]. In this vacuum, until we include the secondsymmetry breaking, B µ is a massless mode. The other two components of A , which wecall A ± , and the modulus of the scalar field φ acquire masses, M A ± = gv , M | φ | = √ λ v . (2.3)– 2 –ell below v the modes A ± are not excited, so they will not appear in the low energytheory. The second term on the right hand side of Eq. (2.2) corresponds to the gauge fieldfor SU(2) magnetic monopoles [11].A straightforward calculation shows that,Tr ( G µν G µν ) = 12 F µν F µν , (2.4)where F µν = ∂ [ µ B ν ] − g ˆ φ · ∂ µ ˆ φ × ∂ ν ˆ φ ≡ ∂ [ µ B ν ] + M µν . (2.5)Then the Lagrangian can be written in the φ -vacuum as L = − F µν F µν + ( D µ ψ † )( D µ ψ ) − λ ψ † ψ − v ) . (2.6)The second term of Eq. (2.5) is the ‘monopole term’. In a configuration where thescalar field at spatial infinity goes as φ i → v r i r , the ( ij ) th component of the secondterm of Eq. (2.5) becomes − ǫ ijk r k gr , which we can easily identify as the field of a magneticmonopole. The flux for this monopole field is πg . On the other hand, a monopole withmagnetic charge Q m produces a flux of 4 πQ m , and thus we find the quantization conditionfor unit charge, Q m g = 1 . The scalar field φ can be written as φ ( x ) = | φ ( x ) | ˆ φ ( x ), where ˆ φ contains two indepen-dent fields (and x ≡ ~x ). So under a gauge transformation ˆ φ has a trajectory on S . Since φ is in the adjoint of SU(2), we can always write φ as φ ( x ) = | φ ( x ) | g ( x ) τ g − ( x ) = | φ ( x ) | ˆ φ ( x ) , (2.7)with g ( x ) ∈ SU(2). Then for a given φ ( x ) , we can locally decompose g ( x ) as g ( x ) = h ( x ) U ( x ) , with h ( x ) = exp( − iξ ( x ) ˆ φ ( x )) , and we can write φ ( x ) = | φ ( x ) | U ( ϕ ( x ) , θ ( x )) τ U † ( ϕ ( x ) , θ ( x )) , (2.8)Here ξ ( x ) , ϕ ( x ) , θ ( x ) are angles on S = SU(2). The matrix U rotates ˆ φ ( x ) in the internalspace, and is an element of SU(2)/U(1), where the U(1) is the one generated by h . If | φ | is zero at the origin and | φ | goes smoothly to its vacuum value v on the sphere at infinity,the field φ defines a map from space to the vacuum manifold such that second homotopygroup of the mapping is Z . Equating φ with the unit radius vector of a sphere we can solvefor U ( θ ( x ) , ϕ ( x )), U = cos θ − sin θ e − iϕ sin θ e iϕ cos θ ! . (2.9)An ’t Hooft-Polyakov monopole (in the point approximation, or as seen from infinity)at the origin is described by U = cos θ e iϕ e − iϕ ! + sin θ ii ! , (2.10)– 3 –here 0 ≤ θ ( ~x ) ≤ π and 0 ≤ ϕ ( ~x ) ≤ π are two parameters on the group manifold. Thischoice of U ( ~x ) is different from that in Eq. (2.9) by a rotation of the axes. Both choiceslead to the field configuration ~φ = v r i r τ i . (2.11)For this case, Q m g = 1 , as we mentioned earlier. A monopole of charge n/g is obtainedby making the replacement ϕ → nϕ in Eq.s (2.9, 2.10). The integer n labels the ho-motopy class, π ( SU (2) /U (1)) ∼ π ( S ) ∼ Z , of the scalar field configuration. Otherchoices of U ( ~x ) can give other configurations. For example, a monopole-anti-monopoleconfiguration [26] is given by the choice U = sin ( θ − θ )2 − e − iϕ e iϕ ! + cos ( θ − θ )2 ! . (2.12)For our purposes, we will need to consider a φ -vacuum configuration with U ( ~x ) ∈ SU (2)corresponding to a monopole-anti-monopole pair separated from each other by a distance > /v . Then the total magnetic charge vanishes, but each monopole (or anti-monopole)can be treated as a point particle.
3. Flux tubes
We started with a theory with SU(2) symmetry and a pair of scalars φ, ψ .
The non zerovacuum expectation value v of the field φ breaks the symmetry to U(1), so that below v we have an effective Abelian theory with magnetic monopoles. The gauge group SU(2)acts transitively on the vacuum manifold S , so the Abelian effective theory is independentof the internal direction of φ . The remaining symmetry of the theory is the U(1), the littlegroup of invariance of φ on the vacuum manifold. This is the group of rotations aroundany point on the vacuum manifold S .There is another scalar field ψ in the theory, a scalar in the fundamental representationof SU(2). After breaking the original SU(2) down to the φ -vacuum, the only remaininggauge symmetry of the SU(2) doublet ψ is a transformation by the little group U(1). Wewill find flux tubes when this U(1) symmetry is spontaneously broken down to nothing.The elements of this U(1) are h ( x ) = exp[ iξ ( x ) ˆ φ ( x )] , rotations by an angle ξ ( x ) aroundthe direction of φ ( x ) at any point in space. This U(1) will be broken by the vacuumconfiguration of ψ . Let us then define the ψ -vacuum by, ψ ∗ i ψ i = v (3.1) D µ ψ = 0 , (3.2)where D µ is defined using A µ in the φ -vacuum, as in Eq. (2.2). Multiplying Eq. (3.2) by ψ † ˆ φ from the left, its adjoint by ˆ φψ from the right, and adding the results, we get0 = ψ † ˆ φD µ ψ + ( D µ ψ † ) ˆ φψ = ∂ µ h ψ † ˆ φψ i , (3.3)– 4 –rom which it follows that ψ † ˆ φψ = constant , (3.4)or explicitly in terms of the components,Tr h ψ † i σ αij ψ j τ α ˆ φ i = constant . (3.5)It follows that the components parallel and orthogonal to φ are both constants. Then wecan decompose ψ † i σ αij ψ j τ α = v cos θ c ˆ φ + v sin θ c ˆ κ , (3.6)where ˆ κ is a vector in the adjoint, orthogonal to ˆ φ . We can always write ˆ κ asˆ κ = hU τ U † h † , (3.7)where h and U are as defined before and in Eq. (2.8).Using the identity σ αij σ αkl = δ il δ kj − δ ij δ kl , we find that ψ is a eigenvector of theexpression on the left hand side of Eq. (3.6). Then writing the right hand side of thatequation in terms of h and U , we find that ψ can be written as ψ = v hU ρ ρ ! , (3.8)where ρ and ρ are constants. Keeping U fixed, we vary ξ and find the periodicity ψ ( ξ ) = ψ ( ξ + 4 π ) . (3.9)This ξ is the angle parameter of the residual U (1) gauge symmetry and in the presence ofa string solution, this ξ is mapped a circle around the string. In order to make ψ singlevalued around the string, we need ξ = 2 χ , where χ is the angular coordinate for a looparound the string. Next let us calculate the Lagrangian of the scalar field ψ . We have D µ ψ = ∂ µ ψ − igA µ ψ (3.10)= ∂ µ ( hU ρ ) − ig h B µ ˆ φ + ig h ˆ φ, ∂ µ ˆ φ ii hU ρ (3.11)= ∂ µ ( U h ρ ) − ig h B µ ˆ φ + ig h ˆ φ, ∂ µ ˆ φ ii U h ρ (3.12)= − iU h τ ρ [2 ∂ µ χ + g ( B µ + N µ )] , (3.13)where h = e − i χτ , ρ i ρ i = v , and we have used the identity U † hU = exp( − iχτ ) . Wehave also introduced the Abelian ‘monopole field’ N µ = 2 iQ m Tr h ∂ µ U U † ˆ φ i , (3.14) ∂ [ µ N ν ] = Q m M µν + 2 iQ m Tr [( ∂ [ µ ∂ ν ] U ) U † ˆ φ ] . (3.15)The first term reproduces the magentic field of the monopole configuration, while thesecond term is a gauge dependent line singularity, the Dirac string. This singular string is– 5 – red herring, and we are going to ignore it because it is an artifact of our construction.We have used a U ( ~x ) which is appropriate for a point monopole. If we look at the systemfrom far away, the monopoles will look like point objects and it would seem that we shouldfind Dirac strings attached to each of them. However, we know that the ’t Hooft-Polyakovmonopoles are actually not point objects, and their near magnetic field is not describableby an Abelian four-potential N µ , so if we could do our calculations without the far-fieldapproximation, we would not find a Dirac string. Further, as was pointed out in [12], theactual flux tube occurs along the line of vanishing ψ , and it is always possible to choose a U ( ~x ) appropriate for the monopole configuration such that the Dirac string lies along thezeroes of ψ . Since | ψ | always multiplies the term containing N µ in the action, the effectof the Dirac string can always be ignored.With these definitions we can calculate L = − F µν F µν + v ∂ µ χ + e ( B µ + N µ )) (3.16)Here, we have defined electric charge e = g and written the magnetic charge as Q m = e .
4. Dualization
Let us now dualize the low energy effective action in order to express the theory in terms ofthe macroscopic string variables. The partition function Z is simply the functional integral Z = Z D B µ D χ exp i Z d x (cid:20) − F µν F µν + v eB µ + ∂ µ χ + eN µ ) (cid:21) . (4.1)In the presence of flux tubes we can decompose the angle χ into a part χ s which measuresflux in the tube and a part χ r describing single valued fluctuations around this configura-tion, χ = χ r + χ s . Then if χ winds around the tube n times, we can define ǫ µνρλ ∂ ρ ∂ λ χ s = 2 πn Z Σ dσ µν ( x ( ξ )) δ ( x − x ( ξ )) ≡ Σ µν , (4.2)where ξ = ( ξ , ξ ) are the coordinates on the world-sheet and dσ µν ( x ( ξ )) = ǫ ab ∂ a x µ ∂ b x ν . The vorticity quantum is 2 π in the units we are using and n is the winding number [28].The integration over χ has now become integrations over both χ r and χ s . However χ r is a single-valued field, so it can be absorbed into the gauge field B µ by a redefinition,or gauge transformation, B µ → B µ + ∂ µ χ r . We can linearize the action by introducingauxiliary fields C µ , B µν and A mµ , Z = Z D B µ D C µ D χ s D B µν D A mµ exp i Z d x (cid:20) − G µν G µν + 14 ǫ µνρλ G µν F ρλ − v C µ − C µ ( eB µ + eN µ + ∂ µ χ s ) (cid:21) , (4.3)– 6 –here we have written G µν = ∂ µ A mν − ∂ ν A mµ + ev B µν . and F µν = ∂ µ B ν − ∂ ν B µ + M µν .Now we can integrate over B µ easily, Z = Z D C µ D χ s D B µν D A mµ δ (cid:16) C µ − v ǫ µνρλ ∂ ν B ρλ (cid:17) exp i Z d x (cid:20) − G µν G µν + ev ǫ µνρλ B µν M ρλ − A µ j µ − v C µ − C µ ( eB µ + eN µ + ∂ µ χ s ) (cid:21) . (4.4)Here j µm = − ǫ µνρλ ∂ ν M ρλ is the magnetic monopole current. Integrating over C µ we get Z = Z D χ s D B µν D A mµ exp i Z d x (cid:20) − G µν G µν + 112 H µνρ H µνρ − v µν B µν − A µ j µ (cid:21) , (4.5)where we have written defined H µνρ = ∂ µ B νρ + ∂ ν B ρµ + ∂ ρ B µν , used Eq. (4.2) and alsowritten M µν = ( ∂ µ N ν − ∂ ν N µ ) . We can also replace the integration over D χ s by an integration over D x µ ( ξ ), repre-senting a sum over all the flux tube world sheet where x µ ( ξ ) parametrizes the surface ofsingularities of χ . The Jacobian for this change of variables gives the action for the stringon the background space time [19, 29]. The string has a dynamics given by the Nambu-Goto action, plus higher order operators [30], which can be obtained from the Jacobian.We will ignore the Jacobian below, but of course it is necessary to include it if we want tostudy the dynamics of the flux tube. Z = Z D x µ ( ξ ) D B µν D A mµ exp i Z d x (cid:20) − G µν G µν + 112 H µνρ H µνρ − v µν B µν − A µ j µ (cid:21) , (4.6)The equations of motion for the field B µν and A µ can be calculated from this to be ∂ λ H λµν = − m G µν − me Σ µν , (4.7) ∂ µ G µν = j µm (4.8)where G µν = ev B µν + ∂ µ A mν − ∂ ν A mµ , and m = ev . Combining Eq. (4.8) and Eq. (4.7)we find that 1 e ∂ µ Σ µν ( x ) + j µm ( x ) = 0 . (4.9)It follows rather obviously that a vanishing magnetic monopole current implies ∂ µ Σ µν ( x ) =0 , or in other words if there is no monopole in the system, the flux tubes will be closed.The magnetic flux through the tube is 2 nπe , while the total magnetic flux of themonopole is 4 mπg , where n, m are integers. Since eQ m = 12 , it follows that we can have astring that confine a monopole and anti-monopole pair for every integer n . Although thisstring configuration could be broken by creating a monopole-anti-monopole pair, there is ahierarchy of energy scales v ≫ v , which are respectively proportional to the mass of themonopole and the energy scale of the string. So this hierarchy can be expected to preventstring breakage by pair creation. – 7 –he conservation law of Eq. (4.9) also follows directly from Z in Eq. (4.6) by intro-ducing a variable B ′ µν = B µν + m ( ∂ µ A mν − ∂ ν A mµ ) and integrating over the field A mµ . If wedo so we get Z = Z D x µ ( ξ ) D B ′ µν δ h e ∂ µ Σ µν ( x ) + j νm ( x ) i exp (cid:20) i Z (cid:26) H µνρ H µνρ − m B ′ µν − m e Σ µν B ′ µν (cid:27)(cid:21) , (4.10)with the delta functional showing the conservation law (4.9). Thus these strings are anal-ogous to the confining strings in three dimensions [31]. There is no A mµ , the only gaugefield which is present is B ′ µν . This B ′ µν field mediates the direct interaction between theconfining strings.The delta functional in Eq. 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