SU(2) gauge field coupled with two massless scalar fields

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2010, Vol. 55, No. 6, pp. 14-19 SU(2) GAUGE FIELD COUPLED WITH TWO MASSLESS SCALAR FIELDS Nguyen Van Thuan Hanoi National University of Education E-mail: thuanvatli@yahoo.com Abstract. From the study of the SU(2) gauge field coupled with two massless scalar fields, we have found solutions to the corresponding Yang- Mills equations. These classical solutions exhibited a form of confinement to an SU(2) gauge charge. Energy expressions of the corresponding gauge and scalar field configurations are obtained. Keywords: gauge field, massless scalar field. 1. Introduction Non-abenlian gauge theories were invented by Yang-Mills. Solutions to the classical field equations, in which field functions are c-number (not operators), play an impotant role in investigating the configurations of the corresponding quantum field theory. Based on these solutions, applying the semi-classical analysis methods, important information about the quantum theory can be obtained, which could not be done with the perturbation theory [1-3]. One important result when studying the classical Yang-Mills equations is to recognise that the extremum of the functional in the Euclidian space do not correspond to the zero-unified field, but correspond to the nontrivial space-time local field configuration known as instanton [4]. In the quantum theory, instantons describe the tunnel effects of the degenerate vacuum states. This result brought a new view to the vacuum structure of the Yang-Mills theory, providing a qualitative explanation of the Quark confinement [5-8]. Researches of classical Yang-Mills equations with external sources are well documented. With these kind of problems, the most interesting result is to find some particular solutions that can produce screening effects with external color charge, similar to the screening effects of the external charge in electrodynamics. Some studies have found that the Quark confinement may have some connections with these screening effects [9,10]. 14 SU(2) gauge field coupled with two massless scalar fields In this paper, we investigate the SU(2) gauge field coupled with two massless scalar fields. In Section 2, we introduce solutions to Yang-Mills equations corre- sponding to these field configurations. It is argued that these classical solutions exhibit the property of confinement that has been looked for in non-abelian gauge theory. In Section 3, energy expressions of the SU(2) gauge field coupled with two massless scalar fields are identified. Finally, we discuss the results and make con- clusions in Section 4. 2. Solutions to the Yang-Mills equations The Lagrangian density of the SU(2) gauge field coupled with two scalar fields is given by: L = −1 4 F µνaF aµν + 1 2 (Dµφa)(Dµφ a) + 1 2 (Dµψa)(Dµψ a) − λ 4 (m2 λ − φ2 )2 − λ ′ 4 (m′2 λ′ − ψ2 +W 2 )2 − 1 2 g2 [ φ2ψ2 − (φaψa) ] , (2.1) where F aµν = ∂µW a ν − ∂νW aµ + gεabcW bµW cν , (2.2) Dµφ a = ∂µφ a + gεabcW bµφ c, (2.3) Dµψ a = ∂µψ a + gεabcW bµψ c. (2.4) HereW aµ , φ a and ψa are the non-alelian gauge and scalar potentials, respectively, and g is the coupling constant. The last term in Equation (2.1) describes the coupling of φa and ψa. This term vanishes if they are of the same direction in group space. The model which we consider here is an SU(2) gauge field coupled with two massless scalar fields. These scalar fields have the same direction in group space. Therefore the Lagrangian density (1) can be rewritten L = −1 4 F µνaF aµν + 1 2 (Dµφa)(Dµφ a) + 1 2 (Dµψa)(Dµψ a). (2.5) We are only interested in static solutions so all the time derivatives will be zero. With this condition the gauge field equations from the Lagrangian density (2.5) are: ∂iF µia + gεabcW bi F µic = gεabc [ (Dµφb)φc + (Dµψb)ψc ] , (2.6) where i = 1, 2, 3 are the space indices. 15 Nguyen Van Thuan For the massless scalar fields the equations are: ∂i(D iφa) + gεabcW bµ(D µφc) = 0, (2.7) and ∂i(D iψa) + gεabcW bµ(D µψc) = 0. (2.8) Further assuming that the gauge field and two scalar fields are radial, we use the Actor ansatz [3] W ai = εaij rˆj gr [ 1−K(r)], (2.9) W a0 = 0, (2.10) φa = rˆa gr J(r), (2.11) ψa = rˆa gr H(r), (2.12) where K(r), J(r), and H(r) are funtions of the radius r, rˆ is unit vector codirectional with ~r. Inserting this ansatz into Equations (2.6)-(2.8) produces three coupled nonlinear differential equations r2K” = K(K2 +H2 + J2 − 1), (2.13) r2J” = 2JK2, (2.14) r2H” = 2HK2. (2.15) The solutions to the above equations are K(r) = Cr Cr − 1 , (2.16) J(r) = A Cr − 1 , (2.17) H(r) = B Cr − 1 , (2.18) where A,B, and C are constants. The only constraint imposed is that A2 + B2 = 1, so that Equations (2.16)-(2.18) involve only two arbitrary constants. Inserting K(r), J(r), andH(r) into the expressions for the gauge and scalar fields of Equations (2.9)-(2.12), we can see that gauge field and both the scalar fields become infinite at the radius r0 = 1 C . (2.19) 16 SU(2) gauge field coupled with two massless scalar fields Further, using these singular gauge potentials to calculate the ”electric” and ”magnetic” field (Eia = F i0 a and B i a = − 1 2 εijkF jka , respectively) it is seen that these fields are also infinite at r0 = 1/C. Therefore a particle, which carries an SU(2) gauge charge, becomes permanently confined if it crosses into the region r < r0. The non-abelian gauge potentials and both the scalar fields of Equations (2.9)-(2.12) also become singular at r = 0, which is true as well for the Coulomb potential of a point charge in classical eletromagnetism. The singularity of all these solutions at r = 0 is of the same character in that they all imply a δ-function point charge sitting at the origin (where the charge of our non-abelian model is SU(2) gauge charge). Just as the singularity at the origin can be taken to be a point source of SU(2) charge, so the singularity at r = 1/C can be taken to be a spherical shell of SU(2) charge. This shell structure is a feature of our solution, and it points to a possible connection with the various models of SU(2) gauge charge bound states. There are two particular cases which can be considered. The first case is for K(r) = 0, solutions to Equations (2.14)-(2.15) are: J(r) = D, H(r) = E, (2.20) where D, and E are arbitrary constants. With K(r) = 0, SU(2) gauge poten- tial W ai = εaij rˆj gr , this is the potential of a point monopole with value of 1/g. The second case is for K(r) = ±1, in this case the solutions to Equations (2.14)- (2.15) are J(r) = G r , H(r) = i G r , (2.21) where G is arbitrary constant. In this case two scalar fields are only different to one imaginary unit. 3. Energy expressions The energy of the gauge and scalar field configurations can be obtained by taking the volume integral of the time-time component of the energy-momentum tensor T µν = F µρaF νaρ + (D µφa)(Dνφa) + (Dµψa)(Dνψa) + gµνL, (3.1) hence E = ∫ T 00d3x. (3.2) 17 Nguyen Van Thuan is the energy of the fields, which is given by E = 4pi g2 ∫ ∞ rc [ K ′2 + (K − 1)2 2r2 + + J2K2 r2 + (rJ ′ − J)2 2r2 + H2K2 r2 + (rH ′ −H)2 2r2 ] dr. (3.3) Notice that the integral has been cut off from below at an arbitrary distance rc, which must be larger than r0. This was done to avoid the singularities at r = 0 and r = r0, since integrating through r = 0 and r = r0 would give an infinite field energy in the same way that the Coulomb potential of a point electric charge yield an infinite field energy when integrated down to zero. Inserting K(r), J(r) and H(r) into Equation (3.3) we obtain E = 2pi g2 (A2 +B2 + 1) (2Crc − 1) rc(Crc − 1)3 . (3.4) For the pure gauge case (A2 = 0, B2 = 0) energy expression is Eg = 2pi g2 (2Crc − 1) rc(Crc − 1)3 . (3.5) Finally, just like the Prasad-Sommerfield solution [11], our solution can be seen to carry a topological magnetic charge when the electromagnetic field is em- bedded into the SU(2) via ′t Hooft generalized, gauge invariant, electromagnetic field strength tensor [12]. 4. Discussion and conclusion Studying the SU(2) gauge field coupled with two massless scalar fields, we have found solutions to the corresponding Yang-Mills equations. These classical solutions exhibited a form of confinement. Any particle that caries an SU(2) gauge charge and enters the region r < r0 = 1/C would no longer be able to leave this region. This is analogous to what happens with the Schwarzschild solution in general relativity, where once a particle passes the event horizon it is permanently confined. If our solution is responsible for the confinement mechanism in non-abelian field theories then it is necessary for the Lagrangian density to always include scalar fields in order for an acceptable solution to exist. Under these assumptions scalar fields become crucial to the confinement mechanism. Still it is encouraging that at the classical level an analytical solution, which seems to exhibit confinement, can be found for a non-abelia gauge theory. Two particular cases of the solutions are 18 SU(2) gauge field coupled with two massless scalar fields considered. 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