Non-abelian classical solution of the Yang-Mills-Higgs theory

JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2012, Vol. 57, No. 7, pp. 100-105 This paper is available online at NON-ABELIAN CLASSICAL SOLUTION OF THE YANG-MILLS-HIGGS THEORY Nguyen Van Thuan Faculty of Physics, Hanoi National University of Education Abstract. In this paper, we investigate an SU(2) gauge field coupled with two massless Higgs triplets. We obtain a non-Abelian exact classical solution of corresponding Yang-Mills equations. We also find the energy expression of this classical solution. Some particular cases of the solution are considered. Keywords: Non-Abelian gauge fields, Higgs triplets, Yang-Mills equation, classical solution. 1. Introduction The Yang-Mills equations for the SU(2) gauge field coupled with the Higgs field suggest many interesting solution types: monopole, dyon, instanton and meron [2, 3, 5, 7]. Physical applications of the classical Yang-Mills-Higgs theory begin with exact solutions. The physical properties of monopoles, dyons, instantons and merons are particularly important. For example, imaginary time solutions of classical theories are usually interpreted as real time tunneling in the corresponding quantized theory. Classical Yang-Mills-Higgs theory can be studied independently of exact solutions, of course. This is an interesting parsuit, because any results may lead to improvements in an integral formulation of quantum field theory [4]. The Yang-Mills equations are nonlinear differential equations. Exact solutions to nonlinear field theories are very difficult to find since there exists no general method for discovering them. The usual approach is to make some guess as to the form of the solution and insert it into the field equations to see if it solves them. For the Yang-Mills-Higgs theory there are some known exact solutions, which are found by this approach [1, 8, 9]. Received July 25, 2012. Accepted September 20, 2012. Physics Subject Classification: 60 44 01 03. Contact Nguyen Van Thuan, e-mail address: thuanvatli@yahoo.com 100 Non-Abelian classical solution of the Yang-Mills-Higgs theory In this article, we consider the Yang-Mills equations for the SU(2) gauge field coupled with two massless Higgs triplets. The exact classical solution of this equations and corresponding non-Abelin field intensities seem to exhibit the property of cofinement, which can be found for non-Abelian gauge theories. 2. Content 2.1. The exact classical solution of the Yang-Mills equations The Lagrangian density for the SU(2) gauge fields coupled with two massless Higgs triplets has the form L = −1 4 F aµνF µνa + 1 2 (Dµφ a)(Dµφa) + 1 2 (Dµψ a)(Dµψa), (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) The equations of motion of the SU(2) gauge fields and two massless Higgs triplets from the Lagrangian density (2.1) are ∂νF aµν = gε abc [ F bµνW νc − (Dµφb)φc − (Dµψb)ψc ] , (2.5) ∂µ(Dµφ a) = gεabc(Dµφ b)W µc, (2.6) ∂µ(Dµψ a) = gεabc(Dµψ b)W µc. (2.7) Assume that the SU(2) gauge fields and two massless Higgs triplets are spherical symmetry. We use the Wu-Yang ansatz [10] W ai = εaij rˆj gr [1−K(r)], W a0 = rˆa gr J(r), φa = rˆa gr I(r), ψa = rˆa gr H(r), (2.8) whereK(r), J(r), I(r) andH(r) are certain functions of the radius r, which satisfy field equations of motion, and rˆa is the unit radius vector. Inserting this ansatz into the field 101 Nguyen Van Thuan equations (2.5) - (2.7) yields four coupled nonlinear differential equations r2 d2K dr2 = K(K2 +H2 + I2 − J2 − 1), r2 d2J dr2 = 2JK2, r2 d2I dr2 = 2IK2, r2 d2H dr2 = 2HK2. (2.9) The exact solution of the classical equations of motion of the SU(2) gauge field coupled with one massless Higgs triplet was found by Singleton [8]. Here we consider two massless Higgs triplets. The exact solution to the above equations are K(r) = Ar Ar − 1 , J(r) = B Ar − 1 , I(r) = C Ar − 1 , H(r) = D Ar − 1 , (2.10) where A,B,C, and D are arbitrary constants. The only constraint inposed is that C2 + D2−B2 = 1 so that the solution of equation (2.10) involves only three arbitrary constants. Inserting K(r), J(r), I(r) and H(r) into the expressions for the gauge fields and two massless Higgs triplets of equation (2.8), we see that the gauge fields and two massless Higgs triplets become infinite at the radius r = r0 = 1 A . (2.11) Using these singular gauge potentials to calculate the non-Abelian electric and magnetic field intensities, we obtain Eai = F a 0i = 1 g [ (2ABr −B) r2(Ar − 1)2 rˆ irˆa − AB r(Ar − 1)2 ( δia − rˆirˆa )] , (2.12) Bai = − 1 2 εijkF a jk = 1 g [ (1− 2Ar) r2(Ar − 1)2 rˆ irˆa + A r(Ar − 1)2 ( δia − rˆirˆa )] . (2.13) These non-Abelian field intensities are also infinite at r = r0 = 1 A . This seems to exhibit SU(2) gauge charge confinement. An SU(2) gauge charge carring particle, which 102 Non-Abelian classical solution of the Yang-Mills-Higgs theory enters the region r < r0, is not able to leave this region. As r → ∞, these electric and magnetic fields fall off like (1/r3), unlike the Prasad-Sommerfield solution, which has a (1/r2) behavior for large r [6]. There are three particular cases which can be considered. The first case is where the spatial component of the gauge fields equals zero (W ai = 0). This corresponds to taking K(r) = 1. In this case equation (2.9) has the solution K(r) = 1, J(r) = E r , I(r) = F r , H(r) = G r , (2.14) where E, F , and G are arbitrary constants, which satisfy the codition F 2 +G2 = E2. The second case is where the time component of the gauge fields equals zero (W a0 = 0). This corresponds to taking J(r) = 0, which implies B = 0 in equation (2.10). The above codition C2 + D2 − B2 = 1 yields C2 + D2 = 1. Therefore we can write C = sinθ,D = cosθ where θ is an arbitrary constant. In this case the solution becomes K(r) = Ar Ar − 1 , I(r) = sinθ Ar − 1 , H(r) = cosθ Ar − 1 . (2.15) The last case is where there are no two Higgs triplets. This corresponds to I(r) = 0, H(r) = 0, which implyC = 0, D = 0 in equation (2.10). The coditionC2+D2−B2 = 1 now requires that B = ±i. The solution becomes K(r) = Ar Ar − 1 , J(r) = ± i Ar − 1 . (2.16) 2.2. Energy of the SU(2) gauge fields and two massless Higgs triplets The energy of the SU(2) gauge fields and two massless Higgs triplets of our exact classical solution 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. (2.17) The energy of the fields is E = ∫ T 00d3x. (2.18) From equations (2,17), (2.18) and using ansatz (2.8) we have E = 4π g2 ∫ ∞ ra [(dK dr )2 + (K2 − 1)2 2r2 + J2K2 r2 + ( r dJ dr − J )2 2r2 + + I2K2 r2 + ( r dI dr − I )2 2r2 + H2K2 r2 + ( r dH dr −H )2 2r2 ] dr. (2.19) 103 Nguyen Van Thuan Notice that the integral has been cut off from below at an arbitrary distance ra, which must be large than r0. This procedure is done to avoid the singularities at r = 0 and r = r0, since integrating through r = 0, r0 would give an infinite field energy. This is similar to the Coulomb potential of point electric charge, which yields an infinite field energy when integreted down to zero. Inserting K(r), J(r), I(r), and H(r) of equation (2.10) into equation (2.19) we obtain E = 2π g2 (B2 + C2 +D2 + 1) (2Ara − 1) ra(Ara − 1)3 . (2.20) From the codition C2 +D2 − B2 = 1, equation (2.20) can be rewitten as E = 4π g2 (C2 +D2) (2Ara − 1) ra(Ara − 1)3 . (2.21) In the case where there is only one massless Higgs triplet (i.e., C = 0, or D = 0), equation (2.21) becomes E = 4πC2 g2 (2Ara − 1) ra(Ara − 1)3 for the case D = 0, (2.22) or E = 4πD2 g2 (2Ara − 1) ra(Ara − 1)3 for the case C = 0, (2.23) For the pure gauge field case (C2 = 0, D2 = 0, B2 = −1), the energy of equation (2.20) becomes zero. This together with the requirement that theW a0 components of this solution are pure imaginary raises doubts about the physical importance of this particular case. If we want to discard the zero energy pure gauge case, then it is necessary for the Lagrangian density to always include the Higgs fields. 3. Conclusion Considering the SU(2) gauge field coupled with two massless Higgs triplets, we have found the exact classical solution of the corresponding Yang-Mills equations. We also obtain non-Abelian electric and magnetic field intensities of this solution. The exact classical solution and field intensities have singularity at r0 = 1 A . It can be seen that a particle, which caries an SU(2) gauge charge, becomes confined if it crosses into the region r < r0 = 1 A . Thus the solution exhibits the property of the SU(2) gauge charge confinement. We investigated three particular cases. First the spatial component of the 104 Non-Abelian classical solution of the Yang-Mills-Higgs theory gauge fields equals zero, leaving only two massless Higgs triplets and the time component of the gauge field. 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