Some properties of SU(2) Yang-Mills theory connecting with φ4 theory

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2011, Vol. 56, No. 7, pp. 72-77 SOME PROPERTIES OF SU(2) YANG-MILLS THEORY CONNECTING WITH φ4 THEORY Nguyen Van Thuan Hanoi National University of Education E-mail: thuanvatli@yahoo.com Abstract. In this paper we consider some properties of the SU(2) Yang- Mills theory connecting with scalar φ4 theory. In the case when scalar field φ is time independent, we find explicit solutions to equations of motion for corresponding φ4 theory. The spatial component of the SU(2) Yang-Mills potential corresponding with this solution is like potential of point magnetic monopole. Keywords: Gauge field theory, scalar field theory. 1. Introduction Non-Abelian gauge theory offer the greatest promise to describe the elemen- tary forces in nature [1]. We investigate here the solutions to the classical Yang-Mills equations, they are non-Abelian field equations. However, solutions to non-linear field equations are notoriously difficult to fine 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 in to the field equations to see if it solves them. Accord- ing to this approach, one have found some exact solutions to the Yang-Mills field equations [2-6]. Besides the above approach one sees that there exists a connection between the SU(2) Yang-Mills theory and scalar φ4 theory. From this connection one reduces the complicated equations of motion of the Yang-Mills theory to the sin- gle equation of motion for scalar φ4 theory. Therefore one can find some properties of the Yang-Mills theory [7-9]. 2. Content 2.1. Equations of motion of the SU(2) Yang-Mills field Yang-Mills fields can be introduced in the following fashion. Consider a multi- plet ψ(x) which transforms locally under the action of some gauge groupG according to the rule ψ(x)→ ψ′(x) = ω(x)ψ(x), (2.1) 72 Some properties of SU(2) Yang-Mills theory connecting with φ4 theory where ω(x) belongs to the relevant representation of G. Let us define a derivative Dµψ of ψ which has this same simple transformation property. We make the ansatz Dµψ = ( ∂µ − igWµ(x) ) ψ(x), (2.2) where Wµ(x) is a matrix function. By assumption, Dµψ transforms like Dµψ → D′µψ′ = ωDµψ, (2.3) where D′µ = ∂µ − igW ′µ(x). This lead to( ∂µ − igW ′µ ) ω = ω ( ∂µ − igWµ ) , (2.4) which can be rewritten W ′µ = ωWω −1 − (i/g) ( ∂µω ) ω−1. (2.5) Wµ(x) is the Yang-Mills potential in matrix form, and eq. (2.5) is the local transformation rule for this potential under the gauge group G. If the potential Wµ(x) is not introduced, then one can not define the covariant derivative Dµψ with its simple transformation property (2.3). Wµ(x) is analogous to the four-potential in electromagnetism. The Yang-Mills forces, or field strengths, are defined as follows Gµν = ∂µWν − ∂νWµ + (g/i) [ Wµ,Wν ] . (2.6) Gµν has the simple transformation property Gµν → G′µν = ωGµνω−1, (2.7) under the gauge group G, as follows from eq. (2.5). Gµν is therefore thenatura generalization of the field strength tensor Fµν in eletromagnetism. Note that Gµν is not invariant under gauge transformations, whereas Fµν is gauge invariant. This is an important difference between Abelian and non-Abelian gauge theories. Let us now particularize the discussion to the gauge group G = SU(2). In the 2× 2 representation Wµ = 1 2 σaW a µ ; Gµν = 1 2 σaG a µν , (2.8) where Gaµν = ∂µW a ν − ∂νW aµ + gεabcW bµW cν , (2.9) 73 Nguyen Van Thuan g is coupling constant; µ, ν = 0, 1, 2, 3 are space-time indices; a, b, c = 1, 2, 3 are SU(2) group indices. A Lagrangian density which is invariant with any local real or complex SU(2) transformation is easily constructed L = −1 4 GaµνG µν a . (2.10) The equations of motion of the SU(2) Yang-Mills obtained from Lagrangian density (2.10) are ∂µGaµν = gεabcG b µνW ν c . (2.11) Here we see two important new features of non-Abelian gauge theories which are absent in electromagnetism: (i) The equations of motion are nonlinear in the gauge potential. (ii) The gauge potential appears explicitly in the equations of motion. It seems that the Yang-Mills potential plays a more basic role than the potential in an Abelian gauge theory. At least, this is true in the conventional formulation of the theory which directly involves the potential W aµ . In electromag- netism one can work exclusively with the field strengths ~E and ~B. Things are not so simple when the gauge group is non-Abelian. 2.2. Connection between Yang-Mills theory and scalar φ4 theory In Minkowski space the ansatz for the SU(2) gauge potential is gW a0 = ±i ∂iφ φ , gW ai = εian ∂iφ φ ± iδai∂0φ φ , (2.12) where φ is scalar funtion. Inserting ansatz (2.12) into eq. (2.11) yields the following equation for φ 1 φ ∂µ2φ = 3 φ2 ∂µφ2φ. (2.13) Equation (2.13) can be integrated once to give 2φ + λφ3 = 0, (2.14) where λ is an arbitrary integration constant. Suppose that solutions of eq. (2.14) are known, and in the ansatz (2.12) these lead automatically to explicit solutions 74 Some properties of SU(2) Yang-Mills theory connecting with φ4 theory of the SU(2) gauge theory. We think that φ can be interpreted as a physical field, and not merely as an ansatz function. When one considers the extension to massive fields, it becomes apparent that φ has a natural interpretation as a Higgs-like field. Let the SU(2) field have real mass m. Then the Yang-Mills equations of motion are: ∂νGaµν = gεabcG b µνW ν c +m 2W aµ . (2.15) These are reduced by the ansatz (2.12) to 1 φ ( 2+m2 ) ∂µφ = 3 φ2 ∂µφ2φ, (2.16) which in turn is satisfied if φ satisfies 2φ− 1 2 m2φ+ λφ3 = 0. (2.17) Equation (2.17) is the equation of motion for the scalar φ4 theory with spon- taneous symmetry breakdown. We now give formulas for several quantities of interest, following from the ansatz (2.12). The Yang-Mills strengths are gEan = gG a 0n = εnam [1 φ ∂0∂mφ− 2 φ2 ∂0φ∂mφ ] ± iδna [ 1 φ ∂20φ− 1 φ2 ( ∂0φ∂0φ+ ∂mφ∂mφ )] ∓ i [ 1 φ ∂n∂aφ− 2 φ2 ∂nφ∂aφ ] , (2.18) gBan = − 1 2 εnijG a ij = ±igEan + δan (1 φ ) 2φ. (2.19) From eq. (2.19) we see that the self-duality condition Ban = ±iEan evidently implies 2φ = 0, or m2 = 0, λ = 0 in eq. (2.17). The field strengths Ean and B a n are in general complex. However their squares are both real, and this means that the energy and Lagrangian densities obtained from the ansatz (2.12) are real, even though the potential W aµ is complex. The Lagrangian density is given explicitly by Ls = 1 2 ( EanE a n − BanBan ) = 1 g2φ2 [ − 1 2 2φ2φ− ∂α∂βφ∂α∂βφ ] + 1 g2φ3 [ − 2φ∂αφ∂αφ+ 4∂α∂βφ∂αφ∂βφ ] + 1 g2φ4 [ − 3∂αφ∂αφ∂βφ∂βφ ] . (2.20) 75 Nguyen Van Thuan For m2 = 0 we have Ls = 1 2g2 [ 2∂α ( ∂αφ φ ) − 3λ2φ4 ] . (2.21) Another quantity of primary interest is the energy-momentum tensor Tµν(x). This is easily shown to be Tµν = 2φ g2φ { 4 φ2 ∂µφ∂νφ− 2 φ ∂µ∂νφ+ gµν [ 1 2φ 2φ − 1 φ2 ∂αφ∂αφ ]} . (2.22) With self-duality field then Tµν = 0 because 2φ = 0. The total energy for the case m2 = 0 is E = ∫ T00d 3x = 6λ g2 ∫ [1 2 ( ∂0φ )2 + 1 2 ( ▽ φ )2 + 1 4 λφ4 ] d3x, (2.23) where we have neglected surface terms at infinite. 2.3. Particular case In the case when scalar function φ is time independent and λ = 0, eq. (2.14) has the spherically symmetric particular solution. φ = ( a r ) , (2.24) where a is constant. For simplicity, we choose + in ansatz (2.12). Then the SU(2) Yang-Mills potential obtained from φ in eq. (2.24) is gW a0 = −i ra r2 , gW ai = −εian rn r2 . (2.25) The spatial component of the SU(2) gauge potential has the potential form of point magnetic monopole with magnetic charge m = 1 g . (2.26) For the field strengths we find gEan = −iδna 1 r2 , (2.27) and Ban given by eq. (2.19), here B a n = ±iEan because 2φ = 0. We easily verify that the square of the electric field intensity is real. 76 Some properties of SU(2) Yang-Mills theory connecting with φ4 theory 3. Conclusion Using specific ansatz for the SU(2) Yang-Mills potential W aµ in terms of the scalar field φ, we reduce the complicated equations of motion of the Yang-Mills theory to the single equation of motion for the φ4 theory. Therefore one can find explicit solutions of the SU(2) Yang-Mills equations by solving the much simpler scalar field equations. Some properties of the SU(2) Yang-Mills theory connecting with φ4 theory are considered. 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