Malhcnuilical and Physical Sci.. 2012. Vol. 57, No. 7, pp. 100-105 This paper is available online at hllp://stdb.hnue.edu.vn
N O N - A B F J J A N ( LASSICAL S O L U T I O N O F THK Y A N ( ; - M I L L S - H I G G S T H E O R Y
N g u y e n Van T h u a n
Faculty of Physics. Hanoi National University of Education
Abstract. In this paper, wc investigate an SU{2) gauge field coupled wiih two massles.s Higgs triplets. We oblain a non-Abelian exact cla.ssical solution of corresponding Yang-Mills equations. We also find the energy expres.sion of this classical solution. Some particular cases of the solution are considered.
Keywords: Non-Abelian gauge fields, Higgs triplets. Yang-Mills equadon, 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 lo find since there exists no general method for discovering them. The usual approach is to make some guess as lo 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: [email protected]
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 properly 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
i = --/'^F-"' + ^(B..<A°)(»v) + \{D„r)(D''r), (2.1)
P^i, = S,-ll';'-9^ll'; + .9e°''W^W';. (2-2) D^^t" = d^if,'+ ge'''^Wlij>\ (2.3)
The equations of motion of the SU(2) gauge fields and two massless Higgs tiiplets from the Lagrangian density (2.1) are
S"F^^ = ge"'[Fl-W"'-(D,,p-)4>''-(D^ij'')rl (2.5)
g''{D^<l,') = ge"^(D^,4l')W>", (2.6) a''(£)^V°) = <7e°'=(I'„'/'')W"'° (2-'') Assume that the SC/(2) gauge fields and two massless Higgs triplets are spheiical
symmetry. We use the Wu-Yang ansatz [10]
w° =
ws - r --
r -
'^'"gr^
gr gr
= -H{r), gr
-K(r)],
(2.8) where K{r) ,J{r), I{r) and H{r) are certain functions of the radius r, which satisfy field equations of motion, and f°, is the unit radius vector. Inserting this ansatz into the field
Nguyen V;in Thuan
equations (2 .S) (2.7) yields four coupled nonlinear differential equations ,-^'''''' 1<(I<'' + U'' I I'-.f'-l),
.'''•' = 2 . / A ' dr-
.y''' - •lll<\
,-'''',' = 211K' (2.9)
dl-
The exact solulion of Ihc classical equations of motion of the Sl'(2) gauge field coupled with one massless Higgs triplet was found by Singleton |8]. Here we consider two massless I liggs triplets. The exact solution to the above equations are
A(r) Hr) = I(r) -- H(r) =
Al - r B C . - I c - l '
D Ar - \
where A, B, C, and D are arbitrary constants. The only constraint inposed is diat C" -i- D'^ — B^ = 1 sothatthesolutionof 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). wc see that the gauge fields and two massless Higgs triplets become infinite at the radius
/• = ro = i - (2.11) Using these singular gauge potentials to calculate the non-Abelian electric and
magnetic field intensities, we obtain
These non-Abelian field intensities are also infinite at r = ro = - j . This seems to exhibit SU{2) gauge charge confinement. An SU{2) gauge charge earring particle, which 102
Non-Abelian classical solution of the Yang-Mills-Higgs theory
enters the region r < ro, is not able to leave this region As r -> oo, these electric and magnetic fields fall off like (1/r^), unlike the Prasad-Soiiinierfield .solution, which has a (1/r^) behavior for large r \6\.
There are three particular cases which can be considered. The first ca,se i.s where the spatial component of the gauge fields equals /.cio [11';' - 0). Thi.s corresponds to taking K{r) = 1. In this case equation (2.9) has the solulion
K{r) = 1, ,/(r) - ';, /(/•) = - , //(/•) = - . (2.14) where E. F. and C are aibitrary conslants. which satisfy the codilion F^ I f/" ^- A".
The second case is where the time component of the gauge fields equals zero (WQ = 0). This corresponds to taking ./(/) 0, which implies 6 = 0 in equation (2.10). The above codition C~ -^- D'^ - B'^ = I yields C^ + D'^ = 1. Therefore we can write C = sinO. D = coti9 where 0 is an arbitrary constant. In this case the solution becomes
The last case is where there are no two Higgs triplets. This corresponds to /(<) = 0, H(r) = 0, which imply T = 0, D = 0 in equation (2.10). The codition C^ + D'^-B'^ = 1 now requires that B = ±i. The solution becomes
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 oiu- exact classical solution can be obtained by taking the volume integral of the time-time component of the energy-momentum tensor
T"' = F'""F^° + {D''<I>')(D''4,'') + (D''^')(D-'^') + g^'L. (2.17) The energy of the fields is
E= fr^'d^x. (2.18) From equations (2,17), (2.18) and using ansatz (2.8) we have
/ dJ Y E = ^ ^4it r°° \(dK\ +^ , ' +~7^+ 2r^ +
/ dl ,\2 / dU y
2r2 "^ r^ 2r2 J dr. (2.19)
103
Nguyen Van Thuan
Notice that ihc integral has been cut off from below at an arbitrary distance r..
which must be large Ihan ,„. Tins procedure is done to avoid the singularities at r = 0 and r -- ;„ since integrating through r = 0,r„ would give an infinite field energy. This is similar lo Ihc Coulomb potential of point ck-cl. ic charge, which yields an infinite field energy when intcgreled down to /.em. Inserting A(r), ./(r), I(r). and H{r) of equation (2.10) inlo equation (2.1')) we oblain
E^'-llLi,i'^C'+IJ'+i) ' ' - ! ' ^ ^ . (2-20)
( y ' ' <i I • ' ' ' (1 '•I
From the codition r^ i I>' - B'= I, equation (2 20) can be rewitten as
In the case where there is only one massless Higgs triple' (i.e., C = 0, or D = 0), equation (2.21) becomes
-ITTC- (2Ar„-l) g^ ra{Ar„ - 1)'
inD- (J.lr,, - 1) / „ ( . l / „ - l ) '
for the case D = 0, (2.22)
for the case C = 0, (2.23) For the pure gauge field case (C" = 0, /)- = 0, B^ = - 1 ) , the energy of equation
(2.20) becomes zero. This together with the requirement that the IVJ components of this solution are pure imaginary raises doubts about the physical importance of Ulis particular case. If we want to discard the zero energy pure gauge case, then il 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 ro = -7. It can be seen that a particle, which caries an SU{2) gauge charge, becomes confined if it crosses into the region r < ro = —. 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 solulion of the Yang-Mills-Higgs theory
gauge fields equals zero, leaving only two massless Higgs triplets and the time component of the gauge field. Second the time component of the gauge fields is zero, leavmg only two massless Higgs triplets and the spatial component of the gauge field. And third, there is the pure gauge solution, where two massless Higgs triplets arc absent, leaving only the time and spatial components of the gauge fields. In pure gauge field case, energy of field equals zero. It follows that the Lagrangian density of the fields always include the Higgs fields so that the zero energy gauge case can be discarded.
REFERENCES
[1] Y. Brihaye, 2001. Dilaionic monopoles and hairy black holes. Phys. Rev. D, Vol.65, p. 024019.
[2] Y. M. Cho and D. G. Park, 2002. Monopole condendation in SU(2) QCD. Phys. Rev.
D, Vol.65, p. 074027.
[3] A. S. Cornell, G. C. Joshi, J. S. Rozowsky and K. C. Wall, 2003. Non-Abelian monopole and dyon soluiions in a modified Einsiem Yang-Mills-Higgs sy.stem. Phys.
Rev. D 67, p. 105015.
[4] D. Metaxas, 2007. Quantum classical interaction through the path integral. Phyi. Rev.
D 75, p. 065023.
[5] M. Nielsen and N. K. Nielsen, 2000. Explicit construction of constrained instantons.
Phys. Rev. D, Vol. 6 1 , p. 105020.
[6] M. K. Prasad and C. M. Sommerfield, 1975. Exact classical solution for the 't Hooft monopole and Julia-Zee dyon. Phys. Lett. 35, p. 760.
[7] A. Rajantie, 2003. Magnetic monopoles from gauge phase transitions. Phys. Rev. D 68, p. 021301.
[8] D. Singleton, 1995. Exact Schwanschild-like solution for Yang-Mills theories. Phys.
Rev. D, Vol. 51, No. 10, p. 5911.
[9] E. Witten, 1977. Some exact multipseudoparticle solution of classical Yang-Mills theory. Phys. Rev. Lett. 38, p. 121.
[10] T T. Wu and C. N. Yang, 1969. In properties of matter under unusual conditions.
Edited by H. Mark and S. Fernbach (Interscience, New York).
105
