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Non-Abelian Magnetic Monopoles in a Background of Gravitation with Fermions Carlos Pinheiro
[email protected]/
[email protected], Cintia G. Pinheiro and F.C. Khanna
[email protected] Universidade Federal do Esprito Santo, UFES. Centro de Ci^encias Exatas Av. Fernando Ferrari s/n0 Campus da Goiabeiras 29060-900 Vitoria ES { Brazil.
Theoretical Physics Institute, Dept. of Physics University of Alberta, Edmonton, AB T6G2J1, Canada and TRIUMF, 4004, Wesbrook Mall, V6T2A3, Vancouver, BC, Canada.
The purpose of this paper is to study static solution con guration which describes the magnetic monopoles in a scenary where the gravitation is coupled with Higgs, Yang-Mills and fermions. We are looking for analysis of the energy functional and Bogomol'nyi equations. The Einstein equations now take into consideration the fermions' contribution for energy-momentum tensor. The interesting aspect here is to verify that the fermion eld gives a contribution for non abelian magnetic eld and for potential which minimise the energy functional. Key-words:
Gravitation; Magnetic monopoles.
I Introduction The study of magnetic monopoles started with the Dirac's paper in 1948 [3] where for the rst time the possibility of existence of this kind of particles in nature was analysed. Some years later t'Hooft and Polyakov again found the magnetic monopoles but in a di erent context, the so called non-abelian monopoles had a scalar eld as a source. More recently Wali [1, 2] has studied the Bogomol'nyi equations and he again obtained the magnetic monopole type of solution. The minimization of the functional of energy gives us the conditions for analysis of Bogomol'nyi equations. Finally Christiansen at all [5] have studied the Bogomol'nyi equations in Gauge theory and they verify conditions on functional of energy which is suitable for obtaining nite energy for a static con guration of eld (gauge and scalar eld) in D = 2 + 1 dimensions. In our case the objective is to study the Einsteingauge- eld-Higgs-fermions system in D = 3 + 1 dimensions and, following of Christiansen and Wali, we wish to know the contribution for magnetic eld and the potential that minimizes the energy functional when
we include fermions in the model. Second we verify the possibility of the existence of non abelian magnetic monopoles in D = 3 + 1 dimension with fermion contributions. The paper is written with the following outline: 1. First we consider the complete system Higgsgauge-fermions and coupling between gravitation and scalar eld. We have established equations of motion for systems with and without fermions. 2. We review some results of Wali at al [1, 2] for Bogomol'nyi equations and for energy functional when fermions and gravitation are not present. 3. Next we consider the contribution of fermions alone but without gravitation. There is no coupling between the scalar eld and gravitation. We obtain here the magnetic eld with fermionic contributions and the potential function that minizes the energy functional. 4. We verify if there is a structure for the magentic monopole when we consider the covariant divergence associated with the gauge eld.
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5. Finally we return to the original problem and consider the lagrangean with all elds and coupling between gravitation and scalar eld. All the equations are solved for this case, but there remain two problems that have no solutions. The rst problem is about the new magnetic eld and the potential function when we account for the fermion contributions and the background of gravitation. The questions is: what is the new magnetic eld and potential function that minimizes the functional of energy if we put fermions and gravitation? Second, what is the meaning of the presence of spinors in the initial lagrangean? What is the physical interpretation for spinors here?
In reality we have been speaking about \fermions". However in a classical problem, the correct description is \spinors". The question is what is the meaning of the spinors in our problem? Anyway we have a partial solution for this problem and it will appear elesewhere. Let us start with a non Abelian Higgs-gauge\fermions" and coupling between scalar eld and gravitation in 3+1 dimensions. We would like to verify the behaviour of magnetic monopole of t'Hooft-Polyakov type in a background of gravitation with presence of fermions. We take the action given by Wali [1] [2] for a system of Einstein-Yang-Mills-Higgs in the form:
c S=
Z
p 1 R 2 ; 1 ;F a 2 + 1 (D a)2 ; ; 2 ; v2 2 d4x ;g 16Gv 2 4 2 4
(1)
d and now we will consider the Dirac Lagrangean D ; m i i a a j L = i i + (T )ij (2) for fermion contribution. The matrix T a are hermitian with null trace. It describes the three generators of SU(2) group, here a = 1; 2; 3 in the N-dimensional representation. The indices i; j = 1; N, where N represent the dimension of given irreducible representation of SU(2). The is only a constant. The signature of metric is given by (+ ; ;;). The indices ; take values from 0 to 3 and the indices i; j assume values between 1 and 3. The indices a; b vary with the representation of a given gauge group. We de ne R as the scalar curvature, F a represents the stress eld associated with the gauge eld Aa given as a = @ Aa ; @ Aa + gf abc Ab Ac F (3)
where f abc represent the structure constants of the group. The gauge covariant derivative associated with Higgs eld a is given by D a = @ a + gf abc Ab c :
(4)
A new covariant derivative associated with the spinor eld is necessary here. The fermion eld derivative i is given by
D i = @ i ; igAa (T a )ij
j
(5)
where g is the coupling constant. ; The term 4 2 ; v2 2 represents the possibility of gauge symmetry breaking. We choose the unit system such that 4Gv2 = 1. Thus the action is written as
c S =
Z
p h d4x ;g ; 14 R 2 ; 41 (F a ) + 12 (D a)2 + U ( i) + i i i D ; m i i a a j + (T )ij :
(6)
d Here we use U ( i); i = 1; 2; 3 in the form given by (1.1). The equations of motions for Yang-Mills and Higgs
elds when fermion are not considered are given respectively by
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c ; p1;g D p;g F a = ;gf abc bD c ; ;p ; 2 a 1 R a 2 p;g D ;g D = ; 2 + ; v
(7) (8)
d and the Einstein equations are given by G = R ; 21 g R = 22 T
where the stress energy-momentum tensor for Einstein Yang-Mills-Higgs is written as
(9)
c
T = g T ; F a F a + D a D a ; 12 2 R ; 21 g R
(10)
d and
The contribution to the energy-momentum due the fermions is given by
; T = 14 F a F a = 21 D a D a + 4 2 ; v2 2 (11)
c ; T = 12 i i D i + i D a D i ; i a (T a ) ; ie a i ij
; i i i + g m +
i
(12)
d Now with fermions, the equations of motion for elds
Aa a and
i
are given respectively by
c h
i
;
d j @ F d = gf acd F a Ac ; (D a ) c + g "i " T ij ; ; ; p1;g @ p;g D a = D (D a ) + R2 + 2 ; v2 a ; i (T a )ij
j
; = D k + m i i ; i a (T a ) ; ; p1;g @ p;g k i k
ik
; (13) (14)
d
II Static Equations and Bogomol'nyi Conditions
technique of Bogomol'nyi, with appropriate boundery conditions [1, 2, 5],
We wish to get only static solutions (time independent) for the system described by (1.6). Thus, using the
D = 0 ; 2 = v2 :
(1)
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The gauge eld without fermions is given by [5] Aai = gv12 "abc b@i c + 1v a Ai (2) where Ai is arbitrary and Fij a satisfy 2 a Fij a = v Fij (3) and the eld Fij is given by 1 "abc a @ b@ c + @ A ; @ A : (4) Fij = gv i j i j j i 2
Only the static abelian gauge eld will survive for long distances. Then, we de ne the \magnetic eld" Bi associated with the monopole for long distance as
Bi = 12 "abc Fjk = 2gv1 3 "ijk "abc a@j b@k c + "ijk @j Ak : (5)
The magnetic charge of the con guration is given by
c
Z Z 1 1 3 i g = 4 d x@i B = 8gv3 2 di"ijk "abc a@j b@k c = ng s1
(6)
d where
1 n = 8v 3
time is given by
Z
1
s2
di"ijk "abc a@j b@k c :
(7)
which is a topological number. Taking the limit g ! 0, the functional of energy obtained from (1.6), without fermions for the at space-
"=
Z
d3x 14 Fij a F ija ; 21 Di a Di a
(8)
that is an energy of Higgs-Yang-Mills eld. For the case with the fermions, we have the energy functional written as,
c Z
h d3 x 41 Fij a F ija ; 12 Di a Di a + i ; i a (T a )ij j + U ( i) :
" =
i ;m ; i k D i + k
(9)
d We are not considering the coupling between the scalar and gravitation here. From eq. (2.8) we can de ne the electric and magnetic elds as: Ei a = Di a ; (10) 1 Bia = 2 "ijk F jka (11) for the case when the fermions are not present and when the at spacetime is considered (R = 0); in other words, the eq. (2.10) and (2.11) are good de nitions for
the case when we don't consider the gravitation background. If we use the radiation gauge, Aa0 = 0, and for the static solution D0 a = 0, from eq. (1.3) and (1.4) it follows that Foi a = @o Aai ; @i Aao + gf abc Abo Aci ; Foi a = 0 = Di a :
(12)
Then we have obtained a solution (null electricaly), because the electric eld is zero in the whole space. We have still [1, 2] the following inequality.
c Z h i2 1 " = 2 d3x (Eia )2 + (Bia )2 Z Z Z = 12 d3x (Eia Bia ) (Eia Bia ) d3xEia Bia d3xEia Bia :
(13)
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d Using now the Bianchi indetity for Fij a , we can write Z Z 1 ijk a a 3 a a 3 d xEi Bi = d x@i 2 " Fjk (14) Comparing eq. (2.14) with eq. (2.3) the surface integral in eq. (2.14) asymptotically takes the value Z v d3x@iB i = 4nv (15) g : On the other hand, if the Bogomol'nyi and equations Eia = Bia (16) are satis ed, then the functional of energy is de nitely minimized. It was shown in [1] [2], that this follows naturely from the relations (2.13) { (2.16). Now, consider the case described by (2.9) where the fermions are present but without the background of gravitation. We can still write suitably the energy functional (2.9) in terms of elds Eia and Bia with the same arguments. However, here it will not have the electric
c
component due to the presence of the fermions. The magnetic eld will be di erent since it shall have the contribution of fermions. The scalars elds will be treated as a condensate of fermions. We shall de ne the following quantities: = m i i ; k D i ; = i i k i a = (T a )ij j :
(17)
We have created three new scalar elds (; ; ). So, our Lagrangean (1.6) with the scalar curvature R = 0 or its form eq. (2.8) with the fermions present, it is not possible anymore to obtain an energy functional that is satured by 4 potential as in [5]. Using now the same prescription as in [5] it is conjectured that it is possible to verify that eq. (2.9) in D = 3 + 1 dimensions may be reduced to the following form:
I Z n 2 p 2 " = ev2 Bka 2e1 diJja "ijk "abc b@k c d3x 12 Bka @k a 2U Z h o p i ; 2e v2 ; j j2 ( + + ) (@k a ) 2U Bka @k a + 21 j(D1 iD2 ) aj2 :
(18)
d Here Bka means the non-abelian magnetic ux. The second term is a surface term and goes to zero at in nity since all elds go to zero at in nity. We need to discover what magnetic eld Bka and potential U( i) will minimize the energy functional or in other words, what are Bka and U( i ) which will saturate the functional, "? If we use the duality condition [5] as in the form D1 a = iD2 a
(19)
and noting that we are considering only con gurations with nite energy the surface integral of current vector is null. In the Bogolomol'nyi limit given by eq. (2.18) the minimum of energy is obtained exactly if " = ev2 Bka : 2
(20)
Since we wish to saturate the functional " the magnetic eld now carrying the fermion's contribution will be given by ; Bka = e2 v2 ; j j2 ( + + ) (@k a ) : (21)
and the new potential will be written as ; U ( a ; ; ; ) = 8e j j2 ; v2 2 ( + + )2 (@k a )4 (22) Clearly the potential is now of type 8 because of fermion contribution. With the de nition of the covariant derivative given by D = @ + i~gA we get D F a = @ F a + i~g [A ; F ]a : Thus, the covariant divergence is given as ~ B a + g~f abc A~ b B~ c = 0 D F a = r
(23)
or still in a compact form, D F a = D~ B~ = 0
(24)
~ + g~f abc A~ : D=r
(25)
where
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Since Bka is given by eq. (2.21) and j j2 is written
the ; scalar eld, gauge eld and condensate of fermions a; Aa ; ; ; . It's sucient to choose or to x the \F" function such as that form to obtain the Gauss law from eq. (2.28).
j j2 = a a
(26) a suitably ansatz for scalar eld may be chosen such as a a = F rr (r ! 1) : (27) Using eq. (2.27) and eq. (2.26) in equation eq. (2.23) or eq. (2.24) above we get r~ B~ ka rg2 : (28)
The energy functional in a curved spacetime Now the same problem is proposed in a curved spacetime. The energy functional (static functional) that is obtained from eq. (1.6) in a curved spacetime is described by.
This gives us a structure of the magnetic monopole where g represents the source of the magnetic eld (nonAbelian magnetic eld) whose source has the origin in
c " =
Z
; p h d3x ;g 41 R 2 + 14 Fij a F ija ; 12 Di aDi a + 4 2 ; v2 2 + i i ;m ; i k D i ; i a (T a ) j : k ij
(29)
d The energy functional can, in principle, be minimized with the gravitation as a background eld only if we introduce a third covariante derivative associated with gravitation, but we need also to transfer the dynamics from the metric to vierbein and to write an appropriate covariante derivative carrying the spin connection such as: D~ = @ + 18 i [ a b ]Bab where
= ea (x) a and g = ea eb ab
are Dirac's matrices and metric respectively written in a local Lorentz coordinate system, ea are the vierbeins and Bab (x) are spin connections, completely determined by vierbeins. Now the gravitation is considered under at space-time given by ab in Minkowski space.
III The basic Equations We choose the spherically symmetric static metric as ds2 = A2 (r)dt2 ; B 2 (r)dr2 ; C 2(r)r2 d 2 : (1)
c
The Einstein tensor and the scalar curvature are easily obtained, 2 1 B2 A 2 B B C 2C 6C G00 = B 2 r2 C 2 ; 1 + r B + 2 B C ; C ; rC ; CC ; 2 2 2C A 2 A Grr = CC + 2r CC + r12 1 ; B C2 + C A + r A ; 2 2 G = CB r2 AA + r1 AA ; AA BB + CC AA ; 1r BB ; BB CC + CC + r2 CC ; G = sen2 G ; C+ R = r22 B12 ; C12 + B22 AA + r2 AA ; BB AA ; 2r BB ; 2B B C 2C A + 2C + 6C + C C A C rC C 0
0
0
0
00
0
0
00
00
0
0
0
00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
00
0
(2)
0
0
(3)
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d Then if we consider the eq. (1.10) and (1.11) together with the eq. (3.2) we take components of the energy momentum tensor for the global system of Einstein-
gauge-Higgs-fermions. The components are given by:
c
"
;
2 2 2 2 2 2 ~T00 = A2 2C A + 2 A ; A B + A ; 1 A + v h ;2 1 + B C A r A A B A 2 A 4 h 2h + 2h + 2h 2C + 2 ; B + 3A h + ; 00 h h h C r B A h " ; 2 2 # 2 2 h ; 1 1 A A h 2B h v 2 2 T~rr = ; 2 A ; A h + B h ; 4 B + 11 ; h2 " 2 2 2 2 ~T = C r2 2 1 A ; h ; h ; h 2C + 2 ; B + B 2 A h h h C r B # ; 2 2 2 2 ; v4 B hh2; 1 + 22 ; T~ = sen2 T~ + 33 00
0
0
0
0
0
0
0
0
0
0
00
0
0
0
0
0
00
0
0
0
0
0
0
0
(4)
d a and ; ; ; are where h = h(r) = vr 00 11 22 33 a
the components of the energy-momentum tensor for fermions given by
c 00 = i i 0 D0 11 = i i 1 D1 22 = i i 2 D2 33 = i i 3 D3
i + g00 m i + g11 m i + g22 m i + g33 m
i i i i
i ; iea i ; iea i ; iea i ; iea
i a D i a D
i ; i ; i a i D ; i a i D ;
i a a (T )ij i a a (T )ij i a (T a ) ij i a (T a ) ij
j j j j
; ; ; (5)
d where e a imply the tetrads or vierbeins. This set of equations together with the Bogomol'nyi equations are the basic equations for our system, Einstein-gauge-Higgs-fermions. Now the next step is to nd solutions for the Einstein equations with the objective to nd the monopoles appearing in this gravitational background when the fermions are present. But on the other hand when we have interaction between spinors and gravitation eld the unique way to consider that coupling in our case is only if we introduce a local Lorentz coordinate system. Thus, we do not have curved space-time anymore.
Conclusions We have analysed two types of systems. One of them with Yang-Mills and scalar elds in at spacetime and other consisting of the Yang-Mills-scalar eld and \fermions". We treated \the fermions" as a condensate of scalars elds. In this case the magnetic eld that saturated the energy functional has contributions from fermions. The potential which minimized the same functional is of the kind 8 and thus di erent from [5] whose potential is of type 4 for 2 + 1 dimension case. For the case of Yang-Mills-scalar eld [1, 2] the Bogomol'nyi equations have a simple solution in at space time. In the present case our conjecture in eq. (2.18)
8 gives us the non abelian magnetic eld much more complicated and potential with fermion's contribution. A structure of magnetic monopoles can be found with a eld given by eq. (2.28). Finally we have assumed the system Einstein-Higgsscalar eld-gauge and we have obtained all the equations for this system but solutions to the Einstein equations are lacking. Magnetic monopoles appear in a new context and it's still necessary to nd in a new magnetic eld Bka and potential U( i) that will be suitable for minimising the functional of energy when the background gravitational eld is considered. At this point we have a con ict between a curve space-time and a local lorentz manifold. On the rst way we have the the complete set of equations plus Bogolmo'ni equations being the basic equations for Einstein-gauge-Higgsfermions system- and, in principle, the energy functional could be minimized with the graviation eld as a background, but the other hand because the spinors we need to introduce a local coordinates system and to proceed the minimization of the functional of energy via vierbeins and spin connection. This will be a part of a separate analysis.
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Acknowledgements:
I would like to thank the Department of Physics, University of Alberta for their hospitality. I would like to thank also Dr. Don N. Page for his kindness and attention with me at Univertsity of Alberta and Dr. Gentil O. Pires for a reading of the manuscript. Finally, C. Pinheiro acknowledges the hospitality of DCP/CBPF, Rio de Janeiro. This work was supported by CNPq (Governamental Brazilian Agencie for Research).
Refer^encias [1] K.C. Wali, Monopoles and Black holes, SU-4240-520 (october 1992). [2] N.A. Viet and K.C. Wali, Magnetic Monopoles in the Einstein-Yang-Mills system, SU-424-558 (octorber, 1993). [3] P.A.M. Dirac, Phys. Rev. 74, 817 (1948). [4] Classical Eletrodynamic-Jackson, John Willey & Sons, inc (20 Edition). [5] M.S. Cunha, H.R. Christiansen and C.A.S. Almeida, Bogomol'nyi Equation in Gauge Theories, hepth/9706206 { Jully 1997.
