Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 1, January -2012 1

ISS N 2229-5518

Dyon Solutions in Non-Temporal SU(3) SU(3) Gauge

Vinod Singh and D C Joshi*

Abs tract - Employing the CabbiboFerrari type non- Abelian f ield tensor w e consider the SU(3) SU(3) gauge theory under the non-temporal gauge conditions and show that the obtained solutions are dyonic and have f inite energy.

Index Term: Dyon Solutions, Non-Abilian Fie ld Tensor, Gauge Field Theory

Introduction

—————————— ——————————
One of the authors (DCJ) has in ear lier paper s11 developed
a Cabbibo-Ferrar i24 type field tensor for non-Abelian fields
In 1930s Dir ac1 advanced the idea that isolated magnetic
poles might exist. The idea of magnetic monopoles got a b oost in 1970s when ’t Hooft2 and Polyakov 3 show ed that in gauge field theor ies in which the symmetry gr oup is spontaneously br oken possess classical solutions w ith the
natur al inter pr etation of magnetic monopoles. Soon the Julia and Zee’s4 conj ectur e was seen as the non-Abelian analogue of Schw inger ’s Ab elian dyons5 . The inter est on monopoles and dyons generated by Dirac 1 , ’t Hooft 2 , Polyakov3 and Julia and Zee4 has r emained undiminished and extensive theor etical and exper imental wor ks on the r elated topics have been under taken 6 -2 1 , 3 0 .
Since, the solutions which w er e inter pr eted as magnetic monopoles w er e or iginally found in SO(3) gauge gr oup and this gr oup being small for unifying electr omagnetic and w eak interactions, lar ger gauge gr oups like SU(3) w er e explor ed8 -1 2 , 2 2 , 23 . A key factor of such theor ies is the twin combination of the choice of gauge and choice of gauge field tensor . Theories have in general follow ed the approach of Julia and Zee4 and employed usual Yang-Mills type field tensor and have used tempor al gauge conditions to arr ive at monopole solutions and obtained dyon solutions in non-tempor al gauge.
In 1960s, Cabbibo and Ferrar i24 developed a tw o potential field tensor for developing a theory of Abelian dyons and Yang Mills type field tensor continued to be
and employed12 -13 it on non-Abelian gauge theor ies w ith electric and magnetic sour ces. Using the same field tensor and the Kyr iakopoulos22 technique w e show in the pr evious paper that the dyon solutions be obtained in the temporal gauge (3 1 ). The Kyr iakopoulos (2 2 ) technique under the temporal gauge conditions r educed the gauge field equations into the fir st or der differ ential equations whose solutions depicted a set of dyon solutions. Extending the analysis in the pr esent paper w e examine the SU(3) SU(3)
gauge under the non- temporal gauge conditions and find that in this case too w e obtain the finite ener gy dyon solutions but unlike the pr evious case they emer ge as the solutions of second or der differ ential equations. The paper has been divided into six sections. Section 2 defines the Lagrangian density, the gauge gr oup of the theory, field equations and matr ix notation .The ansatz for obtaining the solutions has been pr esented in section 3. The s olutions have been shown to have finite ener gy in section 4.the adj oining solutions be obtained in section 5. That the obtained solutions belong to electr ic and magnetic char ges has been shown in section 6 to which then follow the concluding r emar ks.

2. The Gauge Group and the Lagrangian Density

In this section w e br iefly r ecapitulate the steps fr om the pr evious paper (3 1 ).
used for dyon solutions in non-Abelian gauge theor ies.

Vinod Singh, Depa rtment of Physics

Govt. P. G. College Gopeshwa r, Cha moli Utta ra kha nd -246401

The system whose gauge gr oup is descr ibed by the Lagrangian density

SU(3) SU(3) , is

e-ma il:vinodsinghgpr@gma il.com

D.C. Joshi, Depa rtment of Physics,

Amra pa li Institute of Science a nd Technology Ha ldwa ni, Na nita l

Ret. Hea d, H.N.B. Garhwa l University Srinaga r, Ga rhwa l Utta rakhand

   1 Ga

4

wher e(3 1 )

 Ga  1 D a Da  Va a

2

(1)
e-ma il:profjoshi2000@yahoo.com

G a   A a   A a  e f a bcA b A c

      

 1 

2

and its dual

 

B a   Ba  g f a bc Bb B c

(2a)

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ISS N 2229-5518

~

Ga  

Ba  

Ba  g f a b c Bb Bc

in which η and ar e r eal constants with   1 . The

 

is

  

1

 

a  a

a b c

b c

(2b)
fields a

2 6 , 2 7

    A

2

  A

 ef A A

φ may denote the Higgs

tr iplet fields.
in which gauge fields Aa
and Ba tr ansform as
The Euler-Lagrange variations of the Lagrangian

A  UA U1  1  UU1

(3a)
density (1) w ith r espect to Aa
, B, e
and a lead to the

  

e

field equations
and

B  UB U1  1  UU1

(3b)

  

g

 Aa  e f a bcAb Ac  efa bcb D1 c  0

wher e U is a gauge function

U  exp ia Ta

(4)

 

 B  g f B B

 gf

e e

 D   0

(18)
(19)
with

a the r eal functions of space-time and Ta

~ a

a bc

b ~ c

a bc b

g

2 c

g

r epr esenting the gr oup generators of SU3gr oup obeying

1 a a bc b 1 c V

[Ta , Tb ]  i f a bcTc

(5)

 D

 e f

A D

  0

a

(20)
The

f a bc ar e the

SU3

structur e constants with a, b, c

a

and

 D2 a  g f a bcBb D2 c  V  0

(21)

r unning from 1 to 8. Ta , w her e a (a = 1, 2, … 8) ar e

2

 g 

g a

eight Gell-Mann matrices25 .

The

in the Lagrangian density (1) indicates t he
Intr oducing the notation
pr oducts in which the fields have been assumed mutually non-inter acting. As a r esult of this assumption the mutual

A  e Aa Ta

(22a)
inter action terms, i.e. the cr oss-ter ms, disappear leaving
and

B  g Ba Ta

(22b)

~ ~  

a a



a a



a a



(6)

D a Da D1 a  D2 a D1 a  D2 a

and also expr ess the Higgs field φ as

  e  g e g

 D1 a D1 a  D2 a D2 a

 e e  g

g (7)

  (e  g)  a Ta  ea Ta  ga Ta    

(22c)
and

a a a  a a  a

e g e g

e g e g

 a a  a a

(8) a
wher e

e e g g


wher e

Ta with a (a = 1,2,…..,8) the Gell-Mann

2

Aa   Aa   Aa  e f a bcAb Ac

(9)
matrices (25), w e may expr ess the field equations (18) to

      

(21) in matr ix notation as
and

Ba  1  B a

(10)

~ 

  

 A  i[A , A ]  i[ , D1  ]  0

(23)
with

Ba  Ba  Ba  g f a bcBb Bc

(11)

~ 

 e e

~  2

D1  

 e f a bcAb

(12)

B

 i[B, B

]  i[g , D

 ]  0

(24)

D2  

 g f a bcBb

(13)

 D1

 iA , D1  e  V

Ta  0

(25)
and

a  a  a

(14)

 e 

e a

e g

The covar iant der ivative D a which expr essed as

 V

D a  D1 a  D2 a

(15)

 D2

 iB , D2

 g Ta  0

(26)

  e  g

transfor m as

 g 

g a

(D )a  U(D )a

(16)
r espectively. It is obvious fr om the above that 11

 

The potential ener gy

Va  a

in the Lagr angian

a

3. The Ansatz

density (1) descr ibe the self interaction of field
the form

Va  a   a a  a a  2 2

φ and has

(17)
In the pr evious paper 31 the gauge field obeyed the
temporal gauge conditions and her e temporal parts Aand

Bdo not vanish w e wer e r equir ed to have the ansatz 28

e e g g

a

ˆ  x1 7  x2 5  x3 2  ˆ a

2

 ˆ a Ta

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ˆ  4 Ir 2  2ˆ ˆ

3

In ear lier paper defined31

 ~  ~ 

~  ~ 

 2x1x2 1  x3 x1 4  x2 x3 6 x1 2 x2 2 3

  A23 , A31 , A12  P P Q Q R R S S

(39)

A A A A

r 2  3x3 2

(27)

  B01 , B02 , B03 B23 , B31 , B12  P P Q Q R R S S (40)

8 ~ ~

3

~ ~

   

~  ~ 

B B

~  ~ 

B B

wher e

r  x1 2 x2 2 x 2 2 , x1 , x2 and x3 being the

wher e P, Q, R, S
30) and
have been defined in equations (28) to

3

(
components of distance thr ee-vector . W e also intr oduce the

    

~   T

(41)

thr ee-vector functions P, Q, R, S, T and U expr essed by28 P A

r

 

P  ˆ

(28)

~ ~ 1  T 2  U 2

P  r2 R  A A

  r

(42)

Q  1 ˆ

2

(29)

~ 

  Q

  UA A 2

(43)

R  x ˆ

(30)

~ ~

Q  r2 S

  3 TA UA

(44)

S  x ˆ

(31)

A A 3

  

T   x  ˆ

(31)
and

  

 ~  ~ 

~  ~ 

U   1 x  ˆ

2

(32)

D1 e  Pe 4 P Qe 4 Q Re 4 R Se 4 S  4

(45)

 ~  ~

 ~  ~ 

and29

D2 g  Pg 4 P Qg 4 Q Rg 4 R Sg 4 S  4

(46)

1  T  U 

wher e

A  A

r2

T  A U

r3

(33)

~ NTA  MUA

(47)

1  T  U 

e e

e 4 2

B  B T

r2

B U

r3

(34)

2 ~ ~

rN  N

e e


A  R A ˆ  SA ˆ

(35)

r R e 4  Pe 4 2

r

(48)

0

and

r2 r 3


B  R B ˆ  S B ˆ

(36)

~

Qe 4

NUA  2 MTA

r3

(49)

0 r 2 r 3

r 2 ~ ~

rM  M

e e

(50)
wher e
dependent.

TA , TB , UA , UB

RA , SB , RA , SB ar e pur ely r

e 4 e 4 3

The ansatz for the Higgs fields e  g   as befor e2 8 , 2 9

with similar r elations with e  g and A  B .
As shown in the following subsection, the ansatz (33),

N M

  e ˆ  e ˆ

(37)
(34), (37) and (38) allow us to wr ite the field equations (18)–

e

and

r 2

N

g

g r2

r 3

ˆ 

Mˆ

r3

(38)

(21) in terms of field equations without SU 3indices.

W e use the same ansatz and notations as used in the ear lier paper (3 0 ) for temporal gauge. W e also employ the ansatz for non temporal gauge(2 2 )
wher e the coefficients N and M too ar e pur ely r-
dependent. W e also intr oduce the vector
w e can expr ess the space-time component of

~

A and

4. Finite energy Solutions.

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0  ( A10, A20 , A30 )  P0 AP Q0 AQ R0 AR S0 AS (51)

(D  )  iA, D   0

(65)

~  ~

 ~  ~  1 1

e e

Wher e

(D2  )  iB, D2  0

(66)

  

g g

~

P0 A

 R A TA  2SA UA

r2

(52)
Now we first look at the set of eqs. (61), (62) and

(65) that contain the space parts A
of the gauge field A.

2 ~

0 A

~

P0 A

 r R  R

r2

(53)
Using eqs (34) in these equations w e can calculate the individual terms as

~  R A UA  2SA TA

(54)

 T

1  T 2  U 2  

 U

6U T  

0 A r3

     A

 r2

A A A

r4   r3

A A U (67a)

r 

r2 ~ ~

 r S  S

(55)

     

2 2

2  

0 A 0 A 3

 i( A   A  )   (1

TA )1 TA

r 4

U 6T U

A A A T

r 4

and

~ ~ ~ ~

 U 1  T 2  U 2

5

6(1  TA )U

r 5

A TA  

U

(67b)

0  (B 23 , B13 , B12 )  (B 10 , B 20 , B 30 )

(56) 2 2

~  ~

 ~  ~ 

 T R

  A A

 4SA  UA R A SA  

 P0 B P Q 0 B Q R 0 B R S 0 B S

wher e

i[A0 , 0]

  U

R 2

r4

 4SA

T

 T R S  

A A A U

(67c)

~

P0 B

 R B TB  2SB UB

r2

(57)

1 A e

r5

2  4M

e

 4U A N M

 i[ , D  ]   T

2 ~

0 B

~

P0 B

 r R  R

r2

(58)

e e

 U A N e

r 4

 4M 2

e

 4T N

(67d)

M

e U

r 5

 

~  R B UB  2SB TB

(59)

0  i [A 0 0 A] 

0 B 3

~  ~  

(60)

 r 2 R 

 2TA

T R

 2U

A S A

r 4

 2U

U R

 2U A S A

 ˆ

2

0 B

Q0 B

r SB  SB

r3

 r2S

 3T U R

 2T S

 3U

T R

 2S U

Now we look at the field equations 31 (23) to(26) and separ ate their space and time components. Using

  A

A A A A A

r5

A A A

A   (68)

eqyuations (51) and (56) the r espective space and

(D1 )  iA, D1

  

e e

timecomponents of (23) and (24) can jbe expr essed as

 r 2 N

 2T T N

 2U M

 2U

U N

 2T M

    

1

e



A A e

A e

r 4

A A e

A e  ˆ



    ( A   A  )  i [A0 ,

0]  i[e , D e ]  0 (61)

 r 2 M

e

 3T U N

e

 2T M

 3U

e

T N

e

 2U M

e  ˆ

     2

 A A  A 

 r 5

A A 

A 

    (B    A  )  i [B0 ,

0]  i[g , D g ]  0

(62)

(69)

0

 i[A 0

0

A]  0

(63)
Equation (61) is satisfied if the coefficients of

 

and

0  i[B 0

 

~

0 B]  0

(64)

TA , UA , ˆ and ˆ ar e zero that gives the system of nonlinear

differ ential equations
wher e and
ar e (39) and (40) for the space and time

r 2 T  T

T 2  7U

2  1 T R

2  4S 2

parts of eqs (25)and (250, we observe their V = 0 and find that, due to the static natur e of fields and the ansatz (25) and (26) vanish leaving the space parts as

A

 4R

A A

A U A S A

 TA

A A

2  4M

e e

A

2  4U

A

A N  e

M  e  0

(70)

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r 2 U

 U 7T 2  U

2  1 U

R 2  4S 2

4

 

1  T

2  U 2

12 T 2 U 2

 4T R S

 U N

2  4M

2  4T N M  0

(71)

2 dr 4 2T

e

 U 2

A A A A

r 2 r 2

A A A

A  e

 e A  e  e

0   

r2R  2R

T 2  U 2  8T U S  0

(72)

A A A A

A A A

rR  R

2

4 rS  S 2

r2S

 2S

T 2  U 2  6T U R  0

(73)

2 A A A A

A A A A

A A A

 r 3 r

r2 N

 2N

T 2  U 2  8T U M  0

(74)

A A A A A

e e A A

A A e

 2 T R

 2U S

2  U R

r 2

 2T S 2

r2M

 6M

T 2  U 2  6T U N  0

(75)

e e A A

A A e

  2

  2

Similar ly eqs (62),(64) and (66) give the system of

 2 



rN N

e e

r2

4 rM M

e e

3 r2

nonlinear differ ential equations

2 T N

 2U MU N

 2T M

r 2 T  T T 2  7U 2  1 T R 2  4S 2

A e

A e A e

r2

e 



B B B B

B B B

4R U S

 T N

2  4M

2  4U N M

(76)

 0

B B B

B  g  g

B  g  g  

 

r 2 U  U 7T 2  U 2  1 U R 2  4S 2  

4T R S

 U N

2  4M

2  4T N M

(77)

 0

3    1 T

U 12 T U 

B B B B

B B B

4.

2 T 2  U 2

2 2

B B

2 2

B B

B B B

B  g

 g B  g  g

 4  B B

r 2 r 2  

r2R  2R T 2  U 2  8T U S  0

(78)

  rR   R 2

  B B

4 rS  S 2  

  

B B B B

B B B

4

 15 2

dr  2.

3 r 2

r2S  2S T 2  U 2  6T U R  0

(79)

g 2

4  2 T R

 2U S

2 U R

 2T S

2  

B B B B

B B B

 B B

 

B B B B

r 2

B B

 

r2 N

 2N

T 2  U 2  8T U M  0

(80)

 

2 2 

g g B B

B B g

rN  N

  g g

  r 2

rM  M

g g 

3 r 2 

r2 M

 6M

T 2  U 2  6T U N  0

(81)

 2 

2 2 

g g B B

B B g

2 T N

  B g

 2U B M

U

B N g

 2TB M

above system of second or der non-linear differ ential equations (70) to(81) belong to the non-temporal gauge

 

r 2

(84)



conditions Aa  0 and Ba  0

and the ener gy for this case is

0 0 thus our system in the gauge Aa  0 and Ba  0 , is now

0 0

calculated by using the ener gy- momentum tensor T 4 as

  ~ ~

descr ibed by the second or der non-linear differ ential equations (70) to (81) and the ener gy of this system is

m  d3 x T00 d3 x Ga  Ga  Ga  Ga  D a  D a  (82)

0i 0i

0i 0i 0 0

expr essed by eqn (95) . How ever , the ener gy diver ges
at r  0 . Ther efor e , to avoid the singular ity at

r  0 , w e

Using eqs. (9) and (10) the above expr ession yields
impose follow ing b oundary conditions ……..In or der to
avoid the terms becoming singular as r  0 , the following

m  d 3   1

A a A a  1

A a A a  1

D1 a D1 a  Va

boundary conditions ar e r equir ed to be obeyed

 2 0 i

0 i 2 jk

jk 2 i e i e

(83))

 

U 0 1  U 0

d 3 x  15 Ba

Ba  3 Ba Ba  1 D 2 a D 2 a  Va A B

 16 0 i 0 i

8 jk jk

2 i g i g g

T  r1 ,

T  r

1 1

(86a)
Using eqn.(25)31 w e get

A r  0

or

B r  0 1

m  Tr d x A A  A A  D  D 

1 1 1

1 3 1 1

i

U  r  ,

U  r 

e A r  0

B r  0 1

(86b)

 1 Tr g 2

  15

x B0 i

 8

B0 i

 3 B

4

Bjk

 D2

D2

g 

TA 0 1  TB 0

wher e , 1 and ,  1 ar e constants with ,  1  0 . Thus the ener gy (85) becomes finite when its parameters obey

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ISS N 2229-5518

the boundary conditions (86). W hen these boundary conditions ar e obeyed by the solutions of eqs (70) to (81), the same would be the Aa  0 and Ba  0 ,the finite ener gy

4

m  4

0

U N 2

r2

4

dr  4

0

U N 2

dr

r 2

0 0  16 

cosh 2

 16  2

(100)

solutions. Thus our aim now is to obtain the solutions of second or der differ ential eqs (70) to (81) which obey

2 A A

2 B cosh B

g

eqn.(86). For the above pur pose if w e let us put

4. Electric and Magnetic Charge

TA  TB

 FA

 FB

 Me

 Mg

 0 , w e find that out of the

tw elve equations only the following six r emain
In or der to show that the obtained solutions (93) to
(99) having the finite ener gy (100), ar e dyon solutions in

r2 U  U

U 2  N

2  R

2  1 0

non- tempor al gauge, w e shall calculate the electr ic and

A A A

(87)

e A

magnetic char ges. For that purpose we intr oduce the unit

r2 R

 2R

U 2  0

(88)
vectors φˆ a

ˆ

and φˆ a

a

e ,g

defined by28

ˆ a

r2 N

 2N

U 2  0

(89)

a

e ,g

a

e ,g

a

e ,g

1 2 2r

(101)

e e A

Fr om pr evious paper 31

r2 U  U U 2  N

2  R 2  1 0

(90)

1  Ajka  1 a

(102a)

B B B

g B

ijk i

r2R  2R U 2  0

(91)
and

1  Bjka  1 ~

(102b)

B B B

2 ijk g

r2 N

g

 2N

g

U 2  0

(92)
W e intr oduce the field 22
It is inter esting to note that the fir st thr ee equation (87),

a  1 a

e

(103a)
(88) and (89) exactly match the Pr asad and Sommer field equations of motion.32 Similar matching exists for the eqs.
and

a 1 ~

0 i i

g

(103b)
(90) (91) and (92) as w ell and the solutions of these equation comes out as 32
The electr ic char ge q e
using eqs. (102b) and (103a) ) as
may now be calculated by

 r

U  (93)

1 ˆ a a

(93)
(104)

sinh A r

qe

4 φe G0 i ds i

Ne

  cosh A

r coth A

r 1

wher e G0 i can be had fr om eqn (2) and ds i denote the
(94) sur face element o(9f 4t)he sur face at infinity which is also the boundary of the static fields.

RA   sinh A

r coth A

r 1

(95)

q  1

(95) 1

φ [A  

jka

g 4

e oi

ijk B

2

]ds i

 r

U  (96)

1 ˆ a

1 ˆ ~ a

B  φa

0 ds 

φa i ds

sinh B r

4 e e

i 4 e g i

N   cosh 

r coth 

r 1

(97)

a a  4r2

~ ~

g B B B

1 ˆ a R


x ˆ a

ds

1 ˆ a R x ˆ a


ds i

R   sinh 

r coth B

r 1

(98)

8 r

0 A i

e

B i

i 8 r g

1 rR  R 

1 (1  U2 ) 

TA  TB  Mφ  M

 SA  SB  0

(99)

2e

A A x  ds 

r 3

2g

B x  ds r


  2 sinh A  2

(105)

wher e A B A and B ar e arbitrary constants . The finite e g

ener gy corr esponding to these solutions is obtained by
The magnetic char ge likewise is obtained
substituting (93) to (98) above solutions when put in (85)
give

1 1


g 4 g 2 ijk

jka

ds i

1 φˆ a [ 1

A jka 3 a


4 2

a

 Boi ]ds i

2

~ a

1 φˆ a i ds

3 φˆ a 0 ds


4 g e


i 8 g g i

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ISS N 2229-5518

1 (1  U2 )  

3 rR  R  

The ener gy of these solutions can also be calculated

 

2e


A x  ds 

r 4g

B B x  ds r 3

fr om eqn. (85) by putting R

 0  S

, w hich will also

A A


 2  3 sinh B

(106)
admit the boundary conditions (86) r esulting in the

e 4g

Thus the obtained solutions ar e dyonic
follow ing expr ession of ener gy that would be finite
solutions in tempor al gauge with electr ic

m  1 Tr e 2

d 3 x 2A

A jk


char ge  2 sinh A  2

of and magnetic char ge

 1 3

 15  3

2

2   (114)



e g g 2  8 0 i 0 i

B jk B jk

4

D i g D i

g 



of 2  3 sinh θ B .

e 4g

 32

e 2 A

 16

g 2 B

cosh 2

5. Adjoining Solutions

Equations (93) to (98) pr ovide the non-temporal gauge
The electr ic and magnetic char ges for this case can also be calculated as
solutions in which both

a and a

w er e non-vanishing.

q  2

(115a)
How ever , r emaining in the r ealm of non-tempor al gauge g
conditions we can have particular case of (a)

Aa  0

and

q  2  3 sinh B

(115b)

Ba  0


and (b) Aa  0 and Ba  0 . Adopting the pr ocedur e g

0 0 0

e 4g

of pr evious sections, w e show in the following that in these
particular cases of temporal gauge too, the obtained

Thus in the gauge Aa  0 , Ba  0 , w e have finite ener gy

0 0

solutions though ar e finite ener gy dyonic but ar e differ ent
fr om the pr evious section.

Case (a) Aa  0 , Ba  0

dyon solutions (109)-(113) with finite ener gy (114) and dyon char ges (115).

0 0

The vanishing of a
implies the vanishing of

R A and

Case (b) A a 

0 , Ba  0

S A , accor dingly the field equations (87)-(75) become

In this case R B and S B
vanish. The field equation (87)-

r2 U

 U U 2  N

2  1 0

(107)
(89) r emain same, wher eas eqn. (90)-(92) after substituting

RB  0  SB r educe to following two equations

r2 N

 2N

U 2  0

(108)  

e e A

r 2 U  U

U 2  N

g

2  1  0

(116)
r est eqs.(91) – (94) r emain same.
Since, the second or der differ ential equations 107) to

r2 N

g

 2N

g

U 2  0

(117)
(108) again mat ch w ith those of Prasad and Sommer field32 ,

Corr esponding to Aa  0 , w e shall have thr ee equations

the solutions in this case of Aa  0 , Ba  0

ar e the solutions

0 0 viz. (87), (88) and (89). The solutions of these five equations

of field equations (107), (108) and (90)-(92) w hich ar e
written as
yields

U

  r

(118)

UA

 r

sinh A r

(109)

A sinh  r

(109)

Ne

  cosh A

r coth A

r 1

(119)

Ne

 

r coth A

r 1

(110) (110)

RA   sinh A

r coth A

r 1

(120)

 r

UB

(111) (111)

Ng

sinh B r

  cosh 

r coth B

r 1

(112)

UB

 r

sinh B r

(121)

R   sinh 

r coth B

r 1

(113)

Ng

 

r coth B

r 1

(122)

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Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 1, January -2012 8

ISS N 2229-5518

wher e equations (121) and (122) ar e the Prasad- Sommer field32 solutions of equations (116) and (117) and the first thr ee (118-120) ar e the solutions of equations (87), (88) and (89).
The finite ener gy for this case also fr om eqn. (85) on

14. J P Gauntle tt , J A Harv ey & J Liu , Nucl Phys B, 409 (1993)

363.

15. J S mit & A J S ijs v an de r J, Nucl Phys B, 422 (1994) 349.

16. K Zare mbo , Nucl Phys B, 463 (1996) 73.

17. Ake rs Dav id, Int J Theor Phys, 33 (1994) 1817.

18. A Yu Ig natiev & G C Jo shi , Phys Rev D, 53 (1996) 984.

19. J P Gauntle tt, Nucl Phys B, 411 (1994) 443.

substituting equation (99) and

RB  0

and accommodating

20. K Be nso n & I Cho I, Phys Rev D, 64 (2001) 065026.

the boundary conditions (86) becomes

21. C J Ho ug hto n J & E J We inbe rg , Phys Rev D, 66 (2002)

125002.

22. E Ky riako po ulosE, IL Nuovo Cimento, 52A (1979) 23.

m  16 

e2 A

cosh 2

 32 

g 2 B

(123)

23. F A Bais & H A We ldo n, Phys Rev Lett, 41 (1978) 601.

24. N Cabbibo & E Fe rrari “Quantum Elec tro dy namics with

Dirac mo no po le s” Nuovo Cimento, 23 (1962) 1147.

The electr ic and magnetic char ges for this case ar e

25. M Ge ll-Mann & Y Nee man, The Eig htfo ld Way (Ne w Yo rk)

N.Y. 1964.


q   2 sinh A  2 e g

q  2 e

(124a)
(124b)

26. P W Higg s , Phys Lett, 12 (1964) 232.

27. P W Higg s, Phys Lett, 13 (1964) 508.

28. A Chakrabarti, Ann Inst H Poincare, 23 (1975) 235.

29. Z Ho rv ath & L Palla, Phys Rev D, 14 (1976) 1711.

30. F Rahaman , India n J Pure a nd Appl Phys, 40(8) (2002) 556.

31. V. S ing h, B. V . Tripathi, and D. C. Joshi “Euc lidian S pce

Dyo n Solutio ns” Indian J. Pure & Appl. Phy s. 43, 157 (2005).

Conclusion

Using a Cabbibo-Ferrar i type non-Abelian field tensor , the dyon-solutions have been obtained in the temporal gauge. Intr oducing the quantities ˆ and ˆ in
terms of Gell-Mann matrices, thr ee-vectors

32. V. S ing h, B. V . Tripathi, and D. C. Joshi “S tability Analy sis o f Dyo n so lutio ns in SU(3) SU(3) Gauge Theo ry ” Indian J. Pure & Appl. Phy s. 44, 567 (2006)

    

P, Q, R, S, T and U have been defined. The gauge fields have
then been expr essed in terms of these thr ee-vector s which r esults in the r eduction of second or der non-linear field equations into the fir st or der non-linear equations whose solutions employing the self-duality conditions lead to Euclidean space dyon solutions whose ener gy has been shown to be finite. The distinguishing featur e of the obtained solutions is the use of Cabbibo-Ferrar i type non- Abelian field tensor and the temporal gauge.

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