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Exact SU(N) monopole solutions with spherical symmetry by the inverse scattering method
Volume 110B, number 2
PHYSICS LETTERS
25 March 1982
EXACT SU(N) MONOPOLE SOLUTIONS WITH SPHERICAL SYMMETRY BY THE INVERSE SCATTERING METHOD Takao KOIKAWA 1 Department of Physics, Hiroshima University, Hiroshima 730, Japan Received 28 December 1981
The construction of the exact SU(N) monopole solution with spherical symmetry by means of the inverse scattering method is shown. We first derive the Lax equation from the spherically symmetric Bogomolny equation. The Lax equation is equivalent to a pair of equations, the eigenvalue equation and the space evolution equation. The inverse scattering method is applicable to the pair of equations. We also discuss the conserved quantities in connection with the linear manybody problem. After the introduction by 't Hooft [1 ] and Polyakov [2] of the magnetic monopole, much progress has been made in this field. The exact onemonopole solution was found in the BogomolnyPrasad-Sommerfield limit [3,4]. In this limit the energy of the magnetic fields and the Higgs fields in the adjoint representation is bounded below by the topological charge of the magnetic monopole. The bound is saturated if and only if the Bogomolny equation B = Od~,
(1)
is satisfied. The embedding of SU(2) in SU(N + l) enables us to consider the above Bogomolny equation as the one expanded by the SU(N + l) (N + 1) × (iV + 1) matrices. As was shown by Manton [5] this equation can be regarded as the self-dual condition of the time-independent gauge fields by identifying the zeroth components of the gauge fields with the Higgs fields. By assuming spherical symmetry simplicity is brought into the equation. As was shown by Leznov and Saveliev [6], the spherically symmetric Bogomolny equation (SSBE hereafter) with SU(N + l) reduces to a simple form which is related to the one-dimensional (N + 1)-body problem. Starting from the generalized Liouville equation we can derive these equations and other soliton equations as well by choosing the corresponding Lie algebras. 1 Fellow of the Japan Society for the Promotion of Science. 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
In this article we shall try to solve the SSBE by using the inverse scattering method (hereafter ISM) which is well known in the soliton theory. Although the exact solution of the SSBE with SU(N + 1) was found by Bais and Weldon [7], and Wilkinson and Bais [8], their assumptions which limit the form of the solutions seem to lack an obvious justification and might have missed other important solutions. This motivates us for the present work. We show the Lax pair [9] of matrices L and A [see eqs. (18) and (19)] which play a role in the eigenvalue problem and "space" evolution problem, respectively. First we derive the Lax equation from the Bogomolny equation by imposing spherical symmetry, Leznov and Saveliev [6] showed the close relation of the SSBE to the one-dimensional many-body problem, of which a detailed discussion is also found in ref. [10]. Encouraged by the similarity between the SSBE and the linear many-body problem, we try to construct the Lax pair formulation of the SSBE with SU(N + 1) and solve it by the ISM of which the utility is well-established [ 11 ] in the linear manybody problem. Some of the necessary ingredients for the present discussion are briefly recapitulated. We assume the Higgs field • to be a rotational scalar and the gauge potentials W as rotational vectors: [J, ~] = O,
[Ji, Wj] = ieijgWk ,
(2, 3) 129
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PHYSICS LETTERS
where the rotational generators J are given by g=-irXV
+T.
(4)
Here T are generators from the Lie algebra of SU(N + 1) and satisfy an angular momentum algebra. Since the angular dependence of the participating fields are specified we can work on the z-axis. Then from (2) and (3) we obtain
25 March 1982 ~1
0
0
~2 -- ~1
0
0
~3
1
-
-
~2 ,
0
(17)
0
--qJN [T3, q,] = 0,
[T 3, W±I = +We ,
(5, 6)
where We = W1 +iW 2. Next we introduce new functions M by (7)
eW = (M - T) × ¢ /r ,
where ;" = r/r is a unit vector. Then M± = M 1 + iM 2 satisfy [T3,M±] = +114+_.
L = ~ +½i(N, + N _ ) ,
er 2 d(I)/dr -7-1 [M+, M_ ] - T 3 ,
(9)
and e - 1 dM±/dr = +[ep,M±l .
(10)
Let us simplify these equations by introducing ff and N± defined by edP=q/ + T3/r, M± =rg+_ ,
(11,12)
Then eqs. (9) and (10) read dN+/dr = +[@,N+]
(13, 14)
We take the following representation for T3 from SU(N + 1) T3 = diag(N/2, N/2 - 1 ..... - N / 2 ) .
A =½i(N_ - N + ) .
/dr = [A, L ] .
(20)
This turns out to be the Lax equation of the linear (N + 1)-body system with a potential of exponential type if we regard the variable r as time variable and (if/.) 2 as the potential linking the neighbouring particles Eq. (20) is equivalent to the Lax pair equation: LVk=XkO k,
k=l,2
..... N+I,
(21)
dv k / d r = A v k,
k=l,2
..... N+I.
(22)
where the v k are the N + 1 independent (N + 1)-component column vectors, i.e. v k = (vk(1 ;r), Vk(2; r) . . . . . Uk(N + 1 ; r)) t, belonging to the eigenvalue k k. For simplicity we assume that the Xk are not degenerate and satisfy ~kN+1 > XN > --. > X2 > Xl •
(15)
The commutation relation of N±, and ~ with T3 tells us which entries of these matrices are allowed to survive. It is, therefore, reasonable to express them as
011
(lS, 19)
Eqs. (13) and (14) are summarized into the Lax equation
(8)
The Bogomolny equation (1)under the present assumption reads
d@/dr - 1 [N+,N_]
and N_ = Nt+. Let us define the Lax pair [9] matrices L and A by
(23)
As the consequence of the traceless condition of the SU(N + 1) generators and hence qJ, we have another condition for the Xk N+I n=l
Xn =O.
(24)
0 0 .
N+ =
(16)
"fU 0 130
We shall recourse to the well-known procedure of the ISM. Let us first solve the inverse scattering problem of eq. (21). In other words we express the potentials in L, ~j and fj, in terms of the eigenvectors and eigenvalues. The r dependence of these eigenvectors is obtained from eq. (22) with a suitable boundary condi-
Volume 110B, number 2
PHYSICS LETTERS
25 March 1982
tion which we shall discuss later. We assume that the vectors ok *1 are orthonormal:
vectors ok(r ) and hence ak(r ) from eq. (22). The first column of this equation gives us
N+ 1
do k(1;r)/dr = -~t.ally k (2,. r ) .
n=l
vi(n)v/(n) = 60..
(25)
(34)
From this equation, together with the one obtained from eq. (21), we obtain
The completeness relation is N+I
N+ 1
]=1
t~(n )v](m ) = 6 nm
.
(26)
From eq. (21) we obtain
Rvk = ()tk -- )t)-lvk ,
k = 1,2,... , N + 1. (27)
with R = (L - XIN+I) -1 .
dVk(1 ;r)/dr = ( /~=I x ] 4 ( 1 ;r)--)tk ) vk(1 ; r ) .
(35)
The solution to this differential equation is easily found and we get N+I
(28)
ak(r) = ak(O) exp(--2)tkr)l /~1 a](0) exp(--2)tff),
It is easy to see that eqs. (26) and (27) lead to
k = 1, 2 . . . . . N + 1.
N+ 1
Rll= ~.=°~(1)/(;'V-
)t).
(36)
(29)
(30)
This completes the determination of the r dependence of the eigenvectors, ok (r). As an example we show the explicit ISM construction of the spherically symmetric SU(3) monopole solution. It is necessary to take account of the boundary condition so that the Higgs field ep and the magnetic fields B are finite. From eq. (33) we have *2
AN : d e t ( ~ ' - )tiN) ,
(31)
A2IA 3 =~a](r)/()tj - )t),
AN+ 1 = det(L - )tiN+ 1 )"
(32)
where A 2 and A 3 are explicitly written as
On the other hand the same quantity is expressed in terms of the potentials in L from the definition of R, eq. (28): R l l = AN/AN+ 1 , where AN and AN+ 1 are defined by
In (31) the N × N matrix L" is the one obtained from L by removing the first row and column of L. We thus have two ways of expressing R 11, i.e. (29) and (30). Therefore we obtain N+I
AN/AN+ 1 = ]~1"= a](r)/Ot! -- X) ,
(37)
A2 = )k2 +½~l)t -- 1 ~2(ff2 -- t~l) + l f l2 ' A3 = - ) t 3 - l ( ~ l f f 2 - ~ 7 - ~ b 7
(38)
+ f 7 +f2))t
--8 [~l(ff2 - ffl)~2 +f2ff2 -f22ffl] •
(39)
By comparing both sides of eq. (37) we get (33)
where ak(r ) = 02(1 ;r). This formula enables us to exp r e s s f i and Oi in terms o f v i and )ti" We shall next find the r dependence of the eigen*1 In the interests of economy, the argument r of each component ok (re;r) of the vector ok is suppressed hereafter.
½tp 1 = ~ ) t f l / ( r ) ,
(40a)
*2 Hereafter the summation is taken over from 1 to 3 and the suffix k of hk or ak is understood mod 3. 131
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PHYSICS LETTERS
gin. Eq. (45) is shown to hold if we notice that
Z)']+ 2 ()'j -- )'1+1)2a/(r)a/+ 1 (r)
½q~2 = -
(40b)
Z(X/-- Xf+l)2aj (r)af+l(r)
E (~ -- )'j+ 1)2ajaj+ 1 = - C E a j
I 2
(40c)
4 f l = -- E ( X ] -- )']+ 1 )2af(r)a]+ 1 (r) , t
1 2
7If~ = E ) ' j +
25 March 1982
1)'j+2aj(r)+ (E~jaf(r)
= 0,
(46a)
E ()`f -- )'j+ l )2)']+ 2a/a/+ l = - C E ) ' j a ] = 0,
(46b)
E()`j
(46c)
- )'/'+1)
2
2 = )'/+2aflj+l -C2
.
The behavior o f f l and f2 at the origin is as follows
+ E)'/'+ 2 ()'j -- ~'+ 1)2aj(r)aj+ 1 (r))\
r2f 1 -+ 2,
X ( 2 h i + 2 ( ) ' i - Xj+l)2ai(r)ai+l(r)) .
(40d)
r2f 2 -+ 2,
as r-+ 0 .
(47)
As the ak's are determined, the asymptotic behavior of the Higgs field • can be estimated. The result is
Next we need to determine the r dependence ofak(r ) from eq. (36),
e ~ -+ diag(X 1 , )'2, )'3),
ak(r ) = ak (0) exp(--2)`kr)f ~ aj(O) exp(--2)7.r ) ,
under the present assumption; )'3 > )'2 > )'1" The asymptotic behavior of the magnetic fields B is also calculated
k = 1,2,3.
(41)
By substituting these into (40) we obtain the solutions. However, we are still left with an important task to determine the ak's (ak = ak(O) hereafter), k = 1,2, 3. This is done by requiring the condition that the physical quantities • and B should be finite. As is seen from eq. (11) and (15), this condition requires that, at least, ffl should behave at the origin
as 1~1 - + - l / r ,
(42)
as r-+ 0 .
This requirement is satisfied by
~ a ] = O, ~aj~j = O, ~a])' 2= C,
(43a, b, c)
where C is a non-zero constant. The solutions are uniquely determined because we assume [see eq. (23)] )'3 ~ )'2 ~ )'1, and they are given by
ai = C l-I ()'i - X/.)-1 .
(44)
This guarantees that ~2 behaves at the origin as
½~2-+-1/r,
asr-~ 0 ,
(45)
and therefore we have a finite Higgs field qb at the ori-
132
eB.~ -+ - T 3/r 2 .
as r -+ oo,
(48)
(49)
These behaviors of the solution coincide with those found by Wilkinson and Bais [8], though they found the solution by starting with a somewhat artificial assumption on the forms of the fields. We have showed the systematic procedure of finding the SU(N) monopole solution by means of the ISM. In order to apply the ISM we have put the SSBE in the Lax formalism which is equivalent to a pair of equations. One of the pair is the eigenvalue equation and the other is the "space" evolution equation by which we determined the r dependence of the eigenvectors with the boundary condition at the origin. The ISM is well known in the study of non-linear equations and we have found that the method is applicable to field theory. In the remaining we discuss the "conserved" quantities. In the Lax formalism of the Toda lattice [12] it is manifest that there exist a sufficient number of conserved quantities. Though a clear understanding of the dynamical roles of these conserved quantities is not yet obtained, they must guarantee the soliton character i.e. like the solitons preserving their profile even after their collisions. In our case the variable is not time but space and the system is not completely integrable. However, there are some characters which are not exactly common but similar to the Toda
Volume 110B, number 2
PHYSICS LETTERS
lattice which is known to be completely integrable. In fact there exist N + 1 "conserved" quantities with respect to r;
Jn = t r L n,
n =1,2 ..... N+I.
(50)
By taking the space derivative ofJ n we can show
dJn/dr=O,
n=l,2
..... N+I.
(51)
Here we have used the Lax equation (20). For the SU(3) case the conserved quantities are J1 = 0, _1
2.1-2 _ ~ 2 4J3 - ~
1 1~/
+5(
12
2--~1) 2 +5~2--f2--f2, -
1)3 -
+ 53 f2~ 1
32 - 5f~t~2 (52)
In ref. [10], we showed that our system is the Wick rotated version of the (N + 1)-particle system with a potential of exponential type. From this analogy the first conserved quantity is regarded as the conservation of the total momentum. In the particle system this quantity is not necessarily zero except for the case that we take the center-of-mass system. But in the present case J1 is necessarily zero because o f the traceless condition of the SU(3) generator matrices.
25 March 1982
Although it is hard to assess the dynamical meaning of the third quantity, the second quantity is interpreted as the total energy of the three-particle system. The author would like to thank Professor M. Yonezawa for careful reading of the manuscript. He also acknowledges the financial support provided by the Japan Society for Promotion of Science.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
G. 't Hooft, Nucl. Phys. B79 (1974) 276. A.M. Polyakov, JETP Lett. 20 (1974) 194. E.B. Bogomolny, Sov. J. Nucl. Phys. 24 (1976) 449. M.K. Prasad and C.M. Sommerfield, Phys. Rev. Lett. 35 (1975) 760. N.S. Manton, Nucl. Phys. B135 (1978) 319. A.N. Leznov and M.V. Saveliev, Lett. Math. Phys. 3 (1979) 489; Commun. Math. Phys. 74 (1980) 111. F.A. Bais and H.A. Weldon, Phys. Rev. Lett. 41 (1978) 601. D. Wilkinson and F.A. Bais, Phys. Rev. D19 (1979) 2410. P.D. Lax, Commun. Pure Appl. Math. 21 (1968) 467. T. Koikawa, Prog. Theor. Phys.66 (1981), to be published. J. Moser, Adv. Math. 16 (1975) 197. H. Flaschka, Phys. Rev. B9 (1974) 1924.
133
PHYSICS LETTERS
25 March 1982
EXACT SU(N) MONOPOLE SOLUTIONS WITH SPHERICAL SYMMETRY BY THE INVERSE SCATTERING METHOD Takao KOIKAWA 1 Department of Physics, Hiroshima University, Hiroshima 730, Japan Received 28 December 1981
The construction of the exact SU(N) monopole solution with spherical symmetry by means of the inverse scattering method is shown. We first derive the Lax equation from the spherically symmetric Bogomolny equation. The Lax equation is equivalent to a pair of equations, the eigenvalue equation and the space evolution equation. The inverse scattering method is applicable to the pair of equations. We also discuss the conserved quantities in connection with the linear manybody problem. After the introduction by 't Hooft [1 ] and Polyakov [2] of the magnetic monopole, much progress has been made in this field. The exact onemonopole solution was found in the BogomolnyPrasad-Sommerfield limit [3,4]. In this limit the energy of the magnetic fields and the Higgs fields in the adjoint representation is bounded below by the topological charge of the magnetic monopole. The bound is saturated if and only if the Bogomolny equation B = Od~,
(1)
is satisfied. The embedding of SU(2) in SU(N + l) enables us to consider the above Bogomolny equation as the one expanded by the SU(N + l) (N + 1) × (iV + 1) matrices. As was shown by Manton [5] this equation can be regarded as the self-dual condition of the time-independent gauge fields by identifying the zeroth components of the gauge fields with the Higgs fields. By assuming spherical symmetry simplicity is brought into the equation. As was shown by Leznov and Saveliev [6], the spherically symmetric Bogomolny equation (SSBE hereafter) with SU(N + l) reduces to a simple form which is related to the one-dimensional (N + 1)-body problem. Starting from the generalized Liouville equation we can derive these equations and other soliton equations as well by choosing the corresponding Lie algebras. 1 Fellow of the Japan Society for the Promotion of Science. 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
In this article we shall try to solve the SSBE by using the inverse scattering method (hereafter ISM) which is well known in the soliton theory. Although the exact solution of the SSBE with SU(N + 1) was found by Bais and Weldon [7], and Wilkinson and Bais [8], their assumptions which limit the form of the solutions seem to lack an obvious justification and might have missed other important solutions. This motivates us for the present work. We show the Lax pair [9] of matrices L and A [see eqs. (18) and (19)] which play a role in the eigenvalue problem and "space" evolution problem, respectively. First we derive the Lax equation from the Bogomolny equation by imposing spherical symmetry, Leznov and Saveliev [6] showed the close relation of the SSBE to the one-dimensional many-body problem, of which a detailed discussion is also found in ref. [10]. Encouraged by the similarity between the SSBE and the linear many-body problem, we try to construct the Lax pair formulation of the SSBE with SU(N + 1) and solve it by the ISM of which the utility is well-established [ 11 ] in the linear manybody problem. Some of the necessary ingredients for the present discussion are briefly recapitulated. We assume the Higgs field • to be a rotational scalar and the gauge potentials W as rotational vectors: [J, ~] = O,
[Ji, Wj] = ieijgWk ,
(2, 3) 129
Volume ll0B, number 2
PHYSICS LETTERS
where the rotational generators J are given by g=-irXV
+T.
(4)
Here T are generators from the Lie algebra of SU(N + 1) and satisfy an angular momentum algebra. Since the angular dependence of the participating fields are specified we can work on the z-axis. Then from (2) and (3) we obtain
25 March 1982 ~1
0
0
~2 -- ~1
0
0
~3
1
-
-
~2 ,
0
(17)
0
--qJN [T3, q,] = 0,
[T 3, W±I = +We ,
(5, 6)
where We = W1 +iW 2. Next we introduce new functions M by (7)
eW = (M - T) × ¢ /r ,
where ;" = r/r is a unit vector. Then M± = M 1 + iM 2 satisfy [T3,M±] = +114+_.
L = ~ +½i(N, + N _ ) ,
er 2 d(I)/dr -7-1 [M+, M_ ] - T 3 ,
(9)
and e - 1 dM±/dr = +[ep,M±l .
(10)
Let us simplify these equations by introducing ff and N± defined by edP=q/ + T3/r, M± =rg+_ ,
(11,12)
Then eqs. (9) and (10) read dN+/dr = +[@,N+]
(13, 14)
We take the following representation for T3 from SU(N + 1) T3 = diag(N/2, N/2 - 1 ..... - N / 2 ) .
A =½i(N_ - N + ) .
/dr = [A, L ] .
(20)
This turns out to be the Lax equation of the linear (N + 1)-body system with a potential of exponential type if we regard the variable r as time variable and (if/.) 2 as the potential linking the neighbouring particles Eq. (20) is equivalent to the Lax pair equation: LVk=XkO k,
k=l,2
..... N+I,
(21)
dv k / d r = A v k,
k=l,2
..... N+I.
(22)
where the v k are the N + 1 independent (N + 1)-component column vectors, i.e. v k = (vk(1 ;r), Vk(2; r) . . . . . Uk(N + 1 ; r)) t, belonging to the eigenvalue k k. For simplicity we assume that the Xk are not degenerate and satisfy ~kN+1 > XN > --. > X2 > Xl •
(15)
The commutation relation of N±, and ~ with T3 tells us which entries of these matrices are allowed to survive. It is, therefore, reasonable to express them as
011
(lS, 19)
Eqs. (13) and (14) are summarized into the Lax equation
(8)
The Bogomolny equation (1)under the present assumption reads
d@/dr - 1 [N+,N_]
and N_ = Nt+. Let us define the Lax pair [9] matrices L and A by
(23)
As the consequence of the traceless condition of the SU(N + 1) generators and hence qJ, we have another condition for the Xk N+I n=l
Xn =O.
(24)
0 0 .
N+ =
(16)
"fU 0 130
We shall recourse to the well-known procedure of the ISM. Let us first solve the inverse scattering problem of eq. (21). In other words we express the potentials in L, ~j and fj, in terms of the eigenvectors and eigenvalues. The r dependence of these eigenvectors is obtained from eq. (22) with a suitable boundary condi-
Volume 110B, number 2
PHYSICS LETTERS
25 March 1982
tion which we shall discuss later. We assume that the vectors ok *1 are orthonormal:
vectors ok(r ) and hence ak(r ) from eq. (22). The first column of this equation gives us
N+ 1
do k(1;r)/dr = -~t.ally k (2,. r ) .
n=l
vi(n)v/(n) = 60..
(25)
(34)
From this equation, together with the one obtained from eq. (21), we obtain
The completeness relation is N+I
N+ 1
]=1
t~(n )v](m ) = 6 nm
.
(26)
From eq. (21) we obtain
Rvk = ()tk -- )t)-lvk ,
k = 1,2,... , N + 1. (27)
with R = (L - XIN+I) -1 .
dVk(1 ;r)/dr = ( /~=I x ] 4 ( 1 ;r)--)tk ) vk(1 ; r ) .
(35)
The solution to this differential equation is easily found and we get N+I
(28)
ak(r) = ak(O) exp(--2)tkr)l /~1 a](0) exp(--2)tff),
It is easy to see that eqs. (26) and (27) lead to
k = 1, 2 . . . . . N + 1.
N+ 1
Rll= ~.=°~(1)/(;'V-
)t).
(36)
(29)
(30)
This completes the determination of the r dependence of the eigenvectors, ok (r). As an example we show the explicit ISM construction of the spherically symmetric SU(3) monopole solution. It is necessary to take account of the boundary condition so that the Higgs field ep and the magnetic fields B are finite. From eq. (33) we have *2
AN : d e t ( ~ ' - )tiN) ,
(31)
A2IA 3 =~a](r)/()tj - )t),
AN+ 1 = det(L - )tiN+ 1 )"
(32)
where A 2 and A 3 are explicitly written as
On the other hand the same quantity is expressed in terms of the potentials in L from the definition of R, eq. (28): R l l = AN/AN+ 1 , where AN and AN+ 1 are defined by
In (31) the N × N matrix L" is the one obtained from L by removing the first row and column of L. We thus have two ways of expressing R 11, i.e. (29) and (30). Therefore we obtain N+I
AN/AN+ 1 = ]~1"= a](r)/Ot! -- X) ,
(37)
A2 = )k2 +½~l)t -- 1 ~2(ff2 -- t~l) + l f l2 ' A3 = - ) t 3 - l ( ~ l f f 2 - ~ 7 - ~ b 7
(38)
+ f 7 +f2))t
--8 [~l(ff2 - ffl)~2 +f2ff2 -f22ffl] •
(39)
By comparing both sides of eq. (37) we get (33)
where ak(r ) = 02(1 ;r). This formula enables us to exp r e s s f i and Oi in terms o f v i and )ti" We shall next find the r dependence of the eigen*1 In the interests of economy, the argument r of each component ok (re;r) of the vector ok is suppressed hereafter.
½tp 1 = ~ ) t f l / ( r ) ,
(40a)
*2 Hereafter the summation is taken over from 1 to 3 and the suffix k of hk or ak is understood mod 3. 131
Volume 110B, number 2
PHYSICS LETTERS
gin. Eq. (45) is shown to hold if we notice that
Z)']+ 2 ()'j -- )'1+1)2a/(r)a/+ 1 (r)
½q~2 = -
(40b)
Z(X/-- Xf+l)2aj (r)af+l(r)
E (~ -- )'j+ 1)2ajaj+ 1 = - C E a j
I 2
(40c)
4 f l = -- E ( X ] -- )']+ 1 )2af(r)a]+ 1 (r) , t
1 2
7If~ = E ) ' j +
25 March 1982
1)'j+2aj(r)+ (E~jaf(r)
= 0,
(46a)
E ()`f -- )'j+ l )2)']+ 2a/a/+ l = - C E ) ' j a ] = 0,
(46b)
E()`j
(46c)
- )'/'+1)
2
2 = )'/+2aflj+l -C2
.
The behavior o f f l and f2 at the origin is as follows
+ E)'/'+ 2 ()'j -- ~'+ 1)2aj(r)aj+ 1 (r))\
r2f 1 -+ 2,
X ( 2 h i + 2 ( ) ' i - Xj+l)2ai(r)ai+l(r)) .
(40d)
r2f 2 -+ 2,
as r-+ 0 .
(47)
As the ak's are determined, the asymptotic behavior of the Higgs field • can be estimated. The result is
Next we need to determine the r dependence ofak(r ) from eq. (36),
e ~ -+ diag(X 1 , )'2, )'3),
ak(r ) = ak (0) exp(--2)`kr)f ~ aj(O) exp(--2)7.r ) ,
under the present assumption; )'3 > )'2 > )'1" The asymptotic behavior of the magnetic fields B is also calculated
k = 1,2,3.
(41)
By substituting these into (40) we obtain the solutions. However, we are still left with an important task to determine the ak's (ak = ak(O) hereafter), k = 1,2, 3. This is done by requiring the condition that the physical quantities • and B should be finite. As is seen from eq. (11) and (15), this condition requires that, at least, ffl should behave at the origin
as 1~1 - + - l / r ,
(42)
as r-+ 0 .
This requirement is satisfied by
~ a ] = O, ~aj~j = O, ~a])' 2= C,
(43a, b, c)
where C is a non-zero constant. The solutions are uniquely determined because we assume [see eq. (23)] )'3 ~ )'2 ~ )'1, and they are given by
ai = C l-I ()'i - X/.)-1 .
(44)
This guarantees that ~2 behaves at the origin as
½~2-+-1/r,
asr-~ 0 ,
(45)
and therefore we have a finite Higgs field qb at the ori-
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eB.~ -+ - T 3/r 2 .
as r -+ oo,
(48)
(49)
These behaviors of the solution coincide with those found by Wilkinson and Bais [8], though they found the solution by starting with a somewhat artificial assumption on the forms of the fields. We have showed the systematic procedure of finding the SU(N) monopole solution by means of the ISM. In order to apply the ISM we have put the SSBE in the Lax formalism which is equivalent to a pair of equations. One of the pair is the eigenvalue equation and the other is the "space" evolution equation by which we determined the r dependence of the eigenvectors with the boundary condition at the origin. The ISM is well known in the study of non-linear equations and we have found that the method is applicable to field theory. In the remaining we discuss the "conserved" quantities. In the Lax formalism of the Toda lattice [12] it is manifest that there exist a sufficient number of conserved quantities. Though a clear understanding of the dynamical roles of these conserved quantities is not yet obtained, they must guarantee the soliton character i.e. like the solitons preserving their profile even after their collisions. In our case the variable is not time but space and the system is not completely integrable. However, there are some characters which are not exactly common but similar to the Toda
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PHYSICS LETTERS
lattice which is known to be completely integrable. In fact there exist N + 1 "conserved" quantities with respect to r;
Jn = t r L n,
n =1,2 ..... N+I.
(50)
By taking the space derivative ofJ n we can show
dJn/dr=O,
n=l,2
..... N+I.
(51)
Here we have used the Lax equation (20). For the SU(3) case the conserved quantities are J1 = 0, _1
2.1-2 _ ~ 2 4J3 - ~
1 1~/
+5(
12
2--~1) 2 +5~2--f2--f2, -
1)3 -
+ 53 f2~ 1
32 - 5f~t~2 (52)
In ref. [10], we showed that our system is the Wick rotated version of the (N + 1)-particle system with a potential of exponential type. From this analogy the first conserved quantity is regarded as the conservation of the total momentum. In the particle system this quantity is not necessarily zero except for the case that we take the center-of-mass system. But in the present case J1 is necessarily zero because o f the traceless condition of the SU(3) generator matrices.
25 March 1982
Although it is hard to assess the dynamical meaning of the third quantity, the second quantity is interpreted as the total energy of the three-particle system. The author would like to thank Professor M. Yonezawa for careful reading of the manuscript. He also acknowledges the financial support provided by the Japan Society for Promotion of Science.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
G. 't Hooft, Nucl. Phys. B79 (1974) 276. A.M. Polyakov, JETP Lett. 20 (1974) 194. E.B. Bogomolny, Sov. J. Nucl. Phys. 24 (1976) 449. M.K. Prasad and C.M. Sommerfield, Phys. Rev. Lett. 35 (1975) 760. N.S. Manton, Nucl. Phys. B135 (1978) 319. A.N. Leznov and M.V. Saveliev, Lett. Math. Phys. 3 (1979) 489; Commun. Math. Phys. 74 (1980) 111. F.A. Bais and H.A. Weldon, Phys. Rev. Lett. 41 (1978) 601. D. Wilkinson and F.A. Bais, Phys. Rev. D19 (1979) 2410. P.D. Lax, Commun. Pure Appl. Math. 21 (1968) 467. T. Koikawa, Prog. Theor. Phys.66 (1981), to be published. J. Moser, Adv. Math. 16 (1975) 197. H. Flaschka, Phys. Rev. B9 (1974) 1924.
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