P H Y S I C A L R E V I E W D V O L U M E 2 4 , N U M B E R 8 1 5 O C T O B E R 1 9 8 1

Exact Yang-Mills-Higgs monopoles. I

S.-C. Lee Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11 794

(Received 10 March 198 1)

We find explicit expressions for axially symmetric Yang-Mills-Higgs monopoles of arbitrary charge in the Bogomolny-Prasad-Sommerfield limit by using general Neugebauer transformations. The expressions satisfy the reality and asymptotic conditions. However, a general proof of nonsingularity is still lacking.

I. INTRODUCTION

Since Ward published his result1 on two-mono- pole solutions of the Yang-Mills equation, t h e r e has been much progress in th i s field. In part ic- ular , Ward 's resu l t h a s been generalized to mono- poles of a rb i t ra ry charges by P r a s a d , Sinha, and Wang.' In th i s method, it is difficult to obtain explicit expressions fo r gauge potentials for two reasons. F i r s t , the solution is constructed in a gauge in which gauge potentials a r e complex and a complex gauge t ransformation i s necessary to put them in r e a l form. Second, the solution is given by "super potentialsw and fur ther differen- tiations a r e necessary to obtain the gauge poten- tial. Both the gauge t ransformation and the dif- ferentiation could be cumbersome in practice. Recently, Forgacs e t al. showed that the one- and two-monopole solutions can be obtained by Back- lund transformation^.^ This method has the ad- vantage that one obtains the explicit expressions for the gauge potentials directly. In this paper , we general ize the resu l t of F o r g a c s e t al. to mono- poles of a rb i t ra ry charge. In Sec. 11, we obtain expressions for genera l Neugebauer t rans forma- tions. This resu l t is of in te res t in i t s own right. We specialize to the monopole solutions in Sec. I11 and prove that real i ty and asymptotic conditions a r e satisfied. In Sec. IV, we d iscuss briefly the question of nonsingularity, which was solved r e - cently, in Ward's approach by P r a s a d and R o ~ s i . ~

11. GENERAL NEUGEBAUER TRANSFORMATIONS

We begin by writing down the mos t genera l ex- pression for axially symmetr ic monopole poten- t i a l ~ ~ ' ~ :

A:= 9 2 , Ai+i2=(pleinr, A ; = O ,

Aj+j2 = -iVleinv , A;+iz = ip-l(n +q2)eiv , A:::; = -i(V2 +p-'ql)efin+l" , (2.1)

A: Ji =i(-V2 + p ' l q l ) e ~ ' n ~ i ' ~ , where n is the monopole charge.

The self-dual Yang-Mills equations can be inte-

grated partially to get

and

5 v 2 = (vuI2 - ( V V ) ~ , ~ v ~ v = ~ V U VQ.

Let us define6

and

then Eq. (2.3) can be written a s

~ , M , = C ~ M ~ N , , a 1 N i = C ~ N , M , , i = 1 , 2 , 3 ,

(2.6)

where C: a r e constant mat r ices with the following nonvanishing elements:

c;1= c22 = c33 = -c12 = -c21 = -1 2 3

C:2 = C13 = C31= C23 = I (2.7)

1 2 2 - 2 .

F o r any given solution Mi, N, of Eq. (2.6), we can obtain new solutions by solving the total Ric - cat i equations,

dy = (y2 - l ) ~ , d x ' + ( y - l)N,d?, (2.8)

d ( f fy1I2) = [ - y 1 / 2 ~ 2 + c Y ~ ' ~ / ~ ( M ~ -MI)

+ ( C Y ~ - ' / ~ ) ~ M ~ ~ ' / ~ ] ~ X ~

+ [ - y - 1 / 2 ~ l + Q ~ - ~ / ~ (Nl -N2)

+ ( c r y - 1 / 2 ) 2 ~ 2 y - L / 2 ] d ~ 2 . (2.9)

The new solutions Mi, N; a r e then obtained in the following ways:

M;=CY;M~, N; = C Y ; ~ ~ N ~ , (2.10)

with

2200 O 1981 The American Physical Society

E X A C T Y A N G - M I L L S - H I G G S M O N O P O L E S . I 2201

M;=CI.;M@, N;=v ;N~

with

Equations (2.10) and (2.11) define Neugebauer's I, transformation, while Eqs. (2.12) and (2.13) define Neugebauer's S transformation. Neugebauer's I, transformation i s defined to be SIIS. Note that for the I, transformation, we have to solve Riccati equations different from those of I,.

Neugebauer found the following commutation theorem:

~ , ( a ( ~ ) , y (4 ) )~1 (a (3 ) , y'3))~2(a(2),+2))~l(a(1), y'l)) = 1 , (2.14)

where

Using Neugebauer's commutation theorem, the parameters a' ", y'" in the following general Neugebauer transformation I , ( ~ ( 2n), y(2n))zl(a(2n-1), y( '2n- i ) )

X . . z , ( (Y(~) , y (2) )~1(a(1) , 7")) (2.16)

can be obtained algebraically ,6 once a'",y' l' a r e obtained by integrating Riccati equations. To see this, cdnsider the case n = 1. Let a'" = a,, y"' = 71 be solutions of the Riccati equations with some integration constants. Then, according to the com- mutation theorem, the following diagram exists

transformation differ from a,, yl only by the choice of integration constants. We denote the parame- t e r s by a,, y,. Then, from the diagram,

which can be solved to get

This i s called the Neugebauer composition theo- rem. In Appendix A, we prove the following theo- rem.

Theorem. In the transformation (2.16)' the pa- rameters a") and y(i) a r e given by

where A* and B k a r e determinants of order 2k + 1, C, and Dk a r e determinants of order 2k, with

A ~ = ( ~ , Y ~ , . . - , ~:,a~,a~Y~,...,ff~~lb-~(, B k = ly l ,$ , . . . t ~ l b , ~ I , ~ l ~ l , . . - taly:I

(2.21) Ck= I1,y1,.. , ~ f l , a ~ , ( Y l Y ~ , . . . ,al~lh-lI , D k G 1~1~3,. , ~ l k , a ~ , f f ~ ~ ~ , . ,al~:-lI

In Eq. (2.21), we write down only the f i r s t row of the determinants. The ith row can be obtained by substituting a , , y, for a,, y,, where a ,, yi differs from a l , y l only by the choice of integration con- stants.

We should often use the following variables: E j G a i y ; l / 2 ~ ( i ) = ( f ) (4)-112

7 - a Y (2.22)

It i s also convenient to define the following de- terminants:

2 k A i = 11,(Y1w1,w12 ,... ,wl 1 ,

where the single line represents an I, transforma- tion and the double line represents an I, t rans- where w, = y:'2. Then Eqs. (2.19) and (2.20) can be

formation. A , labels the solution in the various written a s

stages of the transformation. $ 2 9 - (zk-2) . . . $2) = ( -1 )k2 , ,,(2* - ~ 2 k

Since the inverse of the I, transformation i s an a Ck Yzr-1

, (2.24) I, transformation and two consecutive I, t rans- formations result in an I, transformation, we see from the diagram tha tA, can be obtained from A, Yzk

by an I, transformation. The parameters of this (2.25)

S . - C . L E E

Let M,,,, N,,, denote the solution obtained from M ,,,, No,,, by the transformation of Eq. (2.16). Then we have

For the solution of Yang-Mills equation, we a r e interested only in the case

y2 = 1 , or equivalently, y2,= 1 . (2 .30)

Equations (2.24)-(2.29) solve the algebraic prob- lem of expressing general Neugebauer transfor- mation in t e r m s of the integral of a given Riccati equation.

No , (2 .27)

>

o 1 ~ ( 7 - l -X;)

111. THE MONOPOLE SOLUTIONS

where 2n

( 2 . 2 8 ) k-1 =fi G(2k-1) (= 6'2" , . 6'4) - ( 2 ) 112

Q! y2k-1 -( 2 k) - ( 2 k - 1 ) . . . ~ ( 3 ) ~ ( 1 )

k = l Q! k-1 Q!

n1 G( 2k) . . -( 4) -( 2)

-C 2 - 1 . , . Q! 3 Q! 2 ,I2) 7

Nn=y-'

Let us write down the gauge potential in t e rms of the variables Mi, N, :

0 fi S( ') $(y-' - YA) k = l

0 0 Y -l

Of particular interest i s the quantity

L 2

since

Following Forgacs et a l . , we take the seed solu- tion Mo,f , No,*, to be

The Riccati equations a r e easily integrated to give

where

and C k , Pk a r e integration constants. We choose the convention

and a s

At this point, it is easy to show that the BHck- lund-transformed solution M , , , N , , , satisfies the asymptotic condition of a monopole of charge 2 n - 1 . However, t o satisfy the reality condition

we shall make the following construction. For even monopoles we s ta r t with M2,,N2, while

for odd monopoles we s ta r t with M2n,,N2n,. Then we choose

where k runs through the appropriate range for even and odd cases, respectively. This choice corresponds to let C2,-rtm for al l k. Next we re-

24 E X A C T Y A N G - M I L L S - H I G G S M O N O P O L E S . I

define

With equations (3.9) and (3.10), we find

where Dk 1 , Ck ( denote Dk, Ck evaluated a t w2,= 1, &,, = 1 , and B, 1 ,Ak 1 denotes Bk,Ak evaluated at wZk+, = 1 , CZk,, = 1. Equations (2,24) and (2.25) then become

1 y ( 2 k ) ==, (Zk-1) - -Ye • (3.13) yk'

In Appendix B, we prove the following formulas:

B ' l C 1 " A' A' Xzn= 1 - -2 I z k-l k-A(w:k-l - m i k ) , A',! Dn ,a c;-1 C;

(3.14)

B;I D;=A;-~ w;k-l - w : ~ y ' =I--- 2 n Abl Ck kd DL-1 WZk-lWZk '

w:k-l-w:k D L ~ C L I -I WZk-l w2k

D)nl W2n-1'

B' D'1 n - l~ : - lx i - i w z k - 1 - &&-CLiDbl y;"-1=+ -z -

A, chi Dk-l Dk w2k-lw2k DLl cil u"l ' (3.15)

where xL,,B;, a r e obtained from A;,,B:-, by substituting d,,, w,, for 62k-1, WZbl. Also we put Ci = 1, Dh = 1. For odd monopoles, instead of using MZp1 ,N2n-1 we shall use

~Sn/l2,,-~ , Nitrl SNZh1 , (3.16)

where S i s the Neugebauer S transformation. Then we have

Consider the asymptotic property f i r s t . Asymp- totically &, - - 1 + exponentially small t e rms s o that

2k-1 2k-1

Note that B',] /A:I can be obtained from Bn/An by letting Cz,-m and similarly for D L [ /Cil .

Combining Eqs. (3.17) and (3.18), we get, asymp- totically,

This i s cor rec t up to exponentially small te rms. We see that the asymptotic condition for mono- poles a r e satisfied.

Next, consider the reality condition. At this point, we make the special choice

for even monopoles and

2204 S . - C . L E E - 24

for odd monopoles. We shall prove that this choice satisfies the

reality condition. Consider the case of even mono- poles f i rs t . We have

& & I = z-l 2 h . 9 ~ z * k - 1 = w,: 9 (3.22)

Note that

s o that BA [/A,: I i s a phase factor. Using Eq. (3.23), we s ee that C U D ; i s r ea l and

Hence we showed that

X$ = X;, . (3.26)

Similarly,

Y:, = Y;, . (3.27)

Equations (2.26) and (2.27), together with Eqs. (3.24)-(3.27) imply that reality condition for even monopoles a r e satisfied.

In the case of odd monopoles, the choice of Eq. (3.21) implies

We have

A ;$ = (wl ' ' ' wZk-2~2k)~2k'2x;-1 9

D;* = (wl . . . )-2k+lC' 2 k

(where k = l , . . . , n - 11,

s o that I c ~ D ; ( = 1 andAA.,/~;-, i s real. It i s also easily seen that

From Eq. (3.29), we get

which implies

xz*n-l = Y;n-l . (3.33)

Similarly,

Yz*n-l =x;n-l. (3.34)

Equations (3.30), (3.31), (3.33) and (3.34), together with formulas for M~,,.,,N&,-,, imply that the reality condition for odd monopoles i s satisfied. We a r e left with n parameters C k for even monopoles and n - 1 for odd monopoles. These a r e t o be deter - mined by the condition of nonsingularity.

Before concluding this section, we shall obtain the formulas for Hzn,H2,,-, on the Z axis. Let us begin by noting the identity

Then

where 8 = $2 + i(s/4) for even monopoles and 6 = $2

for odd monopoles. To lowest order in p,

Define

then

where ~ , = ~ w ~ , w ~ , . . . wim with indices ik al l dis- tinct. It i s also easy to show that

(3.40) Since

24 - E X A C T Y A N G - M I L L S - H I G G S M O N O P O L E S . I

(3.42) we get

(3.43)

Similarly, define

We obtain

Now we consider C;,DL,A',-,, Bi-,. To lowest ord- e r in p, we have

ip (k-2F(l))]. C;;(-~oth0)~II, w , . . . , w : ~ l / [ l + ~ ~

In computing F ( l ) , we put al l w , = 1 so that

~ ( 1 ) = + ( 2 k - I ) , Zk

i~ C:= (coth0)' n (w, - wj) (1 +-) . (3.46) 1 < j

Similarly,

and

(-cothO)'-l 11, wl, . . . , wY-' 1

Zk-1

B:_,= - (coth0)* n (w, - w,) . (3.49) i <j

From Eqs. (3.46)-(3.49), we get

Substituting into Eq. (3.17), we find

Note that in Eq. (3.171, the k = 1 te rm requires special consideration. Also note that we have used

the proof of which i s s imi lar to the proof of Eq. (3.50). Finally, using Eqs. (3.50)- (3.531, it is easy to show that

i(M2n,l +N2n,2)= -2 2np'l, (3.57)

i (M2n-1, +N2n-1,2)z -i(2n - 1)p-' . (3.58)

Compared with Eq. (3.1)~ we see that there exists no singularity in gauge potentials a s p- 0.

IV. THE CONDITION OF NONSINGULARITY

We have obtained monopole solutions which a r e r ea l and nonsingular on the z axis. Now we shall consider the nonsingularity of the solutions on the x-y plane.

At z = 0, we have, for even monopoles,

for p < b,,

R2k-1=R2k= bkE (p2 - b;)ll2, P > C ~ 7 (4.2)

- sinhb, - i cost, - sinhb, - i cost, a2k-l = - coshb, + sinb, , L Y ~ ~ = - coshb, - sinbk- '

We shall prove that unless

t k = i ( 2 m k - l )n (4.3)

for some integer mk, Hz, i s singular. Similarly for odd monopoles on the x-y plane, we have

2206 S . - C . L E E

sin(6, - ak) 6 2 k = - sin(6, - a,) Cia-, = i

1 - cos(6, - a,) ' 1 - c o s ( t k -a,) '

(4.4) f o r p < 6, and

sinhb, - i s in t , - sinhb, + i sinf, aZk-l = - LYZk = - coshb, - cos t , ' cos hb, - cos t , '

In this case , unless

fo r s o m e integer m,, Hz,-, is singular. Consider the even monopole case . F o r notation-

a l simplicity, we shal l assume t l < t 2 < . . . < f, and consider p- 5, f r o m above. The proof f o r other c a s e s a r e s imi la r . Suppose f;,+i (2p - l ) a fo r s o m e integer p. Then c?,,-,, &,, remain non- s ingular a s p 3 b n , while w ,,-, - 0, w2, - a. Let us compute A&,,A;-,, C;, D:, keeping only the leading s ingular t e r m s . Then

F r o m Eq. (4.71, we get

which i s s ingular a s asse r ted . A completely s i m i l a r argument applies to the

odd monopole c a s e and shows that if t ,- ,+pa f o r s o m e integer p , Hz,-, i s s ingular . We conclude that

fo r even monopoles and

f o r odd monopoles. In Appendix C, we give Hz, H, on the x-y plane

and show that fo r Hz, H, to be nonsingular, f , = a / 2 f o r two monopoles and 6, = a for three monopoles. We conjecture that in general , f o r nonsingular solutions,

fo r even monopoles and

f o r odd monopoles. We hope to find a s imple proof of nonsingularity

in th i s approach and to find the relat ion to Ward's approach. A s yet, they a r e s t i l l open questions.

ACKNOWLEDGMENTS

I wish to thank Professor B.-Y. Hou for check- ing some of the formulas and many useful dis- cussions. This work i s supported in par t by the National Science Foundation, under Grant No. PHY 79-06376801.

APPENDIX A: PROOF OF EQS. (2.19) AND (2.20)

We prove by induction. Assume Eq. (2.19) is t rue fo r k = 1, . . . , 1 ; Eq. (2.20) is t rue f o r k = 1 , . . . , I - 1. F i r s t we want to find Q ' ~ " ~ ' , y(21+". This can be achieved by using Neuge- bauer 's composition theorem. To do this, choose

where

and s imi la r ly fo r C; Setting a'2", y"", C Y ( ~ " ' , y(21" into Eq. (2.18), we

get

Note that

D; = - [AlIZl, 1 = - [BlIZl, 2Z+l 9

where [A,] , , , denote the n, mth cofactor o f A , etc . Using Eq. (A3) and a theorem of Jacobi on the

subdeterminant of adjugate d e t e r m i n a t ~ t , ~ we find

the las t equality following f r o m induction hypo- thesis.

A completely s i m i l a r proof shows that Eq. (2.19) is t rue for k =1+ 1 also. Since Eq. (2.19) i s t rue fo r k = 1 , and Eq. (2.20) is t rue fo r k = 0, the in- duction proof i s completed.

E X A C T Y A N G - M I L L S - H I G G S M O N O P O L E S . I 2207

APPENDIX B: PROOF OF EQS. (3.14) AND (3.15)

We shall derive the formula for X2, in detail. All other formulas can be proved similarly. Com- bining Eqs. (2.29) and (3 .12) , we get

Define

C; / = C; evaluated a t c2, = 1 , o,, = 1 ,

C;11= C; / evaluated a t 15,,-, = -1, w,,-, = 1 , A; / =A; evaluated a t &,,+, = 1 , w,,,, , A;jl =A; / evaluated at I?,, = -1 , wlk = 1 ,

and similarly for D; I , Dill, B; 1, B611. Note that

D L 1 - ::I, e tc . c;- 1

Using Jacobi's theorem,' we get

Hence we have

We shall prove

CiAL-1 I (w2k-12 - 1) +A; I C;-, = C; I A ; - ~ ( ~ ~ ~ - , ~ - wZk2) . (B5)

It is easier to prove the following two identities: -

CLAA-~ / = C; l ~ ; , - C; (A;, , (Be) -

A; / C;-, = C; ~ A ; - ~ ( W , , - ~ ~ - 1 )

- C; Iq-,(wzk2 - 1 ) , (B7)

where Eq. ( ~ 6 ) i s obtained from Eq. (B5) by anti- symmetrizing w2,-,, &,,-, and w,,, &,,.

Equation (B6) i s proved by noting

- [C; I I,,,, = [C;Izk,, , 4, = - [C;12k-l,~k,

[~;l12,,,=-[c;12k,, , = [c;hk,2k, LA;, 1 1Zk-1,l =[A;l12kb1, I 9

and using Jacobi's theorem again. TO prove Eq. (B7), f i r s t note that

s o that Eq. (B7) i s equivalent to

A;IC;II=C;IA;II-C;IA;II. To prove Eq. (B8), note that

- c;l =- [A;11,,,, ,+,, C;=[A;I12k-l,2k+l,

and use Jacobi's theorem. Combining Eqs . (B5) and (B4) we get the desired result. The tech- niques a r e completely the same in proving the other formulas in Eqs . (3 .14) and (3.15). We will not repeat them here.

APPENDIX C : H z , H3 ON X-Y PLANE

In Eq. (3.17), B; ( /A;( and D; / /C; I a r e phase factors, so we shall consider only the remaining part which we denote a s H;, , HA,-, in this section. Then we have

Simple computation gives, for p < 5 ,

where

a2 (5' -p2)1/2.

If 5 P $(2p - 1 )n for some integer p, H; diverges a s a - 0. If t =+(2p - l ) a ,

sina

which is manifestly nonsingular a s a- 0. To make su re that g%in2a -a2+ 0 in 0 s a s 5 , we have to choose 5 = n/2.

2208 S . - C . L E E

For p > I ; , we get whereu = p + i a = c e i e , 0 4 0 ~ $ r , and

21;2 sinhb coshb H; = - I + y (u) =u-' coshu, 6 = n, 3n, . . . ,

(C5) = u-' sinhu, I; = 2a, 4a, . . . b~ (p2 -~~) l /~ . For p > L , replace a by i b and change the overall

sign. Note that there is a phase change of n a s we pass One may check directly that H; i s nonsingular p = S 0 near p = 0 and p = I;. For other values of p, we

H; is more complicated. For p i c , we find have to require

~ e y ( & ) I m y ( $ ) + O , O < 0 < + n .

By considering the dependence of ~ e y ( + u ) and 1

(C6) Imy (&) on 0, one can prove that only t = n gives

a 2 Rey(-u) p ' nonsingular ~j .

'R. S. Ward, Commun. Math. Phys. 79, 317 (1981). Rev. D E , 2182 (1981). 2 ~ . K. Prasad, A. Sinha, and L. C. Wang, Phys. Rev. 5 ~ . Rebbi and P. Rossi, Phys. Rev. D 22, 2010 (1980). D 23, 2321 (1981). %. Neugebauer, J. Phys. A 9, L67 (1979). 'P. Forgacs, A . Horvath, and L. Palla, Phys. Lett. IT. Muir, A Treat i se on the Theory of Determinants 99B, 232 (l981). (Dover, New York, 1960), p. 166.

4~-~. Prasad and P. Rossi, preceding paper, Phys.