The ′T Hooft-Polyakov monopole near the Prasad-Sommerfield limit

ANNALS

OF PHYSICS

146, 129-148 (1983)

The ‘T Hooft-Polyakov Monopole Near the Prasad-Sommetfield Limit * CARL L. GARDNER Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received August 2, 1982 A perturbative classical monopole solution for the SO(3) gauge theory is constructed in the limit of small but non-vanishing Higgs potential. This corresponds to the limit $/2Mi. = 1 Q 1, where ,u equals the mass of the scalar particle and M, equals the mass of the intermediate vector particles. The monopole solution and mass are found to involve non-analytic functions of A: \/;i and I In 1. The monopole mass M, is calculated to order .u2/M, as

I. INTRODUCTION A spontaneously broken unified gauge field theory admits magnetic monopoles as finite-energy classical solutions if the residual unbroken symmetry group contains an electromagnetic U(1) factor. The existence of magnetic monopoles in non-Abelian gauge theories was first demonstrated by ‘t Hooft and Polyakov [ I] in the context of the Georgi-Glashow SO(3) model [2]. Subsequently Prasad and Sommerfield [3] discovered an exact classical monopole solution in the limit in which the Higgs potential goes to zero. The purpose of the present investigation is to construct an approximate monopole solution for the SO(3) gauge theory in the limit of small but non-vanishing Higgs potential. This corresponds to the limit ,~‘2/2M2, = A< 1, where ,D is the mass of the scalar particle and M, is the mass of the intermediate vector particles. The Green’s functions for radially symmetric perturbations in the presence of the Prasad-Sommerfield (PS) monopole are also explicitly constructed [4]. The monopole field equations (2.9) and (2.10) are a complicated pair of coupled non-linear second-order differential equations in @J(the scalar field part) and a (the vector field part). The field equations can be written symbolically as

S[@l F[@,a]=l ( o 1; * This work is supported in part through funds provided by the U.S. Department of Energy (DOE) under contract DE-AC02-76ER03069. 129 OOO3-4916/83 $7.50 Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.

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CARL

L. GARDNER

where F is a two-component functional and the source S is a functional of @. Naively we might expect to be able to expand @ and a (at least for r < ,K’) about the A = 0 solutions in powers of A:

This expansion is incorrect. Naive perturbation theory breaks down at the classical level for the monopole problem. The perturbative solution for 2 @ 1 explicitly involves non-analytic functions of il: fi and il In 1. The monopole mass is also a non-analytic function of fi [ 111. Solving the monopole field equations requires a short-distance and a long-distance perturbation method. A natural partition of space occurs in constructing the monopole solution’: (1) (2) (3)

Near (N) region: 1 < r < (21)-“4M;‘, far (F) region: (2A)-“8M,1 < r, intermediate (I) region: (212)-“8M;1

< r Q (2~)-“4M$‘.

In the near region, the field equations can be linearized by expanding about the PS solution. In the far region, the field equations decouple, and the interacting classical theory can be solved perturbatively. The phenomenon which enables us to couple together the long-distance and shortdistance behaviors of the theory is that the far-region and near-region monopole solutions can be matched in the intermediate region. Note that this region of overlap becomes infinite as J + 0.

II. THE SO(3) MONOPOLE

REVISITED

[5]

The fundamental-field spectrum of the SO(3) gauge theory consists of three gauge vector fieids A,, and a (triplet) scalar field 0, in the adjoint representation of SO(3), where a, /3, y = 1, 2, 3. The Lagrangian is

F WW”= +L

- &A,,

+ eemb+brAp3

(4 0, = a,#, + eG3yA4w$y>

(2.1)

where e is the gauge coupling constant and 1 > 0 to ensure that the Hamiltonian bounded below. ’ The values

(2,1m1’*

and (21)-“4

are chosen

for the sake of concreteness.

is

‘T HOOFT-POLYAKOV

131

MONOPOLE

The Higgs potential @/4)(#* -a’*)’ h as 1‘t s minimum at 4’ = a”. 4 is zero vacuum expectation value (4,) = CrJ,, (this choice of direction choice of gauge), thereby spontaneously breaking the SO(3) symmetry down to U(1) rotations about the a = 3 axis. The usual quantum theory about the minimum of the Higgs potential in terms of

given a nonis a partial group of 9 is expanded

In the magnetic monopole sector of the SO(3) gauge theory, (4,) is not set equal to a constant since physically monopoles turn out to correspond to topological knots in the scalar field vacuum expectation value. Instead ‘t Hooft and Polyakov suggested that asymptotically the scalar field approach its vacuum value in the following way:

($Q2 + i2, (D,#),-+O

(2.2)

as r-co.

The existence of a static monopole solution satisfying these requirements demonstrated by ‘t Hooft and Polyakov by means of the Ansatz: $,(x, t)=+ta@(r),

O(r)-+a’e

was

as r-+00

Ami(x, t) = $ ciaD tfi

A,,,(x, t) = 0. The Ansatz (2.3) represents a particular choice of gauge for the monopole solution. Since the monopole solution is stationary, the monopole mass M, = -L = -I d3x9. After substituting Eqs. (2.3), the mass can be expressed as M,=$Jomdr

[ ($ra)2+&(l-r2a2)2+f(g)2

+ r2a2Q2 + +- r2(cP2 - ~2~)~ I

(2.4)

where we have resealed J = X/e2, a = Se. To determine the boundary conditions for @ and a at infinity, we require that @ and a approach their vacuum values as r + co: @--a-+0, a-0

as r-co.

(2.5)

The boundary conditions at the origin depend on the type of monopole solution we are seeking. For instance, the Dirac monopole is an infinite-energy solution of Eqs.

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CARL

L. GARDNER

(2.3), (2.4) and (2.5), which is singular at the origin. We will restrict our attention to finite-energy monopole solutions which are non-singular at the origin. Then we have the additional boundary conditions @+ 0, ra+ 1

as r+O.

(2.6)

The boundary conditions (2.5) and (2.6) suflce to guarantee that the monopole mass is finite. The l/r* magnetic field arises from the long-range l/r part of the vector field A (a-e-’ for L=O, a-e-‘/r for If0 as r --) co by Eqs. (2.13) and (2.19)). As r goes to infinity, 4, +--fa, “, A rri

1 +

-

7

1 Elaj3f/3

(2.7)

7’

For a candidate for the electromagnetic field strength tensor F,,, we require an SO(3) gauge invariant quantity which reproduces the usual F,, asymptotically in the w w (4,) = L IWI. W e may take either (i) ‘t Hooft’s definition or (ii) a definition due to Bogomolny and Faddeev [6]:

where the symbol z denotes the fact that F,,, is only defined asymptotically finite-energy monopole (see Coleman’s 1975 Erice Lectures [7]). Definitions (ii) agree asymptotically,

for the (i) and

(2.8) F Ok+ 0.

The magnetic field of the monopole is Bi = -FJer*, with total magnetic flux g = - 4x/e, sat’ISf ying the Schwinger condition eg = - 4x. The field equations for @ and a are derived by stationarizing the mass functional (2.4): +--$r*-2a*@=l@(@*-u*)--IS[@],

(2.9)

(2.10)

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133

MONOPOLE

Varying the mass functional after imposing spherical symmetry yields the complete set of radially symmetric field equations because the radially asymmetric perturbations integreate to zero [7,8]. For ,I # 0 we expect the fields (CD-a) and a to fall off exponentially at large distances, in which case the field equations decouple: +--$

r(Q) - a) - 2a%(@ - a) = 0,

1 d2 -ra - u2a = 0. r dr2 The solutions to Eqs. (2.11) and (2.12) satisfying the boundary conditions at infinity are ,-Jnar @-a

=A arc-.

d- 22ur ’ epar ur

(2.13)

Exact solutions for @ and a are known only in the PS limit A--$0. For A= 0, the monopole mass (2.4) can be written as a sum of squares plus a boundary term at infinity : 2

r$--+(I-r’a’)

+ $ [@(l - r’a’)] t r$-+-(I

M, is minimized

-r’a’)

by setting each of the squared terms in the integrand equal to zero:

d@o 1 F=;T-a09

2

(2.16)

where the subscript zero denotes the PS case. The derivation of the first-order equations (2.15) and (2.16) are due to Bogomolny [9]. The first-order equations can be cast in a simpler form by setting Q. = #o - l/r.

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CARLL.GARDNER

Then

da,-dr

(2.17)

-%h~

(2.18) The PS solutions (which satisfy the boundary conditions (2.5) and (2.6)) are @,=acothar-1, Cto=-.

r

(2.19)

a

sinh ar

Notice that in the I = 0 case, the scalar field @,, has a long-range l/r part as r-+ co,

O,-az-$

while a,, g 2ae-ar as r + co. The mass of the PS monopole is M,, = (4rr/e’)M,. The constant a-l is a fundamental length in the monopole problem. From this point onwards, we set a = 1 unless explicitly stated to the contrary. Factors of a can always be put back into equations by dimensional analysis. Thus M, = 1, ,u = @, M,, = 4x/e’ in these units. Note on units.

III.

MONOPOLE

SOLUTION

IN THE FAR REGION

In the far region, (CD- a) can be expanded about the free massive solution Ae- wr/pr.

The field equation (2.9) for @ decouples in the far region. Define a dimensionless variable x = fl ar = fl r, and set @ = 1 + t,u.Equation (2.9) takes the form 2

+&XC-y=1W2+TW3.

1

3

Next expand [IO]: ,/,-A

e-x ; A2fi(X) -eeZX+A X

X

3f2(x)

Xe

-3x

+

... .

(3.2)

‘T HOOFT-POLYAKOV

Substituting obtain

135

MONOPOLE

vF into Eq. (3.1) and equating coefficients of equal powers of e-“, we

(3.3)

with boundary conditions fi (x) -+ 0 and f=(x) + 0 as x + co. The solutions are =f

f,(x)

[e3x Ei(-3x)

- eX Ei(-x)],

in which Ei(-x)

= i.’ dt G

is the exponential integral function. Ei(-x)

Ei(-x)=-G Ei(-x)

co

has the following expansions:

cco (-l)kk! 2 k:O Xk

a2 Xk(-l)k = JJ~+ In x + r kYil

k . k!

’

where 3%r 0.557216 is Euler’s constant. Next we argue that A = -fl to leading order. The first term in the expansion (3.2) represents the free Green’s function solution to the spherically symmetric threedimensional Klein-Gordon equation .

1 d= --rr dr2 ly - 2hy = -47LA 6(r).

(3.5)

To determine the leading dependence of A on 1, we require that the free Green’s function Ae-“/x go over to the short-distance massless solution Q0 - 1 = -l/r in the intermediate region. The constant A, then, must equal -9 to leading order. Since A is a small parameter, QF can be expanded for x < 1 in powers of A:

(ln3+2y,-2)

+O(A-‘). I

(3.6)

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CARL L. GARDNER

-+-<++*+++...

FIG. 1. Graphical solution for v(k).

The expansion of (@ - 1) in powers of AePX can be understood more physically by means of a graphical solution of the differential equation (3.1) with an arbitrary source J(x):

The Fourier transform of the operator x- ‘(d*x/u!x*) - 1 is the inverse scalar propagator with unit mass: -(k* + 1). Fourier transforming Eq. (3.7) yields the integral equation:

1 d3k’ d3k” ---Ty(k-k”)y/(k”-k’)ty(k’) + T I (243 (2n)

I

.

(3-g)

y(k) has the graphical solution shown in Fig. 1 obtained by iteration [y,(k) = -J(k)/(k* + l)]. Now set J(x) = -4rrA6(x)/fl as in Eq. (3.5). In the graphical solution for y/(k), for each X there is a factor of A; for each propagator there is a factor of ePx. Thus the first graph corresponds to AePX/x, the second graph to A2fl(x)e-*“lx, and the third and fourth graphs to A3f2(x)e-3x/~. (The graphical solution to Eq. (3.1) represents the tree-level graphs of the quantum theory, according to the loop expansion. The classical theory is equivalent to the zero-loop or tree graphs in the systematic loop-expansion of the quantum theory in powers of A.) Expressed in terms of the variable r, Eq. (3.6) for QF in the intermediate region takes the form’ O,=l+f

(-&+3f$)+ +J$ln(flr)

[-A+$-(ln3.+Zy,-2)] @A +- 2

r + o(A).

(3.9)

* Letf(d, r) be a function of 1 and r, where r is restricted to an interval I which depends on 1. Then f(1, r) = o(g(A)) means that given E > 0, there exists a 6 > 0 such that for 1 < 6 and for all r E I. AnalogouslyJ(1, C such that

If@, 4 G E I &?@)I r) = 0(&)) means that there exist positive constants 6 and If(k 4 < C I gQ)I

for I < 6 and for all r E I.

‘T HOOFT-POLYAKOV

137

MONOPOLE

The order of the terms neglected in Eq. (3.9) is verified in the Appendix. We draw attention to two intriguing features of Eq. (3.9): First, 6@, = Qp,- Cp, = O(J). Second, @ is a non-analytic function of 1. The non-analyticity in A enters the monopole problem at two points: (1) Outside the core (0 < r 5 M;‘) of the monopole, the monopole acts as a point source of non-analytic strength. (2) Logarithms appear in the intermediate-region solution. Since In must be a function of a dimensionless variable and since the only mass pertaining to Eq. (3.1) is the mass of the scalar particle, the solution must involve lnkr). Note that the ,I In(dr) part of QF is generated only when the scalar field equation has interaction terms (right-hand side of Eq. (3.1)). The expression for GF provides a key to the behavior of the vector field in the far region by means of the field equation (2.10). To leading order in 1, QF is approximated by QF = 1 -e-I/r. Set czFto a function B(x) times its asymptotic form: a .(r) = B(x)e-’ F r

(3.10)

*

Substituting this expression into Eq. (2.10) and neglecting a3 in comparison we obtain a differential equation for B(x):

with a,

1 -e-2X

x2

BI

(3.11)

with the boundary condition B(x) --f const as x + co. Equation (3.11) may be solved perturbatively by expanding B as a series in fl: (3.12)

B=B,[1+~B,+2~B,+(21)3’2B3+-~].

Equating equal powers of @.

determines B,, B, ,... . The first term in the series is B, = C exp{Ei(-x)}.

(3.13)

The constant C is fixed by expanding B, for x < 1: B, = fiCeYEr(l

- @r

+ a..).

Matching aF for x < 1 with a,, implies C=

2 eYE@

(3.14)

to leading order. Thus the asymptotic expression for the vector field is (3.15)

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CARLL.GARDNER

In the Appendix, we prove that the ratio B,+l @)/B,(r) of successive terms in Eq. (3.12) goes to zero as Iz goes to zero. Consequently aF decays exponentially throughout the far region.

IV. MONOPOLE

SOLUTION

IN THE NEAR REGION

The field equations for @ and a are a coupled pair of non-linear second-order differential equations, and exact solutions are known only when the source term AS[ @] is set to zero. The equations, however, can be linearized for L e 1 by expanding the fields @ and a about the PS solutions to order 1: @ = Qo + 6@, (4.1)

a = a0 + f&Y. 6@ and 6a equal zero at r = 0 due to the far-region monopole solution that when we linearize the field equations, tional to (fl)’ = A). The monopole mass can be written

the boundary conditions (2.6). We know from the perturbations ?I@, 6a = O(p). Therefore we must keep third-variation terms (proporin the form

dr r*(@* - l)*.

U[@] =$jm 0

Substituting

the expansions (4.1) into M,, we obtain

Mm=$+fbdr$

+$j”

IP (E)

/*

dr 2r2 {a,(&D)* 6a + Q. &P(&Y)* + ao(Sa)3} 0

@+@=

1 d* ---r++ai r dr*

da0 Q.

d z i fiao

9 +a(-’

da0 a0

@=

+nu[Q.+ 601,

(4.3)

(4.4)

r2 1

(4.5) ’

‘T HOOFT-POLYAKOV

139

MONOPOLE

ld ----y 2

p+ =

r2 dr

(4.6)

!

2ao

“(

2

The operator (4.4) multiplying the perturbations squared factors because the PS solution is self-dual. ct”’ is the Hermitian adjoint of p with respect to a weight r’ when the operators act on the space of functions defined on the interval [0, co] and vanishing at r = 0 and r = co. By stationarizing M, in the form (4.3), we obtain the equations

+ (6a)2(y;)+6@6a (-$I. We break the perturbations

(4.7)

into two parts (4.8 1

where F+LQ[E;]=O, QH=O(\/Si)=a,, and &D,, we rewrite the linearized field equations as

where Z is proportional

da, are proportional

to A. Now

to A: Z, = A@,(@: - 1) + 2@,a$ + 4a,@,,,a,,, C, = 2a,@,

+ 6a,ai

+ 4@,@,af,.

(4.10)

In Eq. (4.10) we keep only the terms in (QH)*, cDHaH, and (a,)’ which are proportional to A. CM, and da, can be calculated in terms of a 2 x 2 Green’s function K(r, r’) for the operator p + 8, P+PK(r,

&r-r’) o

r’) = (

We will choose the boundary

condition Wr,r’)-+O

0 6(r-r’)

1’

(4.11)

at infinity as

r-+03.

(4.12)

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CARL

L. GARDNER

If K(r, r’) is to be finite at the origin, then the proper boundary condition K(r, r’) + 0

as r-,0.

is (4.13)

6@, and aa, are given by (4.14) The upper limit of integration in (4.14) must equal 0((2L)-‘12); (21)-‘j4 is chosen for the sake of concreteness. Why does the region of validity of the expansion (4.1) extend from r= 0 to r = 0((21)-“~), but no further? The key to answering this question lies in realizing that the PS solution describes a massless scalar field (so that we expect (4.1) to break down when f - (21)-‘12), but takes into account the massive nature of the vector field (so that we can extend the region of validity of (4.1) into the intermediate region). As far as the perturbations &B, and da, are concerned, limiting r to the interval [0, R] corresponds to a source Z for r < R, zero for r > R (as can be immediately verified by applying 0’8 for I < R and r > R to Eq. (4.14)). Equation (4.14) implies the boundary conditions at the origin d@,(r = 0) = 0,

(4.15)

6a,(r = 0) = 0, which together with 6@ (r = 0) = 0 = 6a (r = 0) imply @& = 0) = 0,

(4.16)

aH(r = 0) = 0. There were two options for the boundary conditions at r = (21)-‘14. We could have chosen inhomogeneous boundary conditions on K(r, r’) at r = (21)-“4 and set the homogeneous solutions equal to zero, in which case the linearization method breaks down. Instead we have chosen to exchange inhomogeneous boundary conditions on K(r, r’) at r = (21)-‘14 for homogeneous boundary conditions on K(r, r’) at r = co, by adding homogeneous solutions aH, aH of @‘+8’ to a@,, 6ac, respectively. QH and aH will contain two undetermined constants (after the boundary conditions (4.16) have been imposed at the origin), which will be adjusted to satisfy the matching of the near-region and far-region monopole solutions. The factorization of the matrix operator on the left-hand side of Eq. (4.11) as Bf@ provides us with a two-step construction method for K(r, T’): (i) First we construct the 2 x 2 Green’s function G(r, r’) satisfying @G(r, r’) = G(r, r') -+ 0

6(r-r’) o as r-+0

0 6(r - r’) ’ and I+ co.

(4.17)

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141

MONOPOLE

The boundary conditions on G follow from those on K. (ii) Then we construct the 2 x 2 Green’s function .F(r, r’) satisfying

0 cc” +.!qr,r’) = 6(r-r’) o fqr-r’) 1. ( K is given by the convolution

(4.18)

of G and F,

K(r, r’) = IomdpG(r, p) .‘@A I’).

(4.19)

The Green’s functions K, G, and .F obey the following symmetry relationships r’K(r,

since F’+c? is Hermitian

r’) = rr2Kt

(r’, r) = rr2KT(r’,

r)

(4.20)

r)

(4.21)

(with respect to a weight r’), and r2G(r, r’) = r’2.Yt

(r’, r) = r2FT(r’,

since P’ is the Hermitian adjoint of 4. The symmetry relation (4.21) implies the boundary conditions on .Y: F(r,

r’) = finite

as r -+ 0,

(4.22)

as r--t a,

Y(r, r’) = 0

(i) To construct the Green’s function G, we solve Eq. (4.17). G(r, r’) can be expressed in compact factorized form by making the following definitions: A,(r)=#,-rai,

C,(r)

A 2(r) = -rcf, @Jo,

C,(r) = 5, fi D,(r) = rQo,

B,(r)

= ai,

= -$oy

(4.23)

B,(r) = aoiol Then G&r,

r’) = d(r’ - r) A,(r)

Cb(r’)

+ 8(r - r’) B,(r)

Db(r’).

(4.24)

(ii) The Green’s function .FJr, r’) is obtained from the symmetry relation (4.21). As a final note on the construction of the Green’s functions, we draw the reader’s attention to the fact that the functions of r appearing in Eqs. (4.23) represent the tirst-

CARLL.GARDNER

142

order perturbations of the PS solutions with respect to translations and scale transformations. Think of do = &(a, r) and a, = a&, r). Then

840 2 -=-ao, g=/,-ra;, ar aa,-- --ao4op aa,-- --Tao Q. ar aa (where after differentiating we have reset a = 1). The last stage in the development of the near-region monopole solution is to construct the homogeneous solutions QH and a,,. After imposing the boundary conditions (4.16), there remain two as-yet undetermined constants GZand 9 in QH, aH. 0’ and 9 (which are O(G)) will be determined by matching the near-region and far-region monopole solutions in the intermediate region. The homogeneous solutions are

The reader may verify these solutions by applying F+b

V.MATCHING

to Eq. (4.25).

OF THE NEAR-REGION AND FAR-REGION MONOPOLE SOLUTIONS

The near-region and far-region monopole solutions are valid over an infinite region of overlap as il goes to zero. This intermediate region provides the setting in which the long-distance and short-distance behaviors of the theory are coupled together. By expanding the near-region monopole solution for (2L) - “’ < r < (2L))I’“, we may match the near-region and far-region solutions in the intermediate region. First we observe that the constant 9 equals zero. To see this, note that QH has a term proportional to (see Eq. (4.26)) 21

3epr

r

I (ZA)-118

$cosh2tZ~e’j0 t

(21)-l/S-r

dt (t 1 r)”

Since the vector field decays exponentially in the intermediate region by Section III and the Appendix, 9 must equal zero. Next we calculate Qp, = Cp, + &D, + CD, to order A. In the intermediate-region,

@o = 1- $ + o(l), cBp, = -a + o(A).

‘T HOOFT-POLYAKOV

I43

MONOPOLE

&D, is given by Eq. (4.14) as

&Do=-

(*A)-“4dr’ K,,(r, r’) Z,(r’) - 1;(211-”dr’ K,,(r, r’) L,(r’). I0

(5.1)

The source C simplifies for r > (2k) -I”: Z,(r) = A@,,(@ - 1) + o(e-2’2~k’m1’8), Z,(r) = o(e-‘2.3’m”*). Evaluating

the integrals in Eq. (5.1), we find 6@, = A[-r + 3 ln((2A)“4r)

Therefore QN in the intermediate

+ 2(2A)-‘I”

- 31 + o(A).

(5.2)

region equals

QN = Q. + A[-r + 3 ln((2A)“4r)

+ 2(2A))‘14 - 31 - l;r + o(A).

(5.3)

Matching the intermediate-region expressions for the scalar field in Eqs. (5.3) and (3.9) determines the constants A and CPI:

+&J!g

1 + o(A3’*),

C7= -+?I-

31 In @I

2 FIG.

595/146/1LlO

2.

Graph

(5.4)

+ (2A)3’4 - 3A ln(3(2A))“4)

- 31y, + o(A).

I

I

I

I

1

I

I

I

1

3

4

5

6

7

8

9

IO

II

I2

curve

is @, .

of Q,.(r)

compared

to G,,(r)

for

A = 0.005.

The

upper

144

CARLL.GARDNER

J

12

FIG. 3. Graph of K+(T) compared to

m,,(r)

for I. = 0.005. ‘The lower curve is

The final expression, then, for the intermediate-region @, = o0 + A[-r + 3 ln(j/%r)] a, = a, - 2 $iI(re-’

+ fi

ra,

monopole solution is

+ 31(ln 3 + YE- 1) + o(A),

- eAr) + o(fl)ewr.

P-5)

Note that while 60,) 6aG, and Gl?depend on the “cutoff’ (212)-“4, the intermediateregion solution is independent of the cutoff. The far-region monopole solution for I = 0.005 is plotted in Figs. 2 and 3. To interpret the graphs, note that ] @(21= (l/e)@ and ]A,i(2 = (2/e*)(a - l/r)*. VI.

MASS

OF THE

MONOPOLE

NEAR THE PS LIMIT

Even at the classical level, the monopole mass is a non-analytic function of A involving 4 and A In Iz. In this section, we evaluate the corrections to order 1 to the PS monopole mass M,, = (4n/e*)M,. The calculation of the monopole mass may be simplified [ 1 l] by differentiating M, with respect to A:

(6.1)

'T HOOFT-POLYAKOV

145

MONOPOLE

The integral

vanishes since @ and a are classical solutions and since the boundary conditions (2.5) and (2.6) are independent of A. Note that (dMJdA)A=o diverges; this reflects the fact that U[ Qo] = co. The evaluation of dM,,,/dl proceeds in two steps: the contribution from the near region and from the far region. The integral from zero to infinity will be split at r0 = (21) - ‘Ia.

In the far region, approximate

Then the contribution

@ by

to dM,,,/dA from the far region is

(%)F+{r;drr2(02-

1)2 (6.2)

+4yE+41n3xo+2

I

to(l),

where x0 = @r,. In the near region, it is necessary only to approximate Q, by Q>, to obtain M, to order 1. The contribution to dM,,,/dA from the near region is

(6.3)

+4r,-41

n r. - 0.96671 to(l).

Combining the contributions to dM,,,/dA from the near and far regions and integrating with respect to A, we obtain for the monopole mass M,=*M,

e2

1 P2 -In +TM:,

--+0.7071-$ ’ M,

W

I

.

(6.4)

ACKNOWLEDGMENTS I would like to express my appreciation to Alan H. Guth and Nick S. Manton for discussing this problem with me and offering many useful suggestions.

146

CARL

L. GARDNER APPENDIX

We prove that the terms which have been neglected in the far-region monopole solution are higher order in ,I by analyzing the long-distance and short-distance behaviors of the solutions to Eqs. (3.1) and (3.11). 1. The Scalar Field

The term in OF. in Eq. (3.2) proportional d2f - 2(n + 1) 2 ----!L dx2

to A”+’

obeys the differential equation

+ n(n + 2)f, = S”(X),

.fxfk -g--r

(‘41)

where f&r) = 1 and i, j, k = 0, 1, 2 ,.... f,(x) is given by f,(x) = - fj:

dx’ emnX’S,(xr) + $?I:

dx’ e-(“+2)“‘Sn(Xt)e

642)

As x + co, 1f, I< C,/x” where C, is a positive constant. This is proved by induction, using the fact that

Thus as x--, co, the leading contribution

to vF. from the (n + 1)th term in Eq. (3.2) is e-(n+l)x e-‘“+l’ Jzr x”+l N ptl .

API+1

As x + 0, rewrite f,(x) f,(x) = +enX

as

/[e2X - 11 jX &’ co

[eecn+*jx’ - epnx’] S,(x’)

+ [e2’ - l] lx dx’ epnx’ S,(x’)

Cc

+ lx dx’[e-(“’

00

‘jX’ - e-nx’ ] S,(x’)

Then a proof by induction establishes the short-distance behavior of f,(x): I.L&>l

< -CL

Ifin+ Lx)1 Q -CL+

In” x2 1 In” x9

I.

(A3)

‘T HOOFT-POLYAKOV

147

MONOPOLE

where C,! is a positive constant. As x + 0, the leading contribution (2n)th term in Eq. (3.2) equals c

In”-‘x

- (fi)*“-’

to wI; from the

ln’-‘(&r-)/r

and from the (2n + 1)th term equals A *n+ I

-

In”-‘x

- (\/2;1)*” In”-‘(@r)/r.

X

2. The Vector Field

The n th term in the expansion (3.12) is determined by equating equal powers of in Eq. (3.11):

fl

--x B,(x)=

e

-1 2x

B,(x) = f

$

3

+ Ei(-x)

+

3(e-*” - 2e-“) 2x2

[ Bn(x)=+,”

2e-’

dt [*+?+ a2

(A4)

9

I

dB

1 -e-l t2

(n > 3).

As x-+ co, B,- 1,3 B, -x-r, B, -xPn (n> 2). Thus the nth term in (3.12)~ (2jl)“‘2/~” = l/r” as x -+ co. As x + 0, B, -x, B, - 1, B, - In x, and B, -x-I (n > 3). Thus the leading contribution from each term in (3.12) as x --) 0 has the following behavior: B, - @r, flBOB, - 2lr, 2LB,B, - (21)3’2r ln( fir), (2,l)“‘*B,B,

- (2,)“’

(n > 3).

Thus the ratio B,+,(r)/B,( goes to zero.

r ) o f successive terms in Eq. (3.12) goes to zero as /I

REFERENCES

I. 2. 3. 4. 5. 6. 7. 8.

G. ‘T HOOFT, Nucl, Phys. B 19 (1974), 276; A. M. POLYAKOV, JETP Lett. 20 (1974). 194. H. GEORGI AND S. L. GLASHOW, Phys. Rev. Lett. 28 (1972), 1494. M. K. PRASAD AND C. M. SOMMERFIELD, Phys. Rev. Lett. 35 (1975). 760. For the full propagation functions in the field of the PS monopole, see P. ROSSI. Nucl. Phys. B 149 (1979), 170. For a review, see P. GODDARD AND D. I. OLIVE. Rep. Prog. Phys. 41 (1978), 1357. E. B. BOGOMOLNY. Soviet J. Nucl. Phys. 24 (1976), 449; L. D. FADDEEV. Lett. Math. Phys. I (1976) 289. S. COLEMAN, Classical Lumps and Their Quantum Descendants, in “New Phenomena in Subnuclear Physics” (A. Zichichi, Ed.), Plenum, New York, 1977. L. D. FADDEEV, Proc. Int. Symp., Alushta, 1976.

’ In this section, the constant C in Eq. (3.13) is set equal to one.

148

CARL L. GARDNER

9. E. B. BOOOMOLNY, Ref. [6]. 10. N. S. MANTON, Nucl. Phys. B 150 (1979), 397. 11. T. W. KIRKMAN AND C. K. ZACHOS, Phys. Rev. D 24 (1981), 999. Kirkman and Zachos indepen dently obtained the leading correction to M,, and the scalar field solution.