Interacting monopoles

Volume 85B, number 4

PHYSICS LETTERS

27 August 1979

INTERACTING MONOPOLES W. NAHM Department of Applied Mathematics and Theoretical Physics 1, University of Cambridge, England Received 7 June 1979

The Bogomolny equation for interacting monopoles of like charge in the case of vanishing Higgs mass is fulfilled by an asymptotic expansion in powers of the inverse monopole distance. Consequently, the monopoles exert no long range forces on each other. However, the expansion is not summable. The degrees of freedom of the multi-monopole system are discussed.

Consider SU(2) monopoles in a classical nonabelian gauge theory with a massless Higgs field ~bin the adjoint representation, such that we can treat it as an additional component of the potential A. The gauge group has been fixed for simplicity, it is easy to generalize our results. After quantization, monopoles are supposed to yield the basic field of the dual sector of the theory, for which a perturbation expansion in terms of the inverse coupling constant e --1 should exist. The behaviour of the classical monopoles can yield important information for the couplings in that sector [1]. Montonen and Olive conjectured that the dual theory looks like the original one, with a new dual gauge group, and monopoles of spin one [2]. However, this idea only seems to be realizable in a slightly modified way in case of the 0(4) supersymmetric model [3]. In particular, Mottola has argued that monopoles have spin zero, as classical one-monopole solutions only have the positional degrees of freedom [4]. His proof of the latter assertion is not quite complete, furthermore additional degrees of freedom might show up for multi-monopole systems, as in the case of instantons. Nevertheless, we shall see below that his conclusions are correct. However, the duality between a theory of scalar monopoles and a conventional Yang-Mills theory would be even more interesting than the duality between two Yang-Mills 1 On leave from CERN, Theory Division, Geneva.

theories, as it would yield a dynamical, non-perturbative way of creating gauge vector bosons out of scalars. The interaction energy of monopoles of positive magnetic charge, Eint = l'-~--f (Fij +eiikDk~) 2 d g , 4e 2

(1)

can be estimated by finding approximate solutions to the Bogomolny equation

Fi/+eilkDk~= O.

(2)

In ref. [5] I have constructed an approximate solution of eq. (2) for which Eint behaves essentially like s - 4 , where s is the minimal distance between the monopole positions rq, q = 1, ..., N. Here I shall show that one can f'md potentials for which Eint goes down like an arbitrarily high power of s -1 , such that there are no long range forces between the monopoles. We use the quaternionic notation A = - i o i A i + dp.

(3)

For each monopole we shall introduce everywhere regular potentials A (q), q = 1 ..... N. Furthermore we shall use an abelian potential A (0) which is singular at the monopole positions and describes the asymptotic behaviour of the system. All the A (r), r = 0 ..... N will satisfy eq. (2). We shall write them in the form of asymptotic expansions in powers of s -1 . If one breaks off at a given or373

der, A(°) - A(q) will decrease like the corresponding power of s -1 if restricted to the region s/3 < [r - rql < 2s/3. Using a smooth interpolation like in ref. [5] one obtains a potential which fulfils eq. (2) up to a correction term which decreases like that power. For all A (r) we fix the direction of the Higgs field in isospace by

r-rq ¢ = 4/141 I r - rq[

27 August 1979

PHYSICS LETTERS

Volume 85B, n u m b e r 4

The bqj also are power series in s - 1 . They are determined to order s - n - 3 , once the aqj are known to order s-n. Now let us define the A(q), q = 1..... N. We require AA(q) - II(q) 4 =

O(Ir-rql-3),

(11)

where near r = rq ,

(ai4)a i G Ir- rq1-1 = 0 . q

(4)

£~A(q) = A(q)- A(q)

(5)

is the deviation from the unperturbed spherically symmetric monopole solution A(q). Together with the background gauge, eq. (2) can be written in the form

This introduces discontinuities of ¢ on some surfaces, which, however, are only gauge artefacts. Take

DAA(q) = ½AA(q) + X AA(q),

A (0) =ioi4 X 3i4

with

(12)

(13)

n

+¢(c-q~l (lr-rql-l+A(q)) ) ,

(6)

\~1.

(14)

where D denotes the covariant derivative in the potential .4(q), and D 0 acts like

where c is the asymptotic value of 141 and A(q) de, scribes the monopole polarization: A (q)= = a q j [ r - r q l - J - 1 Y j

= -ioiD i - 5 ~ ,

riO× = 8(q) X X. (7)

The Yj are the usual spherical harmonics. For simplicity the sum over J3 has been suppressed. The 2 X 2 matrices aqj will be asymptotic expansions in s- 1 without s-independent terms. To each order in s-1 only a finite number of multipoles will contribute. We supplement eq. (2) by the Coulomb gauge condition such that we can write

(lS)

Note that for an abelian field the background gauge reduces to the Coulomb gauge and eq. (13) to eq. (8). With eq. (8) and

AA(q) = ~ s-nAA(q, n) ,

(16)

n=l

I](q) = ~ s-nII(q, n) , n=l

(17)

eq. (13) takes the form

ioiOi ~ A ( q ) = o .

(8)

q

Let II(q) be the Taylor expansions of those terms of A(O) which stay regular at rq: II(q) = - ~

(r - ra)a(t~[)-I

~([r-rml-l+A(m))~r=rq m vaq

,

(9)

where a = (al, a2, % ) is a multi-index. Because of eq. (8) we can write II(q)= i7~o = bqflr-rql s Y , [ ~r)-. rq \

374

n- 1 =1 ~

(18) a a t q , . ) + x a a ( q , . - m ) - Htq,")~4.

2m=l The multipole moments obtained from this equation can be fed back to A(0) putting

"=0

X[~}

£)(AA (q,n) _ l](q,n)4)

(10)

AA(q) - II(q)~b+ A(q)4 ~ e x p ( - c l r - rql),

(19)

order by order. To prove the absence of long range forces, we have to show that the system of equations (9), (11), (13), (19) can be solved iteratively. In fact, all we have to do is to show that eq. (18) with boundary condition (11) has a solution of type (19) with a A(q) of type (7).

Volume 85B, number 4

PHYSICS LETTERS

Note that the last term on the right-hand side of eq. (18) behaves like exp(-clr - rq I). We shall see that to each order in the iteration this is even true for the complete right-hand side. On functions of this type the operator

=-97

-

(20)

has a well-defined inverse which is even known explicitly [6]. Thus we obtain

27 August 1979

For infinitesimal X one has to each order DX - ½(A,4+ X X + X + X ALl) = 0 ,

(25)

where again the index q has been suppressed. The four linearly independent solutions of eq. (23) are the translation modes of the qth monopole, plus the gauge transformation X = D+~ .

(26)

n-1 (21) × ( l~ ~ AA(n)+ X ,~,4(n-m)-II(n)D4~) m=l We have suppressed the index q for simplicity.'Jackiw and Rebbi have analysed the equation

Thus for a system o f N monopoles we obtain 4N degrees of freedom. The 3N translation degrees are easily understood, but the remaining N degrees pose a problem. To discuss them note that the traceless part of eq. (25), which yields eq. (2), coincides with the traceless part of

DX = Y

DX = 0,

LX/I(n) = [l(n)cb + ~ + ( ~ + ) - 1

(22)

in detail [7], using an angular momentum analysis. This is particularly useful as to each order in the iteration only a finite number of multipoles contribute. For the component of X parallel to ~ they found the behaviour of eqs. (7), (10), (19), whereas the components perpendicular to ~ indeed decrease exponentially. This also proves the assertion about the right-hand side of eq. (18). The asymptotic behaviour is easy to understand, as ~2 acts like a mass term c 2 on the components perpendicular to ¢, whereas it vanishes on the parallel component. As the image o f D + is perpendicular to the zero modes of D, one really obtains no J = 1 term in eq. (7). This completes the construction of the A (r), and proves the absence of long range forces. The procedure described above allows an easy identification of the degrees of freedom of the multimonopole system. In fact the only freedom we have is to change the ALl(q,t) to AA(q, 1) + x ( q , 1 ) w h e r e

(27)

where D is the covariant derivative in the new potential A = A + Z£4. The trace of eq. (25) is the background gauge condition in terms of.4, that of eq. (27) is the corresponding condition in terms of A. Thus to each order the equations are gauge equivalent, and we need only consider the simpler eq. (27). Now, because of eq. (20) adapted to D, the ansatz x = D+z

(28)

factors out the sigma matrices in eq. (27). This ansatz is consistently reproduced by the iteration. In fact, with a = ioi0i/2,

(29)

II = itriai~, ,

(30)

one can rewrite the iteration equations in terms of Z, p, ~ alone, without any sigma matrix appearing. In particular the ansatz no

hx(q, 1) = 0 ,

(23)

for each q. The higher A,4(q, n) will change accordingly, but the whole procedure goes through. Zero modes of D can also be introduced at higher orders in the iteration, but it is equivalent to introducing them all at once at the first step. We denote the total change of AA(q) by s - n X ( q , n) = x ( q ) . n=l

(24)

z(q) = ;kq~(q) + D

m=2

Z (q,2) ,

Z(0) = ¢ ~ Caql~(q)l - ta(q)), q

(31)

(32)

suggested by eq. (26) yields a scalar Z and consequently a pure gauge X to all orders in the iteration. If the asymptotic series would converge or be summable e.g. by a Borel procedure, this would yield N solutions of the equation 375

Volume 85B, number 4

DD+Z= 0 ,

PHYSICS LETTERS (33)

with normalizable D+Z. Of course, Z cannot be normalizable itself, nor can the integral of ZD+Z over the surface at infinity be zero. Asymptotically, Z must be proportional to ¢, as otherwise D¢Z would have the same properties with respect to normalization as Z. Thus eq. (33) reduces asymptotically to the Laplace equation, and one sees easily that the only possible solution is Z = ~.

(34)

Thus the asymptotic series cannot be summable. This also applies to the translational modes, as they can be obtained from the scalar solutions of eq. (27) by right multiplication with sigma matrices. If one can find exact multimonopole solutions to eq. (2) it will be interesting to compare this behaviour o f the perturbation expansion of renormalizable field theories, where one has similar problems with summability. Finally note that the number of linearly independent normalizable solutions of eq. (27) is just the index of D, as D + has no normalizable zero modes. However, D is not a Fredholm operator, because

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27 August 1979

our theory has massless excitations. If one introduces a small mass term the usual theory indeed yields an index 4N (see refs. [8,9]). Jackiw and Rebbi have been able to calculate the index of D for a single monopole, comparing the behaviour of the solutions of fixed angular momentum at zero and infinity [7]. They found that a mass term does not matter. Unfortunately, we cannot extend this result to N > 1 by our perturbative approach. I t is a pleasure to thank the staff of the DAMPT, in particular Professor Polkinghorne, for their hospitality.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

N.S. Manton, Nuel. Phys. B126 (1977) 525. C. Montonen and D. Olive, Phys. Lett. 72B (1977) 117. H. Osborn, DAMPT 79/4. E. Mottola, Phys. Lett. 79B (1978) 242. W. Nahm, Phys. Lett. 79B (1978) 426. S.L. Adler, Classical quark statics (Princeton, 1978). R. Jackiw and C. Rebbi, Phys. Rev. D23 (1976) 3398. C.J. Callias, Commun. Math. Phys. 62 (1978) 213. R. Bott and R. Seeley, Commun. Math. Phys. (1978) 235.