IL NUOV0 CIMENT0 VOL. 77 A, N. 3 1 0 t t o b r e 1983

Vacuum Effects on the Static Monopole-Antimonopole

Interaction.

~ . JENGO

In t e rna t iona l School /or Advanced S tud ies - Trieste, I t a l y In te rna t iona l Centre ]or Theoretical Phys i c s - Trieste, I t a l y I s t i tu to Naz iona le d i _Fisiea 1Vueleave - Sezione di Trieste, I t a l i a

M . T . VALLON (*)

In terna t iona l School ]or Advanced S tud ies - Trieste, I t a l y I s t i tu to ~Yazionale d i E i s i ea Nuc lea te - Sezione d i Trieste, I t a l i a

(ricevuto il 18 Aprile 1983)

Snmmary. - - We analyse a static monopole-antimonopole pair interacting with all the loops of charged particles which fill the vacuum. A loop path integral formalism is used and the integrat ion is performed by means of an average description in which circular loops of radius e, weighted with a density function D(~), are taken into account. We determine the behaviour of D(Q) both when ~ -* 0 and when Q -* oo. At the Q --* oo limit we can distinguish between the two eases of noncondensation or con- densation of the loops of electrically charged particles. In the latter case we obtain the confinement of the monopole static pair. The ~ -* 0 l imit gives information about the vacuum polarization divergent contribution. This contribution is shown to be different in sign, with respect to the one of a static electrically charged pair.

PACS. 11.15. - Gauge field theories.

1 . - I n t r o d u c t i o n .

Magnet ic monopoles seem n o t to be fit to a p e r t u r b a t i v e t r e a t m e n t . I n

r ecen t years va r ious vers ions of t he q u a n t u m field t h e o r y wi th po in t l i ke mo-

(*) Address after 1 November 1982: Chalmers Ins t i tu te of Theoretical Physics, G5teborg, Sweden.

17 - II Nuovo Cimento A. 249

9'50 R. JENGO and M. T. VALLON

nopoles (1) and charges have been proposed (%8). These versions can always be writ ten in a loop-path integral formalism. In all these approaches, the perturbative expansion turns out to be ill defined for the appearance of terms tha t break the Lorentz invariance (3) or of arbi t rary 2~ni terms (3) in the inter- action. The theory as a whole, however, is well defined, it maintains Lorentz invarianee, and steps have been done towards a systematic proof of its regular- izability and renormalizability (4.7). I t is, therefore, possible to use these for- mulations for some nonperturbat ive computations. The pa th integration for- malism is suitable to describe a quantum field theory of scalar bosons and we will consider this ease.

Here we have chosen to deal with the particular problem in which the ex- ternal sources are static. We consider a configuration of a static monopole- antimonopole pair as a given (( Wilson ~) loop (3). We will discuss the con- t r ibution of the vir tual loops of the charged particles to the monopole-anti- monopole static interaction, tha t is we will evaluate the expectation value of the Wilson loop interacting with all the loops of charged particles of the theory. We do not discuss the effect of the virtual monopole loops on the monopole- antimonopole interaction, since it corresponds to the usual vacuum polarization effect in QED.

For a static monopole-antimonopole pair, the monopole-eharge interaction can be writ ten as the sum of the difference of the two solid angles under which the surfaces defined by the loops of charges in 3 dimensions are seen by the monopole and the antimonopole t ha t consti tute the static external loop. F rom the dual symmet ry of the theory, the same formulae will describe also the case in which the external loop is made by a static charged particle-antiparticle pair, interacting with all the monopole loops.

The derivation of the explicit form of the interaction is given in sect. 2. The rest of the paper is devoted to an analysis of the behaviour of the Wilson loop in the possible phases of the theory. In particular, we discuss in sect. 4 the well-known result of the area law behaviour (8,9) typical of the confinement of the Wilson loop when the loops of charges, which consti tute the medium in which the Wilson loop is embedded, condense. This effect has been already

(1) P . A . M . DIRAC: Prov. R. Soc. T~ondon Set. A, 133, 60 (1931); Phys. t~ev., 74, 817 (1948). (3) R. BRANDT, F. NERI and D. ZWANZIG~.R: Phys. Rev. Lett., 40, 147 (1978); Phys. t~ev. D, 19, 1153 (1979). (3) G. CALUCCI, R. JENGO and M. T. VALLON: Nucl. Phys. B, 197, 93 (1982); S.I.S.S.A. preprint (1982) to appear in .LYucl. Phys. B. (4) R. B~ANDT and F. NERI: Phys. 2ev. D, 18, 2080 (1978). (s) C. PANAGIOTAKOPULOS: 2Vucl. Phys. B, 198, 303 (1982). (e) W. D~.ANS: ~Vucl. Phys. B, 197, 307 (1982). (7) G. CXLUCCI and R. JENGO: S.I.S.S.A. preprint 72/82E.P. (s) K. G. WILSON: Phys..t~ev. D, 1@, 2445 (1974). (9) A .M. POLYAKOV: Nuct. Phys. B, 120, 429 (1977).

VACUUM EFFECTS ON THE STATIC MONOPOLE-ANTIMONOPOLE INTERACTION 251

studied by NIELSE~ and OLESEN (lo) and b y NA~cBu (11), bu t our approach ,

which is described main ly in sect. 3, is ra ther different. We write the charge- monopole in teract ion in the formal ism of ref. (a) and we in t roduce an average description of the f luctuat ing loops, replacing the p a t h in tegra l with an in-

tegra t ion over the configurations of circular loops of radius ~ (13). This integ-

ra t ion is weighted with a densi ty funct ion D(Q), which we are able to deter- mine a t the l imit for ~ -> 0 and for Q -+ pp.

The occurrence or nonoccurrence of the condensat ion phenomenon deter- mines the behaviour oi D(Q) for ~ -+ ~ . I t is interest ing to see t ha t we obtain

for the str ing tension the propor t iona l i ty to the square of the v a c u u m expecta- t ion value of the Higgs field, wi thout a charge dependence, as in the other approaches (10.11).

The ~ -+ 0 l imit also turns out to be interest ing; in fact , h 'om this region

we curt get informat ion on the divergences of the v a c u u m polarizat ion graph,

which occur when the loop shrinks to a point . This is done in sect. 5. The divergences are discussed and compared to the ones coming f rom the usual

charge-charge interact ion. We conclude t ha t the contr ibut ion to the divergent pa r t of the v a c u u m polar izat ion graph due to a monopole loop will a lways

be of opposite sign, wi th respect to the one coming f rom a loop of charged particles. I f we look a t the consequences t h a t such an effect can have on the renormal izat ion of the electric and the magnet ic charges, we can argue t ha t the polar izat ion of the v a c u u m due to the electric loops, which gives a screen effect

to the electric charge (% ~ e~), will be the c~use of an ant iscreen in the mono- pole case, t h a t is g~ > g~ (7).

2. - T h e W i l s o n loop.

Using the formal ism of ref. (a) we can describe Wilson loops bo th of charges and of monopoles. Since the theory exhibits a dual s y m m e t r y , i t is easy to pass f rom one formulat ion to the other. We will consider a Wilson loop of monopoles which will be confined when the loops of charges of the theory condense, t ha t is in a superconduct ing medium. The same results apply, how-

ever, to the case of confinement of electric charges due to condensed monopole loops.

I n the following, we will specialize the Wilson loop to represent a s tat ic

si tuation, name ly a stat ic par t ic le at space posit ion zl and an ant ipar t ic le a t 53.

A Wilson loop of monopoles, t ha t is the expecta t ion value of an external

monopole loop in terac t ing with all the loops of charges of the theory, can be

(lo) H. B. N~ELSEN and P. 0LESEN: •ucl. Phys. B, 61, 45 (1973). (11) y . NAM~U: Phys. l~ev. D, 10, 4262 (1974). (13) E. GAvA, R. JENGO and C. 0ME]CO: Phys. Lett. B, 97, 410 (1980).

2 5 2 R. ZERO0 a n d ~ . T. v i L n o ~

wri t ten in this way (~):

where ~** is a surface having the external monopole loop as boundary a n d / ~ " is the dual of the field generated by all the loops of charges:

= eu. o A ' ( x ) = - e..aoea f O(x - z) j'("' ( z )dz ,

where [] G ( x - - z) -~ O ( x - z). The sum is extended to all the currents j("~ defined as

(2.2) Jr, = vJCl~, ds (~(x-- q) / . (n}

and /~("~ is the t ra jec tory of the n- th charged particle. In order to evaluate expression (2.1), i t turns out convenient to rewrite the interact ion exchanging the role of the monopoles with the one of the charges. The expression t h a t we obtain in this way can be shown (3) to be equivalent to the previous one when the Dirae quantizat ion condition (1) is satisfied. The Wilson loop can, therefore,

be rewri t ten as

(2 .3) e x p ,

2:( a )

where the sum runs now over the surfaces X (") having the charge loops U ") as boundaries and the field M~: t is generated by the external monopole loop.

In the case we consider as the external monopole loop a stat ic monopole- ant imonopole pair, the magnetic current will be

(2.4) Ku(x) = gOu0[O(x- ~) -- O ( x - g~)].

The field Mu~ generated by this pair is

Me, = 8u W~ - - 8~ Wu (2.5)

with [ ,

W~(x) ~- - - j G ( x - - y)K~(y) d y ,

and we can rewrite the interact ion of expression (2.3) as

2 . ~(a ) 2:(n)

V A C U U M E F F E C T S 0 2 T H E S T A T I C M O N O P O L E - A N T I M O N O P O L E I N T E R A C T I O N 253

Since

fdzoG(X-- z) = G~3)(~ - ~)~

we obtain tha t

I ieg~ f f n ) - emla G ( 3 ) ( 5 _ : dg~(x) ~ zl) -- (51 +-~ z2),

where ~ ) are the spatial projections of the surfaces and the Lat in indices run f rom 1 to 3.

The charge product eg is fixed by the Dirao condition (1) to be an integer multiple of 23. We take i t equal to 23 for simplicity, and the interact ion be-

comes

~ d

where ~ is a unit vector normal to the surfaces. Taking into account that , by definition of (/c3~

= x

volume of all space total closed surface

we can now write the interact ion I in its final form:

(2.6) i

tha t is as a difference of the solid angles under which the surfaces X~n~ are seen from the points 51 and 52. Note tha t , as usually in the monopole theory, the charge-monopole interact ion is detgrmined up to an i r re levant i23n.

The nex t step will be the evaluat ion of the Wilson loop (2.3), tha t is

(2.7) W• --- (exp [I]~

in the various phases of the theory.

3. - The density function D(~) .

The problem of describing the mechanism of condensation in a loop for- malism is not completely solved. We know th a t loops condense when the en t ropy is larger than the self-energy (13,~4) and tha t , as a consequence, the

(la) M. STONE and P. THOMAS: :Phys. Bey. Lett., 4], 351 (1978). (14) T. BA~xS, J. KOGUT and R. MEYERSO~: iVucl. Phys. B, 129, 493 (1977).

254 R. JENGO and M. T. VALLON

contr ibut ion coming from large loops will dominate in this phase of the theory. To analyse this phenomenon we will be, therefore, interested in the be-

haviour of the theory at the limit for large loops. Other interest ing features of the theory will come from the other limit,

t h a t is when the loops are small and divergent vacuum polarization effects are generated.

We will s tudy both limits by following an average description of the fluc- tua t ing loops. The quant i ty (2.7) can be wri t ten as a pa th integrat ion over the f luctuat ing loops (see ref. (8)):

(3.1)

where

,1)

) 1~=)= ds ~('~)(s) + ~-0('~)(s) A~(q) o

and I is given by (2.6). We will replace the pa th integral with an integrat ion over the configurations of noninteract ing circular loops of radius ~ (~2). We have seen in sect. 2 t ha t only the space projection of the loops appears in the interact ion I . We will, therefore, consider circular trajectories in the 3-space.

The configurations ure specified by the positions of the centre, the orientu-

t ion and the radius. The integrat ion over Q is weighted with a funct ion D(~) (density) in which we have t ransferred all the information on the degrees of f reedom tha t we have not explicit ly t aken into account and also on the effect of the charge-charge interact ion among loops. The Wilson loop (2.7), then, turns out to be expressed as

(3.2) W L --~ ~=1 ~ exp [I] _ _

l = l * =

4 ~ (1 - exp [I])]

Here T represents the to ta l length of the four th dimension (of course T -+ c~)

which factors out in our static problem. The rest of this section will be devoted to the p~oblem of determining the

behaviour of the unknown function D(Q) at the two crucial limits ~ - ~ c~

and Q -> 0. I t has been shown (13.15) tha t it is possible to write a theory of in teract ing

oriented loops as a s tandard functional integrat ion over a complex tIiggs field.

(15) S. SAMUEL: Nucl. Phys. B, 154, 62 (1979).

VACUUM EFFECTS ON TH]~ STATIC MOIqOPOL]~-ANTIMOI~OPOLE INTERACTION 2 ~ 5

We will then get informat ion on D(~) by comparing our t r ea tmen t with the equivalent t t iggs field theory in part icular ly simple cases. For example, we could consider a small, constant , external field k coupled to the loops, i.e. a t e rm of the t ype

d~_ (3.3) <exp [if as

where ~ is the vector potential , t ha t is V/~ ~ ---- k. The same si tuat ion can be described in the convent ional way, introducing a funct ional integrat ion

over a complex field T (15). Indeed we can write

d~_

In the formalism on the r.h.s, of relat ion (3.4) the condensed phase is asso- ciated to <~o> # 0.

I f we compute in this formalism the second derivat ive in k for k = 0, we obtain a t e rm of the type

(3.5) 2<q)>*fd'xx~

(having taken k in the x3-direction, i.e. • ----- kXl in the x~-direetion). At the limit in which the length scale L of the dimensions of the 3-space

goes to infinity, this t e rm diverges like TL 5 in the condensed phase, when <~> r 0.

I t is possible to perform the same analysis on the t e rm (3.3). Since

(exp [ifds ~s'~]} ~- exp [-- Tf dr D(Q)fdY'l fd c~ [i(k eosOz~2)]]]

(k cos 0 is the component of k orthogonal to the area of the loop) and the sine par t of the interact ion does not contr ibute, we obtain the expression

Therefore, f rom the evaluat ion of the second der ivat ive in k when b = 0 we get

(3.6)

J5

- - ~ ~La QD(~)~. o

2 5 6 ~ . J'F, NGO a n d ~ . m. VALLO~

Fr om the comparison of expressions (3.5) and (3.6) we have that , in order to reproduce with the loop formalism the divergent behaviour of (3.5) when Z --> c~ and in the condensed phase ((~} ~ 0), we must assume that , at large Q,

D(~) goes in this way:

eo (3.7) D(~)o_~ ~8 ((~0} r O case) ,

where Co is a dimensional constant proport ional to ( ~ . On the other hand, in the case (~) ---- 0 all the derivatives with respect

to /~ of eq. (3.4) behave like TL 3. This indicates an exponential damping of D(Q) for ~ --~ 0. We will argue more explicit ly on i t below.

We can now determine the behaviour of the densi ty also at the other limit, i.e. for Q -> O, which is not re levant in the s tudy of the condensed phase.

We s tar t by observing tha t in relat ion (3.1) the integrat ion variable S has the dimensions of an area. In fact, i t turns out to be related to the average area

of the loops. Here the analysis will be performed in the four-dimensional Eucl idean

space, t ha t is we will analyse the behaviour of a densi ty D(9), in which refers to loops in the 4-space, while in our interact ion (7.6) only the 3-space projections of the loops are involved. Since the average area of a circular loop in four dimensions will be of the same order of its three-dimensional projection, we expect D(9) to have the same behaviour.

To show the correspondence between the variable S and the average area of the loops, we will again consider the constant-field ease. We have for the

potent ia l

(3.8) A.(q)-~A.(qo)-~(q-- qo)~(~A.(q))q=~o+...,

where we will define qo in this way:

B

(3.9) q = qo -F Q, fdQ = O . 0

I f we stop the series expansion (3.8) at the t e rm in Q, the par t i t ion funct ion

of the problem will be (in the Eucl idean space)

~ 1 ~q(o m ~Q(m) xp[__/~SC~,.)]dS(~)

Q(o)=o Q(~):O

,~(m)

r i" 4) ~<~'> �9 e x p L - j - 7

o

/ ~ being ~ A ~ - ~ A . .

(m) (m) t~ /

V A C U U M E F F E C T S O1~ T H E S T A T I C ~ O N O P O L E - A N T I M O N O P O L ] ~ I N T E R A C T I 0 1 ~ 257

From expression (3.10) we have tha t , in order to express the average square area of the loops, we just need to take the second derivative of Z with respect to -- i eF and to set -- i eF -~ O. In fact, an area element has the form

dQ~ 9u -- dQu Q~.

We can now express Q in Fourier series:

(3.11) n~+oo

9 = ~ c, exp [i~, s], h e 0

where ~ ~ 2~rn/S.

I t is, therefore, possible to see tha t the quant i ty

$

0

turns out to be proportional to S 2. We can solve the integration over ~Q in expression (3.10) and choosing the

base in which F v F is diagonal ((FTF)0o --~ (FT~)33 ~- E ~, (FTF)~ ~ (FTF)~ ~ B ~) we obtain (16)

(3.12) z(o) = ~ ~. q~o,.) dS(~) 1 6 x ~ 9- ~(~)

- - (2e)~S~(m)EB exp [--#~S(m)]}. sin ( i cES ('~)) sin ( i eBS Cm))

If we choose to consider the particular case of circular loops, we can relate the variable S with p~. Then the density function could be extrapolated just considering the way the variable S is weighted in expressioa (3.12), where the factor 1/16~r2S ~ is due to the normalization of ~Q. We can now relate the quant i ty

fds (3.13) ~ exp [--#aS] with ~D(p),

and from the proportionality of S to ~ we extrapolate the following behaviour

(1~) j . SCn'WI~G]~R: Particles, Sources and ~ields, Vol. 2 (Reading, Mass., 1970).

258

of D(q):

(3.14)

R. J E N G 0 and M. T. VA_LLON

1 D(q)--, q~

This analysis suggests also that the limit for Q -> oo in the noncondensed case (~ ) = o is

(3.15) D(e) ~ exp [ - - ~ e ~] ( ( ~ ) = 0 case)

apart from powerlike factors. We will see in sect. 5 that with the density of (3.14) we will indeed obtain

the usual UV divergences in the vacuum polarization graph.

4 . - C o n d e n s a t i o n a n d c o n f i n e m e n t .

We can now evaluate the Wilson loop (3.2) at the large-~ limit when (q~) ~ 0, that is in the condensed phase. D(~) is given by relation (3.7), we must now estimate interaction (2.6) at the same limit, that is for d/~ --> O, having called d the distance between the monopole and the antimonopole. I t can be useful to introduce the function J(~, d):

(4.1) f d d'Q [1 - -exp [I]] , J(q, d) =-- ~

which has the dimensions of JL 3. Apart from some exceptional configurations, in general [~(~ ~ -d ) - -

-- Q(5)] --> 0 for d -+ 0. If this is the ease, we can consider the expansion in d:

(4.2) ) t2(~ + d) - f2(~) = - 5 - ~2(~,) + o(a~) .

Accordingly, J(~, d) takes the form

d d J(q, d) - - j 4~

and, considering the series expansion of the cosine, we obtain that at the limit for d]~ ---> 0 the leading behaviour of J is, just from dimensional considerations,

(4.3) J(q , , / ) ,-~ a*q. (a/O--*o)

V A C U U M E F F E C T S ON T H E S T A T I C M O N O P O L E - A N T I M O N O P O L E I N T E R A C T I O N 259

We must now take into account those configurations for which the difference in solid angles is not small when d -+ 0.

A first example is given by all loops tha t are located just (( in between )) the monopole and the antimonopole, tha t is when in the interact ion the two solid angles are summing to each other. This case can be reconducted to the previous one, therefore i t contributes to J with something proport ional to d ~ ~. In fact, the sum of two solid angles (~1 ~-/}~) can be rewri t ten as ~1-- ( 4 s - Q~), since in interact ion (2.6) a 4s t e rm (which is the contribu- tion of these configurations when d -> 0) is just a phase factor equal to one. Therefore, we are in the conditions previously analysed, having the difference of two solid angles which tends to zero, when d -~ 0.

Then there arc other possible configurations tha t can (and in fact do) con- t r ibute to J(~, d) in a different way. This is the case when one solid angle is

ve ry large (nearly 2~) and the other one tends to zero. In the present approx- imation (~ large) this case can be verified only when the surface of the loop is nearly in the direction of d and covers one of the two points with an edge, so us to give a contr ibution of about 2s for the point covered, while the other one sees the surface under a minimum solid angle. For these configurations to occur at the limit dip -+ O, the angle formed by the surface with d must be of order d/~.

Fur thermore , also the ~ (or ~ ~- d in the symmetr ic configuration) compo- nents must be very well tuned: Xs (the component in the d direction) must be Q - dx8 with dx3 varying in a range of values on the d scale. Of the other two components, one must be of order d, while the other one must only be defined in the Q scale.

The integrat ion over ~ within all these limits contributes to define a J(~, d) function proport ional to d s. Therefore, the contr ibution to J(p, d) coming from these configurations is negligible with respect to the leading te rm d~Q.

Accordingly, we can set J(~, d) equal to 2d 2 p (2 constant) for ~ ~ ~, where is a numerical constant . The Wilson loop (3.2), in terms of the function J(o, d), is

(4.4) WL = exp [-- TfdeJ(e,d)D(e)]. The large-~ region gives a contr ibut ion

(4.5) c~r

tha t is the typica l area law (8,9) of the confinement. The string tension is given by the quant i ty co 2]a, which is not well determined, bu t has been, however, shown to be proport ional to (~}~.

260 R. J ~ O O and ~ . T. V~LLO~

The same result has been obtained by ~IELS~.N and OLESE~ (~o) and by ~qA~u (~) in different theories. ~IET,SE~ and O~ESE~ used a vortex model~ while ~ A ~ v re-elaborated the Dirac formalism (1) and did not introduce Higgses in the theory.

I t is interesting to notice that, by means of a loop formalism, and expres- sing the condensation via the function D(~), we indeed obtain for the string tension a term proportional to the square of the vacuum expectation value of the Higgs field, with no dependence on the charges as in the other two ap- proaches.

5. - C o n t r i b u t i o n s to t h e v a c u u m p o l a r i z a t i o n .

In sect. 4 we studied the behaviour of the Wilson loop (3.2) in the large-~ approximation, in order to get information on the confining phase.

I t can be interesting also to analyse the other limit, when the loops are small and, therefore, ~/d--+ O. In general, for ~ smaller than 5, we can use for the solid angle the simplified expression

(5.1) ~2(x) ~ cos 0 X2 '

where 0 is the angle made by the vector 5, defining the position of the centre of the loop, with the direction orthogonal to the surface.

As a consequence, in this region, interaction (2.6), which is a differ- ence of solid angles, will have a ~ dependence (since ~ is factorized). We introduce also in this case the quantity J(~, d), defined in eq. (4.1). From dimensional considerations and from the fact that, for ~ -+ 0, D is proportional to ~ , we can conclude that the leading term will give a contribution of the type ~'/d to J(~, d).

We must now take into account the contributions to J(~, d) coming from those configurations for which approximation (5.1) is not valid. That is the case in which 5 or ~ -~ d are small, of the same order of ~. In these cases the interaction will depend neither on d nor on ~ and the corresponding contri- bution to J(~, d) will be of the type Q3 from the integration over d~. I$ is clear that this is a disconnected-graph contribution, since d does not appear. I f we want to consider the first connected contribution of this type (that is with or 5 -{- d in the ~ scale), we will get a QS/d2 term, clearly negligible with respect

to the other ones. We can conclude that, in the region ~/d --+ O, J(Q, d) can be written as

~4 ~s (5.2) J(o, d) = aoa -~ b--~ -~ e--~ (a, b, v constants).

Q--~O

VACUUM EFFECTS ON THE STATIC MONOPOLE-ANTII~ONOPOLE INTERACTION 261

With the densi ty funct ion of relat ion (3.14) we obtain the usual vacuum

polarization divergences when the circle shrinks to a point.

Tha t is

f f ( limQ~o d~J(~,d)D(o )= d~ ao~ + b~ + v-~ es,

where the hnear divergence is a mass term, corresponding to the fact t ha t in

our problem we are considering static external sources. The logarithmic divergent par t is of the form of the Coulomb interact ion

of the static monopole-ant imonopole pair and i t can be wri t ten as

1

where A~g is a divergent quant i ty , defined modulo a convergent t e rm (we have to imagine an ul t raviolet cut-off 0>0o, for instance 00 is a lat t ice spacing)

fd0 A~a= (23z) 2 - ~ - e x p [ - - # ~ 0 ~] [b[~.

We have taken into account eg = 2z and the fact tha t b ~ 0 for a monopole- ant imonopole since i t comes from the t e rm --1~9(~1)~9(22) in the expansion of 1 -- cos 1[g2(51) - - g2(5~)] (see cqs. (2.6) and (4.1)). This divergence will be reabsorbed in a renormalizat ion of the monopole charge:

g~ = f . (1 + A~g).

We can notice that, if we analyse the problem in the Eucl idean space, the monopole-charge interact ion is purely imaginary, while the charge-charge interact ion turns out to be real (8). We know tha t the divergences come from the quadrat ic terms in the exponent ia l inside the parenthesis of eq. (3.2) or, equivalent ly, in eq. (3.10). Therefore, if we consider the situation in which we have static external charges, the contr ibut ion to the divergences from a monopole loop will have the opposite sign of the usual contr ibution from a loop of charges. I t is t rue that , af ter the Wick rotat ion, the theory in Minkowski space will present different features, since the two interactions will have the same imaginary character . However , in Minkowski space the surface which is coupled to the field in the interact ion t e rm will be in the charge-charge case

of the space-time type , while in the eharge-monopole ease a purely space one. Since the divergent t e rm is proport ional to the square of the surface of the loop ((02) 2 ~-, S 2 in eq. (3.10)), the quant i ty under integrat ion will have a dif-

ferent sign in the two cases.

262 x . f f x ~ o o a n d ~ . r . VALLO~

The difference in r ea l i t y of the two in t e r ac t ions in Euc l ide space shows up

i n Minkowsk i space in the fac t t h a t

? , , P , , = - ~ , ~ , .

I n conclus ion , the c o n t r i b u t i o n to t he v a c u u m po la r i za t ion d ivergences of a

loop of monopoles is a lways different in sign f rom the one of a loop of charged

par* ides . F r o m the c o m p u t a t i o n s of ref. (3) we can s t h a t this c o n t r i b u t i o n ,

a p a r t f rom the di f ferent va lues of the charges, is a lways of the same in

m a g n i t u d e (7).

�9 R I A S S U N T O

Si ~nMizza una coppia stntica monopolo-antimollopolo in inter&zione con lc partieelle virtuMi the fluttu~no nel vuoto. Si usa un formalismo d'integrMe sui percorsi e l ' in- tcgrnzione si f~ per mezzo di una descrizionc in media in cui si considerano percorsi eireolari di r~ggio 0, pesati con unn densit~ D(O ). Si dctermina l 'and~mento di D(e) sin per ~ --~ 0 che per 0 --~ c~. Nel l imite ~ -+ oo si possono distinguerc i due casi di non eondensazione o eondcnsaziolle delle particelle elettrieamente cariehe. Nel se- colldo c~so si ottiene il eonfin~mento della coppia di monopoli stntici. I1 limite 0 --~ 0 dg informazioni sul colltributo divergente della polarizzazione di segno opposto rispetto a quello di una coppia st~tie~ elettricamente caric~.

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