SOME REMARKABLE SPIN PHYSICS WITH MONOPOLES AND FERMIONS

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SOME REMARKABLE SPIN PHYSICS WITH MONOPOLES AND FERMIONS

N. Craigie

To cite this version:

N. Craigie. SOME REMARKABLE SPIN PHYSICS WITH MONOPOLES AND FERMIONS.

Journal de Physique Colloques, 1985, 46 (C2), pp.C2-95-C2-106. �10.1051/jphyscol:1985209�. �jpa-

00224520�

JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°2, Tome 46, février 1985 page C2-95

SOME REMARKABLE SPIN PHYSICS WITH MONOPOLES AND FERMIONS

N.S. Craigie*

Brookhaven National Laboratory, Upton, Long Island, NY 11973, U.S.A.

Résumé - Cette revue couvrira les questions suivantes selon l'évolution histo- rique du sujet : le monopole de IHrac ; le problème de Kazama-Yang-Goldhaber dans la diffusion électron monopole ; le monopole de t'Hooft-Polyakov et spin à partir d'isospin ; l'analyse de Rubakov ; "l'effet Rubakov-Callan",catalysé par le monopole de la désintégration du proton ; le rôle de la théorie des champs à deux dimensions exactement soluble et finalement les conséquences observables.

Abstract - This review will cover the following topics, which follow the his- torical evolution of the subject: the Dirac monopole; the Kazama-Yang Goldhaber problem in electron-monopole scattering; the 't Hooft-Polyakov monopole and spin from isospin; the Rubakov analysis; monopole catalysis of proton decay "the Rubakov-Callan effect"; the role of exactly solvable 2- dimensional QFT's and finally observable consequences.

I am going to talk about an intriguing aspect of the gauge theories we believe might describe the fundamental interactions of nature. As most of us have learnt, such gauge theories have magnetic monopole soliton-like states. The latter have remark- able properties as regards spin and angular momentum. In fact, a charge boson inter- acting with a monopole carrying one Dirac unit of magnetic charge forms a system with half integer total angular momentum, i.e. it behaves like a fermion. The cata- logue of the phenomena of this kind associated with monopoles seems limitless. A detailed study of fermions interacting with heavy monopoles leads to some fascinat- ing lessons in non-perturbative spin physics, through exactly solvable two-dimen- sional quantum field theories. The latter is the first case of such theories being directly relevant to observable elementary particle physics. This is in contrast to condensed matter physics, where there are numerous such examples.

About three years ago, Valerie Rubakov and independently Curt Callan predicted that a remarkable phenomenon would occur if a monopole of a grand unified theory passed through nuclear matter, namely it would cause protons to decay into an electron and pions at rates more characteristic of strong interactions, rather than the "ultra- weak" forces of the grand unified theory, which are known to give rise to baryon number violation. In the last part of this talk I will sketch how these deductions were made and discuss the consequences, but let me begin by describing briefly how Dirac introduced the monopole concept in the 1930's.

THE DIRAC M0H0P0LE

In 1931 Dirac /I/ asked the following question. Since Maxwell's equations are almost symmetrical with respect to the exchange of the electric and magnetic fields, is it not possible to have a point magnetic pole analogue of the electron? This would make the Maxwell system completely symmetrical (see Table I ) . However, he met with a very important technical problem, the solution of which pointed to the monopole having some topological character.

*0n leave of absence from : International Centre for Theoretical Physics, Trieste Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italy

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985209

C2-96 JOURNAL

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Table I

-

Equations governing a Dirac magnetic monopole

I

Modified Maxwellts e q u a t i o n s ( i n u n i t s = c = l )

-1

I

Magnetic f i e l d : = m n l / r 2

-

2n m 6 ( x ) 6 ( ~ )

Vector p o t e n t i a l w i t h s t r i n g s i n g u l a r i t y : J - cos9 i r cos9 (The s t r i n g s i n g u l a r i t y along t h e z - a x i s f e e d s i n 4nm u n i t s of

magnetic f l u x . )

Dirac q u a n t i z a t i o n c o n d i t i o n :

( I n o r d e r t h a t t h e s t r i n g s i n g u l a r i t y b e i n v i s i b l e i n an Aharanov experiment )

P

e $ I A . ~ I = 2 n , around

s t r i n g

I

^I

^l

The d i f f i c u l t y I am r e f e r r i n g t o i s t h a t once one t r i e s t o w r i t e down t h e corre- sponding v e c t o r p o t e n t i a l A ( r e q u i r e d f o r a gauge i n v a r i a n t d e s c r i p t i o n of W D ) , t h e n one f i n d s t h a t it mustU have a s t r i n g s i n g u l a r i t y . One can t h i n k of t h i s s t r i n g a s a s o l e n o i d f e e d i n g i n a s o t o speak t h e monopole's

4mn u n i t s of magnetic f l u x . I n o r d e r t h a t t h i s f i c t i t i o u s s o l e n o i d b e un- o b s e r v a b l e , f o r example i n an Aharanov e l e c t r o n i n t e r f e r e n c e experiment, t h e l i n e i n t e g r a l of t h e v e c t o r p o t e n t i a l around it should be a m u l t i p l e t of 2a. This has t h e immediate consequence t h a t t h e e l e c t r i c and magnetic charges a r e quantized ac- cording t o eg = 1 / 2 . Notice t h a t t h e l i n e i n t e g r a l maps t h e U(1) gauge group around a c i r c l e ( i . e . i n homotopy t h e o r y t h i s i s denoted by r i l ( U ( l ) ) = Z = ' 0 , 1 , 2 , 3 , .

. . l )

and t h i s i s what I meant above by "the monopole has a t o p o l o g i c a l c h a r a c t e r " .

THE KAZAMA-YANG-GOLDHABER PROBLEM I N ELECTRON-MONOPOLE SCATTERING

Another unforeseen f e a t u r e o f a Dirac monopole emerges when one c o n s i d e r s an elec-.

t r o n s c a t t e r i n g o f f it. The t o t a l angular momentum i s made up o f t h r e e p i e c e s :

where

2

i s t h e u s u a l o r b i t a l p i e c e , S + = 1/2543 i s t h e s p i n of t h e e l e c t r o n , w h i l e t h e e x t r a p i e c e i s due t o t h e i n t e r a c t i o n of t h e charge of t h e e l e c t r o n w i t h t h e monopole magnetic f i e l d . This i s given by

By v i r t u e o f t h e Dirac c o n d i t i o n eg = 1 / 2 , T a l s o corresponds t o h a l f a u n i t of t h e a n g u l a r momentum. Thus t h e lowest v a l u e of t h e t o t a l a n g u l a r momentum J can t a k e i s z e r o , when t h e two s p i n 1 / 2 p i e c e s e x a c t l y compensate one a n o t h e r (L = 0 wave) o r t o g e t h e r compensate an L = 1 o r b i t a l a n g u l a r momentwn. However, Kazama, Yang and. Goldhaber /2/ p o i n t e d out a problem. I f one i n s p e c t s

3,

one n o t i c e s t h a t i t s

-+ -+

s i g n depends on t h e d i r e c t i o n r , i . e . it changes s i g n i f r p o i n t s i n i n s t e a d of o u t . This means a s an e l e c t r o n p a s s e s t h e c o r e ( s a y i n t h e S-wave), t h e n i n o r d e r t o conserve a n g u l a r momentum, e i t h e r

e + -e ( c h a r g e exchange, i . e . T f l i p ) o r

Sz -+ -S

z

( h e l i c i t y f l i p , i . e . S f l i p ) i . e . t h e r e a p p e a r s t o be a n ambiguity.

The problem can be viewed i n a n o t h e r way. I f one examines t h e Hamiltonian of t h e system, namely:

then a s it s t a n d s it does not g i v e r i s e t o a deep s e l f - a d j o i n t Hamiltonian s c a t t e r - i n g problem. It h a s t o b e modified o r e q u i v a l e n t l y one must supplement t h e problem w i t h a s p e c i a l boundary c o n d i t i o n a t r = 0. However, t h e r e i s an a r b i t r a r i n e s s i n doing t h i s , which can be c h a r a c t e r i z e d by a phase a n g l e - 9.

To be e x p l i c i t , l e t u s r e c a l l some work o f Kazama and Yang /3/ on t h e p a r t i a l wave a n a l y s i s of t h e e q u a t i o n

H J , = E J ,

By v i r t u e of t h e f a c t t h a t t h e t o t a l a n g u l a r momentum, which commutes w i t h H , has two s p i n 1 / 2 p i e c e s , t h e o r b i t a l s e r i e s f o r a monopole w i l l , i n f a c t , run over i n t e g e r v a l u e s

where = eg w i t h eg = 1 / 2 f o r a Dirac monopole. The r e l e v a n t harmonics a r e of t h e Jacob and Wick t y p e o r e q u i v a l e n t l y t h o s e proposed by Yang ( s e e Ref.!3./ and r e - f e r e n c e s t h e r e i n ) , namely Y ( 8 , 'P ) . Let us c o n c e n t r a t e on t h e lowest p a r t i a l

i,m,q

wave, s i n c e only t h i s s e c t o r h a s t h e i n t e r e s t i n g p h y s i c s , due t o t h e f a c t t h a t t h e corresponding wave f u n c t i o n i s non-vanishing a t r = 0 . We can w r i t e t h e S-wave f u n c t i o n i n t h e form

where

qa i s a two-component s p i n o r and i s a s o l u t i o n of t h e equation

The Dirac e q u a t i o n i n two component form X a =

[ Y ]

can be m i t t e n i n t h e form

- - a

where H = i q y

2~

⁺ y0 m.

The s o l u t i o n s a r e t h u s c l a s s i f i e d a c c o r d i n g t o t h e e i g e n v a l u e s v = ? 1 of

..

and t a k e t h e form ( f o r m = 0 ) : qY 5

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This means t h a t t h e y decompose i n t o e i t h e r in-movers o r out-movers, i n p r i n c i p l e w i t h no outgoing wave f o r a given incoming wave.Onehas t o impose a boundary c o n d i t i o n t o r e l a t e t h e i n and o u t s e c t o r s . I n o r d e r t o f i g u r e o u t what would b e t h e a p p r o p r i a t e one, l e t us c o n s i d e r t h e s e l f - a d j o i n t n e s s o f

R,

i . e . c o n s i d e r

i 8

Thus o n l y i f we choose x + ( o ) = e ~ ~ ( 0 ) do we o b t a i n a s e l f - a d j o i n t Hamiltonian and consequently w e l l d e f i n e d p h y s i c s . However, t h e p h y s i c s depends i n an important way on t h e v a l u e of t h i s a r b i t r a r y a n g l e 8 a s h a s been demonstrated /4/ by t h e work of Yang, Wu and o t h e r s . For example t h e fermion ground s t a t e ( i . e . t h e Fermi s e a around t h e monopole) v a r i e s a s we change 8. F u r t h e r t h e p o s s i b l e bound s t a t e spectrum depends on i t s v a l u e .

When t h e r e a r e more t h a n one f l a v o u r of charged fermion i n t h e problem, t h e n more complex boundary c o n d i t i o n s emerge and consequently q u i t e a d i f f e r e n t p h y s i c s .

't HOOFT-POLYAKOV MONOPOLE AND SPIN FROM ISOSPIN

C e r t a i n non-Abelian gauge t h e o r i e s i n t h e i r Higgs can have monopole c o n f i g u r a t i o n s of t h e i r gauge f i e l d s / 5 / . These a r e t o p o l o g i c a l l y e x c i t e d s t a b l e s o l i t o n s o l u t i o n s of t h e e q u a t i o n s o f motion, which c a r r y one o r two Dirac u n i t s o f magnetic charge, depending on t h e gauge group. The o r i g i n o f t h i s phenomenon l i e s i n t h e s e l f - i n t e r - a c t i o n s of t h e gauge f i e l d s . The u n d e r l y i n g non-linear system o f e q u a t i o n s admit s o l i t o n s o l u t i o n s , which a r e c h a r a c t e r i z e d by a t o p o l o g i c a l charge. This discovery i n 1974 caused c o n s i d e r a b l e excitement, because o f t h e s u c c e s s non-Apelian gauge t h e - o r i e s were having, by p r o v i d i n g a d e s c r i p t i o n o f a l l elementary i n t e r a c t i o n s i n n a t u r e .

To d e s c r i b e such a monopole, l e t u s c o n s i d e r an SU(2) gauge t h e o r y , couple t o a charged s c a l a r f i e l d , chosen t o b e i n a Higgs p h a s e , i n which o n l y a l o n g range U(1) gauge f i e l d remains m a n i f e s t a t l o n g wave l e n g t h s . The s e t o f e q u a t i o n s governing t h i s systeni i s g i v e n i n Table 11, t o g e t h e r w i t h ^{I t} Hooft-Polyakov monopole s o l u t i o n . I f t h e vacuum e x p e c t a t i o n v a l u e o f t h e s c a l a r f i e l d i n t h e Higgs phase <

41"

> = $ na t a k e s on t h e hedgehog c o n f i g u r a t i o n shown i n F i g . 2 , i n which t h e i s o s p i n p o i n t s 0 along t h e r a d i a l d i r e c t i o n everywhere, t h e n a s t a b l e monopole c o n f i g u r a t i o n o f t h e gauge f i e l d s s i t s a t t h e c e n t r e . For t h o s e f a m i l i a r w i t h homotopy t h e o r y i n mathe- m a t i c s , t h i s s i t u a t i o n i s c h a r a c t e r i z e d by II2(sU(2)/U(1)) which i s e q u a l t o

II ( U ( 1 ) ) = 2, i . e . t h i s i s a mapping o f a s u r f a c e o f a s p h e r e i n r e a l space i n t o t h e

~ h ( 2 ) group, w i t h winding number Z = 1 , 2 , 3 , .

. .

The l a t t e r d e f i n e s t h e s t a b i l i t y c l a s s o f t h e monopole, because it cannot b e unwound by quantum f l u c t u a t i o n s of t h e system. The U(1) gauge group t h a t i s l e f t m a n i f e s t a l o n g way from t h e c e n t r e , corresponds t o r o t a t i o n s around t h e r a d i a l d i r e c t i o n s as i n d i c a t e d i n Fig.2. Refer- r i n g t o t h e s e t of e q u a t i o n s governing t h i s system i n Table 11, we n o t i c e u n l i k e a Dirac monopole t h e v e c t o r p o t e n t i a l does n o t have a s t r i n g s i n g u l a r i t y . T h i s i s due t o t h e presence o f t h e s c a l a r f i e l d , which i n a s e n s e r e p l a c e s t h e Dirac s t r i n g . By v i r t u e of t h e f a c t t h a t t h e s e non-Abelian monopoles a r e non-singular a t r = 0 and have f i n i t e c o r e r a d i u s , t h e r e should b e a unique s o l u t i o n s t o t h e Kazama-Yang- Goldhaber problem, even a s we l e t t h e c o r e r a d i u s ( i . e . l / m a s s ) go t o z e r o . Let u s c o n s i d e r ND S U ( ~ ) d o u b l e t s of left-handed two-component fermion f i e l d s , couplea t o t h e sU(2) monopole system we have j u s t described. The fermion p a r t of t h e a c t i o n i s given by:

Nn

where a' =

(4

, o i ) ; A;(x) i s t h e monopole s t a t i c p o t e n t i a l given i n Table I1 and "'(X) i s t h e quantum f l u c t u a t i o n of t h i s c o n f i g u r a t i o n . I f we do a p a r t i a l

wave a n a l y s i s o f fermions s c a t t e r i n g o f f t h e monopole c e n t r e d a t r = 0 , t h e n we s e e t h a t t h e t o t a l a n g u l a r momentum i s made up of t h e t h r e e p i e c e s :

where L i s t h e u s u a l o r b i t a l p i e c e , S = 1 / 2 0fi i s t h e fermion s p i n and T = 112 f 4 i s h a l f a u n i t of a n g u l a r momentum coming from t h e i n t e r a c t i o n o f t h e fermion charge w i t h t h e monopoles s t a t i c magnetic f i e l d . Thus t h e monopole promotes i s o s p i n t o s p i n . This means, j u s t l i k e t o t h e D i r a c c a s e and u n l i k e t h e u s u a l c e n t r a l p o t e n t i a l problem, i n which t h e lowest p a r t i a l wave i s J = 112, f o r a monopole t h e lowest p a r t i a l wave i s a J = 0 , i . e . S-wave. The l a t t e r i s b u i l t up e i t h e r %y L 0 , S

+

T = 0 o r L = 1, S

+

T = 1. T h i s i s r e f l e c t e d i n t h e two independent f u n c t i o n s g ( r ) and h ( r ) i n t h e f o l l o w i n g decomposition o f t h e S-wave f i e l d

/6/

where a r e f e r s t o s p i n and i t o i s ~ s p i n . ~ I f we c o l l e c t t h e f u n c t i o n s g and h i n t h e form of a two component s p i n o r f = [ h

1,

t h e n f ( t , r ) s a t i s f i e s a f r e e Dirac equation on t h e h a l f space

(t

,r ) ( 0 r < m ) , namely

where so f a r we have o n l y t a k e n i n t o account t h e monopoles s t a t i c f i e l d . Thus we appear t o have a f r e e two-dimensional fermions, except f o r a s p e c i a l boundary con- d i t i o n a t t h e monopole c o r e r = 0. The l a t t e r i s given by t h e s o l u t i o n t o t h e c l a s - s i c a l s c a t t e r i n g problem and i f we e x p r e s s each fermion doublet i n terms o f an upper component o f charge Q =

+

1 and a lower component of charge Q =

-

1, t h e boundary c o n d i t i o n corresponds t o p u r e charge exchange Q + -Q. I f we s t a y w i t h i n t h e one p a r t i c l e s c a t t e r i n g system, t h e n t h i s charge 2e must go somewhere. It h a s been known f o r some t i m e t h a t a ' t Hooft-Polyakov monopole h a s a l s o a dyon degree of freedom, which when e x c i t e d corresponds t o a monopole w i t h b o t h e l e c t r i c and ma- g n e t i c ch rge. However, t h e energy r e q u i r e d t o e x c i t e t h i s degree of freedom i s of o r d e r 10'' mp. On t h e o t h e r hand, i n o r d e r t o conserve p r o b a b i l i t y i n t h e S-wave s e c t o r some p r o c e s s must t a k e p l a c e . C l e a r l y t h e answer i s p a r t i c l e Creation o c c u r s and one needs a f u l l quantum t r e a t m e n t o f t h e problem.

Fig.1

-

P o t e n t i a l i n Higgs phase

Fig.2

-

Hedgehog c o n f i g u r a t i o n of Higgs f i e l d around a monopole

C2-100 JOURNAL DE PHYSIQUE

Table I1

-

^The ' t Hooft-Polyakov monopole system Lagrangian

8

⁼

-

^Fayv

^{+ l} /

D : ~ c$b

l2 ^-

~ ( ( ~ c $ ~ )

,

'G

uv 2

1

where

2 2 v ( $ 2 ) = '&2

-

( o ) C l a s s i c a l e q u a t i o n of motion

ab b

= D.

l +

1 % Hooft-Polyakov monopole s o l u t i o n

tla(x) =

g

^c$o

1 n ^J

n a ( x ) _I. =

-

_2e

E

_{a l j}

. . -

_r (A; = 0 )

The monopole's magnetic f i e l d

Magnetic charge

i . e . e.g. = 1 / 2 ( t h e Dirac c o n d i t i o n )

THE RUBAKOV ANALYSIS

I n o r d e r t o study t h i s Rubakov noted / 7 / t h a t t h e quantum electro-dynamics of S-wave fermions i s governed by t h e f o l l o w i n g a c t i o n

where x = ( t , r ) and Yp = ( 0 3 , i o l ) . The b.c. a t r = 0 i s ( l + j 5 ) f ( 0 ) = 0;

A The Rubakov system i s i n f a c t e x a c t l y i n t e g r a b l e and i s v e r y s i m i l a r t o t e = 2

i"?.

dimensional QED model of Schwinger. They d i f f e r i n two e s s e n t i a l r e s p e c t s . F i r s t l y f o r t h e monopole t h e fermions l i v e on a h a l f space w i t h a s p e c i a l boundary

c o n d i t i o n a t r = 0 and secondly t h e dimensional coupling c o n s t a n t of t h e Schwinger model i s r e p l a c e d by e / r 2 , which becomes i n d e f i n i t e l y l a r g e a s we approach r = 0.

These d i f f e r e n c e s a r e t h e ones t h a t a r e r e s p o n s i b l e f o r t h e unexpected new p h y s i c s . The system i s s o l v e d by simply making a r o t a t i o n o f t h e fermion f i e l d so it becomes a f r e e f i e l d , i . e .

where t h e f u n c t i o n s a ( x ) and @ ( X ) a r e chosen t o c a n c e l t h e gauge i n t e r a c t i o n . The a p p r o p r i a t e choice i s a ( X ) = 6 a v a ( x ) +

a

b ( x ) , where t h e second term i s simply a gauge choice. Thus theFLonly n o g y t r i v i a l dynhnical v a r i a b l e i s a ( x )

.

The a c t i o n ,

s p l i t s i n t o two p a r t s S = ~ ( a ) + s ( f 0 ) , where S ( f o ) i s t h e a c t i o n o f f r e e fermions and ~ ( a ) i s given by

m 0

a a

where =

- - -

a t 2

ar ^{2 '}

The f a c t t h a t t h i s a c t i o n i s Gaussian means t h a t t h e system i s i n t e g r a b l e and can be s t u d i e d non-perturbatively. I n f a c t , t h e u n d e r l y i n g d i f f e r e n t i a l e q u a t i o n s can be s o l v e d e x a c t l y and consequently a l l fermion Green's f u n c t i o n s e x p l i c i t l y computed.

By t h e s e means Rubakov was a b l e t o demonstrate t h a t t h e p a i r i n g o r condensate para- meter c f l ( r ) f 2 ( r ) >

*

r-l f o r t h e c a s e o f two fermion d o u b l e t s . This c o r r e s p o n d s t o

N,

an anomaly i n t h e fermion number c u r r e n t J~ =

2 F ' ~ ~ -

f ( k ) , namely a s

U

Yu

k = l

I n t e g r a t i n g t h i s e q u a t i o n g i v e s t h e change i n f e r n i o n number, namely

On can show t h a t t h e above condensates correspond t o i n s t a n t o n - l i k e c o n f i g u r a t i o n s i n t h e monopole's ~ ( 1 ) r a d i a l quantum gauge f i e l d . This i n t u r n corresponds t o t h e monopole being i n a s u p e r p o s i t i p n o f s t a t e s w i t h d i f f e r e n t fermion number c h a r a c t e - r i z e d by m u l t i p l e s o f ND.

The condensate

c o r r e l a t e s fermions w i t h t h e same h e l i c i t i e s and i s a s s o c i a t e d w i t h v i r t u a l fermion number v i o l a t i n g p r o c e s s e s o c c u r r i n g i n t h e vacuum around t h e monopole. These have p r o b a b i l i t i e s which f a l l o f f very slowly a s we move away from t h e core.

To s e e t h e i m p l i c a t i o n of t h e above phenomenon, l e t u s c o n s i d e r t h e c a s e of a mono- p o l e i n t h e S U ( ~ ) grand u n i f i e d t h e o r y o f Georgi and Glashow, which i s t h e minimal one t h a t c o n t a i n s t h e S U ( ~ ) X S U ( ~ ) X ~ ( 1 ) gauge t h e o r i e s o f s t r o n g , weak and e l e c - tro-magnetic i n t e r a c t i o n s . T h e l i g h t e s t g e n e r a t i o n of fermions i . e . e , v , ull u2, u3, dl, dg, d3 form t h e following S u ( 5 ) m u l t i p l e t s

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where we have i n d i c a t e d t h e S U ( ~ ) , S U ( ~ ) and lepto-quark t r a n s i t i o n s , r e s p e c t i v e l y . The corresponding gauge bosons form a 5 X 5 m a t r i x which, i n a d d i t i o n t o t h e 8 s t r o n g i n t e r a c t i o n m a s s l e s s bosons, t h e 3 heavy weak i n t e r a c t i o n W and X bosons and t h e pho- t o n s c o n t a i n 1 2 very heavy lepto-quark bosons.The l a t t e r f o r example can t u r n a b q u a r k i n t o an e l e c t r o n . T h i s e n l a r g e d gauge symmetry o n l y becomes m a n i f e s t a t mass s c a l e s of 1015 a t which a l l t h e i n t e r a c t i o n s a r e s a i d t o be u n i f i e d i n t o a s i n g l e m a t r i x gauge f i B i , which i s coupled t o a r e a l s c a l a r f i e l d $ i n t h e same 24 r e p r e s e n t a - t i o n . The l a t t e r h a s a v e r y l a r g e vacuum e x p e c t a t i o n v a l u e , which l o o s e l y speaking breaks t h e ~ ~ ( 5 1 symmetry t o t h e observed low energy S u ( 3 ) @ S u ( 2 ) @ ~ ( 1 ) symmetry of n a t u r e . The o n l y e f f e c t of t h e heavy gauge bosons i s t o v e r y o c c a s i o n a l l y cause a nucleon t o decay and t o g i v e r i s e t o monopoles a s r e l i c s o f t h e very e a r l y uni- v e r s e . The l a t t e r a r i s e from a p a r t i c u l a r lepto-quark subgroup, e.g. i n t h e space

(?i3,eF) a s i n d i c a t e d above, f i n d i n g i t s e l f i n t h e t o p o l o g i c a l l y e x c i t e d s t a t e . I n t h i s c a s e t h e l i g h t fermions form t h e f o l l o w i n g 4 S u ( 2 ) monopole d o u b l e t s :

A remarkable f e a t u r e o f t h e monopole, which i s n o t y e t f u l l y understood i s t h a t it breaks t h e S U ( ~ ) ~ s t r o n g i n t e r a c t i o n gauge symmetry t o ~ ~ ( 2 1 @ ~ ( 1 ) ~ ~ . I n f a c t , t h e S-wave fermions e x p e r i e n c e t h e following SU( 2 ) c ~ ( l ) ~ c ~ 7 1 ) ~ gauge i n t e r a c t i o n s , i n which t h e SU(2le i n t e r a c t i o n s t u r n t o t h e two U-quark d o u b l e t s i n t o one another. The l a s t ~ ( 1 ) ~ f a c t o r i s t h e monopoles electromagnetism and it i s made up o f t h e o r d i n a r y e l e c t r l c charge g e n e r a t o r Qem p l u s xhe S U ( ~ ) , s t r o n g i n t e r a c t i o n hypercharge Yc. I f we o n l y t a k e i n t o account t h i s ~ ( 1i n t e r a c t i o n s , t h e n one can ) ~ simply r e p e a t Rubakov's a n a l y s i s w ' t h ND = 4 and d i s c o v e r t h e baryon number v i o l a t - i n g condensates < u l u ~ d j e - > ^% r-' surrounding t h e c o r e o f t h e monopole. T h i s cor- responds t o t h e v i r t u a l p r o c e s s p + e-. The f i r s t attempt t o t a k e i n t o account t h e o t h e r interactions,made by Callan

/ B / ,

i n which he r e p l a c e d t h e above gauge group by

~ ( 1 ) ~ ~ - X U ( l ) y X ~ ( 1 ) ~ and used t h e so c a l l e d b o s o n i z a t i o n t r a n s l o r m a x l o n confirm- '-

ed t h z i t h e c a t a l y s i s p r o c e s s w i l l occur unhindered and t h a t t h e monopole can b e con- s i d e r e d t o b e i n a s t a t e of i n d e f i n i t e baryon number. Subsequently, Rubakov, Nahm and myself / g / showed t h a t one can t r e a t t h e S U ( ~ ) , 2-dimensional f i e l d t h e o r y and we a l s o confirmed t h a t t h e above a d d i t i o n a l i n t e r a c t i o n s do n o t i n p r i n c i p l e s w i t c h o f f t h e e f f e c t . However, t h e y do impose sQme important s e l e c t i o n r u l e s , which could i n p r i n c i p l e have c o n s i d e r a b l e consequences once we i n c l u d e t h e h e a v i e r fermion g e n e r a t i o n s observed i n n a t u r e /10/.

Very r e c e n t l y Nahm and I r e a l i z e d t h a t t h e 2-dimensional SU(2) gauge t h e o r y above i s a c t u a l l y a l s o e x a c t l y s o l v a b l e a s f a r a s i t s r e l e v a n c e t o t h e S-wave fermion-monopole dyamics i s concerned. The reason i s S u ( 2 ) gauge group has a hidden g l o b a l symme- t r y under U I a u

+

b a u * , w i t h (a?

+

/ b 1 2 = 1, d e s c r i b e d by c u r r e n t s s a t i s y i n g t h e c o n s e r v a t i o n c o n d i t i o n s

ap

J"(x) P = aPV

a v

=

o

and t h e Kac-Moody a l g e b r a

[ s a ( x ) , J b ( y ) : =

tabc

~ ~ ( x ) 6 ( x - y ) + k 6ab 6 ' ( x - ~ ) / h n

which has c e n t r a l charge k = 1. Such a l g e b r a i c systems have been e x t e n s i v e l y s t u d i e d i n r e c e n t y e a r s by mathematicians and from t h e i r r e p r e s e n t a t i o n t h e o r y we know t h a t we a r e d e a l i n g w i t h a unique dynamical system, which can be e q u i v a l e n t l y d e s c r i b e d by t h e following 2-dimensional non-linear sigma model

where F 2 = 112% and t h e m a t r i x f i e l d g ( x ) i j i s an element o f a g l o b a l

S u ( 2 ) @ SIJ(2) symmetry group. The above f e r m i o n i c c u r r e n t s a r e d e s c r i b e d e q u i v a l e n t - l y by J = gL1(x)dg(x)/dx. I n t h e c a s e of f r e e fermions, t h i s non-Abelian g e n e r a l i - z a t i o n of b o s o n i z a t i o n w a s r e c e n t l y p o i n t e d o u t by Witten

I l l / .

^However,

Polyakov and Wiegmann /12/ have given arguments t h a t i n d i c a t e t h a t t h e above non- l i n e a r sigma model r e p r e s e n t s a wider c l a s s of fermion t h e o r i e s and i s e x a c t l y solv- a b l e . Thus t h e f u l l s U ( 5 ) monopole-S-wave fermion a c t i o n s e p a r a t e s according t o

where ~ [ u ] i s t h e Rubakov a c t i o n ; S [ e , d ] i s e s s e n t i a l f o r t h e f r e e e l e c t r o n d quark a c t i o n ( i . e . t h e r e a r e no i n s t a n t o n - l i k e e f f e c t s i n t h e i r remaining ~ ( 1 ) i n t 3 r - a c t i o n s ) ; sUE[gl i s a 2-dimensional non-linear sigma model a c t i o n d e s c r i b i n g t h e di-U-quark z e r o mass S U ( ~ ) , s i n g l e t bound s t a t e s , which i n t e r a c t w i t h t h e monopoles core.

The b a s i c c u r r e n t s s a t i s f y t h e c o n s e r v a t i o n c o n d i t i o n

a + j -

^a

-

^{2 gab? a}

+

^J

-

⁼

+ ^m

^{a 6a3} f o r t h e U; system and

a + J " d

-

=

L a u

f o r t h e e,d system P

These anomalous c o n s e r v a t i o n c o n d i t i o n s c a n be s o l v e d i n a ' f a c t o r i z e d way and t h e e x a c t s o l v a b i l i t y o f t h e above a c t i o n s can be used t o compute a r b i t r a r y m a s s l e s s S-wave fermion Green f u n c t i o n s i n t h e c a s e o f an S U ( ~ ) monopole [ 1 0 ] .

SUMMARY

I f one t r i e s t o embed t h e observed S U ( ~ ) ~ ~ ~ @ [ s u ( ~ ) Q ~ ( l ) system l ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

o f f o r c e s i n some semi-simple grand u n i f i e d gauge group, e.g. S U ( ~ ) , ~ 0 ( 1 0 ) o r S ~ ( 1 6 ) , which has no elementary ~ ( 1 ) t o b e g i n w i t h , t h e n i n e v i t a b l y t h e s e u n i f i e d t h e o r i e s have magnetic monopole c o n f i g u r a t i o n s o f t h e gauge f i e l d , which a r e topolo- g i c a l l y s t a b l e . I f t h e s e t h e o r i e s have anything t o do w i t h t h e observed elementary p a r t i c l e spectrum, it i s c l e r t h a t t h i s h i g h e r gauge symmetry can o n l y become mani- f e s t a t mass s c a l e s M 10'' GeY. The corresponding monopole s t a t e h a s a mass of o r d e r M/e and can b e thought of a s a dense coherent s t a t e o f t g e v e r y heavy gauge bos quanta. The r a d i u s of such a system i s very t i n y indeed namely l e s s t h a n 10-g' cm., i n f a c t one could even contemplate monopoles, which s i t i n s i d e t h e i r Schwarzchild r a d i u s , s o t h e y would i n a d d i t i o n b e b l a c k holes. Leaving t h e l a t t e r p o s s i b i l i t y a s i d e , f o r most p r a c t i c a l purposes one c o u l d imagine t h a t t h i s t i n y non- Abelian monopole would i n t h e l a b o r a t o r y l o o k l i k e a Dirac monopole. This i s c e r t a i n - l y t r u e a s r e g a r d s i t s l o n g range electro-magnetic f o r c e , however a t t y p i c a l number s c a l e s t h e y behave i n q u i t e a d i f f e r e n t way. A s Rubakov p o i n t e d o u t i n 1981 t h e s e monopoles w i l l c a t a l y z e proton decays a s t h e y p a s s through n u c l e a r m a t t e r , a t r a t e s t y p i c a l of s t r o n g i n t e r a c t i o n s , r a t h e r t h a n t h e ultra-weak e f f e c t s a s s o c i a t e d w i t h t h e massive gauge boson, which a r e r e s p o n s i b l e f o r o r d i n a r y spontaneous proton decay. The l a t t e r causes a p r o t o n t o decay o n l y once every 1 0y e a r s o r so. ~ ~ O f c o u r s e , a mono- p o l e i s a v e r y u n l i k e l y c o n f i g u r a t i o n t o o c c u r i n t h e f i r s t p l a c e . However, once it

C2-104

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e x i s t s , it h a s b o t t l e d up i n i t s c o r e a system, which v i o l a t e s baryon and l e p t o n con- s e r v a t i o n and t h i s profoundly changes t h e s t r u c t u r e of t h e fermion s e a around i t s core f o r some S i s t a n c e . We have t r i e d t o d e p i c t t h i s s i t u a t i o n i n Fig.3.

CORE OF THE MONOPOLE a dense condensate o f very heavy l e p t o -quark gauge bosons r < 1 0 - ~ ~ CM DISTORTED FERMI SEA f ermion p a i r i n g o u t s i d e t h e c o r e

REGION OF QCD AND QED MAGNETIC FIELDS

i n t h i s r e g i o n S-wave e l e c t r o n and quark s t a t e s reach t h e c o r e of t h e monopole and t h e system

can undergo a fermion number v i o l a t i n g t r a n s i t i o n

Fig.3 d e p i c t s a monopole.

Needless t o say due t o t h e Rubakov-Callan e f f e c t , an abundant s o u r c e of t h e s e v i r - t u a l l y i n d e s t r u c t i b l e o b j e c t s w i l l have some q u i t e a s t o n i s h i n g consequences, n o t t o mention a mind boggling new s o u r c e o f n u c l e a r energy. Let me end t h i s t a l k by men- t i o n i n g t h e l i m i t s on t h e f l u x o f monopoles i n t h e u n i v e r s e t h a t have been o b t a i n e d on t h e b a s i s of t h e Rubakov c r o s s - s e c t i o n

'~ubakov fl(v/c )-2 m i l l i b a r n s

The f i r s t k i n d of l i m i t comes from t h e g i a n t p r o t o n decay d e t e c t o r s . If a magnetic monopole o f t h e t y p e we have been c o n s i d e r i n g above p a s s e s through such a chamber, t h e n every 1 0 cm t o 1 0 m e t e r s , depending on t h e above c r o s s - s e c t i o n , t h e r e w i l l b e an induced p r o t o n decay, corresponding t o one o f t h e f o l l o w i n g i n t e r a c t i o n s :

1 ) . p

+

M + M + e + p i o n s

The s i g n a t u r e being suggested h e r e i s shown i n Fig.4 and a l l t h e p r o t o n decay expe- r i m e n t s have put l i m i S on t h e monopole f l u x [ 1 3 ] , c l o s e t o t h e s o - c a l l e d P a r k e r bound, namely < 10"' cmm2 sr-l sec-l. The l a t t e r corresponds t o t h e m a x i m f l u x of magnetic monopoles, t h a t t h e observed g a l a c t i c magnetic f i e l d s can t o l e r a t e b e f o r e being quenched. These l i m i t s a r e summarized i n Fig.5 t a k e n from t h e IBM p r o t o n decay experiment [14].

Other l i m i t s come from a s t r o p h y s i c a l o b s e r v a t i o n s . By v i r t u e o f t h e Rubakov-Callan e f f e c t o b j e c t s l i k e n e u t r o n s t a r s and w h i t e dwarfs and a r e v e r y e f f i c i e n t monopole d e t e c t o r s . The e s s e n t i a l p o i n t i s t h a t t h e monopoles c a p t u r e d by t h e g r a v i t a t i o n a l f i e l d s o f t h e s e o b j e c t s , can cause them t o c o n s i d e r a b l y h e a t up and r a d i a t e . From t h g o b s e r v a t i o n of r a d i a t i o n from known n e u t r o n s t a r s and w h i t e dwarfs, l i m i t s o f

between

1od

and 1 0 - l 0 s m a l l e r t h a n t h e P a r k e r bound have been o b t a i n e d 1151. I f c o r r e c t , t h i s means t h a t monopoles a r e v e r y r a r e indeed and u n l i k e l y t o be observed on e a r t h .

/

~ i ~ . h

-

A schematic drawing of a monopole p a s s i n g through a g i a n t proton decay d e t e c t o r

Fig.5

-

^Upper l i m i t s on t h e monopole f l u x a s f u n c t i o n of i t s v e l o c i t y

Bm ^{v / c} o b t a i n e d by l o o k i n g f o r m u l t i p l e i n t e r a c t i o n s i n t h e IBM.

Let me end by s a y i n g t h a t t h e phenomenon we have j u s t d i s c u s s e d i s one o f t h e most f a s c i n a t i n g examples of s p i n p h y s i c s , i n which we saw t h a t a &-dimensional problem i n elementary p a r t i c l e p h y s i c s can be reduced t o an e x a c t l y s o l v a b l e 2-dimensional

&FT problem. The l a t t e r e n a b l e s us t o p r e d i c t some r e a l l y remarkable new phenomena.

JOURNAL

DE

PHYSIQUE

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