RECENT RESULTS IN SU(3) LATTICE QCD WITH FERMIONS

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RECENT RESULTS IN SU(3) LATTICE QCD WITH FERMIONS

G. Martinelli

To cite this version:

G. Martinelli. RECENT RESULTS IN SU(3) LATTICE QCD WITH FERMIONS. Journal de

Physique Colloques, 1982, 43 (C3), pp.C3-278-C3-286. �10.1051/jphyscol:1982356�. �jpa-00221911�

JOURNAL DE PHYSIQUE

CoZZoque

C3,

suppl6ment au n o

12,

Tome

43,

de'cembre

1982

page

C3-278

RECENT RESULTS

I N

~ ~ ( 3 1 L A T T I C E QCD WITH FERMIONS

G. M a r t i n e l l i

Laboratori NazionaZi di Froseati,

I N F N ,

Frascati, I t a l y

'1. I n t r o d u c t i o n . - I n t h e l a s t y e a r a b i g p r o g r e s s h a s been made i n l a t t i c e QCD by s t a r t i n g t h e computation of t h e spectrum o f t h e hadrons by Monte Carlo t e c h n i q u e s . Although many problems remain s t i l l open, t h e f i e l d i s r a p i d l y developing and I b e l i e v e t h a t many of t h e p r e s e n t d i f f i c u l t i e s w i l l be overcome i n t h e n e x t f u t u r e . I n t h i s t a l k I w i l l r e c a l l t h e b a s i c i n g r e d i e n t s f o r t h e s e computations, I w i l l d i

IfYS

t h e l i m i t a t i o n s of Monte-Carlo t e c h n i q u e s and r e p o r t t h e more r e l e v a n t r e s u l t s

.

F i n a l l y I w i l l b r i e f l y d e s c r i b e a r e c e n t computation o f t h e proton anomalous magnetic moment on t h e l a t t i c e .

2 . L a t t i c e QCD w i t h fermions.- A r a t h e r g e n e r a l form, widely used i n Monte-Carlo s i - mulations, f o r t h e a c t i o n of i n t e r a c t i n g quarks and gluons on an e u c l i d e a n l a t t i c e i s [ 2 ] :

c o l o u r rind s p i n i n d i c e s have been omitted f o r s i m p l i c i t y , $ i s t h e quark f i e l d w i t h f l a v o u r f , U (x)

v

i s t h e gauge f i e l d i n t h e y - d i r e c t i o n . S

f ~ )

i s one of t h e p o s s i b l e pure gauge f ~ e l d a c t i o n s on t h e l a t t i c e . For example, i n t h e Wilson f o r m u l a t i o n [ 3 ] , SG(U) h a s t h e form :

where N i s t h e number of c o l o u r s a n d B = 2N/g 2 w i t h go = l a t t i c e coupling c o n s t a n t . U i s t h e product of l i n k m a t r i c e s over an eyementary p l a q u e t t e .

P

I n e q . ( l ) , f o r r = o we have a Kogut-Susskind [ 4 ] l i k e a c t i o n plagued by t h e fermion doubling problem [2,5] ; f o r r=l we o b t a i n t h e a c t i o n o r i g i n a l l y proposed by K. Wilson [ 3 ]

.

The main problem w i t h fermions i s t h a t t h e fermions degrees of freedom a r e anticommuting v a r i a b l e s t o bhich u s u a l Monte-Carlo methods cannot be a p p l i e d .

£ 1 ) For l a c k of space I w i l l d i s c u s s o n l y t h e more r e c e n t r e s u l t s f o r S U ( 3 ) c o l o u r gauge t h e o r i e s . Other p a p e r s r e l a t e d t o t h i s s u b j e c t a r e l i s t e d i n r e f . [ l ] ( s e e a l s o t h e t a l k by C . Rebbi a t t h i s c o ~ ~ f e r e n c e )

.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982356

The e x p e c t a t i o n v a l u e of some gauge i n v a r i a n t Operator depending on quark and gluon f i e l d s can be w r i t t e n a s

< 6 ( u , $, V ) =

L t u l

~ [ $ I ~ [ T I ~ - ~ ( ~ ~ ~ ~ T '

o i u , i , i )

/a bid

e-s'ul'r')

JlulTf

B e t d f ( ~ !

I 5

[u, A;'(U)

I

-

t h e formal i n t e g r a t i o n over fermions i s p o s s i b l e because t h e a c t i o n i n e q . ( l ) i s q u a d r a t i c i n t h e fermionic f i e l d s .

The determinant of A ( U ) c o n t a i n s t h e e f f e c t s of t h e fermion loops on t h e gluon Green d u n c t i o n s - f s diagrammatically shown i n f i g . 1 . For a f i x e d gauge f i e l d f c o n f i g u r a t i o n

{u/ ,

A ( U ) i s t h e quark propagator i n presence of t h e e x t e r n a l f i e l d

U ( s e e f i g . 2 ) . Many techniques [ 6 , 7 ] have been proposed i n o r d e r 2 t q compute f L''(U) and d e t b f ( U ) . The enormous number of elements of A ( U ) (32N V

,

where V i s

f f n

t h e t o t a l volume; t y p i c a l l y t h i s number i s of t h e o r d e r of l o y ) makes very d i f f i c u l t t h e computation of i t s d e t e r m i n a n t f o r computer time o r memory problems; on t h e o t h e r s i d e t h e computation o f A-I ( U ) appeared much more f e a s i b l e by using w e l l known r e l a x a t i o n t e c h n i q u e s [ 7 ] . f

A l l t h e r e s u l t s r e p o r t e d below were obtained p u t t i n g d e t A ( U ) = 1 (quenched approximation). We have good r e a s o n s t o b e l i e v e t h a t t h i s i s a goo3 approximation of t h e h a d r o n i c world a t l e a s t i f one excludes very s p e c i a l c a s e s ( l i k e f o r example t h e width of t h e p + 2 s ) . I t i s c l e a r however t h a t t o o b t a i n f u l l y meaningful r e s u l t s it w i l l be n e c e s s a r y i n t h e f u t u r e t o i n c l u d e t h e e f f e c t s of quark loops. I n t h e quen- ched approximation eq. ( 3 ) becomes :

-

1

The quark propagator A f ( U ) has been g e n e r a l l y computed, f o r a f i x e d gauge f i e l d con- f i g u r a t i o n generated by u s u a l Monte-Carlo a l g o r i t h m s , w r i t i n g t h e r e c u r s i v e equation:

B ( U ) : B ( U ) = I - B ( U ) ; I i s t h e i d e n t i t y m a t r i x and 0 an a r b i t r a r y v e c t o r . The f i x e d p o l n t s o l u t i o n ( X f = Xn =

x*)

of eq. ( 5 ) i s :

n+ 1

and g i v e s t h e n t h e i n v e r s e o f A ( U ) . f

3. Computation of hadron masses.- Let u s d e f i n e o p e r a t o r s with t h e same quantum num- b e r o f t h e p a r t i c l e s we want t o measure t h e masses :

I t i s t h e n ' s t r a i g h t f o r w a r d t o compute t h e c o r r e l a t i o n f u n c t i o n s o f t h e s e o p e r a t o r s . For example i n t h e pion c a s e :

JOURNAL DE PHYSIQUE

F i g . 1 - T y p i c a l diagram c o n t r i b u t i n g t o d e t A f ( U )

F i g . 2

-

Quark p r o p a g a t o r i n an e x t e r n a l f i e l d

Flg. 3

-

T h i s f i g u r e shows t h e t y p i c a l behaviour o f t h e s q u a r e masses o f t h e IIand of t h e p a g a i n s t t h e quark mass m ( i n a r b i t r a r y u n i t s ) . The c r o s s e s r e p r e - s e n t t h e measured p o i n t s . The v a l 8 e o f t h e quark mass c o r r e s p o n d i n g t o m = 140 MeV ( m = m *) i s i n d i c a t e d by a d o t . The minimum v a l u e of t h e q z a r k m a s s a t waich Y i n i t e volume e f f e c t s a r e n o t t o o l a r g e and it i s p o s s i b l e t o c m u t e t h e p i o n and r h o masses l i e s v e r y f a r from t h e phy-

P P s i c a l p o i n t m

.

9

G. M a r t i n e l l i

G f ( x , o ) i s t h e propagator of a quark of f l a v o u r f between 0 and x. I n e q . ( 8 ) d e t A ( U ) = 1.

f

I f t h e r e i s only a s i n g l e p a r t i c l e s t a t e propagating we e x p e c t t h a t :

~ ( t ) =

C _-

^{~ ( x ) % e-mt}

X

t f i x e d

On a p e r i o d i c l a t t i c e w i t h p e r i o d T e q . ( 9 ) should be r e p l a c e d by :

G ( t ) % e o s h [ m ( t - ~ / 2 )

1

(10)

The c o e f f i c i e n t m measures t h e mass of t h e hadron i n l a t t i c e u n i t s . Because o f r a d i a l e x c i t a t i o n s and many p a r t i c l e s s t a t e s p r o p a g a t i n g simultaneously w i t h t h e l o w e s t l y i n g s t a t e e q . ( 9 ) becomes t r u e only f o r t ⁺

-.

For f i n i t e t , t o t a k e i n t o a c c o u n t f o r t h e propagation of many d i f f e r e n t s t a t e s and p a r t i a l l y c o r r e c t t h e s e s y s t e m a t i c

f 2 ) e f f e c t s it i s convenient t o p a r a m e t r i z e t h e propagator (on a p e r i o d i c l a t t i c e ) a s

.

G ( t ) = iil cos h [ m l ( t - T/2)1

+

a 2 COS h [ k 2 ( t -

T / ~ ) I

_{m2 > ml} ⁽¹¹⁾

ml i s t h e mass of t h e l o w e s t l y i n g s t a t e and, f o r mesons, a i s r e l a t e d t o f T , f _{1 .}

...

BY f i t t i n g t h e p r o p a g a t o r s one o b t a i n s t h e masses of t h e p a r t r c l e s i n u n i t s of t i e l a t t i c e spacing. T o t r a n s l a t e t h e s e r e s u l t s i n p h y s i c a l u n i t s one h a s t o f i x a funda- mental s t r o n g i n t e r a c t i o n s c a l e and a mass parameter f o r each quark f l a v o u r . T h i s i s n o t however t h e end of t h e s t o r y : one should r e p e a t t h e computation of t h e masses a t s e v e r a l v a l u e s of t h e coupling c o n s t a n t B i n t h e p e r t u r b a t i v e r e g i o n and v e r i f y t h a t A l l t h e masses s c a l e inr$he w e l l known, p e r t u r b a t i v e l y computable way, e - g . :

No s y s t e m a t i c a n a l y s i s [ w i t h t h e e x c e p t i o n of t h e hopping parameter expansion by A. Hasengratz e t a l . [ I ]

1

h a s been y e t done on t h e s c a l i n g behaviour of t h e masses, a l b e i t t h e r e s u l t s , w i t h i n l a r g e s t a t i s t i c a l and s y s t e m a t i c a l e r r o r s , a r e compatible w i t h s c a l i n g . The l a c k of s c a l i n g found f o r e x c i t e d g l u e b a l l s t a t e s [9] i n d i c a t e s t h a t a more c a r e f u l s t u d y of t h i s p o i n t i s r e q u i r e d .

A r e l e v a n t source of s y s t e m a t i c and s t a t i s t i c a l e r r o r s i s due t o t h e f a c t t h a t , a t l e a s t f o r hadrons made by t h e l i g h t e s t (up and down) q u a r k s , it i s n o t pos- s i b l e t o compute t h e hadron masses d i r e c t l y a t t h e p h y s i c a l p i o n mass. I n f a c t , f o r t h e v a l u e s o f t h e parameters commonly used i n t h e l i t e r a t u r e t h e i n f r a r e d c u t o f f Na ( N i s t h e l i n e a r s i z e of t h e l a t t i c e ) i s s m a l l e r o r e q u a l t o t h e p i o n Compton l e n g t h l/mV. Because of f i n i t e volume e f f e c t s t h e minimum p o s s i b l e quark mass t u r n s f 2 ) I t has been r e c e n t l y shown [8] t h a t , even u s i n g a p a r a m e t r i z a t i o n l i k e t h a t o f

eq.11, t h e a c t u a l computations of t h e masses a r e a f f e c t e d by s y s t e m a t i c e r r o r s %lo-15%.

C3-282 JOURNAL DE PHYSIQUE

o u t e m p i r i c a l l y t o be such t h a t :

l / m K

;

⁺ m 500 700 MeV (13)

The s i t u a t i o n i s s c h e m a t i c a l l y r e p r e s e n t e d i n < i g . 3.The measurementsare taken f a r from t h e " p h y s i c a l " r e g i o n corresponding t o m = 140 MeV and an e x t r a p o l a t i o n t o small v a l u e s of t h e quark masses i s needed.

~ g i s

e x t r a p o l a t i o n g i v e s a s y s t e m a t i c e r r o r and a m p l i f i e s t h e s t a t i s t i c a l e r r o r of t h e measured p o i n t s .

4. R e s u l t s f o r hadron spectroscopy i n SU(3).- T h f 3 y e s u l t s f o r hadron spectroscopy, taken from r e f s . [10.11.121 a r e l i s t e d i n t a b l e I

.

I n t h e f i r s t 5 rows t h e tvoe of

-

^L

l a t t i c e fermion a c t i o n ( r ; s e e e q . ( l ) ) , t h e s i z e of t h e l a t t i c e (Volume), t h e avera- ge number of gauge f i e l d c o n f i g u r a t i o n s ( N ) and t h e v a l u e s of t h e coupling c o n s t a n t

( 6 )

used bo o b t a i n t h e r e s u l t s argo?gported t o g e t h e r w i t h t h e t o t a l e s t i - mated computer time ( t ) s p e n t f o r t h e s i m u l a t i o n i n u n i t s o f CDC 7600 CPU time.

The group of r e f . [ l O ] f i x e s t h e s t r o n g i n t e r a c t i o n s c a l e by f i x i n g t h e mass i s predkcted assuming t h e s c a l i n g law o f e q . ( 1 2 ) . I n t h e o t h e r

i s f i x e d s o t h a t t h e p mass i s p r e d i c t e d . I n a l l t h e c a s e s Smasses were f i x e d by f i x i n g t h e pion mass t o i t s p h y s i c a l v a l u e ( i n t h e approximation m

up=mdown)

.

Some d i f f i c u l t i e s have been e n f a y n t e r e d ( a t l e a s t f o r r#o) i n computing t h e masses of t h e 6 ( 9 8 0 ) , A1 gnd

B

mesoys.

.

The reason i s probably due t o t h e use of l o c a l o p e r a t o r s (e.g. 'A = u ( x ) Y yld(x) ) f o r p-wave e x c i t a t i o n s f o r which it is more n a t u r a l t o use s p a t i a l l y extended o p e r a t o r s a s f o r example [I31 1 :

I t h i n k t h a t f o r t h e s e s t a t e s t h e s i t u a t i o n should be f u r t h e r c l a r i f i e d b e f o r e making a comparison w i t h experimental numbers.

The p h y s i c a l r e s u l t s should be independent of t h e a c t i o n we choose on t h e l a t t i c e i n t h e continuum l i m i t a ⁺ 0 (g + 0). Then we e x p e c t t h e r e s u l t s of t a b l e 1 t o be compatible Rmong them. However weOnotice s e v e r a l d i s c r e p a n c i e s :

i ) f T from r e f . 12 ( f o u r t h column i n t h e t a b l e ) i s s i g n i f i c a n t l y below a l l t h o t h e r s . This r e s u l t s seems r a t h e r s t r a n g e i n v i e w o f t h e f a c t t h a t t h e v a l u e s of

- f

f _P a r e a l l compatible f5)

.

ii) The continuum r e n o r m a l i z a t i o n group i n v a r i a n t quark mass [14] t h a t one o b t a i n s s t a r t i n g from t h e r e s u l t s of r e f s . [10,111 i s incompatible w i t h t h e r e s u l t of r e f . [12]

.

T h i s d i s c r e p a n c y c o u l d be due t o l a r g e f i n i t e b a r e coupling e f f e c t s ( t h e measurements were taken a t d i f f e r e n t v a l u e s of

B)

s i n c e t h e b a r e coupling cons- t a n t i s s t i l l of o r d e r u n i t y .

iii) The mass of t h e barvons ( p r o t o n and A

++

) f o r Susskind fermions (second column) t u r n s o u t t o be exceedingly h i g h ( m +mA++)/2 % 1700 MeV.

P

I d o n ' t s e e any p o s s i b l e e x p l a n a t i o n of t h i s r e s u l t . A l l t h e o t h e r r e s u l t s -1

.

f 3 ) The v a l u e s of f K and fl, I n t h e f i r s t column were n o t p u b l i s h e d i n r e f . [ l o ] .

£4) For a d e t a i l e d d i s c u s s i o n s e e t h e o r i g i n a l papers [10,11,12].

f 5 ) Notice however t h a t a d i f f e r e n t o p e r a t o r was used i n r e f [12] t o compute f T .

G. Martinelli

, ¹ - ¹

1 F. FUCITO I H. HAMBER I ^{H. HAMBER} I D. WEINTGARTEN 1 i

I ( G. MARTINELLI ( G. PARIS1 ( G. PARIS1 1 I I

I C. OMERO I I I I I

I G. PARIS1 I I I I I

I R. PETRONZIO I REF.ll I REF.ll I REF.12 1 I

I F. RAPUANO I I I 1 I

I REF. 10 I 1 ^I

I I I I

I I

1

I

0.5 I

1

I r I

I

I I I I I

I 32

1 8 1

28 I

⁸

I ^{Nconf I}

5.6 I

^4.0

1 -4

I 6.0 1 6.0 1

6.0

1 ^5.55 1

R

I

T(o# ^{970 MeV 1 950}

^L

^{100 MeV I} 7 3 0 I ~ ~ ~ M e V 1 m6(g80) I

I I

I I r r r l r r r l l - / I l l

I

I

j 220

5

90 MeV lyl

I

%

200 MeV I 2 9 0 1 i ; w I rn ++

-mp

I

I I ^A

.

. . .

J I

^.

1

TABLE I

JOURNAL DE PHYSIQUE

seem t o be ( w i t h i n l a r g e s t a t i s t i c a l e r r o r s ) compatible. One should n o t i c e t h a t t h e c m u t e d f T and f - l a r e t o o l a r g e when compared t o t h e i r experimental v a l u e s and t h e A ?+P

-

p r o t o n mass $ l i t t i n 9 t o o small.

We b e l i e v e t h a t t h e s e s y s t e m a t i c d i s c r e p ~ n c i e s a r e connected t o t h e f a c t t h a t t h e l a t t i c e spacing i s s t i l l t o o l a r g e compared t o t h e s i z e of t h e hadrons and t h e y should become l e s s s e r i o u s a t l a r g e r v a l u e s of 6. The s y s t e m a t i c over-estimation o f baryon masses, a l t h o u g h t h e Monte-Carlo r e s u l t s a r e compatible w i t h i n t h e e r r o r s with t h e experimental v a l u e s , a p p e a r s t o a r i s e from l a r g e f i n i t e volume ( i n f r a r e d ) e f f e c t s . These e f g e c t s could be removed by going ( a t f i x e d 0) t o l a r g e r l a t t i c e s .

I n t a b l e s I1 and I11 I l i s t o t h e r p r e d i c t i o n t a k e n from r e f s . [I11 and [I51 f o r t h e charm and s t r a n g e quark spectroscopy.

I n conclusion t h e r e s u l t s a r e very encouraging b u t a t p r e s e n t we have pro- blems coming from u l t r a v i o l e t and i n f r a r e d l i m i t a t i o n s of t h e l a t t i c e .

5. The computation of t h e anomalous magnetic moment

1J61 .-

The mass of a Dirac par- t i c l e i n presence of an e x t e r n a l weak magnetic f i e l d 3 i s given by :

m _n = m _o

+ -

_m (n

+ T )

¹

+

⁺

u .

⁺

0

m i s t h e mass i n absence of t h e f i e l d + ; e i s t h e e l e c t r i c charge ; n i s an i n t e g e r l z b e l l i n g d i f f e r e n t Landau l e v e l s and IJ i s t h e magnetic moment. Eq. (15) shows t h a t t h e measurement of t h e magnetic moment can be reduced t o t h e computation of t h e mass of a p a r t i c l e i n an e x t e r n a l magnetic f i e l d . On t h e l a t t i c e a uniform, c o n s t a n t ma- g n e t i c f i e l d i s i n t r o d u c e d by r e p l a c i n g t h e c o l o u r l i n k U ( x ) by :

u uw

( x ) +

u

( x ) x

u

( x ) ^EXT

u u

U~~~ (XI = eiaX Y

which corresponds t o a magnetic f i e l d p o i n t i n g i n t h e d i r e c t i o n with s t r e n g t h :

e l ~ I =

+

a/a ² (171

We t a k e a small enough s o t h a t e q . ( 1 5 ) h o l d s and we d e f i n e :

G(+,+) i s t h e s p i n up, s p i n down ( p r o t o n o r neutron) propagator,+summed over t h e s p a t i a l d i r e c t i o n s a s i n eq. ( 9 ) a t f i x e d d i s t a n c e t. For s m a l l

I H I

and t l a r g e enough we expect :

Using e q . ( 1 9 ) a t t = 4 , 5 we measured t h e gyromagnetic f a c t o r f o r t h e p r o t o n and t h e neutron q p , N , d e f i n e d a s :

- 2 m p , ~

.

^'P,N

'P,N = e

G. M a r t i n e l l i

e i s t h e u n i t e l e c t r i c charge ;

up

^and

\

a r e t h e measured p r o t o n , n e u t r o n magne- t i c moments (eq. 19) and masses r e s h e c t i v e l y :

A t t = 5 we o b t a i n e d :

gp = 2.96

+

^{0.58 (exp}

*

2.79) gN = -1.93

+

^{0.45 (exp}

*

1.90) lgp/gNI = 1.60 ? 0.15 (exp

*

1.47)

i n f a i r l y good agreement with t h e experimental v a l u e s .

Remember however t h a t t h e s e r e s u l t s a r e a f f e c t e d by r a t h e r l a r g e s y s t e m a t i c e f f e c t s because of e x c i t e d s t a t e s propagating simultaneously w i t h t h e lowest l y i n g s t a t e s f o r s m a l l t.

CHARMED MESONS

m (2980) 13000

"

^{30 MeV}

^[

I ^{? I}

^I

I 1

3400 2 50 WeV

I

I

^m,, (3414)

I I

I

^{m, (3507)}

1

^{3500 f 50 MeV}

I ^pc i

1 I

+- 1

3600

+

^{100 MeV}

¹

I I

TABLE I1

STRANGE HADRONS

i

^mK*

1

890

+

70 MeV

j

I I

I

I

^m$

I

990 t 50 MeV

i

I I I

JOURNAL

DE PHYSIQUE

References.. -

[I] -

HAMBER H. and PARISI G., Phys. Rev. Lett.

47,

1792 (1981) ;

MARINARI E., PARISI

G.

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314 (1982);

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D.,

Phys. Lett.

m,

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.

[41

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See for example SUSSKIND L., ~ h y s . Rev.

G,

3031 (1977)

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[5]

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E ,

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[6]

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T.

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2-4,

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[lo]

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[ 11

1 -

HAMBER H., PARISI G., Brookhaven Report, BNL 31322 (May 1982)

.

[I21 - WEINTGARTEN

D.,

Indiana Univ. Preprint IUHET-82 (June 1982)

.

El31 - ^KUNTZ.

Z,

talk delivered at the " ~ 1 t h Worshop on Current problems in High Energy Particle Theory", Firenze, June 2-4 (1982).

[I41

-

GONZALES ARROYO A., MARTINELLI G. and YNDURAIN F.J., Frascati report

~ ~ ~ - 8 2 / 3 3 (May 1982) to appear in Phys. Lett. B.

1151

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MARTINELLI G., OMERO C., PARISI G., PETRONZIO R., CERN-TH 3335 (June 1982) to appear in Phys.Lett. B.

[I61

-

MARTINELLI G., PARISI G., PETRONZIO R., KAPUANO F., CERN

-

TH 3334 (June 19821 to appear in Phys. Lett. B.