IL NUOV0 CIMENT0 VOL. 23 A, N. 2 21 Set tembre 1974

Scalar Multiplets and Asymptotic Freedom (*).

R. DELBOURGO a n d ABDUS ~ALAM

.Physics Department, Imper ia l College - London

J. STRATHDEE

International Centre for Theoretical Physics - Miramare (Trieste)

(ricevuto 1'11 Aprile 1974)

S u m m a r y . - - In a gauge theory of scalar mesons interact ing with gauge vector particles, we show that scalar-meson four-point Green's functions are free of infinities, provided that i) the scalar mesons do not couple to any fermions in the model and ii) there is no direct 2~4-term in the Lagrangian. The proof relies on a ~ gauge approximation >> technique which systematically exploits the information provided by Ward-Takahashi identities. For non-Abelian gauge theories, the significance of this result is tha t a large class of scalar mult iplcts which can induce spontaneous symmetry breaking do not affect the issue of asymptotic freedom. In order to exploit the information of Ward-Takahashi identi t ies we use the <( axial gauge ~), in which there are no fictitious scalar particles and the identities assume their naive form, with Z~ = Z~ for all mat ter and gauge fields.

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

I t is c o m m o n l y be l i eved t h a t i nc lu s ion of H i g g s - K i b b l e m u l t i p l e t s i n n o n -

A b e l i a n gauge theor ies as a ru le (1) works a ga i n s t a s y m p t o t i c f reedom, if these

m u l t i p l e t s are used to i m p l e m e n t s p o n t a n e o u s s y m m e t r y b r e a k i n g . I n th i s

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (1) D. GROSS and F. WILCZEK: Phys. Rev. D, 8, 3633 (1973); T. P. CHE~G, E. EICHTEN and LING-FoNG LI: SLAC preprint .

237

2 ~ R. I )ELBOURGO, A.. SAL&M a n d Z. S T R & T H D E E

p a p e r we wish to argue t h a t the issue of a s y m p t o t i c f reedom (2) is unaffected b y the inclusion of wide classes of scalar representa t ion , p rov ided such scalar mul t ip le t s do not in te rac t wi th fermions or wi th themselves via direct 2~v4-1ike couplings.

The conflict be tween scalar fields and a s y m p t o t i c f reedom comes abou t in the following way. Fi rs t ly , the inclusion of scalar mut l ip le ts affects the sign of fl~, the Ca l lan-Symanz ik funct ion (3) for the vector particles associated wi th the var ia t ions 3/~g: the cont r ibu t ion of scalars and spinors to fi~ is posi t ive so t h a t in this respec t scalar mul t ip le t s are on a par with any fermion mul t ip le t s in the model . I n order to secure the negat ive definiteness of fig - - a necessary condi t ion for a s y m p t o t i c f r e e d o m - - i t becomes i m p o r t a n t to l imi t the n u m b e r ans size of scalar mul t ip le ts , t hough one finds t ha t in prac t ice this is seldom a serious ba r to the i r in t roduct ion. The more dangerous effects of including scalar mul t ip le t s comes abou t th rough a second Ca]lan-Symanzik func t ion fl~ associa ted wi th var ia t ions ~ / ~ in the effective p a r a m e t e r for scalar-scalar sca t te r ing 2~ 4 which represents self-interactions among these mul t ip le ts .

W h y is the p a r a m e t e r 2 needed in the renormal iza t ion group equat ions in addi t ion to the gauge field coupling p a r a m e t e r g? To be sure, f undamen ta l t e rms of the t y p e ~ 4 are not essential for genera t ing nonzero expec ta t ion values ( ~ in the t heo ry ; the ( ~ ¢ 0 effects can be produced rad ia t ive ly as shown b y COLEMAN and WE[NBERG (4) withou t any direct 2~ 4 terms. The reason why one is accus tomed to in t roduce intrinsic ~ 4 couplings is t h a t such t e rms are supposed ly t ied in wi th the renormal izabi l i ty of the theory , the convent iona l a r g u m e n t being tha t , even if no such t e rms are included in the Lagrang ian f r o m the outset , the logar i thmic infinities associated with ~-~ sca t te r ing graphs will br ing t h e m back (5) as counter - te rms. I n the renormali- za t ion group language, i t is the logar i thmic dependence on the cut-off of the ~4 ampl i tude which necessi ta tes the in t roduc t ion of a new independent ~-para- m e t e r in the theory .

I n this pape r we wish to argue t h a t in a gauge t heo ry of scalar and vec tor mesons, wi th no fermions and no ;t~ 4 coupling, the ~4 ampl i tude is ac tual ly ]inite and independen t of the cut-off a f ter the convent ional renormal izat ions of mass , wave func t ion and gauge charge g have been performed. This means t h a t in such theories ~ can in principle be compu ted in t e rms of the renor- mal ized g, and the ~4 Green ' s funct ion is on pa r wi th all higher n-point Green 's

(2) D. POLITZER: Phys. Rev. Lett., 30, 1343 (1973); D. G~oss and F. WILCZEK: Phys. t~ev. Zett., 30, 1346 (1973). (a) C. CALLAN: :Phys. Rev. D, 2, 1541 (1970); K. SYM)~NZIK: Comm. ]lath. Phys , 18, 227 (1970); L. V. OSVIANNIKOV: Doklady Abad. Nauk SSSR, 109, 1112 (1956). (4) S. COLEMAN and E. WEINB~.RG: Phys. Rev. D, 7, 1888 (1973). (5) P. T. MATT~IEWS: Phys. Rev., 80, 293 (1950).

SCALAR MULTIPL;ETS AND ASYMPTOTIC FREEDOM ~ 3 9

functions. And since there is no intrinsic dependence of ~4 on the cut-off, we have only fly ~/~g appear ing in t he Cal lan-Symanzik equat ion. Hence this class of scalar mul t ip le t has no bear ing on the p rob lem of a s y m p t o t i c f r eedom except for its re la t ively mild effect on the sign of fl~.

The ma jo r result, crucial to the a rgument , t h a t in gauge theories the re exists a consis tent (nonper turba t ive) scheme of c o m p u t a t i o n whe reby O-0 sca t ter ing is au tomat i ca l ly finite i) p rov ided no bare ~ in te rac t ions exis t and ii) p rov ided t ha t there are no fermions in the theo ry (or equ iva len t ly t h a t there is some group- theore t ic a r g u m e n t which forbids di rect ~ 0 inter- actions), was in fact demons t r a t ed b y the au thors (e) in some fo rgo t ten paper s of 1964. We shall briefly recap i tu la te our work in the nex t Sect ion which concerns the s imple case of scalar e lec t rodynamics and the idea of the (( gauge app rox ima t ion ~) technique. I n Sect. 3 we show how the t echn ique can be general ized to non-Abel ian gauge models. Because the gauge a p p r o x i m a t i o n makes h e a v y use of Ward -Takahas h i identi t ies, we choose to work in the noncovar ian t (~ axial ~) gauge whose mer i t s have been s t rongly extol led b y LEE. This gauge is character ized b y the absence of fictitious scalar par t ic les , has t he grea t v i r tue of respect ing Z~ ~-Z2 and leaves all gauge ident i t ies of one- part icle irreducible vert ices in thei r s imple naive fo rm; thus gauge t heo ry is exhib i ted a t its least myster ious . ( Indeed we believe t h a t in spi te of its noncovar i~nt s t ructure , this is the gauge of choice whenever one needs to work with Ward -Takahas h i identit ies. This is set out in detai l in an Append ix to the paper . ) Section 4 is devoted to a discussion of our results in the l ight of the renormal iza t ion group techniques for eva lua t ing h igh-energy behav iou r ~nd the issue of a sympto t i c freedom.

2. - Sca lar c l e c t r o d y n a m i c s .

I n this Sect ion we shall s tudy the interact ions of pho tons wi th charged mass ive scalar mesons and will r ap id ly review the re levant pa r t s of our 1964 pape r (6) to explain the (~ gauge a p p r o x i m a t i o n )) t echnique and how it renders finite scalar-scalar scattering. I n the following Section we shall generalize the work to non-Abel ian gauge theories.

On account of gauge invar iance, the Green 's funct ions of scalar electro- dynamics sat isfy Ward -Takahas h i ident i t ies connect ing processes wi th in- creasing numbers of photons. I f F (') denotes the one-part ic le i r reducible n-poin t funct ion, we can ex t rac t coupling cons tan t factors to define, in m o m e n t u m

(6) A. SALAM and R. ])ELBOURGO: Phys. Rev., 135, B 1398 (1964); J. STRATHDEE: .Phys. Rev., 135, B 1428 (1964).

2 ~ 0 1~. D]~LBOUI%GO, A. SAL~kM ~nd J. STRATttD:EE

space,

Fc2)(p) A-~(p) ,

/~(~(p' ; p) ~ gF~(p'; p ) ,

F~(p ' , k' ; p, k) ~ g2 C ~,(p', l~' ; p, k) ,

the inverse scalar propagator /1 -~ ,

the proper ver tex funct ion Fx ,

the Compton par t C~,,

in the older terminology. All quanti t ies appearing above are renormalized and the Ward identi t ies among t h e m read

(1) (p ' - - p)~ F~(p' ; p ) = A-~(p ') - -A -~ (p ) ,

(2) k~' c. , , (p' , k'; p, ~) = F . (p ' ; p ' + k') - F . (p - k', p ) ,

k" Q . ( p ' , k'; p, k) = C ( p + k, p) - - C ( p ' , p ' - - k ) ,

etc. These identi t ies app ly equally well to the bare functions. In fact the convent iona l gauge-invar iant Feynm~n per tu rba t ion series is generated by taking the free p ropaga tor A ~ ( p ) - ~ p ~ - - m ~ as the s tar t ing approximat ion and using the corresponding (longitudinal) solutions to (1) and (2), viz. F a = ( p ' ~ p ) a , C =2~],z with all higher /'('~>=0 as input in a closed set of equat ions (e.g. the Dyson-Schwinger equations for A, F and C) together with a skeleton expansion. Quite clearly a different (nonperturbat ive) solution of the gauge theo ry could be genera ted self-consistently b y s tar t ing with another A-l(p), r a the r t h a n the free A-~(p), with longitudinal parts of /~ and C de te rmined by (1) and (2). This is the basic idea which underlies the <( gauge approx imat ion ~) technique.

The procedure relies upon finding suitable solutions to the gauge iden- t i t ies (1), (2) etc., solutions which supply the most significant par ts of F, C, etc. in te rms of the p ropaga tor funct ion zJ. So far as high-energy behaviour is concerned, the identi t ies, crudely speaking, tell us t h a t

F ~ ~d-l l~p , C,'~ ~FI~ p ~ ~2 ~-11~p2, e tc . ,

indicat ing t h a t the asympto t ic behaviour of A can play a critical role in shaping the high-energy characterist ics of the F<% The self-consistency of the scheme of course needs to be guaran teed th rough the asymptot ic behaviour of the s tar t ing A, which is itself a funct ional of A, F and C. We shall not go into a discussion of this s tabi l i ty problem, referring the reader instead to the 1964 paper, bu t will only r emark t ha t the self-consistency criterion s ta ted there in (i.e. vanishing of wave funct ion renormalizat ion constant) apparen t ly holds ident ical ly for renormalizable gauge theories.

SCALA~I~ MULTIPLETS AND 2~SYMPTOTIC FREEDOM 241

To show how the technique of gauge a p p r o x i m a t i o n works, consider t he ~ lowest order )). The general solution of i den t i ty (1) is

(3)

F~ -~-F~ , where

F~(p' ; p) ---- ( p ' + p)~ [A-~(p') - - A-l(p)]/(p ' ~ - p2) ,

, 2 ] F[(p' ; p) = [ (p ' - -p)~(p ' - -p) , , - - (p - -p ) U~. ( p ' + p ) " T ,

and T is a scalar invar ian t funct ion, s y m m e t r i c under the in te rchange of p,2 with p~. The decomposi t ion into t r ansverse and longi tudinal pieces of the v e r t e x funct ion is unique since F ~ is defined to ca r ry no k inema t i c fac tor p ' - - p . Iqote that

( p ' - - p ) ~ F ~ = 0 and ( p ' - - p ) ~ F ~ = ~J-'(p') -- LJ-'(p) .

We can likewise make the decomposi t ion

C~ L~ C ~r4- C ~L4- T~

where C ~L is ent i rely de te rmined f rom the p ropaga to r , whereas the t r ansve r se components br ing in funct ions which are unde te rmined b y the gauge ident i t ies (for details see eq. (II.94) ref. (~)). More precisely, the expl ic i t ly d - d e p e n d e n t

app rox ima t ion C LL which satisfies (2) is g iven b y

(4) [A-l(p -~- k) - - A- l (p ' - k)]

c ~ ( p ', k '; p , k) = 2w~, (p + k) ~ - ( p ' - - kV - -

( p ' ÷ p ÷ k ) ~ ( p ' ÷ p - - k ' ) . [A-~(p + k ) - - A - ' ( p ' ) A - ~ ( p ' - - k ) - - z l - l ( p ' ) ]

( p ÷ k p - - ( p ' - - k p [ ( p ÷ k ) ~ - - p '~ - - ( p ~ - - k ) ~ - - p ~ J - -

( p l + p - - k ) ~ ( p l + p --~ ~f)~t [ A - - l ( p -~- k ) - - A - - l ( p ) A - i ~ ~-~(p)] (p+k)~--(p'--k)~ [ ~+k)~--p~ (p--~)~--p~ ]"

This displays the Bose s y m m e t r y of the ampl i tude under the separa te inter- charges p ~-~ - - p ' ; k,/z ~-~ - - k', ~. A similar procedure could be used to genera te

all the longi tudinal higher F (~) in t e rms of A. To find the s ta r t ing A the general p rocedure suggested in ref. (~) is to use

the gauge approx imat ions F ~ and C ~ in the Dyson-Schwinger equa t ion for A, which, on account of this approx imat ion , is now an equa t ion for A alone. After we have found A, F ~ and C Lr etc. can be c o m p u t e d b y means of the Dyson-Schwinger equat ions for F and C. This t hen pe rmi t s a r e -eva lua t ion of A and so on. The procedure is sys t emat i c t hough tedious.

Now our interest in this pape r is to d e m o n s t r a t e the finiteness of ~4 in the gauge approx imat ion . We shall do so in the lowest order using a s impler dispersion approach to compu te the stal~ing A, ins tead of the more c u m b e r s o m e

242 R. DELBOURG0, A. SALAM and J. STRATHDEE

a p p r o a c h b a s e d on t h e D y s o n - S c h w i n g e r e q u a t i o n s . Cons ide r t h e t w o - p a r t i c l e

u n i t a r i t y c o n t r i b u t i o n to I m z1-1 w i t h F a p p r o x i m a t e d b y i ts l o n g i t u d i n a l

p iece , F z of eq. (3):

I m ~ -~ (p ) = ½ e~fI~(p, p + k ) F ~ ' ( p + k , p)( i+[(p -~k) ~ - m ~] I m k ~ D~'~(k) d4k/(2z) ~ .

I n a c o v a r i a n t g a u g e spec i f ied b y

k~ Dl,~(k) = -- ~]i,~ Jr- (1 -- a) kl, kv/k2

o n e e a s i l y sees t h a t t h e a b o v e c o n t r i b u t i o n l e a d s to t h e d i s p e r s i o n r e l a t i o n

1 e2(3 - - a) ~ (p,Z ÷ m 2) alp,2

E q u i v a l e n t l y we h a v e a n i n v e r s e p r o p a g a t o r s p e c t r a l f u n c t i o n ( I m A - ~ - - -

- - I m ALIA] 2) w h i c h b e h a v e s (7) a s y m p t o t i c a l l y as

I m gl-~(p) ~ e2(3 - - a)p~/16:~ ln2(p~/m 2) .

W e o b s e r v e t w o i m p o r t a n t consequences , e v e n a t t h i s m o s t r u d i m e n t a r y l e v e l :

i) F~(p'; p) = ( p ' + phflm 3 - , ( s ) ds/r~(s -- p~)(s -- p,2) P

I n p a r t i c u l a r , for p ~ 2 = m ~ a n d p ~ - - > - - c ~ ,

$ '~(p ' ; p)]---- ( p '÷p )2z l -~ (p ) / ( p 2 - m 2) --> ( p ' ÷ p ) a / c in( - -p~/m~) ,

w h e r e c ~ e2(3 - - a ) / 1 6 z 2.

ii) F o r p~ =- p,2 __ m ~

2 i5) c ~ ( p ', ~ ; p , k)l

c In ( - - k2/m 2) 2 [ ( p ' ÷ p)~(p' + p)~,-- k~k~]

÷ ck2 In 2 ( - - k2/m 2)

T h u s as one m o m e n t u m k in F or C b e c o m e s a s y m p t o t i c E u c l i d i a n , we d i s c o v e r

a d a m p i n g f a c t o r (ln k2) -1 for F a n d C. Th i s is in a c c o r d w i t h w h a t one w o u l d

e x p e c t f r o m c r u d e l y d i f f e r e n t i a t i n g A -~ once or t w i c e t o o b t a i n a p p r o x i m a t e F

a n d C r e s p e c t i v e l y .

(7) One may also wri te a dispersion relat ion for A -I provided due care is taken wi th a n y possible CDD zeros. See footnote (22), ref. (6), where it is shown tha t such zeros do not influence the high-energy behaviour of A -1.

SCALAR MU LTI PL E T S AND ASYMPTOTIC F R E E D O M 243

Let us now compute meson-meson scattering. I n the lowest-order gauge approximat ion we draw Dyson ' s skeleton diagrams in order e 4 and replace each line by A (as calculated above) and each vertex by eF L or e 2 C LL as appro-

priate. Basically then we have a set of one-loop integrals to evaluate, as depicted in Fig. 1. So far as infinities are concerned, we note t ha t whereas z]

is more divergent t han dj by a power of In (p~/m2), the ver tex insertions F and C are more convergent t han /'1 and Ct. These damping factors more t h a n counterbalance the diverging behaviour of ~, and it turns out t ha t the scalar-

a) b) c) d.) e)

Fig. 1. - Contributions to ~4 in the lowest-order gauge approximation.

scalar scattering ampli tude is indeed finite. To prove this assertion it suffices to place the mesons on their mass shells. We can work in an a rb i t ra ry pho ton

gauge, neglecting vacuum polarizat ion insertions at this stage. There are the five diagrams of Figs. la) to e), plus their crossed companions, to contend

with; retaining the leading behaviours in the integrands, one finds tha t

Ma,~, M b ~ _ 1 M ~ , ~ _ ½ M ~ M ' / 2 ( 3 4- a s) ~ -- i e ' f d ' k / ( 2 ~ ) ' z]2(k) k"

or altogether

M oc - - i f d ' k/k" ln~( - M / m 2) ~ f d q 2 / q 2 In 2 q~,

which is finite (s). This means tha t we have no need of any ( ~ ) ~ infinite counter-Lagrangian if we work within the gauge approximation. I n fact the a rgument carries through for A(p ) - ,~p -2 ( lnpS ) 8, F x , ~ p a ( l n p 2 ) -~ providing s > 1.

We have so far considered just the (~ lowest order ~> in the gauge approxi-

mation. I n ref. (s) it was shown t h a t the finiteness of meson-meson scat ter ing

(s) To see that the diagrams exhibited do provide a gauge-invariant mass shell ampli- tude, write a dispersion integral in momentum transfer: the absorptive part (obtained by cutting the two-photon lines) is the product of two mass shell gauge-invariant Compton amplitudes. In other channels, the prescription for gauge invariance is more elaborate.

244 R. DELBOURGO. A. S~_LA~ aJId J. STR~THD]~

persists in (( h igher orders ,) of the a p p r o x i m a t i o n technique which are sys tem- a t ical ly cons t ruc ted b y using the Dyson-Sehwinger equations. We refer the r eader to ref. (6) for details.

3. - Scalars in n o n - A b e l i a n g a u g e theor ies .

Mussless ¥ang-Mil l s gauge fields W~ obey qui te compl ica ted Slavnov- Tay lo r ident i t ies (9) in eovar ian t gauges specified by

(6) ~ v . . = [--~],v-~- ( ] - - a ) ] ~ k / k e]~ab/k2,

owing to the occurrence of fictit ious par t ic le t e rms (~o). At the same t ime these fictit ious contr ibut ions des t roy Z~- -Z~ equalit ies which exist in the Abel ian case ( though, happi ly , Z~/Z2 is source independent , even if the rat io

is infinite in p e r t u r b a t i o n theory) . The gauge a p p r o x i m a t i o n relies heav i ly on solving Ward -Takahash i iden-

t i t ies. I t is clearly desirable to seek some formula t ion in which the gauge ident i t ies are not obscured b y fictitious t e rms bu t assume the convent ional fo rm

(7)

(p'-- p)~ I'~(p' ; p) = A-~(p ') T ~ - T~ A-~(p) ,

k l ' F b a l - ' k ' i ~baF ~ ' b , T ~ _ _ ~ F b = F ~ ( p , p ' ~ - k ' ) T ~(l, k ' ; p ) , . , , i , , ; p, k) ] ~(p ; p) +

e t c . ,

which any unsuspec t ing theor i s t would wri te down. This can be achieved in a noncova r i an t gauge which sets W o - - 0 . The

f o r m a l fea tu res of this gauge are summar i zed in the Appendix . Here we would like to highl ight some of the more prac t ica l consequences, since they remove m u c h of t he m y s t e r y cloaking the subjec t of non-Abel ian gauge theory at the

presen t day :

a) The gauge ident i t ies for one-par t ic le irreducible Green 's funct ions are t he na ive ~Vard-Takahashi set (7) which one would obta in b y canonical methods . I t is re la t ive ly easy solve these in the gauge approx imat ion .

b) The dynamics is curried ent i re ly b y the spatial components of the Green ' s funct ions . This is because ex te rna l polar iza t ion vectors are pure ly spacelike, while for in terna l lines the t e m p o r a l components of the gauge field

(9) A. SLAVNOV: Kiev ITP-71-83 (1971); J. C. TAYLOR: Natl. Phys., 33 B, 436 (1971); B. W. LEE and J. ZINN-JusT~N: Phys. Rev. D, 3, 3121, 3137, 3155 (1972). (10) R. P. FEYNMAN: Acta Phys. Polon., 26, 697 (1963); B. D]~W~TT: Phys. Rev., 160, 1113 (1967); L. FADEEV and V. N. POPOV: Phys. Lett., 25B, 29 (1967).

SCA.LAI~ MULTIPLETS AND kSYlVIPTOTIC FI~F~EDOM 2 ~

D a b ~ - ab _ _ propaga tor vanish identically. Specifically, Do~ o~ D ~ o - 0 and

(s) a b . D~, (k) : ((5,-- ki k~/k~o) (~ab/k2

character izes this axial gauge. Al te rna t ive ly s tated,

(s') Dab{k) - - k, -~- ,~b/k2 ,

where K-- - - (ko ~, - - kko2 ) . One needs to in terpre t the s ingular i ty at ko----0 as a principal value for the following reason: since the Yang-Mills par t ic le has two physical degrees of freedom, the absorpt ive par t of the free p ropaga to r must equal

o r

= f .

This is consistent with the propagator , obta ined b y inver t ing the Lagrangian ~ z ½ [~ A . ~ A ~ (a.A)2], which equals

~ , j ( x ) =fexp [ i k . x ] ( (~ i J - ki ~ / k 2 ) / ( k 2 - i ~ ) ,

provided the noncovar iant pole at ko----0 has no absorpt ive par t and is therefore t aken as a Cauchy principal value.

Note the impor tan t point tha t , for large k, D behaves like 1/k ~ so the theory is renormalizable in the axial gauge.

c) Although the off-shell Green's functions are noncovar iant , all on-mass- shell S-matr ix elements are necessarily Loren tz covariant , and this includes the inverse propagator .

d) The pe r tu rbs t ion inJinit ies of F e y n m a n graphs have a covariant structure, a result which is perhaps not ve ry surprising in view of c), since the infinities have to be associated with counte r - te rm Lagrangians and this will affect mass shell elements. In any case, as we shall see, Z~ str ict ly equals Z2.

e) The crucial point for inf ini ty-counting and renormal iza t ion purposes is t ha t even though the Yang-Mills p ropaga tor is noncov~riant , it exhibits the scaling proper ty , D-~(~p) = ~ D-~(p) under p -* 7tp. The homogene i ty p rope r ty in p and Po for high-energy behaviour carries over to the Green's funct ion F('*); it sets the character is t ic dimensional scale ~4-~ (times some log- ar i thmic dependence on Jt due to in ternal loop integrations) at large Eucl idian momen ta despite noncovar iance of the a m p l i t u d e - - a fac t which has obvious bearing on the use and applicabil i ty of renormal izat ion group methods.

246 R. ]:)~]LBOURGO, A. SALAM and j. STRATHDEiE

One can br ing out all of these points b y working out in some detai l the example of the sca lar -meson self-energy / / . :Naturally H is noneovar ian t ; we shall regard it as a func t ion of the covar ian t p~ and the noncovar ian t p~:

ig2e2(R) ( d ' k ( 2 p - - k ) . ( 2 p - - k ) - - K . ( 2 p - - k ) k . ( 2 p - - k ) (9) / / ( p ~ , p ~ ) - - ~ j k 2 ( P - - k ) 2 - m 2 ,

where Ca(R) is the value of the Casimir opera to r wi thin the scalar-part icle mul t ip le t . Using dimensional regular izat ion to discard cer tain quadra t ic infinities, we can s impl i fy H to read

ig 2 C~(R) f d4k (2p - - k) ~ ÷ (p~- - m 2 ) K .(2p - - k) (9') II(p2, p~) - (2~) 4 k 2 ( P - - k ) ~ - m 2

:Now / / ( p ~ p~) is v is ibly covariant at p 2 = m 2, its dependence on p~ vanish ing on the mass shell. :Next, consider the u l t ravio le t infinite pa r t

i g 2 C 2 ( R ) ( d 4 k [ ( k2)__p2 (8 4 k 2 o ~ ~[16k~ 4)]

Since f4q~dqo/q" = f d q o / q 4 = i z / 2 ] q l 3, w e see t h a t

(10) IId~.(P ~, P~o) = g~ C~(R)(6P ~ -- 3m~)In A / 8 ~ ~ ,

being independen t of p~, is a covariant infinity. I n fact the infinite pa r t of the wave func t ion renormal iza t ion is

(11) Z2 = 1 - - 3g 2 C2(R) In A / 4 ~ 2

to this order and the renormal ized p ropaga to r is

(12) ~-~(p) -~ Z2(p 2 -- m ~ ~- ~m 2) -- l I ( p ~, p~) .

H a v i n g isolated the infinities, we can m a k e the expansion

i i (p2 , p~) = A m 2 ~_ Bp~ j r (p2 _ m ~) [H0~(p~) -~ (p~ -- m2)YL(p ~, p~)] ,

ar rang ing z2 and ~m 2 to cancel the infinite constants A an B so as to give the finite answer

(13) A-~(p) = (p2 _ m ~) [1 --I lof(p~) -- (p~ -- m 2 ) II~(p ~, p~)] = (p~ -- m ~) Z ( p ~, p~).

Thus our par t ic le p ropaga tes wi th the correct relat ivist ic e n e r g y - m o m e n t u m relat ion, corresponding to a covariant pole in A; but the residue Z--~(m~,p~)

S C A L A R M U L T I P L E T $ A N D A . S Y M P T O T I C Ft~]£]~DOM ~ - 4 7

at the pole, as defined in eq. (13), is the noncovariant q u a n t i t y 1 ~-IIo~(p~o), reflecting the choice of gauge. I n second-order p e r t u r b a t i o n theo ry the res idue at p 2 = m ~ is

Z - i ( m 2, p~) = 1 + 4u ~ (p~-- J P(~ _ 4 _ ~ ~ _ ~ _ m 2 ] ,

which is real for p ~ < m ~. An i m p o r t a n t point for fu ture discussion will be the quest ion of dispers ion

integrals for the p ropaga to r . He re a t last we have to face up to the disad- van tage of the axial gauge, its noncovar iance . Le t us examine the s ingular i t ies of the s e l f - e n e r g y / / as a guide to t he general p roper t ies of LJ. The denomi- nators of the in tegrand in

ig 2 C2(R) ~ d4k (2p - - k) ~ ~- (p2 _ m ~) (k ~ _ 2p. k -~ 2k~ - - 4poko)/k~ (9") H ( p 2 , p ~ ) - (2~) ~ - j ~- ( P - - k ) ~ - m 2

vanish when

a)

b)

and

c)

ko = ~ [k[ :y i~ ,

ko --~ - - Po • [m 2 -~ (P -4- k)2] t T i s

ko = ~: ie ,

where in accordance with the pr incipal -value prescr ip t ion s t a t ed nbove one mus t average over bo th signs of is in c). Confluence of a) and b) s ingulari t ies corresponds to a covar iant discont inui ty , while confluence of b) and c) singu- larities is noncovari~nt . One can show tha t , provid ing IPol < [ m 2 ~ (P -~ k)~] ½, the second kind of confluence cannot arise; therefore so long as po 2 < m 2, t h e usual covar iant Cutkosky d iscont inui ty formula holds and we m a y wri te

(14) //(p2, p~) =_ ~ f I m l I ( s , p ~ ) d s s _ p 2 _ i s ,

where

(15) ?

SJr ~ I m I I (p 2, p~) = g2 C2(R)jd 4 k ~+(k 2) 6+[(p -- k) 2 - - m2] •

• (2p -- k)~ [5,j -- k~ kj/k~] (2p - - k)j

is given b y the t ransverse-vec tor states. A s imple calculat ion yields the resul t

(16) I m / / ( p ) - - g2C2(R) [ Po l n P 0 - ]P]] 4~ (v2-m2)o(p2-m2) - 2 + ~ ~ o + ~ ] =

__ g2C2(R) 27~ (p2--m~)O(p~--m~)[1 + {P2/P~°--I}-~ tg-~ {P2/PI--I}½] "

2 4 8 R. DELBOURGO, A. SALEM and a. STRATHDEE

Once again we wish to draw a t t en t ion to the homogene i ty of behav iour of I m / / ( p ) in the var iables P0 and lpl f rom the point of view of scaling p - ~ p . I n fact , I m / / ( ~ p ) ~ 2 ~ as 2 - ~ c ~ in this order. Generalizing (14), the full p r o p a g a t o r m a y be expec ted (11) to sat isfy a dispersion relat ion of the t ype

= f ~ ( s , p ~ ) d s Z-~(m2, p~) + ~ ~(s ,p~)ds (17) A(p) .]p~--s + i s - - p ~ - - m 2 .) p2 - - s + i s '

where q(p~, po 2) or a(p 2, p~) are funct ions of p2 and p~/p2 and behave as ~2 under t he scaling p - > ~p for l~rge 2.

I t is s t r a igh t fo rward to eva lua te other one-loop diagrams. For instance, the p roper scal~r-vector ve r t ex p a r t F~ is g iven by the sum of five graphs. A careful compu ta t ion , which we shall no t exhib i t here, shows t h a t the infinite pa r t s of the ve r t ex fo rm a covar ian t whole, i.e.

FOd,~ = ( p ' + p ) o ( Z ~ - - l ) and F~d~= ( p ' + p ) ~ ( Z l - - 1 ) ,

where Z~ = Z~ and is g iven b y (11). The same a rgumen t applies to the Yang- Mills self-energy and ve r t ex pa r t s : a simple, though tedious, calculation shows t h a t the cont r ibu t ions f rom vector and scalar loops (in the no ta t ion of GRoss and WILCZEK) add up to

Z3 = 1 + [1198 C2(G)/247t ~ -- ~, g~ T(R)/24~ 2] In A , R

which equals the Yang-Mills ve r t ex func t ion renormal izat ion. Al though no ficti t ious part icles occur in the axial gauge, the negat ive contr ibut ion f rom the meson loop to Z3 m u s t be adduced to the spat ia l longi tudinal pa r t (ks kj/k 2) of the p r o p a g a t o r

31j - - k ik j lk 2 kiki lk 2 (8") D . ( k ) - - k s + k~- .

The perspicacious reader will have not iced t h a t the expressions for the infinite Z coincide in order g~ wi th those ob ta ined in the covar ian t gauge a ~ - - 3 , the special gauge in which Z I = Z~ is respected. Whe the r one can a lways choose a covar i an t gauge a = - 3 + ](g2) so t h a t Z1 = Z2 is preserved in all orders and coincides wi th the noncova r i an t formula t ion we do not know at the present t ime.

(11) We believe that the correct form of (17) should be a double dispersion integral since beconles complex for p~ ~ m 2 (e.g. see (16)). However (17) is quite sufficient for our purposes as we shall be integrating over Euclidian four°momenta where p~ is negative, Q is real, 2 is large and only the high-energy behaviour of Q is pertinent.

S C A L A R M U L T I P L E T S A ND A S Y M P T O T I C F R E E D O M 2 4 9

Return ing to the gauge app rox ima t ion , the fac t t h a t t he renormal ized p ropaga to r depends on p~ and p~ s epa ra t e ly means t h a t t he solut ion of the Ward -Takahash i iden t i ty (7) will no t lead to a covar ian t v e r t e x (except for its infinite part) . Howeve r i t is good to r e m e m b e r t h a t only the spa t ia l p a r t F~ of F~ is r e l evan t to the dynamics , since th is is the only p a r t which couples to wave funct ions and propagators . A solut ion of

( p ' - - p)~ Fa(p' ; p) ---- (p': -- m ~) Z ( p '~, p'o ~) -- (p~ -- m ~) Z ( p ~, p~) :

_~ ½ (p,2 _ p~) [Z(p'~, p,o~) ~_ Z(p~, p~)] ~_ ½ (p,~ + p~ _ 2 m ~) [Z(p'~, p'o~) _ Z(p~, p~)]

is

(18) F ~ ( p ' ; p ) - ~ ( p ' + p ) ~ { l [ Z ( p ' ~ , p ' o ~ ) + Z ( p ~ , p ~ ) ] +

÷ ½ (p'~ -~ p~ - - 2m ~) [Z(p '~, p~) -- Z ( p ~, p~)]/(p'~ -- p~)}.

Wi th one part icle on mass shell

Z ( m , Po ) - - ~ Z ( m , P o ) ] • (19) F~(p'; p)I~ . . . . ~-~ ( p ' ÷ p)~[Z(p',° Po)2 ..]_ ~ ~ ,2 ~ ~

The behaviour of Z ( p 2, p~) and Z ( m ~, p~) is the same under p --> ~p and P0 -~ ~P0, name ly logar i thmic dependence on ~. To e s t ima te the h igh-energy behav iou r of the solution in the gauge approx ima t ion , we commi t no er ror in d ropp ing the last two t e rms on the r ight of (19) and re ta in ing only Z ( p 2, p~). This approxi - ma t ion has the mer i t of solvabil i ty, i t being a lways possible to ob ta in the com- plete solution b y successive i terat ions. F r o m the two-par t ic le un i t a r i t y relat ion

8u ~ I m A-~(p) ~ g2 C~(R) fd 4 lc 5+[(p - - k) 2 - - m 2] ~+(k~) •

• r (p; p - k ) F T ( p ; p -

we obta in the result

I m Z-~(p ~, p]) g C (R) [2 --P° In po--IP[] L lPl

Correspondingly as 2--> c~, up to p ropor t iona l i ty factors, Z- l (22p ~, 2~po ~) -~ in 28, A ( 2 p ) . ~ ( l n 2 ~ ) / 2 ~, F ( 2 p ' ; 2 p ) . ~ 2 / I n 2 ~. More par t icular ly , to this order of the gauge approx imat ion , F~(p ' ; 2p)[~ . . . . ,.-~ (p' + 2p) i / in ~ ~.

We can go on to cons t ruc t the CL~; apa r t f rom lack of covar iance (which is largely i r re levant in the a s y m p t o t i c Eucl idian domain) and the occurrence of in ternal s y m m e t r y group factors , the result is as before, i.e. C~(2p) .~- -2(~i j /c ln22, where c oc g2 C2(R). I f we insert F ~ and C z~ into the skeleton diagrams of Fig. 1 to com pu t e (?¢F)~, the calculat ion is ident ical

17 - l l N u o v o C i m e n t o A .

2 5 0 1~. DF.LBOURGO, A.. SALAM ~I id J . STRATYIDY, F~

to the one g iven in Sect. 2, apa r t f rom t r iv ia l in ternal s y m m e t r y factors, its

final fo rm being the ]inite in tegral

M ~ 6g,fd~ (To, m ~} × (T : , T O } l ) . ~ ( l n ,1,2) 2167~ 2 c 2 .

Once again no infinite coun te r - t e rm ~[(~+~s) 2 is needed. I n all the above work we have not considered the v a c u u m polar izat ion

insert ions in to gauge vec to r -meson p ropaga to r s D r . I n rcf. (6) it was shown t h a t the convergence of meson-meson sca t te r ing Green 's functions is unaffected prov id ing all A ~nd D are self-consis tent ly computed in a one-loop Dyson- Schwinger skele ton expansion, where in each d iagram we subs t i tu te Fa and Cs, of the gauge approx ima t ion . The details are compl ica ted and will not be

repea ted here.

4 . - D i s c u s s i o n .

Using the n o n p e r t u r b a t i v c gauge app rox ima t ion technique, we claim to have shown t h a t meson-meson sca t te r ing ampl i tudes are finite in gauge theories, if no direct ~t~ 4 or scalar- fermion couplings are present. The me thod relies on a self-consistent c o m p u t a t i o n of scalar- and vec tor -meson propaga tors A and D, wi th the use of W a r d - T a k a h a s h i identi t ies to e s t i m a t e / ' , in the contex t

of Dyson-Schwinger equat ions. Since, for non-Abel ian gauge theories, a s y m p t o t i c behav iour of A and D,

as well as of F(p'; p) (with both m o m e n t a p and p ' becoming large), has also been c o m p u t e d using renormal iza t ion group methods , one m a y inquire how the resul ts compare in the two approaches . I n part icular , could we use the a s y m p t o t i c es t imates for A, D and F obta ined f rom renormal iza t ion group methods , in conjunc t ion with Dyson-Schwinger skeleton diagrams, to confirm the finiteness of meson-meson sca t te r ing ampl i tudes? (One would have to ob ta in F(p' ; p) for one m o m e n t u m on the mass shell and the other large using our gauge a p p r o x i m a t i o n technique on the renormal iza t ion group A(p).) Our feeling is t h a t a hybr id p r o g r a m m e of this type , where on-mass-shell meson- meson sca t te r ing m a t r i x e lements are eva lua t ed by means of the renormali- za t ion g roup es t imates to be used as inputs for loop integrat ions, is l ikely to be in t r ins ical ly iacons is ten t wi th the v e r y assumpt ions under ly ing the renor- ma l i za t ion group method , where all m o m e n t a are normal ly t a k e n as large. Also, p r e sumab ly , the renormal iza t ion group me thod adds up pieces of more t h a n one-loop Dyson-Schwinger graphs in an involved way. Clearly a synthesis of t he two me t hods is desirable b u t has no t ye t been achieved.

Assuming t h a t such a synthesis can be carried out, wha t is the re levance of our resul ts to the p rob lem of a s y m p t o t i c f reedom? To il lustrate, we consider

SCALAR MULTIPLETS AND ASYMPTOTIC F R E E D O M 251

the concrete example of a recent ly proposed (12) gauge model of weak, s t rong

and electromagnetic interactions which exhibits severe group-theoret ical limi- tat ions on the permissible Yukawa couplings of Higgs-Kibble particles; so much so tha t only a very small number of the Higgs-Kibble mult iplets are

permi t ted to interact with fermions and are thus relevant for the problem of asymptot ic freedom, according to the results of this paper. The model is

the one proposed by PATr and SALAIVI (la)~ which utilizes a 16-fold of fermions

which t ransform under a semi-simple group s t ructure S UL4 ×SU~4 × S U w as follows:

u v ,

There are three gauge multiplets W z , W R and V corresponding to the three

gauge t ransformations U~, U R and U v. Three Higgs-Kibble mult iplets are introduced, A , B and C, t ransforming as

A -~ U L A U~ 1 , B -> U R B U~ 1 , C -~ U v C U~ 1 .

The significant point is tha t B and C cannot Yukawa couple to fermions

because no such renormalizable invar iant interact ion can be wri t ten down. The multiplet A however can couple with fermions th rough TLA~R. Thus, if we do not introduce interactions B 4, C 4 and B 2 C ~ or direct couplings A 2 B ~

and A sC 2, the multiplets B and C would not seem to affect the issue of

asymptot ic freedom. One m ay raise the quest ion: if B and C do not occur in a ~4-1ike potent ial t e rm and do not give expecta t ion values at a classical level, why introduce them at all, when, in any ease, one is going to resort to

the Coleman-Weinberg mechanism for generat ing vector and other masses? We believe tha t the introduct ion of scalar fields like B and C is of advantage , because in their absence the Coleman-Weinberg method would lead us to

consider expectat ion values of bilinear fields like < ~ > ~ 0 or <Ws Ws> ~= 0, etc. I n tha t event there is always the suspicion tha t scalar particles would occur

as bound states and composite fields of B and C var ie ty would emerge from the theory. The natura l conclusion of this a rgument is tha t if the mult iplet A

is uncoupled with fermions and with itself (so tha t its only coupling is to the gauge mesons) the Pat i -Salam model is asymptot ica l ly free.

(12) See S. WEINBERG: High Energy Physics Con]erence (Aix en Provence, 1973) for ~t good summary of the status of gauge theories. (13) j. C. PA~I and A. SALAd: Phys. Rev. D, 8, 1240 (1973).

252 ~ . D E L B O U R G O , A.. SAL~klK and Z. S T R A T I t D E E

APPE~NDIX

Renormalization in the axial gauge.

The re m a y be considerable advan t age to working in the axial gauge since no fictit ious scal~r par t ic les are needed there . This causes the Ward iden t i t i es to r e t a i n the i r classical form. On the o ther hand, the Green's funct ions will involve a f ixed four-vec tor and so lack mani fes t covariance.

Consider f i rs t ly some genera l aspects of the quant iz~t ion programme for a gauge sys tem A~(x), ~v~(x) which is governed , a t the classical level, b y an i n v a r i a n t ac t ion funct iona l

(A.1) S ( A , 90) =fd x ~ ( A , 90).

This func t iona l is supposed to be i n v a r i a n t wi th respect to the local t ransfor- mat ions

(A.2) ~A~ ~ V~ ~ / g ~b~ b ~ / A~, ~f2 ~ ÷ ~, ~ / g ,

(A.3) 390 ~ = 3Y2~(T~)$90 ~ ,

where the s t ruc tu re constants/~b¢ and genera tors T ~ sat isfy the usual algebraic condit ions. I nva r i ance of S can be s t a t ed in the infinitesimal form

~S 3S (A.4) 0 ---- V~ 3A a ~90~ (T~)~ 90~,

which is the p r o t o t y p e Ward iden t i ty . Because of it the classical equat ions of mo~ion

a (A.5) 3S/3A~, : 0 and 3S/~q~ ~ - - - 0

are no t i ndependen t . The axia l gauge is def ined by the cons t ra in t

n A~----0, (A.6) ~ a

where n , is a fixed four-vector which m a y be e i ther t imel ike , l ightl ike or spacelike. I t is easy to ve r i fy t h a t the f ict i t ious-part icle cont r ibut ion is tr ivial . This t e r m in the effect ive act ion is def ined b y the integral

(A.7) exp [-- i W ( A ) / ~ ] 3(n '~A~).

On the subspace where (A.6) is satisfied this in tegra l can be eva lua ted because

f(df2) ~(n~'A~) : f ( d ~ ) ~ ( n . A -~ n .V~ /g + ...) ---- ]det n'V1-1

S C A L A R M U L T I P L E T S A N D A S Y M P T O T I C F R E E D O M 2 ~

equals a constant i ndependen t of A (but subject to n . A ~-0) . Thus, since W(A) is a constant on the subspace (A.6), we are e n t i t l e d to set i t equal to zero.

The genera t ing funct ional of the axia l gauge Green ' s f u n c t i o n s - - t h e v acu u m t rans i t ion ampl i tude in the presence of ex te rna l cu r r en t s - - i s defined b y the p a th in tegra l

(A.S) exp [iZ(I, J , K)/h] ~--

i = f (dA d~dC) exp [~S (A , cf) ÷ fd'x mon "A ~ ÷ I~,qJ' ÷ j~ .A ~ + KaCa)]

where a (~ source ~> Ka(x) goes wi th the Lagrange mul t ip l i e r field C~(x). At the end set K----0 to recover condi t ion (A.6). Define the effect ive act ion /" b y the Legendre t rans format ion

(A.9) Z(I , J , K) ~ F(~, A, C) ÷ f d 4 x (Cn'A~, + Iq~ ÷ J~'A~, ÷ K C ) ,

where the mapping which is inverse to

(A.~o) a q~:'---- ~Z/3I~, , A~, ~-- ~Z/3J ~' , C a : ~ Z / ~ K a

can be expressed by the formulae

(A.l:l) I~,-~ -- ~F/~cf ~ , J~' ~ -- ~T'/~A~, -- n ~ C , K ~-- -- ~F/~C -- n. A .

The funct ional F, the effect ive action, is the cen t ra l objec t which we a im to compute . I t can be shown t h a t 3F/3C vanishes, and this means t h a t F is independent of the Lagrange mult ipl ier . This is in fac t a g rea t simplification, for it means t h a t we can ignore C in the Dyson equat ions , excep t in so far as it plays an implici t role in de te rmin ing the correla t ion funct ions th rough equat ions like

( A . 1 2 ) G:,~(x,x')=--S~[F+fd'xCn.A]/~'<x)Sq~(x') with F now conta ining the classical ac t ion S plus its q u a n t u m corrections.

Consider now the Ward-Takahash i ident i t ies satisfied b y /L To find t h e m first make an infini tesimal gauge t rans format ion on the in tegra t ion var iable in (A.9). Since the measure and domain, as well as S, are assumed to be invar ian t , one obtains

0----fd4xf(dA dq~dC)[Cn.~A ÷ I ~q~ ÷ J . 3 A ] exp[...]----fd,xf(dA d~ dC).

• [Cn.(~ ~ / g ÷ A × ~f2) ÷ I ~ ( T ) q ~ ÷ J . (~ ~ / g ÷ A × ~2)] exp [ . . . ] ,

or, in tegra t ing by par t s where necessary, and ex t r ac t ing the coefficient of 3f2(x),

----f(dA d~ d e ) [ - - n ~ V, C/g -- V~'J~,/g + I (T )~ ] exp [ . . . ] . (A.13) 0

9 ~ :R. DELBOURGO, A., SA.LAM and J . STR~.THDEF~

Only one term appears to be nonlinear in the fields, viz. --n~'A~, X C, and this can be put in the equivalent form

C × nt'At, = - - ih(~/3K) × n"At, = ilg(3/3K) × K = - - i~K × (~/3K).

The ident i ty (A.13) therefore reads

(A.14) 0 ~ / n , ~ ~Z ~Z 1 g . ~ + K × S-~--~ ~ , j ~Z I(T) ~ I

Going over to the Legendre ~r~nsform by means of (A.a0) ~nd (A.11) one obtains

O ~ - - - g n ' ~ C - - ~ + n -~.g ~ + n C + A × -~-~+nC --

8/' 1 3/" 8/" 8/' 8/" ~(~r )~=g~ .~-~+A.x~+Cx~v ~ (T)~,

but, since F is independent of C, this reduces to

(A.]5)

which is the basic Ward ident i ty satisfied by F. I t has precisely the same form as the ident i ty (A.4) which is satisfied by S.

To construct the bare propagator consider the bilinear terms

(A.16) f d ' x [ - - ½ ( ~ , A ~ ) ( 3 ~ ' A ' ) + ½ ( ~ . A ) 2 + C n . A + J . A + K C ] .

These give rise to a linear sys*em of equations

O~As,--Ol,~A, + n ~ , C = - - J ~ , , n " A l , = - - K ,

which are easily solved:

1 [__ ~]~, n'~" + n'3" n ~ "] ~ K C -- ~ ' J Ai' ~- ~ + n.~ (n-8)~J J - n . ~ ' n.~ "

This defines the bare propagators

(T(A~C)} : ibm, In" ~ ,

(T(CC)} =o,

n" ~ + n' ~/~ n~ ~' ~1 1

only the first of which has any real interest because the Lagrange multiplier

SCALAR MULTIPLETS AND ASYMPTOTIC FREEDOM 255

f ield C does n o t e n t e r a n y of t h e ve r t i ce s . I n m o m e n t u m space we h a v e

i~ [ n~'k ~ 4- n~k ~' n~k"k ~] <T(At'(k)A"( - k))} = ~-~ - - V ~ 4- n - k ~ ? ~ J "

I f m i x i n g occm% t h e n t h e p r o p a g a t o r s a re m u c h m o r e c o m p l i c a t e d . G e n - erally~ one wil l f ind a set of fields ~ w i t h the s a m e q u a n t u m n u m b e r s as n'~A~ a n d t he b i l i n e a r t e r m s will '~ake t h e fo rm

[-- (~j,A~)(~ A ) 4- ½ (~.A) ~ 4- Cn .A 4- ½ ( ~ 4- MA) 2 4- I~ 4- J . A -{- KC]

i n s t e a d of (A.16). One m u s t t h e n solve t h e s y s t e m

n"At, = - - K , - - O~,(~"z~ 4- M A ~) = - - I .

A f t e r some t e d i o u s l a b o u r one f inds

1 [_ , , ,~ + n" ~ 4- n ~ ~ n' ~' ~] M(n'~n~--n%/'~) ~ K , A~, ---- ~ 4- M ~ _ n. ~ ~ J J" -5 ( ~ 4- M~)(n. ~)~ ~,,I--

1 [ i ~ n ~ ] M M(n"n~- -n~ '~ ) ~t,J,, + ~ M~ 1 4- I 4- K = ( ~ + M'~)(n. ~)~ + - - _ (~ .~)~] ~ '

C = - - (~ .J 4- M I ) / n . ~ ,

t h e p r o p a g a t o r s a re eas i ly e x t r a c t e d . The r e l e v a n t p a r t s are f rom which g iven b y

i?i [ nl'k ~ 4- n~k '~ n~kt~k ~] < T ( A ' A ~ ) > - - k 2 - - M 2 - - ~ ' ~ + n . k ~ . ~ ) 2 j ,

?~M nt'n " - n2v ~" <T(A"~)} -- k~,

k ~ - M ~ (n .k) ~

<T(~)> k~-- M~ [ ~ . 7 ) ~ J "

• R I A S S U N T O (*)

Si dimostra, in una teoria di gauge di mesoni scalari che interagiscono con particelle vettoriali di gaugc, che le funzioni di Green di quattro pun t i del mesone scalare sono prive di infiniti, purch4 i) i mesoni scalari non si accoppino ad alcun fcrmione nel modello c ii) non vi sia alcun termine diretto ~T 4 nel lagrangiano. La prova si basa s u u n a tecniea di approssimazionc di gauge che impiega sistematicamente le informazioni fornite dalle ident i t£ di Ward-Takahashi. Per teorie di gauge non abeliane, questo risultato significa ehe un ' ampia classe di mul t iple t t i sealari ehe possono indurre ~tna spontanea infrazione della simmetria non influiscono sulla questione della libert~ asintotica. Allo seopo di s l rut tare le informazioni delle idcnti t~ di Ward-Takahashi si usa una (~ gauge assiale ~, in cui non vi sono particelle scalari fittizie e le identi t~ assumono la loro form~ origi- naria, con Z I = Z 2 per t u t t i i campi materiali e di gauge.

(*) Traduzione a eura della Redazione.

256 It . DELBOURGO, A. SALAM and J . 8TRATHDEE

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