Volume 31B, number 9 P H Y S I C S L E T T E R S 27 April 1970

T H E P R I M I T I V E G R A P H S O F D U A L - R E S O N A N C E M O D E L S *

D . J . G R O S S , A . N E V E U **, J . S C H E R K ***, J . H . SCHWARZ Joseph Henry Laborator ies , Pmnceton Universi ty , Princeton, N.J. 08540, USA

Received 17 March 1970

We show that the problem of construct ing and, if necessar~ r e n o r m a h z m g general dmgrams m d u a l - r e s - onance models can be reduced to the study of only four pmm~tlve graphs, three of which are tadpoles, the last one being a se l f -energy graph. We discuss consequences and appheat lons of this reduction.

T h e p r o c e s s of u m t a r i z m g a f a e t o r l z a b l e d u a l - r e s o n a n c e m o d e l r e q u i r e s c o n s t r u c t i n g F e y n m a n - h k e d i a g r a m s c o m p a t i b l e w i t h d u a h t y and f a c t o r ~ - z a U o n [1,2] . As w a s s h o w n m r e f . 1, t h e r e e x i s t d i f f e r e n t F e y n m a n - h k e d i a g r a m s w h i c h a r e c o n - n e c t e d by d u a h t y . F o r i n s t a n c e , any p l a n a r d i a - g r a m can be r e d u c e d to t he l t e r a t m n of a s i n g l e p l a n a r t a d p o l e . T h i s r e d u c t i o n s x m p h f l e s t he p r o b l e m of r e n o r m a h z i n g p l a n a r d t a g r a m s : one h a s j u s t to r e n o r m a l i z e t he p l a n a r t a d p o l e o p e r - a t o r , and to c h e c k t h a t t he l t e r a t m n of t h i s r e - n o r m a h z e d o p e r a t e r d o e s no t c r e a t e new i n f l m - t~es. It ~s t h u s i n t e r e s t i n g to look f o r a s i m i l a r r e d u c t i o n of n o n - p l a n a r d i a g r a m s .

T h e s e t of a l l F e y n m a n - h k e d m g r a m s r e l a t e d by d u a h t y f o r m an e q u i v a l e n c e c l a s s , c a l l e d a d u a l i t y d i a g r a m . In r e f . 2 the e n u m e r a t m n of d u a h t y d m g r a m s w a s shown to be e q m v a l e n t to t he c l a s m f l c a t m n of c o m p a c t t w o - d l m e n m o n a l m a n i f o l d s w i th b o u n d a r i e s [e .g . 3]. By u s i n g t he p r o p e r n e s of t h e s e m a n i f o l d s , we show t h a t any d u a h t y d i a g r a m c a n be b u i l t up by a t t a c h i n g to a t r e e d i a g r a m , one o r m o r e t i m e s , only fou r g r a p h s : a s eK e n e r g y g r a p h a n d t h r e e t a d p o l e s (fig. 1). In o t h e r w o r d s t he fou r g r a p h s m f ig . 1

(la) (ib) (Ic) (id)

Ftg. 1. The four pr imi t ive graphs of a dual theory.

r e p r e s e n t a l l t h e p r o p e r n - p o i n t f u n c t i o n s m a dua l t h e o r y , a l l o t h e r g r a p h s b e i n g one p a r t i c l e r e d u c i b l e .

* Supported m par t by the Air Force Office of Smen- t l lm Research, under contract AF-49(638)1545.

** P r o c t e r Fellow. *** NATO Fellox~, on leave of absence from Labora-

tmre de Physique Theorlque, Orsay, France .

592

Any d u a h t y d i a g r a m [2] can b e pu t m a o n e - t o - o n e c o r r e s p o n d e n c e w i th e i t h e r a s p h e r e o r a c o n n e c t e d s u m of for1, o r a c o n n e c t e d s u m of p r o j e c t i v e p l a n e s w i th w indows a n d p o i n t s on i t . S o m e of the p o i n t s d e m a r c a t e r edes of the w~n- dows , w h i l e t h e o t h e r s c o r r e s p o n d to c l o s e d l o o p s of r e l a t e d F e y n m a n - h k e d ~ a g r a m s . T h e r e d e s of t he w i n d o w s c o r r e s p o n d to t he e x t e r n a l l e g s of t he dua l i t y d i a g r a m . A g i v e n t r i a n g u l a - t i on of t he s u r f a c e i s r e l a t e d to a w e l l - d e f i n e d F e y n m a n - l i k e d i a g r a m [ 2 ] ? . In t he c a s e of a p l a n a r d i a g r a m ( s p h e r e w i th one window and p o i n t s ) , i t i s known t h a t t h e r e e x i s t s a t r i a n g u - l a t i o n w h i c h c o r r e s p o n d s to t h e i t e r a t i o n of a p l a n a r t a d p o l e . W e show t h a t m the o t h e r c a s e s a s p e c i a l t r i a n g u l a t i o n can b e found w i t h m m f l a r p r o p e r t i e s .

We s t a r t f r o m a s u r f a c e w i th w i n d o w s and N p o i n t s . Le t us f i r s t r e d u c e t h e n u m b e r of p o i n t s : a po in t c an be b r o u g h t n e a r a g i v e n s i d e of a window, and we can a b s o r b i t t o g e t h e r w i th t h a t rode by the t m a n g u l a t l o n of f ig . 2a. T h i s t r m n g u -

(~ a Q3

(2a) (2b) Fig. 2a) Trlangulat ton of a point and the rode of a wm- dow. b) The corresponding Feynman-hke diagram.

l a t m n m e a n s t h a t a p i a n a r t a d p o l e h a s bee . , i n - s e r t e d on t he e x t e r n a l l eg c o r r e s p o n d i n g to t he rode of the window t h a t we h a v e u s e d (fig. 2b). W e a r e now b r o u g h t b a c k to t he c a s e of t he s a m e s u r f a c e , w~th t he s a m e t y p e of window, but w i th N - 1 p o i n t s . By r e p e a t i n g t h i s p r o c e s s N t i m e s , we c a n ge t r i d of a l l t h e p o i n t s , and i n s e r t N p l a n a r t a d p o l e s on e x t e r n a l l i n e s . N o t i c e t h a t

? See footnote ? on next page

Volume 31B, number 9 P H Y S I C S L E T T E R S 27 April 1970

t h e s e N t a d p o l e s c an be d i s t r i b u t e d in a n a r b i t r a - r y f a s h i o n on t he e x t e r n a l l e g s .

L e t us now r e d u c e the n u m b e r of s i d e s of a g i v e n window. Th~s i s done by the t r m n g u l a t l o n of f ig . 3a, w h i c h m e a n s t h a t we h a v e a r r a n g e d

Br

(3a) (3b) Fig. 3 a) Tr iangula t ion of a window.

b) The cor responding Feynm an- hke diagram.

t he e x t e r n a l l i n e s c o r r e s p o n d i n g to t he s i d e s of t he window in a m u l t l p e r i p h e r a l t r e e - c o n f i g u r a - t ion (fig. 3b). W e c a r r y out thxs p r o c e d u r e f o r a l l t h e w i n d o w s of t he s u r f a c e , and t h u s a r e l e f t w i th the s a m e s u r f a c e , w~th t he s a m e n u m b e r of w i n d o w s , e a c h of t h e m h a v i n g now only one rode (/39 m f ig . 3a).

W e s h a l l now r e d u c e the n u m b e r of o n e - r e d e d w i n d o w s . We b r i n g two of t h e m t o g e t h e r and p e r - f o r m the t r i a n g u l a t m n of f ig . 4a. L i n e s y l and

J,) ,.

(4a) (4b)

fig. 4 a) Triangulation of two one-sided windows. b) The corresponding Feynman-hke dlagram

2 correspond elther to external lines or to hnes a n a l o g o u s to ~ (fig. l b ) o r / 3 9 (fig" 3b). T h e two h n e s 7 3 a n d y 4 ( f i g . 4b) a r e t w l s t e d t . We e n - c o u n t e r h e r e t he s e c o n d t ype of i r r e d u c i b l e g r a p h , a s e l f - e n e r g y c h a g r a m wi th t w i s t e d h n e s . A f t e r t h i s t m a n g u l a t l o n , t he two w i n d o w s h a v e b e e n a b s o r b e d in to a new l a r g e r o n e - s i d e d w i n - dow.

We r e p e a t t h i s p r o c e s s by a b s o r b i n g t h i s new window wi th a t h i r d one . If t he w i n d o w s a r e on a s p h e r e , they can a l l be o b s e r b e d , a n d t he t r i - a n g u l a h o n i s c o m p l e t e d . It g i v e s a c h a i n of s e l f - e n e r g y g r a p h s i n s e r t e d b e t w e e n m u l t l - p e m p h e r a l t r e e s o r on e x t e r n a l l e g s , p l u s p l a n a r t a d p o l e s w h i c h a r e h o o k e d m a r b i t r a r y p l a c e s on t h e t r e e s .

We reca l l that to each t r iangle of fig. 2a, 3a, 4a, 5a, 6a cor responds a ver tex in fig. 2b, 3b, 4b, 5b, 6b, respect ively . For the re la t ion between a twist and the or ienta t ion of a t r iangle , see ref. 2.

If t he s u r f a c e i s no t a s p h e r e , t h i s p r o c e s s of t r i a n g u l a t i o n s t o p s when we a r e l e f t w i th on ly one window. So, we now c o n s i d e r t he two o t h e r p o s - rub le t y p e s of s u r f a c e s , i . e . , a c o n n e c t e d s u m of t o m wi th one o n e - r e d e d wmdov¢ o r a c o n n e c t e d s u m of p r o j e c t i v e p l a n e s wi th one o n e - s i d e d w i n - dow. In the h r s t c a s e we u s e the c a n o n i c a l p o l y - gon r e p r e s e n t a h o n of t he c o n n e c t e d s u m of n t o m wi th one window w [3].

4)1~ 1(~ ; 1~ l l w 0 2 ~ 2 (~21~21 . . . . . ~nt2nC~;l~n-1 A tmangulatlon of the surface for the par t icu lar c a s e n = 3 i s s h o w n in f ig . 5a. T h e F e y n m a n - h k e

(5a) (5b) Fig. 5 a) Triangulat ion of a sum of tom with one-s ided

window. b) The corresponding Feynman-l ike diagram

d i a g r a m c o r r e s p o n d i n g to t h i s t r l a n g u l a h o n i s s h o w n m f ig . 5b. A s m u l a r t r i a n g u l a t i o n e x i s t s f o r a l l n. T h i s c a n o n i c a l t r i a n g u l a t i o n g i v e s a F e y n m a n - h k e d i a g r a m w h i c h i s an i t e r a t i o n of o u r t h i r d one p a r t i c l e i r r e d u c i b l e g r a p h , a n o n - p l a n a r o r i e n t a b l e t a d p o l e wi th two loops .

T h e r e m a i n i n g t ype of s u r f a c e ~s a s u m of m p r o j e c t i v e p l a n e s wi th one o n e - s i d e d window w ' . T h e c a n o m c a l po lygon r e p r e s e n t a h o n of t h i s s u r - f a c e i s

X1X1 w' X2X 2 X3X 3 . . . . . XmX m.

T h e c a s e rn = 4 i s d r a w n m h g . 6a, t o g e t h e r w i th

X ~ 4 X3 X4

X~_~X~ W'

(6a~ (6b) Fig. 6 a) Triangulat ion of a sum of project ive planes

with one one-s ided window. b) The cor responding Feynman- l ike d iagram.

a t r i a n g u l a t i o n of t he s u r f a c e . T h i s t m a n g u l a t l o n i s e a s i l y g e n e r a h z e d to a l l m. I t g i v e s a F e y n - m a n - l i k e c h a g r a m (fig. 6b), w h i c h i s an i t e r a t i o n of the f o u r t h and l a s t one p a r t i c l e i r r e d u c i b l e

593

Volume 31B. number 9 DHYSICS L E T T E R S 27April 1970

graph, the non-planar non-or len tab le one- loop tadpole.

We have now proved the following theorem: any dual dmgram can be reduced to the inser t ion on a t r e e of the four p r imi t i ve one -pa r t i c l e i r - reducible graphs of fig. 1. The tadpoles can be moved to any place on the t r ee , and there may be any number of them. Non-planar s e l f - ene rgy graphs a r e locked between t r e e s or on ex te rna l hnes , and there a r e at most L - 1 of them if t he re a re L externa l legs . The two tadpoles m fig. l c and ld never occur together in this reductmn: ~f the dmgram is non-or len tab le , only lc occurs , ~f it is or lentable , only ld occurs . The ~dentlty of fig. 7b shows that a F e y n m a n - h k e d m g r a m

(7a)

(Tb) F~g. 7. Two identities between Feynman-like diagrams.

containing tadpoles l c and ld can in fact be drawn by using only the tadpole l c . The re is a one - to - one cor respondence between these reduced graphs and the set of duahty d i ag rams as defined m [2], so that no two of these reduced graphs a r e r e - ducible to each other by a duahty t rans format ion .

This reduct ion of F e y n m a n - h k e d iag rams to a canomcal fo rm is vahd for any dual mode l s and has two impor tant appl ica tmns.

1) The r eno rmahzabHl ty of a dua l - r e sonance model reduces to the study of the renormal izab~-

It ~s evident that, due to its topological nature, thls theorem will remaln valid in the presence of inter- nal quantum numbers. The irreducible operators that appear in fig. 1 will however carry additional labels referrlng to the quantum numbers of the one particle states of the model.

lity of the four i r r educ ib l e ope ra to r s c o r r e s p o n d - mg to the graphs in fig. l a -d , and of the i r i t e r a - t ion. Notice that the non-planar s e l f - e n e r g y graph m hg. l a need not be i t e ra t ed if the i t e r a - tion runs over planar and non-planar tadpoles . On the other hand, thanks to the identify of hg . 7a, the i t e ra t ion of planar tadpoles can be turned ~nto the ~terat~on of s e l f - ene rgy graphs, if the re is at l eas t one of them. It is r e m a r k a b l e that the r e n o r m a h z a t m n of dua l - r e sonance models can be exp re s sed only as a mass r eno rmahza t lon .

In the pa r t i cu la r case of the gene ra l i zed Venezlano model, we have a l ready found [4,5] that the planar tadpole of fig. lb and the non- or len tab le tadpole of fig. lc can be r e n o r m a l i z e d by using elhpt~c functions. However , we have found that graph l a exhibits pecul ia r branch points due to the fact that i ts d ive rgence depends on its momentum. It is worth noting that, if we include in terna l quantum numbers or draw quark hnes , this graph occurs only in channels w~th vacuum quantum numbers . If the P o m e r a n - chuk ~s to appear in the model, it is c l ea r that new s ingu la r i t i e s would be expected to occur in this graph, but it is not c l ea r that the ones we have found a re physical ly acceptable .

2) Another apphcat ion of the reduct ion of dual d m g r a m s ~s the poss lb lh ty of a fo rma l s u m m a - tion of the whole per turbat ion s e r i e s . In ref . 5, we show that the set of one-loop d m g r a m s sa t l s - h e s pe r tu rba t ive uni tar l ty in second o rde r if each i r r educ ib l e d iag ram is counted once. If the same counting p rob lem can be solved for m o r e comphca t ed d i ag rams , the pe r tu rbanon s e r i e s could be fo rmal ly summed.

Rcfe~'ences 1. K. Kikkawa, B. Sakita, M. A. Vlrasoro, Phys. Rev.

184 (1969) 1701. 2. K. K1kkawa. S. Klein, B. Saklta, M. A. Virasoro,

Wmconsin Preprint C00-268. 3. W.S. Massey, Algebralc topology, (Harcourt, Brace

and World, Inc., New York) 1967. 4. A. Neveu and J. Scherk, Phys. Rev., to be publlshed

T. H. Burnett, D. J. Gross, A. Neveu, J. Scherk and J. H. Schwarz, Princeton preprint.

5. D.J. Gross. A. Neveu, J. Scherk and J. H. Schwarz. to be pubhshed.

594