LETTERE AL NUOVO CIMENTO VOL. 1, X. 3 16 Genna io 1971

Scale Invaxiance, Goldstone Bosons and the f' Trajectory (').

C. B. Cri~"

Cali /ornia Ins t i tu te of Technology - Pasadena, Cal. Argonne Nat ional Laboratory - Argonne, I l l .

Y. F u J H (**) a n d W. W. WADA (***)

Argonne Nat iona l Laboratory - Argonne, l l l .

( r i cevu to il 14 D i c e m b r e 1970)

As was e m p h a s i z e d by GELL-MANN, in t he t h e o r y of par t ic les , t he scale i n v a r i a n c e is fulfi l led e i t h e r in t he l im i t of zero mass for all pa r t i c l e s or b y t he ex i s tence of a mass less scalar par t ic le , Go lds tone boson, t h a t allows t h e masses of all o the r par t ic les to be nonzero (~). In th i s work we cons ider t he poss ib i l i ty t h a t t he ac tua l wor ld of h a d r o n s is close to t he sca le - invar iance l imit . Then , to give au a p p r o x i m a t e desc r ip t ion for the h a d r o n mass spec t ra , i t appea r s t h a t we need a t leas t two Golds tono bosons : one, h a v i n g the cha rac te r i s t i c s of an S U a s inglet , dc s igna t ed as 0,, l if ts all m e m b e r s of t he same SUa mul t i p l e t to a c o m m o n mass, whi le the o ther , de s igna t ed as 0x, spli ts t he masses w i t h i n each nml t ip le t . These Golds tone bosons m u s t be nea r ly massless for t he scale i n v a r i a n c e to hold. T he fac t t h a t zero-mass scalar mesons h a v e n o t been obse rved e x p e r i m e n t a l l y does n o t necessar i ly i m p l y t h a t a large sca le - invar iance v io l a t ion occurs in n a t u r e . T h e crucia l p o i n t he re is t h a t such Golds tone bosons m i g h t p a r t i c i p a t e in s t rong i n t e r a c t i o n only in te rna l ly . W e a d h e r e to such a view, a n d propose t h a t t h e ~ nonsense s t a t e ,) on t he f ' t r a j e c t o r y a t ~t' = 0 should be iden t i f i ed as the. Go lds tone boson for t h e mass sp l i t t i ng w i t h i n each 8 U a m u l t i p l e t a n d shall re fe r to i t as t he 0a-meson. The s i t ua t i on for t h e l i f t ing of the c o m m o n masses for the mu l t i p l e t s is no t c lear a t t h i s stage. Our specu la t ions on t h i s p o i n t will be p re sen ted la ter . In t h i s no t e we m a i n l y c o n c e n t r a t e on t h e discussion of t h e 0a-meson as proposed , a n d inves-

t i g a t e t he consequences t h a t follow. As m e n t i o n e d above , t he 0a-meson is no t obse rved expe r imen t a l l y . T h e n the

r e s idue func t i on of t h e f ' t r a j e c t o r y m u s t con ta in a ghost -k i l l ing zcro a t af, = 0.

(') ~,Vork performed under the auspices of the IY.S. Atomic Energy Commissiol).. (**) On leave of absence from tile hmtitute (if I)hysics, College of General ].:duetrtion, 17niversity of Tokyo, Tokyo. (***) I'crm~ucnt ~ddress: Department of I)t~ysiea, Ohio ~tate University, Columblm, O. (t) .~[. GELL-MANN: Lectures Delivered at Summer ,~chool o/ Theoretical Particle l'hysi~s, l'niveI~ity of H~waii, Itonolulu, tlawaii (1969).

110

SCALE INVARIANCE, GOLDSTON~ BOSONS, AND THE ~ TRAJECTORY 111

We adopt the usual nousense-choosing mechanism (~-), i .e . for a general sca t ter ing process a + b - ~ c § in the t-channel, the residue funct ion for the exchanged f' t ra- jec tory near :e~,--0 has the behav ior 7 ~ , ~ ~ ' , Through the analy t ic i ty and the factor izat ion proper t ies of the residue funct ion, one finds tha t near a~,= 0,

(1)

where toa is the square of the mass of the Oa-meson. F r o m the assumpt ion of exchange degeneracy and the physical masses of f' and 7, i t is plausible t ha t te~ ~ O. Al though the O~-meson is no t a par t ic le in the usual sense, let us see how this meson can serve as a Goldstone boson.

Consider the s t ress -energy-momentum tensor 0 m, in the l imi t of scale invar iance . Fo r a spin-O meson 0),

(2) 1

(p2]Op~lPl) . . . . . . . . . . . [2 . ,~2F(K 2) - - 3K2G(K2)] = 0 , %/-4EI E s

where K = P l - P~ and t = K ~. We assume m ~ 0. Fo r a moment , we ignore the complicat ion of eq. (I). Sa tu ra t ing G ( K ~) by the Goldstone bosons 0~ and 0~, wc get

(3) 1 1

where go~a and go, a are the couplings of the 0a and 0~ part icles to the vacuum. In our model, the Goldstone boson 0a is on the f' t r a jec tory . Wi th the assumption of the ideal mixing, we have the relat ion / ' = (1/~/3)(1o--~/21s), where 1o being the 8Ua- singlet component a n d / 8 the SUa-octet isosinglet component (a). So f' has the ),X quark conten t (').

N e x t consider a par t icular S U 3 mult ip le t . Any of the members wi th the nonstrange quark conten t ( e . g . . ~ o ~ . (1/~/2)(p~ ~-n~) ) decouples f rom 0a; in turn the correspond- ing first te rm on the r i gh t . hand side of eq. (3) vanishes. For this meson eqs. (2) and (3) lead to m ~- = {Yo,M.~t90,o. The mass spl i t t ing be tween the o ther members and this one is given by

(4) A m ~ - - m2~ m2 -- ~ Y o ~ , ~ t g o z , .

To expl ici t ly account for the complicat ion of eq. (l), bo th 70a.~.~ and g0~a must depend on t. The former is then the Regge ve r t ex funct ion, while the la t tc r is, for example, l inearly re la ted to the g rav i ton form factor. We wri te 70~.~.~(t) = v / t ~ to~ go;.~l~(t), where goa~u(t) is regular near t -- 0. Clearly, in order to a r r ive at a finite resul t for the r igh t -hand side of cq. (4) we must have

(5) Fob(t) y _ _ ~

goao(t) = -%/~ _ toa

( ') M. GELL--~IANX: Pro:'eedings o] International Conlerence on lligh-Energy Physics (('}:I~,N, 196o), p. 539. (=) This idctt[ m ix ing folh)ws direct ly f rom the assumpt ion of exchange degeneraeF. Scc, for example , ( ' . B. Ct~tu and J . FINKEL~TELN': Phys. Lett., 27, B 510 (1968). ( ') Sce, for example , J. J . J . I~OKKEDEE: The Quark Model (New York, 1969).

112 c . B . CIIIU, Y. F U J I I a n d w . w . WADA

with F~a(t ) being regular and finite a t t = 0 (~). The fact tha t the form factor G ( K ~)

has a pole at t ~ 0 for t0a~ 0 reflccts the <,induced )) scalar in te rac t ion between the hadron cur ren t and the scalar component of the grav i ta t iona l field (e.~). Express ions ident ica l to eq. (4) are also found to hold for higher-spin bosons. F r o m eqs. (2) and (4) we obtain the general ized g rav i t a t iona l (, Goldberger -Tre iman relat ion ))

(6) Agn2[ 2 o -- m ~ - - m" = ~Fo~.(O) go~..~s~(O) ,

where m now stands for the common mass of the mul t ip le t l i f ted by 0~, for example , m~, m o, and m~, etc. , and g0~r~(0) is p ropor t iona l to the coupling for the hel ic i ty-non- flip ve r t ex at t = 0. For baryons, a s imilar procedure gives

(7) Am n : m~ - - m = 3]0a(0) go~B(0) ,

wi th m being m ~ , mh, etc. Ill the physical world, tea is posi t ive and deviates from zcro slightly. However , by assuming PCDC (~), eqs. (6) and (7) cont inue to hold approxi- mate ly . Since we have not been able to ident i fy 0~ with a near ly massless par t ic le nor a nonsense state, only those relat ions for the 0~-meson, eqs. (6) and (7), will be con- sidered.

Ear l ier , GELL MA.'~.'~ conjec tured (i) tha t the energy densi ty for hadrons could be expressed by 0oo := ~oo ~- 5 + u o + cu s, where 0oo is scale invar ian t , 5 violates the scale iuvar iance while conserving S U s • 3 symmet ry , and u o and u s belong to (3, 3 ' )~- + ( 3 " , 3) violate S U s • a and the scale invariance. Also, G E L L - M A N N , 0AKE3 and RENN;~;Ir showed tha t in the l imi t c = - ~ 2 , the te rm u p + cu s conserves 8 U z •

and leads to the zero-mass pion is). We observe tha t our proposed scheme differs from this. Since ] ' = ( l / v / 3 ) ( ]o - - .V / 2 ]8 ) and the mass of the 0a-meson is approx imate ly zero, in our model scale iuvar iance is approx imate ly main ta ined for u 0 - ~/2u s.

Now we proceed to der ive mass relations. Since r., p and f0 do not conta in ~ , the coupling constants g0a~=, g0~0o, and g0ae~o must vanish. Fol loqing the usual pro- cedure, a s imple quark count ing for K ~ Ks, r, s and 0 a leads to

(8) goa~,,~,. 4

g0~tKK 3

(6) In tile real world t0a r 0, so eq. (5) xr imply , for example , a s ingular i ty associated with the

g rav i t a t ion from lactor near l- 102. l lowever , the coupl ing be tween an o~-shell grav i lon a]~d the 0a-meson is finite. U) We ment ion here t h a t R. II . I)[CKE l l ~ discussed the possibi l i ty of in t roducing a scalar g rav i ta t iona l field in addi t ion to the Einste in g rav i t a t iona l me t r i c tensor in a different c o n t e x t . . ~ e c 1t. Y. ( 'HIu and W. F. I[OFF:~kNN: Gravity a~d Relativity (New York, 1964), p. 143. (~) As is discussed in ref . ( ') , scale invar iance implies the conservat ion of the di lat ion ehal~'e 1), i.e. dD/dl-~ O. From the c o m m u t a t i o n re la t ion for the operator D, i.e. [D, Po]-- - i P o . i t can be shown t h a t P , [ I + i e D I H ) =ps i1 -~-e)[1 ~- ieDl l l t ) for an infini tesimal E. This impl ies t h a t [1 ~ - i e D ] ' H ) is ano the r c igens ta te of t'o wi th a s l ight ly larger energ) ' , which in tu rn suggests t h a t this new s ta te [1 T i e D ] IH) m i g h t correspond to a physical s ta te conta in ing * soft ,) 0). particlt 's. If we a ~ u m e t h a t the mass of the 0Fmeson is s l ight ly larger t han zero, it wil l decay into a grav i ton . We suggest t ha t the c o u t i n n u m st~tes genera ted b)" the di lat ion operator to be the a sympto t i c stoics of the h~drons plus

soft ~ gravit, otts. We wish to Ell&Ilk Prof. 1[. J. TJlt'KtN for cal l ieg out att~cntion to t.llis a p p a r e n t a[ tomaly ~ s o c l a t e d wi th the c o m ~ z t a t i o n rel~ti~:~: for the operator D. i s) M. (]I~LL-MANN, ]~.. J. OAKFS aD.(]. U. RENNER: t)hyS, l~e~., 175, 2195 (1968).

SCALE INVARIANCE, GOLSDTONE BOSONS, AND THE fl TRAJECTORY l l 3

artd s imilar ly for t he none t s of t he vec tor and the tensor mesons. A subs t i tu t ion of eq. (8) in to eq. (6) gives

(9) m 2 2 2 2 2 _ _ ~ 2 4

m ~ - - m = m E . - - m p A~

in ag reemen t w i th t he G e l l - M a n n - 0 k u b o (G.0.) mass relat ions. Not ice t h a t w i th the assumed s y m m e t r y for t he couplings t he quadra t i c mass re la t ions for mesons follow

automat ica l ly f rom eq. (6). F r o m similar cons idera t ions , eq. (7) leads to t he l inear- mass equal -spacing rules,

(10) m Z - - m ~ , = m = - - m ~ and m a - - r a ~ . = m ~ . - - m v ~ = m y ~ - - m A ,

which are the same as t he G.0 . mass re la t ions in t he l imi t m A ~---m Z (~). We men t ion also some consequences t h a t follow f rom the a s sumpt ion of (~ univer-

sal i ty )) in the coupl ing of t he 0 k par t ic le to t he pseudoscalar , t he vector , and the tensor SU~ mult ip le ts . This a s sumpt ion leads to

(11) ],2 __K**2 = K*.2__]02 = ~2 K*2 ~ K * 2 ~02 = K2__~2 , etc.

These are satisfied reasonably well by the da t a and can also be obta ined , for example ,

f rom the Veneziano r ep re sen ta t ion (~o). W i t h the assumpt ion of S U 3 s y m m e t r y , t he universa l i ty in the 0 x coupling implies a s imilar un iversa l i ty for o ther memb er s of the t ensor nonet , such as among t h e couplings fffo~, g~n:.~., a n d g ~ . . ~ . . , etc.

F u r t h e r m o r e , let us compare our model p red ic t ion and h igh-ene rgy to ta l cross-sect ion da t a (1L12), Assuming S U z s y m m e t r y and the ideal mix ing along the t r a j ec to ry , f rom eqs. (6) and (7) we get

g~o~+=- 4 g f~K 2 gO~K~ 4 m ~ - m~ (12) . . . . . .

gf,p~ 3 g~~247 3 go~z@+ 3 m Z - - m A.

Deno te F(=~(~) and F ( ~ f f ) to be t h e a p p r o p r i a t e combina t ions of t h e i nva r i an t am- pl i tudes for T:~ and J~'J~ fo rward sca t te r ing . F o r our convent ion , t he optical t heo rem is g iven by

2m~, Im FUcJ~') 4m~v Im F(2~'2/') (13) a ~ ( ~ ) ~ and a ~ ( J ~ ) =

2 q W 2 q W

F r o m eqs. (12) and (13), the ra t io of t he fo con t r ibu t ion to t he r and J ~ S to ta l cross-

(9) It is interesting to make connections between our model and the result of yon ]~IPPEL and l~I)[ (F. VON HIPPEL and J. KIM: Phgs. Rev. D, l, 151 (1970)). Since in our scheme the 0 k particle is respoasible for the mass splitting of the �89 baryon octet, if we denote 0oto -- u0 - %/2 u8 as the energy density operator, then m E - m ~ -- (Xluo-- %/2u81~) -- (XluolZ) -- (r etc. The fact that (xlu~lz) = 0 can be checked easily in the quark model. We replace the left-hand side by the SU~-averaged mass-splitting �88 [m~ + m A + m Z - 3m~] - 202 MeV. This is the expectation value of uo with respect to the ~+ baryon o~tet which is essentially tile same as their result. We quote that the expectation value of u~ for decouplet is 146.7 MeV. (t,) See, for example, M. JACOB: Schladming Lecture Series (March, 1969).

I14 c . B . CIIII;, Y. FUJII and ~ V . ~'V. "WAI)A

sections is given by

//b K - - ~ l~ (14) 1~'= : - - -- 0 .69,

and through the assumption of S U 3

Equations (14) and (15) are to be contrasted with those ratios R= and ]?K obtained from �9 simple quark counting scheme, i.e. ~ and �89 respectively (~). The plot of the experimental total cross-sections with the vacuum quantum number in thc t-(.hannel for ~J~~ and K ~ " scattering vs. that for ~\%V scattering (~m-.) is illustrated in Fig. I.

16 14 ~-o---~ ~ '~

22, ~ ' ' ' ' ' ,

! -

L-- I I I I I l I 1 43 45 47 49 51

O- (M ' J ~ ) ( m b )

Fig. 1. - Our p r e d i c t i o n s vs. t h e dattt (,,.~,). For t he s o l i d a n d t h e d a s h e d l ines s ee t e x t . T o t a l cross - s e c t i o n s , (~.#') -- �89 [ (= 'p) + (~-p)], (JVbr = �89 |(pp) -i- (pp)] a a d (K~f') ~ ~ [(K§ -'.- (K+n) + (/s F (K-n) ] .

I n d i ca t e4 ]mmb cr s are Plab in GeV]c. N o t e t h a t w i t h o u r a s s u m p t i o n s a~ ~ 12~a~ + t r ar V. BORECKA and if . J . LIPKIN ( W c i z m a n n p r e p r i n t , s u b m i t t e d to K i e v Conlerence ( 1970 ) ) f irst s h o w e d t h a t s t r a i g h t l i n e s a p p e a r in s u c h plot~ . A l t h o u g h p o i n t s are t a k e n a t t h e s a m e v a l u e of P lab , t h e u s e of s or v m~kes l i t t l e d i f ference a t t h e s e energ i e s .

Within thc approximation of the fo dominance for the energy-dependent part, the slope of the data should be given by R= and R K, respectively. Our theoretical predictions for these two cases using the numbers from eqs. (14) and (15) are illustrated by the solid lines in the same Figure. If we replace the difference m~-- -m v in eqs. (14) and (15) by an SLr~ aver~tgc ~[m,~+my--mA--3m:v], we obta.in R n ~ 2 I ? ~ 0.79. The cor- responding predictions are illustrated by the dashed lines in the same Figure. ~Ve find

(~x) W. (]ALBRAITtI, E. W. JENKINS, W. F. }QYCIA, ]]. A. LEON'TIC, T~. H. ])IIILLIPS, A. L. ]~EAI) a n d R. I~UBINSTEIN: ]~h!/s, I ter. , 138, l~ 913 (1965), ( " ) I{. .J . I~'OLEY, }~. ~. JONI,'S, ~. J. LINDENBAU.~I, ~V. A. I,OVE, ~. OZAKI, ]']. I). I'LAT.N'ER, C'. A. QUARLES aIid E. H. t~VILLI.:N: l)h718. ROt ~ Lef t , , 19, 330 (1967).

SCALE I N V A R I A N C E , GOLDSTONE BOSONS, AND THE ff T R A J E C T O R Y l l 5

that the agreement is remarkable. The discrepancies perhaps could be at t r ibuted to other energy-dependent contributions, such as for examples Regge cuts.

Let us summarize the advantages of our proposal. The masses of , and f' suggest that 0k could be massless. While the 0x is only a nonsense state, it however serves the role of a Goldstone boson for scale invariance. We have seen that it naturally gives rise to the linear equal-mass splitting for the baryons and the quadratic equal-mass splitting for the mesons. I t predicts mass relations in accordance with the G.O. mass formula. At this stage, the couplings obtained from the generalized gravitat ional << Goldberger-Treiman relation >> cannot be directly tested. However, upon invoking the S U 3 symmetry, the fo couplings to various multiplets can be obtained. Comparisons have been made for some of these couplings with the data. Results are found to be very reasonable. Furthermore, from the mass formula, we have indicated that the universality of the 0 k coupling to hadrons are reasonably well satisfied.

We speculate that the common-mass lifting Goldstone boson, 0s, mentioned in the beginning should also exist at least internally. This 0s particle which has the characteristics of an S U3 singlet should couple to all hadrons except for the pseudo- scalar octet in accordance with the smallness of the pion mass. The decoupling of 0z to the pseudoscalar meson octet immediately leads to c = - - V / 2 in the term % + cus, and thus to m~ = 0. In our model, the finiteness of the pion mass could be at tr ibuted to either a slight deviation from the ideal mixing angle for the 0 k particle, or a weak coupling of 0, to the pseudoscalar octet. In the work of GELL-MANN, OAKES and RENNER, in connection with the (3, 3")+(3" , 3) algebra and chiral symmetry, the former choice is implied (s). Furthermore, we observe that the S U 3 common masses of the baryons, of the vector mesons, and of the ninth pseudoscalar meson ~' lifted by 0~ are more or less of the same order of magnitude. What kind of an object should this 0 z be? For this, let us consider the description of the mass of the ~'. In the Hamiltonian H o of ref. (s), which conserves SU3• we need a part that violates U~• U 3. One can show that this part cannot be given in terms of bilinear forms of q and ~. Therefore, we suggest that 0~ should contain some exotic component.

Finaily, how the matr ix clement of 0~,~ taken with respect to the hadron states is modified in the presence of the 0~ particle? One can easily verify that this has the same expression as what one would obtain should there be a physical massless Goldstone boson. I t gives rise to an induced scalar component in 0~, which necessarily modifies the linearized version of Einstein's theory of gravity.

$ $ $

We wish to thank Dr. R. C. ARNOLD for the hospitality extended to us at the High Energy Division of the Argonne National Laboratory during the summer of 1970. We also thank Prof. H. J. LIPKIN for discussions, and for informing us the form of the plot presented in Fig. 1 in advance of publication.