L]~TTERI~ AL NUOVO CIMENTO VOL. 16, N. 6 5 Giugno 1976

Diquarks as an Explanation for ~'s, R, and Everything Else.

~/[. PAVKOVIC

Stan]ord University School o/~lediciq~e, Welch Road 701, S~tite C-1 - 1)alo Alto, Cal. 94304

(ricevuto 1'1 Marzo 1976)

Despite the intensive efforts of the experimentalists to detect the presence of charm in the final-state products of the annihi lat ion of e+e - pair into hadrons, no particle carriers of this new conjectured quantum number have been so far unambiguously identified. This is generally regarded as a source of major embarrassment for the charm model (~) which predicts a copious production of charmed particles above E ~ 4 GeV. To make things worse, the observed values of

a(e+e - -+ hadrons) R =

a(e +e--+ ~+t~-)

above this threshold are way too large for the theoretical number R ~ la$, and the predicted radiative decays of ~'s into their pseudoscalar kin d/1--+~,-~/0- are yet to be observed.

If we take all evidence into account, it is fair to state that there arc as yet no com- pelling reasons to accept the charm scheme as the final answer to the riddle that SPEAR has presented us with its spectacular discoveries during the past year.

Other quark models tha t have been proposed (3) do not fare better, mainly because they all predict the existence of new quantum numbers tha t have so far defied detection.

Is it possible to construct a model tha t would possess much needed new degrees of freedom for the explanation of ~'s, R and related phenomena, and yet be conserva- tive as far as the new quantum numbers are concerned?

The intent ion of this short note is to present a candidate for such a model. The discovery of ~'s and the unexpectedly large values of R have created a need for the new degrees of freedom. Among other things, one must have enough elementary

comparable with the data. As charges e~ to make the theoretical number R : ~ e i

(1) J . D. BJORKEN and S. L. GLASHOW: Phys. Leti., l l , 255 (1964); S. L. GLASHOW, J . ILIOPOULOS a n 4 L. M A I ~ I : Phys. Rev. D, 2, 1285 (1970); M . K . GAr[S,~RD, B. W. LEE and J . L. ROSNER: ReV. Mod. Phys., 47, 277 (1975); T. APPELQUmT a n d D. POLITZER." Phys. Rev. Left., 34, 43 (1975); T. APPEL- QUIST, •. DE R~JULA, ]3[. D. POLFrZER and S. L. Gr,ASHOW: Phys. Rev. Left., 34, 365 (1975); E. EICHTEN, K . Go2"rFRIED, T. KINOSHITA, J . KOGUT, ~ ' . D. LANE and T.-I~I. YAN: Phys. Rev. Let., 34, 369 (1975). ( ') H. HXR)~RI: Phys. Lett., $ 7 B , 265 (1975); H. FR1T~SCH a n d P. MINKOWSKI: Phys. Lett., 5 6 B , 69 (1975); R . M. BARNETT: Phys. Rev. Left., 34, 41 (1975); F. G~RSEY, P. I~A~IOND and P. SIKIWE: Y a l e U n i v e r s i t y p r e p r i n t (May 1975); M. SuzuKI: Phys. Left., $6 B, 165 (1975); H. FRITZSCH, M. GELL- MANN and P. MINKOWSKI: Cal. Tech. p r e p r i n t 68-503 (1975).

168

D I Q U A R K S A S A N E X P L A N A T I O N F O R + ' S , R, A N ] ) E V E R Y T H I N G E L S E 169

a response to this urgent need, new quarks have been invented and added to the already existing set (1,2). However, the new quarks lead to larger symmetry groups (SU4 in the case of charm), and the larger symmetry groups predict new families of particles whose presence in nature has not been confirmed.

Our approach is different. We do not bring new elementary objects from the out- side, but instead manufacture them from the elemcntary objects tha t ah'eady exist in the quark model. This is the idea of diquarks. One generates new elementary objects of spin-parity 0 + and 1 + by forming quark pairs qq and assuming as a working hypoth- esis tha t these pairs are pointlike and just as elementary as the objects tha t partic- ipate in their construction�9 In detail, our assumptions are the following. We start with the three triplets of fractionally charged quarks tha t differ in color (red, white and blue, say). The need for color has already been expressed before (a), and we find the hypothesis of color to be quite cssential in our scheme as well. By applying Pauli exclusion principle to the hilt, we form diquarks only from the quarks tha t differ in color. Accordingly, we will have red-whitc, white-blue, and blue-red diquarks, but not red-red, white-white, or blue-blue diquarks. How many diquark states are generated

�9 1 + in that way? First, two spm-~ objects can couple into the spins 0 + and 1 +. Consequently, we will encounter the 0 + (scalar) diquarks and the 1 + (axial vector) diquarks. Each variety will contain thrce differcnt color combinations, as explained earlier. Finally, the S U a degrees of freedom will enter the scheme through the tensor product 3 • 3 =

3 + 6. In other words, aside from the described triplets of quarks, we will also have the antitriplets and the sextets of diquarks. In total 3 • = 9 quark states and 2 • 3 • (3 + 6) == 54 diquark states.

In the annihilat ion channel e+e -~had rons the masses of quarks and diquarks are important parameters, directly related to the loc~tions of thresholds for their production. The production of (the pointlike) axial-vector diquarks is associated with the steep quadratic growth of R with energy (a). Since nothing of tha t sort has been observed in the covered energy range E ~8 GeV, we must assume that the masses of axial-vector diquarks are larger than m v ~ 4 GeV, which makes them inaccessible to the present energics at SPEAR.

In order to explain the observed behavior of R, we must assume furthermore that the masses of quarks are small, say mq40.6 GeV, while the masses of scalar diquarks should lie at the beginning of the threshold at E ~ 3 . 4 GeV, i.e. m, ~1.7 GeV. With the listed assumptions about thc quantum numbers of quark and diquark states, we are now prepared to explain R, ~'s and the related phenomena.

a) R. In the region above the resonances and below the threshold for the produc- tion of scalar diquarks, 1 GeV 4 E 43.4 GeV, the only contribution to the total cross- section will come from quarks. Hence (e~ are quark charges)

R=Ro ,rk,=E E 4 colour quarka

in a reasonably good agreement with the data. In the energy interval 3.4 GeV ~ E ~ 5 GeV, various channels with scalar diquarks open up. Without a detailed knowledge of the pat tern of mass spli t t ing among diquark states and the forces

(a) *Color ~ was first proposed, u n d e r a d i f ferent n a m e , by O. W. GREENBERG: Phys. Rev. Lett., 13, 598 (1964). The r ecen t emphas i s on i ts i m p o r t a n c e a n d t h e n a m e ((color * is due to W. BARDEEN, H . •RITZSCH, ~/L GELL-MANN ~nd H. LEUTWYLER. See, for example , H . FRITZSCH, M. GELL-~ANN and H. LEUTWYLER: Phys. Left., 47 B, 365 (1973). (4) lYI. PAVKOVIC: to be publ i shed .

170 ~. P.~VKOVI C

among them we cannot a t tempt to explain the observed behavior of /~ in this energy range. Beyond E ~ 5 GeV, however, all diquark channels are opened, and /~ is the sum of two components,

where (4) (e d are diquark charges)

1 1 16 7 E E 4=Tx3xT=4.

color dlquarkB combinations

Hence, for 5 GeV ~ E ~ ? GeV, /~ : 6, again in reasonable agreement with the ex- ist ing data.

b) ~'s. ~'s are the bound states of diquark-antidiquark type. We assume that all hadrons are color singlets, ~'s included. Symbolically,

new mesons = ~ dij d-~j, ~,J

d~j : qt q~ - - q~ q~ (i, j = red, white, blue) .

The fact tha t the constituents have spin-zero has the profound consequences on the spectroscopy of new ~b-like meson states. In particular, only the (~ natural ~> spin-parity sequence 0 +, 1-, 2 + . . . . can be obtained from the two 0 + objects in a bound state, while the complementary (( unnatura l ~) series 0-, 1 +, 2-, ... does not occur in the model. This feature differentiates the colored version of the diquark model from any quark scheme tha t builds ~/s from the fermionic constituents.

Since the 1- spin-pari ty assignment of ~/(3100) and ~(3700) has been firmly established, these states must correspond to the p-wave spatial arrangement of diquarks in the bound state. The p-wave configuration of diquarks in ~'s helps one to explain the un- usually narrow decay widths of ~'s. We claim that this conspicious phenomenon does not have a single cause, but is a cumulative effect of at least two factors, namely the selection rules based on the duali ty diagrams that suppress the decay of qqqq quark configurations in general (4,5), and the depletion of the ~ wave function at the origin because of the centrifugal barrier. Objects in the p-wave configuration have difficulties interacting and annihilat ing each other, a feature which results in a considerable reduc- t ion of the decay rate. Rough calculations, performed within the boundaries of a non- relativistic potential model confirm tha t the suppression factors of 10--100 can easily be obtained by the described mechanism (6).

I t should not pass unnoticed that in order to explain the narrow width of ~/s we did not need to invoke any extraneous dynamical assumptions. The narrow width of ~'s is an unavoidable consequence of the model.

(5) P . G. O. FREUND, I~. WALTZ a n d 5. lC~OSNER ; NUCl. Phys. , 13 B, 237 (1969). (6) Th is a r g u m e n t works u n d e r t he t a c i t a s s u m p t i o n t h a t t h e effect ive v o l u m e wh e re t h e ann ih i l a t ion process dd- -~hadrons t a ke s p lace is sma l l c o m p a r e d w i t h t h e t o t a l size of t h e sys t em. Since t h e d e c a y w i d t h s of O's go ing in to t h e e+e - pa i r s a re of c o m p a r a b l e size to t h e cor responding decay wid ths of k n o w n v e c t o r mesons p, ~ an d ~, we m u s t conclude t h a t t he descr ibed cen t r i fuga l b a r r i e r has no effect on the e l e c t r o m a g n e t i c ann ih i l a t ion process d d ~ e + e - t h a t p r e s u m a b l y r equ i re s a l a rge v o l u m e , c o m p a r a b l e in size to t h e dd bound s t a t e s y s t e m .

DIQUARKS AS AN EXPLANATION FOR d/'S, R, AND EVERYTHING ELSE 171

From the point of view of uni tary symmetry we conjecture tha t d/'s are the neutral members of the representations 1 and 8 from 3• 3 = 1 + 8. The diquarks from the S U 3 representation 6 either do not part icipate in the formation of bound states at all, or may perhaps be found in the s-wave bound states where the forces among the constituents are stronger because of the larger overlap among their wave functions. Tentat ively we may identify d/(3100) with the S U 3 singlet and d/(3700) with the I = 0 member of the octet. These assignments are suggested by the observed decay modes of these resonances (~).

The s-wave states d/0+ lie lower in mass than the corresponding p-wave states d/l-, and therefore we expect the radiat ive transitions of the type d/1-~ 7 + d/0 +. I f the di- quarks from the S U 3 representation 6 do not participate in the formation of d/o + bound states, the maximal number of neutral d/o + states is 3. Otherwise, this number can be 5 t imes larger.

The d-wave states d/s+ lie higher in mass than d/l-, and so the radiat ive transi- tions d/1--->y+d/~+ are energetically forbidden. There is a possibility, however, that the 2 + recurrence of d/(3100) lies lower in mass than d/(3700), in which case the cascade decay

d/1-(3700) -+ Y+ d/s +

I__> y+ d/1-(310o)

is possible. The charge conjugation quantum number of the neutral d/~- states is C = --1, while

the neutral d/o + and d/s+ states are characterized by C = + 1. More generally, our model admits the series jPe = 0++, 1-- , 2 ++, 3- - , ..., and no other states. This is a strong restriction.

c) In the energy interval 1 GeV ~ E ~3.4 GeV the angular distribution of the final-state jests should follow the law 1 + cos20, as is appropriate for the spin-�89 quark- partons. In this energy range the channels with diquarks are not opened as yet. Beyond E N 5 GeV the angular distribution is a combination of two components, the quark component and the diquark component. We expect 1 + =cos20, ]a[ < 1, where the real parameter a measures the relative strength of these two components. There are two hypothetical extreme cases, ~ = 1 if the diquark component is not present, and c r if the quark component is not present. The experimental data for a, ~r = 0.78 -4- 0.12 at E = 7.4 GeV (s), are not in contradiction with our model, although a more isotropie angular distribution in this energy range would be more in line with the model. Quanti tat ive predictions are difficult to make, since we do not know whether quarks and diquarks fragment in the same way.

There are three critical tests of our model. Firs t and foremost, no +-like states of (( unnatural ~> spin-parity 0-, 1 +, 2-, ... should be observed. Their possible discovery in the radiat ive decays of d/'s would bring an early end to the model. For comparison we mention tha t the states 0-, 1 +, 2-, .... forbidden by our model, will be present in all models that build ~'s from fermionic constituents.

The diquark model is equally vulnerable to the possible discovery of charm or similar quantum number in the final-state products of e+e--+hadrons. Our model does not predict charm, nor can the charm be introduced in the model without an immediate conflict with the existing data on R.

( ' ) S e e , f o r e x a m p l e , H . L . LYNOH: e+e - interactions, L e c t u r e s d e l i v e r e d a t Cargese S,~mmer School, ~argese, France, July 1975, S L A C - P U B - 1 6 4 3 . (a) G. HANSON et al.: S L A C - P U B - 1 6 5 5 .

172 ~. FAVKOVIC

Finally, a dramatic confirmation of the val idi ty of basis premises of the diquark model would come from a discovery of a sudden upswing of R with energy, R ~ E 2. This behavior of /~ is characteristic for the axial-vector diquarks, and can hardly he explained by any other means. A non-discovery of the described effect, on the other hand, does not constitute evidence against the diquark model. I t merely asserts tha t the axial-vector diquarks are heavy and beyond the reach of the presently available energies.

Aside from the predictions listed in this article, the diquark model also has definite predictions concerning mass spectra, magnetic moments, scaling, etc. No conflict with the data has been encountered thus far, and in many instances the data fitting is much easier than in the quark model. The details will be reported elsewhere.

The idea of diquarks in the role of partons is due to Prof. BJORKEN. I thank him for a number of i l luminat ing discussions on the subject of diquarks and their possible relevance to the ~ new physics ~>. My thanks also to G. F~LD~A~ and other members of the SPEAR group for valuable information concerning the interpretation of some aspects of SPEAR data.