L]~TTEtCE AL NUOVO CIMENT0 VOL. 5, ~. 10 4 Novembre 1972

Internal Symmetries and Han-Nambu Partons.

T. H. CI~A~G mid D. K. CHOUDHURY

Department o] Theoretical Physics, University o/ Ox]ord - Ox]ord

(ricevuto 1'11 Set tembre 1972)

In the 7:~ ~'~" decay, the charge of the fundamental fields has become an impor tant problem due to the Bell-Jackiw-Adler (1) anomaly. Only when the integrally charged quarks as in Han model (5) or the three t r iplet of fractionally charged quarks as sug- gested by FRITZSCH and O~ELL-MANN (3) a r e inserted in the tr iangular diagram, one can obtain the correct decay data. I t is perhaps quite natura l to ask if the deep inelastic region can also shed some light on the charges of the fundamental fields, since they are directly related to the experimental ly measurable structure functions.

Wi th this spirit, we have noticed recently (~) that the main features of the deep inelastic lepton-hadron scattering as the bounds for the cross-sections of ep-+ ex and vp-+ ~x can as well be explained gy an integrally charged quark pa t ton model as by the usual quark-pat ton model of fractional charge (5). The basic assumptions were tha t the patrons are Han-Nambu quarks and tha t in the weak and electromagnetic interactions su 3 is maximal ly violated so tha t the (~ charme ~ states presumably ex- cited can redis t r ibute themselves to give final-state zero charm hadrons.

The much concerned ratio 2~=(x)/F~(x) in all different pa t ton models is related to the charge square of the f lmdamental fields and imposed internal symmetries. The aim of this note is to present the predictions of this ratio from our model by invoking different internal symmetries in the deep inelastic region.

I t has been pointed out by NACHTMANN (~) and CALLAN et al. (s) that , in the scaling l imit , the structure functions of the deep inelastic lepton-hadron scattering obey certain posi t ivi ty restrictions. This can be obtained either from the current hadroa scattering in the light-cone current algebra approach or from the par ton-hadron scattering in par ton models, as a consequence of a presumed strong-interaction internal symmetry. For our analysis we follow the language of par ton model and s tar t with su 2' x su 2" as the

(1) ft. S. BELL a n d R . JACKIW: 2Vuovo Cimento, 60 A, 47 (1969); S. ADLER: Phys . Rev., 177, 2426 (1969), F o r review, see S. L. ADLER: Lectures at the 1970 Brandeis Summer Ins t i tu te , vol . 1 (Cambr idge , Mass . , 1970). (2) H . Y. HAN a n d Y. NAMBU: Phys . Bey., 139 B, 1006 (1965). (3) H . FRITZSCH a l ld M. GELL-MANN: in Proceedings o] the 1971 Internat ional Conlerence on Dual i ty and Symmetry , Tel A v i v (to be pub l i shed) . (4) R . BUDNY, T. I t . CHANG a n d D. K . CHOUDttlYRY: O x f o r d p r e p r i n t re f . 5/72 (to be pub l i shed in Nucl . Phys . ]3). (1) O. NAC~TMANN: Nucl . Phys . , B 38, 397 (1972); Phys . Rev. D, 5, 686 (1972) a n d re fe rences the re in . (6) C . G . CALLAN, M. GRONAI~, A. PALS, E. PASCEOS a n d S. B. TREIM-~I~: N A L prelar in t Th-29 (1972).

727

728 T. it. CHANG and D. K. CI-IOUDHURY

relevant symmetry group. Assuming that partons are Han-Nambu quarks, all the structure functions can be expressed in terms of 18 diagonal operators, matrix elements of which between the proton state are symbolically denoted by

(1) { 2r = (plt:~(x)ti~(x)[p~,

for pat tens and antipartons respectively. Here tTk(x) and ti*+(x) correspond to the an- nihilation and creation operators of a parton with su~ index j and su~ index - - k in the notation of GOUI~DIN (7). Factorizing these amplitudes into su~ and su~ components, we see that while the su~ par t is expressed in terms of three positive reduced ampli- tudes, su~ par t is characterized by only two such quantities. This distinguishing feature is a direct consequence of the isosealar nature of the nucleon in su~ space. The su~ posi- t ivi ty restrictions give the inequalities for any k

(2)

22Y~-~(x) - - _hr~k(x) ~> 0,

:V;k(x) > 0 ,

2V~(x) > 0 ,

- - ~V~_I (x) + 22V~_2(x)/> 0 ,

:V*_~(x) > O,

2v*_~(x) > o ,

while the su~ par t gives the relations

{ hr~_~(x), :V2_~(x),

(3) 2V71(x ) , ~72(x), hr3_~(x) > 0 ,

NTs(x) >~ 0,

for any ]. Fur ther isoscalar nature of the nucleon in su~ yields the identity

{ .~l_j(x) = N~(x), (4) :V71(x ) = N;2(x).

These matrix elements can be identified with the Bjorken-Paschos (8) notations by the tollowing simple correspondence:

(5) :~;~(x) ~ P~lT~(x) n~ ~ .

Here ]~k(x) is the probabil i ty of finding n7 k number of t~ ~ partons with a fraction of momentum x and Px denotes the probabil i ty for the 2Y-parton configuration. We also notice that eq. (4) will imply that the number of independent diagonal matrix elements characterizing the structure functions is reduced to twelve from eighteen.

(') M. GOUR1)IN: Erice Summer School lec ture notes, 1971 CERN, ref. TH 1384. (8) J. D. BJORIi~EN and E. PASCrTOS: Phys. Rev., 185, 1915 (1969).

INTERNAL SYMMETRIES AND HAN-~TAMEU PARTONS 729

The structure functions for the process e~?-+ ex can be expressed in terms of the densities of partons and antipartons. Assuming the Callan-Gross relation F~(x)= xF~(x), we have

(6) + ~V~_~(x) + ~_~(~) + _~_~(x),

F~"(x) = ~V~(x) + N~(x) + ~v~(x) + ~V~(x) + N~(x) +

Using eqs. (2), (3) and (4) and noting tha t

~ ( x ) ~ ~F(x) ~< G~ + -~~ ~'

we get

Ul~ (7) 0 ~< ~,~:5__ ~< ~ . (x)

Equat ion (7) is the general result of the integral ly charged Han-Nambu par ton model with the minimal symmet ry of su" x su~. The corresponding results from the quark par ton model with su 2 and s% invariance were respectively

(s) ~<F?(~)<~ and -~<~F(~)< 3

This last result (*) also corresponds to the predict ion of the Han-Nambu quark par ton model if su~ is an exact symmet ry in the weak and electromagnetic interactions (0).

I t is also interesting to note tha t eq. (7) remains to be true even if su~ is an exact symmet ry for the elcctromagnetic and weak interactions. This can shown by decom- posing the charge density Q for the HamNambu nonet:

(9) = tl~3t~-3 ~_ -2 + -2 --1 + --1 • --1 _ _

[ t ~ t ~ a + I • -2 -2 + -~ _ _ t - l + t - l ~ t • 1~1 + ~(tx t l - - t 2 t2 ) + �89 1 2 2 J - - 3 3 ,J

+ [�89 -2 + t ~ + t ~ ~) - - �89 ( t ~ + t ~ ~ + t~-~+t~-~)].

The second bracket transforms like an su~ isovector and hence does not contribute to the nucleon matr ix element if su~ is an exact symmetry. Then the effective charges of the partons become l , 0 , - - 1 , + �89 a n d - - � 8 9 Since the domain of the ratio F~"(x)/FVP(x) depends only on the minimum and maximum charges carried by the fundamental fields, we may conclude tha t eq. (7) remains unchanged even if su~ is str ict ly conserved.

(*) The lower bound �88 was also ob ta ined recen t ly b y PALLUA an d RENNER assuming absence of exotic t -channel exchanges a nd van i sh ing of decuple t (10) contr ibut ions in the scaling limit. See ref. (10). (~) H. J . LIPKIN: Phys. Rev. Left., 28, 63 (1972). (10) S. PALLUA and B. I~ENNER: Phys. Lett., 28 B, 105 (1972).

7 ~ 0 T . H . CHAI~G and D. K. CIIOUDHURY

II On the other hand, if one considers another version of the model (4) where sua is

maximal ly broken in the final state, while sv' 3 x su~ acts as the relevant strong-interaction symmetry , the par ton (antiparton)-hadron scattering ampli tudes hrjk(x) (~Vk_j(x)) are expressible in terms of 6 independent diagonal matr ix elements only instead of twelve, since the su~ singlet nature of the nucleon gives the following condition on the parton- hadron ampli tudes:

(10) { N_~(x) = N~_j(x) = ~V'~/x),

NT'(x) = ~VT~(x) = N?~(x),

for any j. In this case, the ratio w yp H 1 /F~ now has the domain (~*)

2 / ~ (x)

Any deviat ion from eq. (11) is an su~ x su~ invariant strong-interaction model will imply significant su'~ nonsinglet mixture to the physical nucleon state. The present experi- mental value of the ratio is smaller (~2.~a) than the lower bound of eq. (11). Since an appreciable su~ nonsinglet mixture to the nuclear state will disturb the low-energy results of the usual hadron spectroscopy, therefore we feel tha t this hypothesis is rather spec- ulative. If the experimental t rend is such tha t the ratio will go even below the lower bound of eq. (8) as x-+ 1, then our result eq. (7) will indicate tha t the fundamental fields car ry integral charge and the choice of su ' x su~ as the internal symmetry group in the deep inelastic region is a reasonable one.

To conclude, we also point out tha t the Han-Nambu quark par ton model can fit the neutrino da ta (14) from its electroproduction predictions so long as one assumes tha t the addit ional internal symmet ry of su~ or su~ cannot differentiate the weak and electromagnetic interactions, i.e. the diagonal quantum number (charm) associated with the group has either to be s t r ic t ly conserved or violated maximal ly in both class of interactions in the final states. A pa t ton model formulated from CAmBBO, MAIANI and PRErARATA (~5) three-tr iplet model allows the breaking of su~ in the electromagnetic interactions but not in the weak interactions and hence i t predicts too low a cross- section (a) for the neutrino-induced reactions, contrary to experiment.

We are grateful to Prof. R. H. DALITZ for helpful comments. We also thank Profs. J. D. BJORK~ and C. H. LL~WELLYN SmTH for very useful correspondences in clarifying the roles of internal symmetries. One of us (T.H.C.) acknowledges financial support from the Science Research Council and the other (D.K.C.) an 1851 overseas scholarship.

(11) C. If . LLEWELLYN" S~ITH: I n v i t e d t a l k in t he I V International Con]erence on High-Energy Col- isions, O x f o r d U n i v e r s i t y (Apri l 1972). (,2) E. D. BLOO~: SLAC P U B 796 (1970). (1~) j . D. BJORKEN: t&lk p r e s e n t e d a t t he Con]erence on Particle Physics, I rv ine , Cal. , December 1971, SLAC-PUB-1017 . (14) ~ . ]~E~DALL: 1971 International S y m p o s i u m on ElectrOn and PhotOn Interactions at High-Energies, (Cornell , 1971); G. MYATT a n d D. H . PEr t a inS : Phys . Lef t . , 3 4 B , 542 (1971). (16) N. CABIBBO, L. MAIA~I a n d G. PREPARATA: Phys . Left . , 2~ B, 132 (1967).