L~TEaE AL NU0V0 CIMEN~0 V0L. 14, ~. 1 6 8ettembre 1975

A s y m p t o t i c S c a l i n g L a w for I n c l u s i v e P r o d u c t i o n

in Q u a r t e t Q u a r k M o d e l .

S. IWAO

Department o] Physics, College o] Liberal Arts, Kanazawa University - Kanazawa

(ricevuto il 5 Luglio 1975)

It is currently fashionable to assume thc newly observed ~(3095) (1.a) to be an odd parity and charge conjugation parity 1-- c~ bound state (c being the fourth member of the quartet quark) of unitary ( 1 5 + 1)-plet meinber of SU~ symmetry. The ~(3684) (~) reported soon after the first discovery of ~(3095) is, then, considered to be a radially excited state of the latter due to the strong (nmin) decay mode of ~(3648)-~ ~(3095)+2v: in s-wave two pions (~) but with extremely narrow total width similar to +(3095). It is usually assumed that the smallness of thc width is due to the so far unclurified 0kubo-Zweig-I izuka rule in analogy to the ~-meson decay. Along this

(~) J . J . AUBERT, U. nECKER, P. J . BI~68, J . BUR(~ER, M. CI[EN, G. EVERHART, P . GOLDltAOEN, J . LEONG, T. McCO~r T. G. RtlOADES, M. I~OIIDE, S. (~. (!. TING, S. L. WU ~nd u Y. LEE." Phys. Rev. Lett., 33, 1404 (1974). (2) J . - E . AU(~USTIN, A. M. ]~OYARSKI, M. ]3REI1)ENBA('tl, F. BUL()S, Z. T. DAKIN, G. J . FELDMA~ ~, G. E. FISCHER, D. FRYEERGER, G. tIANSO~, B. JEAnne'-MARIE, 1)~. [{, LARSE.'g, V. L()TH, H . L. LY~(:H, D. LY()]~, (!. C. MOREHOUSE, J . M. PATERSOn, M. L. PERL, B. ]~ICItTER, P. ]~&PIDIS, R . F . SCII~V1TTERS, W. iW. TANENBAUM, F. VANNUCCI, G. S. ABRAMS, D. BRIGGS, %V. CllINO%VSKY, (~'. E . FRIEDBERG, G. GOLDtIA:BER, ]~. J . HOLLEBEEK, J . A. KADYK, ]3. LULU, F. PIERRE, G. H. TRILLING, J . S. ~VI[ITAKER, J . WISS ~lld J . E. ZIPSE: Phys. Rcv. Left., 33, 1406 (1974). (a) (J. BACCI, l~. BALDLNI (JELIO, J~. BERbrARD1NI, (]. (JAI,ON', R . DEL FABBRO, ~r GRILLI, E. IAROCCI, L. JONES, M. LOC~;I, (!. MENCITCC[NI, G. P. ~r (J. PENS(), G. SALVINI, M. SPINETTI, M. SPANO, U. STELLA, V. VALENTE, B. BARTOLI, D. BISELLO, B. ESPOSITO, F. FELICETTI, P . ~r M. NIGRO, L. PAOLUZZI, I . [PERUZZI, G. PIAI~O MORTARI, ~ . PICCOLO, F. RON'G~A, F. SEEASTIANI, L. TRAS&TTI, F . VANOLI, G. BARBARI~rO, G. BAREIELLINI, C. BEMPORAL, I~. ]~IA~CASTELLI, F . CEVENINI, ~1. CALVETTI, M. CASTELLA~O, F. COSTA~rTINI, P . LARICOIA, e . PARASCAS~rDALO, E. S&ssI, ~ . SPENCER, L. TORTORA, U. TROYX a~14 S. VITALE: Phys. Rev. Lett., 33, 1408 (1974). (~) (~. S. ABR&MS, D. BRIOGS, ~V. (JHI~OWSKY, C. E. FRIED]]ER(~, G. GOLDttABER, ]~. J . I-IoLLEBEEK, J . A. KADYK, A. LITKE, B. LULU, F. PIERRE, ]~. SADOULET, G. H . TRILLING, J . S. WHITAKER, J . ~VIss, J . E. ZIPSE, J.-E. AUGUSTIN, A. ~L BO~rARSKI, M. I{ICEI1)ENBACH, F. BULOS, G. J . FELDSIA2~ r, G. U. FISCHER, D. FRYBERGER, G. HAI~SO~, B. JEAn-MARIE, ]r 1~. LAltSEN, V. LOTIt, H . L. LYI~CH, D. LYOn, (J. (J. MOREHOUSE, J . M. PATERSOn, M. L. PERL, l~. RICtITER, P. RAI'IDIS, R . F. SCHWITTER,S, W. TA- NENBAUM a ~ d F. VAN~UCCI: Phys. Rev. Letl . , 33, 1453 (1974). (~) G. S. AB[r D. 1). BRIGGS, W. CItINOWSK~, (;. E . FRIEDBERG, (J. GOLDHABER, J . A. KXDYK, A. M. LITKE, 1]. A. LITLU, F. M. PIERRE, B. SADOULET, G. H . TIr J . S. WItITAKER, J . E. ~VIss, J . E. ZIPSE, A. M. BOYARSKI, M. BREII)ENB&CII, F. ]~ULOS, (.~. J . FELDMA~ r, (~. E. FISCItER, D. FRY- I~ERGER, ({. ~tANS()]~, B. JEAN-MARIE, 1{. R . LARSEN, V. LUTI[, H . L. LY~CII, D. LYON, C. C. MOREI[OUSE, J . M. PATERSON, M. L. [)ERL, P. I{AI'IDIS, ]~. I{ICIITER, R . F. SCItWITTERS, W. TANE~BAUM ~n(1 F. V ~ r ~ u c c i : Phys. Rev. Left. , 34, 1181 (1975).

2 S. :WAO

line of thought the third resonance with broad width at 4.15 GeV (r may also be inter- preted as a radially excited state of c~ and the broadness of the width may be due to the opening of a new channel for charm-conserving strong decay (7). The S U 4 symmetry seems to be the best candidate to explain the earlier information of NIU et al. (s) and the recent data by CAZZ0LI eta l . (9,10) and that of JAIN and GIRARD (~), interpreting them either as the member of 20-plot baryons or as the member of the regular representation of p.s. (or vector) mesons in that symmetry. The detailed (but still qualitative) work prior and after the discovery of the ~-particles was pursued by LEE and his colla- borators (1~) as a continuation of the unified and renormalizable (non-Abelian) gauge theory of weak and electromagnetic interactions of W~INB~G and SALAM and of GLASHOW, ILt0POVLOS and MAIANI (~a) along that line. Owing to the large symmetry breaking of S U a (I4.i~) the charmed particles including ~'s are expected to be produced copiously in high-energy collision and the simple quasi-two-body production study seems to wait for some time. Above all, the high masses of the new particles require to haw~ a lot of open channels so that the expected weak-decay lifetime ( ] 0 -11 tO l0 -14 S) is in the range of enmlsion study. Under these circumstances GA:SS~R and HALZEN (iv) have estimated the inclusive cross-section of ~-particles by making use of the so-called scaling law developed earlier by SH~N and BERKEY (18) in connection with cosmic-ray study. A slightly modified version of the scaling law has been proposed by HAYKSttI and myself (~9) in order to relatc the observed energy spectra of relative multiplicity of electrons (positrons) with large transverse momenta to that of negative (positive) piers, as the decay products of ~-particles, obtained at three different laboratories (~o).

(~) J . -E . AUOUSTIN, A. ]Yl. BOYARSKI, M. BREIDENBACtt, F. BULOS, J . T. ])AKIN', G. J . FELD~IAN', G. E. FISCIIER, D. FRYRERGER, (~. HANSON, ]3. JEAN`-MARIE, 1~. R. LARSEN`, V. L(~TH, 11. L. LY'NCtI, ]). LY()!~ r, C. (J. MOREHOVSE, J . M. PATERSON`, M. L. 1)ERL, B. RICtlTER, 1)~. F. SCHWITTi~,RS, ~'. VAN']~'UCCI, (~. S. _A_RRA.~IS, D. BRIGOS, W. (~tIINOWSKY, (I. E. I?RIEI)RERG, G. (~OLI)]IABER, R. J. HOI,[,Em,~leK, J . A. KAI)YK, G. H. TRILLIN'(~, J . S. WJCITAKI,:R a nd J . E. ZIPS~,:: P h y s . ReD. Le t t . , 34, 764 (1975). (~) M. SuzuKI: P h y s . Rev. L e l t . , 3 4 , 1412 (1975). The po in t discussed in tho t c x t is not yot f i rmly estab- l ished exper imenta l ly . (s) K . NIu , E. MIKU)IO aud Y. MAEDA: Prey . Theor. P h y s . , 46, 1644 (1971); K . IIOSHtN'O, S. KURAMATA, K . NIU, K . MIWA, S. TASAKA, E. MIKUMO all(]. Y. MAEDA: Oa the X part icles , prep r in t (1974). (9) E. G. (JAZZOLI, A. M. CN'OPS, P. L. (~ONNOLLY, R. I . LOUTTIT, M. J . MURTAGIt, 1~. S. PALMER, N. P. SAMIOS, T. T. TSO aRd H. I-I. WlLLI&MS: P h y s . ReD. Le f t . , 34, 1125 (1975). (10) B. W. LEE: A comment on the BNL event vp -~ ~t-A~ , F E R M I L A B - 7 5 / 3 8 - T H Y (May 1975). He proposes a new i n t e r p r e t a t i o n of t h a t process as vp--~ ~-A~ ~ The e s t i m a t e d F + m a s s is 2.29 GeV. L ~ + ~ + ~ - . (~x) p . L. JAIN" and B. GIRARD: Phys . ReD. Lo l l . , 34, 1238 (1975). (:~) M. K . GAILLARD, D. W. LEE aD.4 J . L. ROSN'ER: Search ]or charm, F E R M I L A B - P u b - 7 4 / 8 6 - T H Y (August 1974); A d d e n d u m Search for charm, F E R M I L A B - P u b - 7 5 / 1 4 - T H Y ( J a n u a r y 1975). I shal l use t he s ame no ta t ion as t h a t of th is papcr t h r o u g h o u t m y paper . (aa) B. ~V. LEE: Perspective on theory of weak interactions, iR Proceedings o! the X V I Internat ional Con- ]erence on High-Energy Phys ic s , Vol. 4, ed i ted by J . D. JA(~KSON', A. ROBERTS au d R. DONALDSON" (1972), p. 249, and references t he r e in conta ined. (~4) S. BORCHA~DT, V. S. MATHUR and S. OKuno: P h y s . Rev. Le f t . , 34, 38 (1975); V .S . MAT~IUR, S. OKUBO and S. BORCHARDT: SUa symmet ry and the new resonances, COO-3065-106 ( J a n u a r y 1975). (is) S. OKUB(): SU4, SUs mass formulae and weak interactions, COO-3065-112 (1975). (~) Z. MAKI, K . OIINISH1, T. T],~'SHIMA aD.4 I . UMEMURA: The <, ~ *> part ic les in the quartet model,

RIFP-215 (Februa ry 1975). (~) T. K . GAISSER a nd F. HALZEN: Produclion mechanism of +(J) particles in hadrore collisions,

COO-435 (December 1974). (as) C. S. Sm,ZN an4 G. B. BERKEY: P h y s . RED., 171, 1344 (1968). (1~) H . HAYASHI and S. IWAO: Inclusive product ion of ~ part ic les in pp coll ision and scaling law, t t P I C K - 0 8 (Feb rua ry 1975). (~0) f . ~V. B(~SSER, L. CAMILLERI, L. ])I LELLA, B. (]. POPE, A. M. ,~M1TH, B. J . BLUMEN'FELI), ~. N. WItlTE, A. F. ROTHENBERO, S. L. SE(~LER, M. J . TANNENBAUM, M. 11AN'N'ER, J . B. (~HEZE, J . ~. HAMEL, H . KASHA, J . P. 1)AN'SART, (~. ,~MADJA, J . TEIGER, 11. ZACCO~E an d A. ZYLBERSTEJN`I P h y s . LOll. , 53 B,

212 (1974).

ASYMPTOTIC SCALING LAW FOR INCLUSIVE PI4ODUCTION ETC. 3

:Rccently, L B L and F N A L groups havc repor ted inclusive p0 produc t ion in n -p in- t e rac t ion at 205 GeV/c (~). This exper iment tells us t ha t about one- thi rd of pions are decay products of rho-mesons at t ha t energy. This resul t suggests tha t the scaling law ut i l ized by earl ier workers (mul t ip l ic i ty is invcrscly propor t ional to the mass square of t ha t par t ic le inclusively produced) should be modified. The purpose of this paper is to cons t ruc t a new scaling law which accommodates the above new observat ion as well as avai lable informat ion (see la ter discussion) including +-particle inclusive produc- t ion (2~), w i thou t damaging apprec iably a good aspect of the old scaling law, in te rms of the qua r t e t version of the quark model. The physical significance of the a sympto t i c scaling law in the new version is also discussed to a cer ta in extent .

L e t me begin wi th the mass formula for pseudoscalar mesons to the lowest order in chiral SUa breaking, as a s t ra igh t forward extension of the chiral SUa-breaking model (~.~a'~a), viz.

7%2 2 2 (1) - _ _

where u, d, s and c indicate up, down, s t rangc and charmed quark, respect ively , D and F arc new members of p.s. mesons in the regular representa t ion of the S U c g r o u p (12). The above relat ion as well as the addi t iomd pa rame te r

(2) R = me--re, 79? m - ~l u

plays a decisive role for de te rmin ing S U 4 mass relat ion in the lowest order of s y m m e t r y breaking. By making use of t he first equa l i ty of eq. (i) and R = 20.7 (14), assuming m , = me, I find

m, m~

The smallness of the mass-mixing parantc ter

(4) e ~ ( 4 R - 2) - l _~ 0.012

guarantees the re l iabi l i ty of eqs. (1) and (3). Le t me s ta te tirst a new scaling law. l assume tha t (~ at ex t remely high energy

the mul t ip l ic i ty of the part icle, hence its inclusive single-part icle product ion cross- section, is inversely propor t iona l to the sum of its cons t i tuen t -quark masses, p rov ided by the re la t ive st icking p robab i l i ty due to two and three cons t i tuents in t hc na ive quark model )r. The asympto t i c scaling law s ta ted here will hold at ve ry high energy where the large mass differenccs among quarks can be neglected. Such a supposi t ion m a y

(21) F. {'~. XVINKEI~IANN, II . H. BINGlfAM, I), 3[. CItEW, B. Y. [)AUGI,]RA,r ~V. 13. I~'ltETTER, G. (]OLDIIAI]ER, ~V. It. GRAVES, A. I). JO1LN'gO~, .]. A. KAI)YK, L. STUTTF, (I. }[. TRILLING G. l ). YOST, D. ]~()GERT, 1~. IIANFT, F. [r HUSON, S. KAII.N', 1). LJUA~(J, (I. PASCAUD, ~. I~R[!SS ti~]I(]_ ~V. M. ~MART: L B L - 3 3 9 0 (November 1974).

W) F. W. BCTSSER, h. (~aMn,LmtI, L. DI LV, LLA, B. G. ])el'E) A. M. SMITIt, l~J. J-. BLUMENFELD, S.N. ~:HITE, A. F. ROTItENBERG, .~. L. SE(HA~R, ~/[. J . TAN'.'VENBAU~I, ~{. BANN-ER, J . B. CHEZI~, J . L. ~AMEL, H . I(ASHA, J . p . pA~rSART, G. SMADJA, J . TEIGER, H . ZACCO.N'E a n d A. ZYLBERSTEJ~: Obse**atlon of high masseleetron-posilron~)airs produced i~ proton-proton collisions at the C E R N I S R , p r e p r i n t (Apri l 1975).

(2a) M. OELL-MA:~'.~,', R . J . OAKES ttltd. B. JiE.~'~'I.;R: Phys . Rev. , 175, 2195 (1968). (.m) S. L. GLASHOW ~nd S. WEIN~BERG: Phys . Rcv. Lcl t . , 20, 224 (1968).

4 s . ~ w x o

not be unreal is t ic , since at an ex t remely short dis tance one may expec t an a sympto t i c free condi t ion by Yang-Mills (massive) vec tor -g luon b inding p ic ture of f undamen ta l cons t i tuents (25.26) and now taken as the basis of quark confinement , the so-called bag theo ry (27.2s). Thus, I expec t t h a t the s i tuat ion will depend upon the absolute va lue of quark masses, s t ruc ture of the part icles (gluons, sea quarks n qq, n = 0, 1, 2 . . . . . b ind ing energy of produced particles, etc.) and another unfami l ia r factor such as a s t ruc tu re of the quark, if i t exists, at least at f inite energies. I shall no t be concerned wi th all compl ica t ions arising from these quest ions in this phenomenologica l analysis, excep t for making a c o m m e n t on quark masses.

Some t ime ago L]~UTWYLER (~o) argued t h a t the mass of the nons t range quark is as smal l as m~ = 5.4 MeV. Combining this va lue wi th eq. (3), I find m o = 2.6 GeV which is no t inconsis tent wi th 1.5 GeV surmised by GAILLARD and L}~E (,2) before the discov- ery of the ~. In my opinion i t is dangerous to assume a too l ight mass va lue for the l igh tes t quark, as we have exper ienced in the old-day nuclear physics where the nucleus was supposed to be cons t ruc ted f rom protons and electrons, f rom the size of hadrons in cont rad is t inc t ion to the Compton wave leng th of the cons t i tuent quarks. I f I choose a guessed va lue m~ = 350 MeV from the anomalous magnet ic m o m e n t of nucleon, mo be- comes larger. F u r t h e r m o r e if m ~ > 1 GeV wi th bag theoris ts I should assume a huge va lue as a sympto t i c energy. I n any case a s imple-minded es t imate of quark mass al- r eady suggests us w h a t ex t r eme ly h igh energy should be in a new vers ion of the scaling law. Ano the r aspect of the scaling law is t h a t all the part icles on the Regge t r a j ec to ry should be produced wi th near ly equal mul t ip l i c i ty at an asympto t i c energy.

The mass factor of the old and new scaling laws is compared in Table I in re la t ion to the pion mul t ip l ic i ty . The mix ing states of the pseudoscalar ( 1 5 + 1)-plet of S U 4 are shown for two choices, e i ther ~, X ~ or ~, E wi th ~r according to two possible assign- ments (8~). The re la t ive st icking p robab i l i ty of baryons is indica ted by P . Some of the masses of cha rmed baryons are es t imated by making use of the squared-mass formula (,2).

Before enter ing into the discussion of most recent da ta i t would be wise to recal l the s t andard analysis per formed by ANTINUCCI et al. some t ime ago (3~). The average mul t ip l i c i ty of a nonst range hadron of t ype i can successfully be f i t ted by the s imple formula f rom PS to IS]~ energies:

(5)

o r

(6)

< n i ) = A + B l n s + C s - i ,

<n~} = A ' ~ - B ' i n s + C's-�89 i n s ,

where A ( A ' ) , B ( B ' ) , C(C ' ) are numer ica l constants . I t is known tha t eq. (5) fits wi th a smaller ; /-square re la t ive to eq. (6). A t ex t reme high energy only the

(25) D. J. GRoss an4 F. WILCZEK: Phys. ReV. Lett., 30, 1343 (1973). (~s) H. D. POLITZER: Phys. Rev. Left., 30, 1346 (1973). (~') A. CHODOS, R. L. JAFFE, K. JOHNSOn, C. B. THORN and V. F. WEISSKOPF; Phys. Rev. D, 9, 3471 (1974). (2s) W . i . BARDEE~, M. S. CtIANOWITZ, S. D. DRELL, M. WEI.~STEI~ a n d T.-M. YAN: Phys. Rev. D, l l , 1094 (1975). (39) j . KOGUT and L. SUSSKI~D: Phys. Rev. D, 10, 3468 (1974). (a0) H. LEUTWYLER: Phys. Left., 48B, 431 (1974). (al) R. F. DASHEN, I. J. MUZINIOlt, B. W. LEE an4 C. QuIG(~: Coherent production and decay modes o] a pseudoscalar partner o] the ~(3105) boson, COO-2220-39 and. FERMILAB-PUB/75/18/THY (January 1975); B. W. LEE an4 C. QuI(~(~: Some considerations on ~r FERMILAB-74/l10-TttY (December 1974). (a2) M. ANTIIWUCCI, A. BERTI~, P . CAPILUPPI, M. D'AGOSTIIWO-BRU~O, A. M. ROSSI, G. VANNINI, G. GIACOMELLI an4 A. BUSSII~RE: Left. NUOVO Cimento, 6, 121 (1973).

ASYMPTOTIC SCALING LAW FOR INCLUSIVE PRODUCTION :ETC. 5

T A B L E I . - - Comparison o] old and new stalin 9 taws (relative multiplicity in units o] that o] the charged pion).

Particle Old New

~:+ 1 1

po 0.033 1

0.065 1/9.28

X o 0.021 1/15.7

~q 0.065 1/13.9

E 0.0097 1/11.1

K 0.080 2/25

K* 0.025 2/25

0.019 1/24

I) 0.0042 1/239

F 0.0040 1/250

~(q~o), ~qo 0.0020 1/477

J~', ~N" 0.022 2P/3

A 0.013 2P/3

A 0.016 3P/26

E 0.014 3/)/26

0.011 3P/49

0.0070 P/24

C 0.0023 /~

S, A - - P/167

T - - /)/175

Xu 0.0011 P/318

X. - - /)/326

(ccc) - - P/477

relative magnitude of B (B ' ) i s of interest. I t is given to be 0.84=~0.07, 0 .87~0.05, 0.13 ~= 0.01, 0 . I1 • 0 . 4 • and 0.059 ~= 0.006 for r:+, re-, K +, K - , p and ~, re- spectively. As we shall see we cannot compare directly these values with the predic- tion of the asymptotic scaling law, since these numbers are the superposition of decay products from p, K*, A etc. However, the minimum-maximum of the relative sticking probability, defined in Table I, may be estimated to be

(7) O . I O ~ P < I .

As far as the relative multiplicity of pions and kaons is concerned, there appears no dif- ference between the old and new scaling laws. Now, let us turn our at tention to recent information.

6 8 . IWAO

In tha t course, i t would be more appropria te to specify the to ta l centre-of-mass energy ~/~ as a reference value. WI~KELMANN et al. (21) compare a(p)/~(r~) in p ~ = 24 and 205 GeV/e ~-p interactions. The numerical values are ~ 2 4 and (34.45=8.7)% at -~/~= 4.0 and 19.6 GeV, respectively. If the ratio increases l inearly wi th ~/s-, for instance, we need ~/s = 43 GeV in order to reach a fifty-fifty value. The corresponding pion labora tory momentum p ~ 1 0 0 0 GeV/c. The same rat io for pp collisions at p ~ = 24 GeV/c ( V ' s = 6.8 GeV) is ~ 15%. The channel dependence of the rat io will be diminished with the increase of energy. Another interest ing observation of tha t paper is the centre-of-mass rap id i ty dependence of tha t ratio. A relat ively large rat io is obtained in the central region. I t would be interesting to check such a correlation at ISR energies.

Another interest ing information is the ~/7:- ratio observed at large p~ in 300 GeV/c p-Be collisions (~8). The ratio is 0.082, 0.055, 0.069 and 0.103 for p ~ = 2.05, 2.48, 2.91 and 3.33 GeV/c, respectively. These values seem to favour the new scaling law (see Table I). The asymptot ic energy seems to be high due to the presence of strange quarks so tha t the inclusive cross-section at tha t energy is not expected to sa turate the asymptot ic scaling law.

The th i rd interesting work is the measurement of the inclusive cross-sections for the production of neutral strange particles a t 300 GeV/c ( ~ / ~ = 23.8 GeV) pp inter- actions (34). The results arc summarized as a(K~)= (7.35=0.6)mb, ~(A~176 (3.6 5= 5= 0.4) rob, a(A~ ~ = (1.0 5= 0.3) mb and ~(~:o) = (129 5= 9) mb. There are 10 V events which cannot be interpreted as known neutral particles. I f I assume tha t all these part icles are the l ightest charmed neutral pseudoscalar D 0, for simplicity, then the experiment tells us ~(D~ (75+~)~b.

The fourth example is the production of A++(1236) at p ~ = 102 GeV/c (~ /~= 13.9 GeV) and PL= 400 GeV/c ( v / s = 27.4 GeV) in pp collisions (~s). The corresponding cross- sections are a(A + + ) - (1.65 5= 0.13) and (1.59 =~ 0.15) rob, respectively. The inclusive cross-section observed here is roughly energy independent in the range of energies studied. The production ratio A ~ tells tha t the production occurred pe- r ipheral ly with I = 1 exchange in the t-channel, so tha t the energy seems to be far from the asymptot ic one. A similar s tudy for u+p and pp interactions at 100 GeV/c (3~) leads to result a consistent with the above study.

The last example is the 11 events of high-mass electron-positron pairs observed at ISR for different ~ / s - - 30.6, 44.8, 52.7 and 62.4 GeV. The rap id i ty dis tr ibut ion of these events is nearly flat. Except one (or two) event, the remaining seem to he a decay product of ~(3095). By making use of the most recent value of B~ = 4.8/69 (~) and

(a,) j . A . APPEL, M, H . BOURQUIN, I . GAINES, D. C. HeM, L. M. LED]~RMAI~ r, H . P . PAAR, J . - P . I~EPELLIN' I~. D. SNYDER, J . M. WEISS, J . K . YOH, B. C. BROWN, C. N. BReWer, J . -M. GAILLARD, J , R . SAUER a n 4 T. YAI~IANOUCttI: Search ]or o meson produced at high Pc in p-Be collisions at 300 GcV, F E R M I L A B - P u b - 7 5 / 3 7 - E X P , 7100.070 (May 1975). (st) A . SHENG, V. DAVIDSON, A. FIRESTONE, F. NAG:y, C. PECK, F. T. DAO, R . H A ~ T , J . LACH, E. MALAMUD, F. NEZRICK, A. DZIERBA a n d R . POSTER: p p interactions at 300 GeV/c: y and neutral strange

1)article production, F E I t M I L A B - P u b - 7 4 ] 1 0 5 - E X P , 7200.037 (December 1974). (35) j . p . DEBRION, C. BROMBERG, T. FERBEL, T. JENSEN, R . SCIIINDLER, P . SLATTERY, A. A. SEIDL a n d J . G. VANDER VELDE: Phys . Rev. Lef t . , 14, 910 (1975). (aa) J . ERWlN, R . D. KASS, J . H . ]~LEMS, W . KO, R . L. LANDER, D. •. PELLETT, P . M. YAOER and. ]Y[. ALSTON-GARNJOST: Inclusive A § production in ~+p and p p interactions at 100 GeV/c and the average mult ipl ici ty o! r~+~ - scattering, UCD-PPL-4 -15 -75 (1975). (aT) A. M. BOYARSKI, M. BREIDENBAC]t, F . BULOS, G. J . FELDMAN, G. E . FISCHER, D. FRYBERGER, G. HANSON, B. JEAN-MARIE, R . R . LARSEN, D. LUKE, V. LOTH, H . L. LYNCH, D. LYON, C. C. MOREtIOUSE, J . M. PATERSO]~ r, M. L. PERL, P. •APIDIS, B. RICHTER, R . F . SCHWITTERS, W . TANENBAUM, F. VA~NUCCI, G. S. ABRAMS, D. D. BRIGGS, W . CHI~O~VSKY, C. E. FRIEDBERG, G. GOLDHABER, J . A. ]~ADYK, A. M, LITKE, B . A. LULU, F. ~r PIERRE, B . SADOULET, G. H . TRILLn~rG, J . S. WHITAKER, F . WINKELMA~N a n d J . E . WISS: Phys . Rev. Le f t . , 34, 1357 (1975).

ASYMPTOTIC SCALING LAW FOR INCLUSIVE PRODUCTION ETC. 7

by in tegra t ing over the r ap id i ty in t e rva l y : - - - 0.5 to 0.5 for s impl ic i ty (exper imenta l ly Ym1~=--0"165 and y ~ , . = 0.318), the average cross-section becomes a ( ~ ) = (0.11 • ~b.

Fo r definiteness, all in format ion discussed above, excep t ~, is compared wi th t h a t of the old and new sealing laws in Table I I . I n this compar ison the va lue of the pion cross-section ca lcula ted f rom cq. (5) (32) was nml t ip l ied by ~} and a1~1=32.5 nab (38) was assumed so as to sat isfy the exper imenta l resul ts (21). The compar ison should be t aken as a ve ry t e m p o r a r y one due to the low-energy charac ter of the avai lab le in format ion re la t ive to the surmised asympto t ic l imit . Especia l ly a(~) in the last l ine has no detect ion correct ion etc. so one should not take i t seriously. I suppose t h a t the scaling l imi t for each hadron, including resonances, wil l be g radua l ly satisfied wi th ~he increase of energy. There will exist var ious complicat ions discussed earlier, i nhe ren t to hadrou dynamics , bu t stil l the new scaling law seems to work be t t e r t han the old onc.

TABL]~ I I . - Comparison o] ]inite-energy data with the asymptotic scaling law.

Par - Inci- ~ (GeV) a(i) (nab) t i t l e den t (cxp) t y p e i channel

A s y m p t o t i c a(i) (rob) Rcf.

new old

~+ pp 23.8 79 65 65 (32)

rr- pp 23.8 65 65 65 (32)

K + pp 23.8 11 5.2 5.2 (32)

K - pp 23.8 6.9 5.2 5.2 (3~)

pp 23.8 4.0 4.3 _ 4 3 1.4 (3~)

po ~ - p 19.6 13.5 65 2.1 (21)

K ~ pp 23.8 7.3 :t:0.6 5.2 5.2 (3,)

A /Z ~ pp 23.8 3.6 4-0.4 0 .75-- 7.5 0.98 (34)

A /Z ~ pp 23.8 1.0 4-0.3 0 . 7 5 + 7.5 0.98 (34)

D O pp 23.8 ~ ~7~ +~176 (a4) . . . . . . o.06s 0.27 0.27

A +§ pp 27.4 1.594-0.15 4.3 + 4 3 0.85 (3~)

A ++ ~+p 13.7 1 .04~0 .10 4.3 + 4 3 0.85 (36)

qJ(~o) pp 30 .6+62 .4 (0.11 4-0.04). 10 -3 0.13 0.13 (22)

Final ly , let me quote Giacomel l i ' s s u m m a r y ta lk on recent exper imenta l results f rom the C E R N - I S R (39) tha t (~ the exper imenta l s tudy of the proper t ies of the asymp- to t ic region may v e r y well have to wa i t for an even higher-energy accelerator ~>. The last ten l ines of co lumn 3 (marked New) of Tab le I should be mul t ip l ied by ~}, hence the numbers of lines 8 and 9 of co lumn 5 of Tab le I I should be mul t ip l ied by ] .

(a,) ~]. BRACCI, J. P. DROULEZ, E. FL~II~tO, J. D. HANSE~ and D. R. O. M()RR~SO~: Compilation o/cross- sections. III - p and p i~duced reactions, CERN/tIERA 73-1 (January 1973). (*~) G. GIACOMELLI: Recent experimental results ]rom the CERN-ISR, FERMILAB-Pub-74/106.EXp, 2009.000 (1974).