IL I~UOVO CIMENTO VOL. 9 A, N. 4 21 Giugno 1972

Differential Cross-Sections for Charge Exchange: Reactions with t-Dependent Residues (').

1~. DELBOUR~O and P. ROTELLI

Phys~s Department, Imperial College - London

(ricevuto il 18 Novembre 1971)

S n m m ~ l T o - - A model incorporating biquark and triquark Regge propa. gators in Reggeized supermultiplet theory is applied to some of the highest-energy meson-baryon charge-exchange data available. The fits are good within the limitations of the model and the standards prevalent in this field of phenomenology. Renormalization of some of the theo- retical amplitudes improve the fits and indicates S U a and ~U6 breaking of about 10% in the residues.

1 . - I n t r o d u c t i o n .

Two of the most popular theories of the last decade have been Regge theory and supermult iple t theory. The former provides a high-energy parametr iza- t ion of two-body scat ter ing, while the la t te r categorizes and correlates the inter- actions of the par t ic ipat ing hadrons. However , each theory suffers separate ly f rom well-known difficulties: Regge phenomenology, for instance, is only useful if there is dominance b y the leading t r a j ec to r i e s - - the re is evidence f rom fits to to ta l cross-sections and f rom other sources tha t this assumption breaks down below about 8 GcV/c, while a t ul tra-high energies, a t finite mom e n tum transfers, the assumption is cer ta in ly incorrect because of u l t imate Regge-eut dominance (1); likewise, supermult ip le t theory incorporates exac t

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (1) This complicates matters only in the region where pole and cut contributions can compete effectively, and at large enough energies there will always be a value of mo- mentum transfer where this occurs.

504

D I F F E R E N T I A L CROSS-SECTIONS F O R CHARGE E X C H A N G E ETC. ~ 0 ~

symmetries like 17~ | ~ra or 8 U ~ which are at best only approximate sym- metries in reali ty--without an adequate model for the internal- and spin- symmetry breaking all one can do is to provide * rules of thumb ,, culled from experience, for passing from the symmetric to the physical world.

Obviously, Regge and supermultiplet theory in some sense complement one another~ and in recent years a number of Reggeized supermultiplet models have appeared (~). Our aim in this paper is to apply the most recent version (3) (having acceptable extrapolation properties) which features t-dependent residues, to some of the best and highest-energy charge-exchange (CEX) meson-baryon data available. Section 2 contains a resume of the model, which is applied in Sect. 3 to meson-baryon scattering. Our fits are given in Sect. 4 and used to estimate the amount (about 10 %) of symmetry breaking involved. The conclusions follow.

2. - Regge residues and supermuitiplets.

Our model consists in dividing any process into three parts, the two reggeon-particle-particle vertices and the exchanged Regge-pole propagator. We use ~2 traces for the vertices to provide 8U6~ symmetry, while for reg- geon exchange we employ biquark (for mesons) or triquark (for baryons) ~ propagator, v/z.

and

= (4sin.)-1 ~ (p + ~)~'(p + ~)~'(p + ~)~'r[~-~,(f)](~'~8)~ ~,')-, perm$

where p and m are the meson and baryon supermnltiplet masses, respectively, and % and % are the corresponding trajectory functions. I t must be stressed that the propagator , numerators , become projectors only at % = 1 (mesons) and ~v = ~ (baryons).

In this paper we shall apply the model to two-body meson-baryon reac- tions in which only pseudoscalar mesons participate so that the processes are well described by vector-tensor Regge exchange. (One fortunate circumstance

(~) ~R. C. ARNOLD: Phys. Rev., 153, 1506 (1967); 162, 1334 (1967); P. G. 0. FREUND: Phys. Re~., 157, 1412 (1967); R. D~.LBOURGO and A. SALEM: Phys. Left., 28B, 497 (1969); Phys. 2~ev., 186, 1516 (1969); S. A. ADJ-.I, P. A. COLLINS, B. X. HARTLEY, K. J. M. MORIARTY and R. W. MOOR~: I.C.T.P./70/19. (a) l~. DELBOURGO and P. ROTELLI: Phys. Lett., 35 B, 65 (1971).

506 R. D~.I,BOUROO and P. ROTELLI

is that the physical O and co masses are nearly degenerate with p, the mean meson supermultiplet mass, so we can equate the vector-tensor par t of Reg- geized supermultiplet exchange whith its physical counterpart.) Taking con- stant residues with our multiquark propagators, we obtain polynomial residues in the conventional Regge formalism which extrapolate correctly from the time- like resonance region to the spacelike scattering region (4).

In order to compare our theoretical results with experimental data we proceed as follows:

i) Treat the trace calculation in the (degenerate mass) exact symmetry limit, employing the additive quark rule (a)

3p -~ 2m with p2 ~ 0.6 (GeV/ea) ~ .

ii) Assume all p~ dependence of the amplitude occurs in the traces and

the (almost) conventional Regge structure (e)

F[1 - - g(p2)](a's) ~(~'~ • signature,

i.e. take the remaining residue factor fl to be constant.

iii) Break the amplitudes into orthogonal t-channel helicity contributions, the natural ones associated with Regge exchange and SUew, e.g., take the A' and B amplitudes in 0-1+ reactions as given by the supermultiplet model. Thus write T = ~ ' [ A ' ~ KB]u, where K is dimensionless and involves the mo-

menta and supermultiplet masses m, p.

iv) Descend to the physical world by replacing the momenta and (dimen- siouless) wave ]unctions by the physical values without reinterpreting m amd ~u

whenever they arise in the helicity amplitude T.

Obviously these rules are to a certain extent ad hoe. However, the alterna- tive course of replacing all supermultiplet masses by their physical values is to some extent ambiguous when all the hadrons are different and in any ease produces unacceptable large amounts of symmetry breaking through factors of the type m/p, particularly where pions are involved. Another source of ambiguity concerns the possible dimensionality of coupling constants; fortu- nately, again this problem is obviated when we compare the dimensionless

vector-pseudosealar couplings with the residue ft.

(4) This success is particularly striking for A exchange in backward ~J~ scattering. (5) This means we take m = 1.16 GeV/c, which lies between the physical J~ and A masses, and is rather lower than the mean baryon supermultiplet (56) mass. (s) That is, we are adopting the Gell-Mann mechanism for nonsense zeros, which is also suggested by Veneziano models.

DIFFERENTIAL CROSS-SECTIONS FOR CHA_RGE EXCHANGE ETC. ~0"~

3. - Meson-baryon charge exchange .

We begin with the higher-symmetry amplitude

(1) T = mac. . ' ~%~<~p, :o~,(-p, ~)>{~, ~}~:,

which, in the limit of exchange degeneracy, provides the following 0-�89 -+ 0- �89 elements:

(2) T = [(~u)BA'+ (~Ku)D+j~B ] -fl/~[1--a](a's) ~-1-

�9 [�89 (1 - -exp [izce])(r -k �89 (1 + exp [iza]) {q0~0)h ] .

Our kinematics is specified by assigning incoming momenta �89 + q, --�89 + q' to the baryons and mesons, and outgoing momenta --�89 +q , �89 +q ' . Thus ~ =p~', s= (q + ql) ~ and

(3) K = 2%~p#q,q'~Ts//a(4m ~ - t) ,

and the t-channel nonflip and flip amplitudes are given at large s by

~ = ~ + p + i - ~ + ~ - a ~ ~'

( )[ ~ t 1 5__m_m -t- 2t B= ~ - - ~ + ~ ~ + ~ .

We now extrapolate (2), (3) and (4) to the physical values of momentum. The differential cross-section for high-energy scattering off nucleons reads

t , t sb da 1 ( l _ ~ ) [ , a , ._~_~{2m#(l~t /4m,) F], (5) d-{ ~

or in detail

(6) dad_.t Io~ ~ ~ 1 (1 -- 4mt--~x) {fl/'[1 -- ~(t)] (~'s)~")-I (2-~--/~)}*"

~,,+4--~+~-~--~,! -~-~ 1+ 9 [1 - - cos ~ ( t ) ]

�9 1

�89 + cos ~(t)]

"ql ' l #+~}jj for 7rp -->To~ for K-p--~ K~ for 7c-p -~ ~n.

508 R. DELBOURGO a n d P. ROTELLI

We m a y a t t e m p t to normalize the ampl i tude (i.e. to fix ~) b y considering ~-p-+7~~ near t = #2, where T is domina ted b y the p-meson pole. Thus as

10m [1 - - ~ ' \ - K ] 3(1 + 1~[2m)~r

to be compared wi th e lementa ry p exchange which has the charge contribu-

t ion as

2 ' n T ' P 2 m - 2 nq"TP

b y p universal i ty . We deduce t ha t

, 3fl 1 --I- Pl2m (7) g P ~ = ~ - 4m~ '

and with this v a h e of fl m a y now go to t : 0 in eq. (6) to see if the extra- polation gives reasonable values for the observed cross-sections. We shall adopt a l inear exchange-degenerate t ra jec to ry to reproduce the main features of the react ion 1:-p-~=~ and must (7) therefore choose

(8) a(t) ~ 0.5 + 0.85 t .

Then near the forward direct ion

(9)

The exper imenta l cross-sections near t----0 lead one to gp~ ~ 27 and this

fails wi thin the p-decay width es t imate g ~ - - - - - 2 5 - 30! On the other hand, wi th our t r a jec to ry in te rcep t of 0.5, SU3 provides the

addi t ional relat ions

(10) da ~n)

which are not well obeyed by the data.

(7) The slope and intercept parameters a ' ~ 0.85, a(0)~ 0.5 are fixed by two condi- tions, a(0.6)= 1 and 2~(-- 0.1) - - 2 ~ --1.17, the first because p~----0.6 is the (mean meson mass) S , the second because of the observed energy dependence of the forward ~-p--> non peak.

DIFFERENTIAL CROSS-SECTIONS FOR CHARG]~ ]~XCHANGE F, TC. ~

The other CEX processes we shall examine will be the production of baryon* resonance. The 0-�89 + elements provided by (1) are

(11) T = C%.,aq.q~,p ~/).f ir(1 - - ~ ) ( : t ' s f - ' -

�9 [�89 (1 - exp [ i = a ] ) ( ~ ) , + �89 (1 + exp [iza])(~q,)~]

with

(12) C = 1 A - S m + 2 t 1 + . m2~

We thereby arrive at the cross-section

d~ I (13) ~-,Ic~x ~ - -

t 1 - - t /4mxm a 96~m ~ ms#s {•F[1 -- ot(t)](a's)"('~-'} ~.

I [ 1 - cos ~ ( t ) l �9 (1 -1- 5m + 2t mt~s I

1 \

511 + COS ga(t)]

for ~+p -->~~

for K+p->K~ ++,

for ~:+p --> ~A ++ .

Again it is to be noted that the S Us predictions are not respected by the experi- mental numbers.

4 . - F i t s , i n t e r p r e t a t i o n a n d e o n e l u s l o n s .

We can test our expressions directly against experiment taking

(14) m s = 1 . 3 5 (GeV/cS) s , #2 = 0.6 (GeV/c') s

and the residue and trajectory from formulae (7) and (8), There are no other ~ parameters , in our model�9 In units of mb/(GeV/c) s the differential cross- sections for 0- 5 + ~-> 0- 5 + processes read

(15) d ~ ~- (PB -~ P B )o~x ~ D ( o . 5 - - o. 850 (0.858)-,+,.,,.

(1 -}- sin 0.85~t) �9 ( 1 - - 0 . 3 t ) [ 1 - - 3 0 ( t + t s + �89 1

}(1 -- sin 0.85~;t)

for 7:,

for K ,

for ~](2y),

where we have multiplied the 7~--p-+~n expression by the branching ratio (~-->2y)/(~-->all) ~, 0.38. In the case of baryon-resonance production 0-5+--> -~0-~ + the counterpart equation is

(16) d ~ ~- (PB ~ PB*)o~ '~ -- 35t/"s(0.5-- 0.85t)(0.85s) -I+Im.

I 1 + sin 0.85~t

�9 (1--0.2011+0.68t] s 1 �89 - - s in 0.85~t)

f o r ~

for K ,

for ~ .

5 1 ~ R . D E L B O U R G O and P . R O T E L L I

10 o

10 o

1 0 o"

10

a)

10 o

10 -I

10 -z

10

~EJO-'

b "lJ

10 o

10-1

lO-2

10 -2

'~ 1 1~ ~

1 ~ s 10-3 10 -~

10 -4 10~! -4 ~ �9 �9 I I ! I ; J J ' I I ~ 0 - 4 10" P L ~ ~ ~ ~ I ' ' ' '0 .5 1.0 1.5 o o.5_t I~Gev/c12 ] lo 15 -t[cG,v/,-)~

F%. 1. F%. 2.

Fig. 1. - ~-p-->~~ at a) 9.8, b) 13.3 and e) 18.2 GeV/c. SACLKY-ORSAY COLLABORA- TION: Phys. Rev. Lett., 14, 763 (1965).

Fig. 2. - *, K - p - , K O n at a) 9.5 GeV/c and b) 12.3 GeV/c. CERN-ZuRICH COLLABO- EATZO~: Phys. Left., 23, 396 (1966). +, K+n-+K~ at c) 12.0GeV/c. A. FIRESTONE e$ al.: Phys. Rev. Lett., 25, 958 (1970).

D I F F E R E N T I A L CROSS-SECTIONS F O R CHARGE E X C H A N G E ETC. 5 1 1

In fitting to experiment we shall withstand the temptat ion of applying our model at energies less than about 8 GeV/c where we have grave doubts about the validity of assuming the dominance of leading Regge poles. The H-p --~H~ curves at the stated energies are shown in Fig. 1 and are characterized by a zero at t ~ - 0.6 because of the nonsense residue mechanism used for

t 0~ 10 -I

10 -2

10 -~

~u

~ I 0 - '

b

10 -~

l0 -2

- t _ i

, , , , , , , , , I , , , , j o ~ 10~0 0.5 1.0

Fig. 3. - ~-p--~(2T)n at a) 9.8, b) 13.3 and c) 18.2 GeV/c. SACLAY-ORSAY COLLA- ]3ORATZON: Phys. Ldt., la , 200 (1965).

519. R. D~.LBOURGO a n d P, ROTELLI

the p trajectory; this zero will obviously be changed to a dip when secondary (cut, daughter, ...) contributions (8) are superimposed. Because of the afore- mentioned 8U3 violation, the predictions for K-p -~K~ contained in eq. (15) do not fit the data; because the theoretical numbers are consistently above the experimental points, we have chosen to take account of gross 8U3 breaking by simply renormalizing all symmetry values for K by a factor 0.55, and then e•hibit these cross-sections fits in Fig. 2. Roughly speaking this corresponds to a 25 % reduction relative to ~ p of the KKp coupling constant, reminiscent of a similar SUB violation in ~ - p o m e r o n and KK-pomeron couplings (0). Simi- larly, in Fig. 3 where the ~--p--~(2v)n curves are drawn we have renormalized down the theoretical prediction by the factor 0.45 to take approximate account of EU3 breakdown. Turning next to the baryon resonances we find i t necessary to enhance (16) by the factor 1.4 to obtain a satisfactory fit with the ~+p-~~ which corresponds to a violation of the spin par t of ~Ue at the baryon vertex; see Fig. 4. However, we maintain the relative ratio ~(K)/~(7~)

0.55 and ~(~)/~(~) ~ 0.45 to take account of the ~U8 breaking at the meson

10

b

~10-2

Id 0 0.5 1.0

Fig. 4. - ~+p~~ at 8.0 GeV/e. R. D. MATTHI~WS: ~ . Phi8., 11 B, 339 (1969).

(8) Or by adopting a complex trajectory or pair of trajectories generated by a pole-cut collision. (9) Even after averaging out any pomerou violating contribution one finds that at(K-p) ~- 6t(K+p) ~ 36.6 rob, while at(n+p) ~- a~(lr-p) ~ 44.8 mb as s-~ co, corresponding to a 10% discrepancy in the n~ and KK couplings to the pomeron.

DIFFERENTIAL CROSS-SECTIONS FOR CHARGE EXCHANGE ETC. 51~

v e r t e x and fits for the reac t ions K + p - ~ K~ ++, ~:+p-->~iA ++ are depic ted in Fig. 5 and 6 a t the s t a t ed energies (~o}.

Tak ing s tock of our resul ts we see evidence of SU3 breakdown (15 %) and ~Ue b reakdown (10 %) about the mean in the res idue which conspire in opposi te direct ions for ~J~--> ~A and K3~'--> KA re la t ive to ~ J ~ - > 7:J~. To improve

10 t - 10 o

~'~ I0-I ~) I0-~

b - 9 ~ "d 10 10 -2

10 -2 , , , 0-2

Fig. 5.

0 10

I "G

0 0.5 1.0

-t [IG,v/c) Fig. 6.

Fig. 5. - K+p--~K~ ++ at a) 7.3 and b) 12.7 GeV/e. D. J. I~ELLEMA: University of California report No. U.C.L.A.-1047 (7.3 GeV/c); R. S. HOLMES et aZ.: University of Rochester report No. UR-875-313 (12.7 GeV/c).

Fig. 6 . - ~+p--~A ++ at 8.0GeV/c. Private communication from AACHEI~-BERLX~- CERN COLLABORATIO~ to M. KRAMMER and U. MAol~: ~uavo Cimento, 50 A, 963 (1967).

compar ison wi th expe r imen t (11) i t is necessary to include addi t ional (cut) contr ibut ions , no t jus t to fill in the s ignature zeros bu t also to account for the nonzero polar izat ion da ta available. However , considering the success a l ready achieved b y our model, including the cross-over zero, we can p resume to suggest t ha t these addi t ional t e rms are small a t p r e sen t energies. Never-

(lo) Actually there is inconsistency in the normalization of the experimental data for K+p-+K~ ++ at 7.3 and 12.7 GeV/c and that predicted by simple Regge-pole theory. (n) Our curves would look more impressive, even without cuts, if we had used a steeper slope in the trajectory for ~ < 0, but we are loth to introduce any further parameters in the trajectory function to effect this improvement because secondary corrections have yet to be added to our model.

34 - I I Nuovo Oimento A.

5 1 4 R. DELBOURGO and P. ROTELLI

theless, there are values of t in cer ta in processes such as ~ C E X where the magn i tude of leading pole and all o ther cont r ibu t ions are comparab le (e.g.,

near the t : 0.6 zero) giving po ten t ia l ly large in te r ference effects l ike polari- zat ions, which are crucially dependen t on the deta i led phase s t ruc tu re of the nonleading te rms . At this s tage all we can ven tu re to say is t h a t (without the s ignature factor) a t p r e sen t energies, the cut /pole ra t io var ies f rom abou t 3 0 % a t t ~ - - 0 . 6 to abou t 1 5 % a t t ~ 0 .

A t this t ime i t is difficult to see how to choose be tween our model and other a l te rna t ives l ike the weak and s t rong cut models a l ready in the l i te ra ture . Since cut ampl i tudes grow fas te r t h a n pole ampl i tudes for t :~ 0 i t m a y be possible to find a reg ime at high enough energies where cuts domina te so t h a t one can exper imenta l ly de te rmine the i r s t r eng th and s t ruc ture . I n this

way ex t rapo la t ing down to p resen t energies will answer p a r t of the con t roversy

be tween weak and s t rong cuts which we now join.

�9 R I A S S U N T 0 (')

Si applica ad aleuni dei dati di seambio di carica mesone-barione di altissima energia disponibili un modello clue include propagatori di Regge di biquark e triquark nella teoria del supermultipletto reggcizzata. L'accordo ~ buono tenendo conto delle limitazioni dcl modello e degli standard prcvalenti in questo campo della fenomenologia. La rinor- malizzazione di alcune delle ampiezzc teoriche migliora l'accordo ed indica una rottura di SUa e S U e di circa il 10% nei residui.

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