DE:UTSCHES ELEKTRONEN-SYNCHROTRON DE SY 
DESY 74/30 
June 1974 

Dual Light-Cone Model Predictions for Total and Inclusive 

e+ e Annihilation 

by 

G~ Schierholz 

II. Institut far Theoretisahe Physik der Universitat Hambu.~ 

and 

M. G. Schmidt 

:rnstitut filr Theoretisahe Physik der Universitat HeideLberg 

2 HAMBURG 52 • NOTKESTIEG 1 



Dual Light-Cone Model Predictions for Total and Inclusive + 
e e Annihilation 

by 

G. Schierholz 
II. Institut flir Theoretische Physik der Universitat Hamburg, Germany 

and 

M.G. Schmidt 
Institut flir Theoretische Physik der Universitat Heidelberg, Germany 

Abstract 
Predictions of a recently proposed dual scaling model for the total and inclu-sive e+e- cross section are discussed taking particular interest in preasympto-tic corrections. Good agreement with available data is found. 



Recently, we have presented a dual light-cone model for deep inelast
ic 

electroproduction and annihilation! which led to an explicit ansatz 
for the 

structure functions F2
(x) and F

2
(x). In particular, we found a generalized 

Gribov-Lipatov reciprocity relation
2 which connects F 2

(x) and F
2

(x) in their 

physical regions so that the continuation to the deep inelastic anni
hilation 

reg>on becomes exceedingly simple. Once our model has been tested in
 the scat-

tering region, we thus can predict the annihilation structure functio
n F

2
(x) on 

firm grounds of the electroproduction data. 

Preliminary SPEAR results show
3 

that, at present energies (q
2 ~ 25 GeV2 ), 

the inclusive cross section does not scale in the annihilation reg
ion except 

'C 

near x "' I which is in sharp contrast to the experimental finding of
 early 

scaling in deep inelastic electroproduction and raises the question, 
whether 

scaling has not yet been reached or even is broken in the timelike re
gion. In 

order to decide this question and before one draws any conclusions again
st 

. k 1" f . . 
2 h 

BJor en sea >ng or pos>tlve q , one as to examine closely what cur
rent seal-

ing models predict in the annihilation region and to look for preasy
mptotic 

corrections which we have good reasons to believe play an important 
role at 

SPEAR energies. This may reveal that the SPEAR results do not contra
dict asymp-

totic scaling though (contrary to many augurs), perhaps, they are no
t the ex-

perimentum crucis for probing parton structures. 

In this paper we shall discuss the predictions of our model
1 

for total and 

inclusive e+e- annihilation taking particular interest in nonleading
 (scale 

breaking) contributions. We shall see that the preliminary SPEAR dat
a can be 

well described in terms of this (scaling) model. 

The (scattering) structure function F 2
(x) was given by

1
•
4 

(x = 

= N x-a(O)+I F
2

(x) 

+I c 

I ( 
•I I 

dB' 1:8 J 

-I 

-c +c'+a(0)-2 
2 2 2 I I 

( (l+x) -~1-x) 8' l 

2 
-q /2v) 

( I ) 

where c 1 
and cj are determined by 

netic) target form factor and the 

the asymptotic behavior of the (electromag-
+ -

(2 ) + (I ) transition form factor respecti-

"'''""''"'' ., .. , ... , ... "'" '' .. , .. ,.., .•.. '"'"'""'"'"" "'"" '''"''' '''""'~'" '"''""""''"'"''""" ,.,,.,,, . ''"'' '"""'''''"''' ',, ,, ''"'"'"'"''" ''"'""' 



2 

vely. The normalization of the structure function (i.e.,N) is provided by the 
Adler sum rule (a(O) < I) 

I 

f 
0 

dx F (x) = 
X 2 (2) 

being a consequences of the current algebra constraints with actually led to 
scaling. The (annihilation) structure function F

2 (x) could most simply be ex-
pressed in terms of F

2
(x) by means of the reciprocity relation (which holds for 

arbitrary c 
1) 

=X 

2c '-I 
I 

In the following we shall set cj 

(3) 

2 5 2 in accordance with other models ' . We 
furthermore take c

1 
= 0 for mesons (monopole form factor) and c

1 
= I for nucle-

ons (dipole form factor) respectively. 

Assuming SU(3) (exchange degeneracy), nonexoticity in all channels and the 
photon being a U-spin scalar, we obtain for the nondiffractive part of the 
pseudoscalar meson octet the (physical) scaling functions 

5 T) 5 
= - F (x) = - F (x) 

3 2 9 2 (4) 

and similarly for the annihilation structure functions. !n the case of the 
baryon (antibaryon) octet we have to allow for a 10 (10) representation in the 
baryonic channels. If we assume that the decuplet does not contribute in the 
scaling region, we would get the analogue of Eq. (4). This might be justified 

"' for x"' since the ~(1236) + N transition form factor shows a slightly faster 
decrease

6 
than the nucleon form factor which suggests a suppression near x = 1 

yia the Drell-Yan relation. But, in general, there is no doubt that the decuplet 
contributes to the scaling functions.

7 
Another solution (which treats N and 

& on the same footing) would be the SU(6)/quark model resultS 

+ 
F p,~ 

2 
3 

(x) = 2 3 F ~ > ~ (x) 
2 

F 
2
(x) , (5) 

A detailed analysis of the various symmetry aspects allowing for a different 
threshold behavior of the decuplet contribution which also explains the rather 
small F2n(x)/F2

P(x) ratio will be given in a forthcoming paper. 9 



3 

The pomeron contribution cannot be integrated in the duality scheme so far 

developed, but has to be added by hand. It is tempting to assume the same 

ansatz (Eq.(l)) for the diffractive term as for the nondiffractive part (here 

~(0) = I of course). This can be motivated in our model (having nonlinear tra-
jectories) by replacing the s-channel trajectory by some background trajectory 

without resonances being dual to the porneron ~n accordance with the Harari-

Freund conjecture. Here, the threshold behavior (i.e., c
1

) is, however, no longer 

determined by the asymptotic behavior of any form factor as in the nondiffrac-

tive case. But following the general belief that the background corresponds 

) . . 
10 2 d 3 (at least to a four and f1ve quark ass1gnment we conclude c

1 
= an c

1 
= 

for mesons and baryons respectively. Hence, the pomeron contribution is strongly 

suppressed near threshold compared to the nondiffractive part which seems to 
. . d 1 d . . II be supported by compar1son of neutr1no an e ectropro uctlJn exper1ments. 

In the following we take the SU(6) solution for the nucleon octet. We further-

more assume that the pomeron be a SU(3) singlet. Then, the normalization of 

the pomeron contribution is the only free parameter. The intercept of the Regge 

trajectory is taken to be a(O) = 0.3 

would rise linearly up to the A
2

, f 

we have drawn F
2
P(x) taking N 

pomeron 

(what would come out if the trajectory 

resonance with a'(O) = I GeV- 2 ). In Fig. I 

= 2. which gives a good fit to the 12 data. 

Also indicated is the pomeron contribution. This is significant only at small 

x as is to be expected. 

In the case of the meson octet there is no ambiguity as far as the SU(3) 

structure is concerned. Here the pomeron coupling is via factorization deter-

mined by the ratio anP /opp = 2/3 (note that the x ~ 0 limit of the scaling 
tot tot 

functions does not depend on c
1
). The resulting scaling function is shown in 

Fig. 1. Due to the different threshold factor, F2T
I(x) increases much faster 

near threshold than the proton scaling function. The pomeron contribution 

again is negligible for x ~ 1 but is considerably larger for medium x (x ~ 0.2) 

than in the case of the proton. 

The annihilation structure functions can be deduced from 

For large x we have F2 (x
) "' 

Fig. 1 employing 

xa(0)+ 2 for the the reciprocity relation (3). 

nondiffractive term and F
2

(x) 
3 

~ x for the pomeron contribution. 

So far we have concentrated on the p1on and nucleon (octet) structure func-

tions only. But any other particle h which gives rise to scaling in the deep 



4 

2 inelastic scattering region, will contribute a portion to the large-q total 
+ -

inclusive e e cross section as well. If we had exact SU(6), e.g., we would 
get 35/8 times as many signals as if there were only pions. In the case that 
the particle h is unstable (broken SU(6)) this gives of course, a contribu-
tion to the inclusive spectrum of its decay products as shown in Fig. 2a. 

In the imaginary part of the (annihilation) Compton amplitude this process 
(Fig. 2a) corresponds to the double h exchange diagram as drawn in Fig. 2b 
for p decaying into two pions. As is well known, this diagram (q

2 
> 0) gives 

rise to anomalous singularities
13 

which certainly are not included in the pion 
structure function F

2
TI(x) so far considered. The reason being that our model 

only accounts for normal threshold singularities. Hence, it is plausible to add 
these anomalous singularity contributions to the pion (annihilation) structure 

- TI 
function F

2 
(x), whereas the normal threshold part can be thought of being al-

ready included in our model. 

In the scaling limit (q
2 

+ oo) the p production diagram gives, in the zero 
width approximation, rise to the (anomalous) cut contribution

13 
(supplementary 

to the normal threshold pion scaling function F
2

TI(x)): 

3 
2 X 

2 m 

I 8(m2 2 p dn n :F o (.!) + m _n_ ) (6) p 2 n p 'IT I-n 
Pcm 

and similarly for any other resonance (if. say the p would decay into a pion 
and a different particle of mass m, formula (6) had to be devided by 2 and the 
8 function be replaced by 8(m~-m~n+;2 n/I-n)). In this approximation the p is 
forced to be on the mass shell so that the (spin averaged) p structure function 
appears under the integral. For Fi(x) (as well as for the other SU(3) partners) 
we make the same ansatz as for the pion structure function (apart from a 
possibly different c

1
) which leads to the analogue of Eq.(4). It is tempting 

to assume c
1 

= 0 also (for the nondiffractive part) as one would expect from 
SU(6). Other resonances like, e.g., the A

1 
which prominently decay into three 

and more particles can be handled in a similar way. In case of the A
1

, e.g., we 
would have another pion rung in Fig. 2b due to the cascade A

1 
+ pTI + 3TI. 

2 
For finite q preasymptotic corrections resulting from kinematical factors 

become very important in expression (6), especially for large x, even at the 



5 

highest SPEAR energy (q
2 

= 25 GeV
2
). This happens because the dominant contri-

- p 
but ion 

F/ (x) 
to the integral (6) 

a(0)+2 
~ x ), whereas 

comes from large arguments of F2 
(x) (remember 

the somewhat higher resonance mass gives rise to 

threshold factors which sensitively cut off this region. Thus, in order to make 

substantial predictions at SPEAR energies, we definitely have to include these 

effects. In the following we shall quote the results only, but will g~ve a de-

tailed analysis of these corrections in a more extensive paper.
9 

We shall now assume the validity of the Callan-Gress relation in order to 

define the scaling function F 1 
(x) (in Ref. I we have argued that the Callan-

Gress relation should hold irrespective of the spin of the constituents). In 

the scaling limit (q
2 ~ oo) the Callan-Gress relation also is maintained for the 

. . ( . . . . 14 f cut contr~but~ons 6) wh~ch then leads to the ~nclus~ve cross sect~on or 
+ -

e e ~ h + X: 

2 doh 
q 

dx 
= (I -

- h 
) F 

2 
(x) 

where x 
If 2 2 2 2 2 2 

(7) 

2 
= 25 GeV ), how-= 2p/V q~ and sh = 1-4~ X /q . For finite q (even q 

9 
ever, the Callan-Gress relation no longer holds for the resonance terms but 

threshold factors become involved which have a considerable effect on Eq.(7). 

In Fig. 3 we have drawn our prediction for the (total) charged inclusive 
+ -

e e ~ TI + X cross section taking into account (besides the pion contribution) 

the lowest lying resonances 6, 

commonly assuming c 1 
= 0 which 

p, A
1 

and B and their octet 

may be justified upon symmetry 

15 
partners and 

arguments. In the 

small x region (large x) the resonance terms provide by far the dominant 

contribution, whereas they are negligible for~~ 0.5. Hence, we are not sur-

f 
16 - < 

prised that scaling breaks down at SPEAR energies or x ~ 0.5. In fact, the 

shape and order of magnitude of the predicted inclusive 

agreement with the preliminary SPEAR data.
3

•
17 

A precise 

cross section is in 
. 18 

predict~on depends, 

however, on the effect of the higher (excited) resonances like A2
, f, etc. 

although we believe that their contribution is small at SPEAR energies due to 

the increasing mass and a likely more suppressing Dtell-Yan threshold factor. 

We also have looked at the angular 

stantial deviation from the asymptotic 

distribution is absolutely flat in the 

distribution of the 
2 

form ~ !+cos 8. At 

pion 
2 

q 

and find a sub-

25 GeV
2 

the pion 
- < 

region x ~ 0.3 (which accounts for most 

of the events) and beyond that region gradually turns over into the asymptotic 



6 

2 form. For increasing (decreasing) q the boundary between these two regions is 
shifted towards lower (higher) ~ being consistent with the asymptotic distri-
bution. 

The total cross section is given by the energy conservation sum rule 

R 
+ -

cr(e e + hadrons) 
2"1. 

I dx (8) 1 =-I 2 h X 
where the sum 1s over all participating hadrons. We now assume that all the 
available energy is carried away by pions and kaons and ealculate the total 
cross section from the inclusive spectrum of these particles (note that in our 
model the neutrals are produced with the same strength as charged particles). 
The kaons are found to contribute roughly 15 % to the total cross section. The 
results is shown in Fig. 4 and compared with the world's data. At higher q

2 

the predicted cross section is substantially smaller than the experimental cross 
section which is somewhat surprising. We would have expected that both roughly 
agree since the predicted inclusive cross section is consistent with the pre-
liminary SPEAR data. 

3
•

17 
In order. to explain this (energy crisis

19
) a large 

fraction of the energy has to be carried away by neutrals or by particles other 
than pions and kaons (perhaps baryons). The general feature that the ratio R 
increases with energy, however, is well reproduced. 

So far we have assumed perfect scaling for the structure functions. Neglect-
ing curvature of the Regge trajectories, we obtain that the preasymptotic cor-
rections to our scaling functions can, in first order, be accomodated by 
rescaling of the variable x: 

X+ X
1 = X 

a 
l+ 7 
1
_ bx 

q2 

1 
a= (a(O)+ 2) I a'(O), 

(9) 

1 
b = <2 + c 1 - a(O)) I a'(O) 

where a is meant to be the trajectory in the photon channel (note that a and 
b do not depend on the external masses). The variable x' 
by Bloom and Gilman

7 
(here a = 0) and by Rittenberg and 

has long been 

R b
• . 20 

u 1nste1n 

advocated 

and for 
electroproduction allows the concept of scaling to be extended down to very low 

, .. ,, '"" ,,.,, ,,,,.,,,, ''"""''"'"''' ....... , ,,,, '''""' ·~··~·"'' '' 



7 

values of q
2

. For proton targets we would predict (a(O) = 0.3) a= 0.8 GeV
2 

and 
2 . 21 .. 

b = 1.2 GeV being in not too bad agreement with exper1ment. In the annlhlla-

tion region the substitution (9) leads to higher effective x values and, hence, 

to higher cross sections because of the singular behavior of the scaling func-

tions at x + oo. The resonance contributions which account for most of the cross 

section will, however, not very much be effected by these corrections since 

they are confined to smaller x in the resonance structure functions. 

In our model (s,u) terms do not contribute to the current algebra fixed pole 

and, hence, do not survive in the scaling limit. However, they may give r~se to 

nonnegligible contributions at finite q
2

, especially in the very large x 
22 

region as has been emphasized by Satz. From SU(3) and nonexoticity we derive 

the general relation 

0 
-n 

vW
2 

(s,u) = -

(I 0) 

We see that neutral particles contribute with a sign opposite to charged par-

ticles. This sets, of course, an upper limit to the (s,u) terms because the 

total structure function has to be positive definite. Hence, we cannot expect 

that the (s,u) terms contribute much to the total cross section since most of 

the contributions cancel out. But they might have a substantial effect on the 

neutral to charged ratio and lead a way out of the energy crisis. 
19 

The most challenging question now 1s that of the asymptotic value of R. In 

order to draw any conclusions from our model we have to include the higher 

mesons and the baryons which might give a sensible contribution at very high 

energies. For a finite number of resonances (being in the spirit of our model 

with nonlinear trajectories) R will tend to a constant what we would expect if 

QED continues to hold at very small distances. 



8 

References 

!. G. Schierholz and M.G. Schmidt, Phys.Lett. 48B, 341 (1974); 
SLAC Report No. SLAC-PUB 1366 (1974), to be published in Phys. Rev. 

2. V.N. Gribov and L.N. Lipatov, Phys.Lett. 37B, 78 (1971); Yad.Fiz. 12• 
781 (1972) [Sov.J.Nucl.Phys. 12• 438 (1972)) ; for scaling models and 
further literature see: G. Schierholz, Phys.Lett. 47B, 374 (1973). 

3. B. Richter, talk presented at the Conference on Lepton Induced Reac-
tions, University of California, Irvine (1973); G. Goldhaber, talk 
presented at the Frlihjahrstagung der Deutschen Physikalischen Gesell-
schaft, DESY, Hamburg (1974). 

4. We shall not take into account corrections on the daughter level which 
were necessary to obtain a selfconsistent fixed pole residue (for 
details see Ref. 1). We have convinced ourselves in the case of nucleon 
Compton scattering that these corrections are indeed negligible. On the 
other hand, the model should not be taken too seriously on this level. 

5. G. Benfatto, G. Preparata and G.C. Rossi, Rome Report No. INFN 472 
(1973). 

6. J. Gayler, Proceedings of the Daresbury Study Weekend, 11-13 June, 1971. 

7. E.D. Bloom and F.J. Gilman, Phys.Rev.Lett. 25, 1140 (1970). 

8. M. Chaichian, H. Satz and S. Kitakado, Nuevo Cim. 16A, 437 (1973). 

9. G. Schierholz and M.G. Schmidt, to be published. 

10. S.J, Brodsky and G.R. Farrar, Phys.Rev.Lett. ]l, 1153 (1973); 
C. Alabiso and G. Schierholz, SLAC Report No. SLAC-PUB 1395 (1974), to 
be published in Phys.Rev. 

11. D.H. Perkins, Oxford Report No. 67-73 (1973). 



9 

12. For a(O) = 0.5 we would have to allow a higher normalization of the 

scaling function to obtain a fit of comparable quality. 

13. For a thorough discussion of this diagram in respect to the scaling 

functions see: R. Gatto, P. Menotti and I. Vendramin, Ann. Phys. 22• 
I (1973). 

14. 
- h 

The scaling function F2 
(x) 1s always understood spin averaged. If the 

sp1n of particle h is not detected, Eq.(7) has to be multiplied by 

2J + I. 

15. Particle Data Group, Rev.Mod.Phys. ~. Part II, I (1973). 

16. Had we included kaons also, we would have obtained an energy splitting 

up to larger values of x. 

17. Note that the kaons account for 10 % of the data, whereas the nucleon 

contribution is found to be very small. 

18. There is also the possibility that the normalization of the scaling 

functions deviates from that prescribed by the Adler sum rule due to 

SU(3) and exchange degeneracy breaking effects. 

19. C.H. Llewellyn Smith. CERN Report No. TH. 1849 (1974). 

20. V. Rittenberg and H.R. Rubinstein, Phys.Lett. 35B, 50 (197]). 

21. F. Brasse et al., Nucl.Phys. 39B, 421 (1972). 

22. H. Satz, Nuovo Cim. 12A, 205 (1972). Our (s,u) terms are, however, 

bounded for large x. But they depend very much on the imaginary part 

of the trajectories. 



10 

Figure Captions 

Fig. I 

Fig. 2 

Fig. 3 

Fig. 4 

The nucleon and pion scaling functions. The shaded area corre-
sponds·to the SLAG electroproduction data. 

The resonance production diagrams. 

Prediction of the charged pion inclusive cross section. 

Predicted total cross section compared to the world's data. 



0 

X 
z 
0 

z 0::: 
0 w 
0::: ~ 
w 0 I.C'! 
~ 0.... 0 
0 -0.... ;:) 

0 ..c ..c - -::: 3 
I 

I \ 
I \ r-I Ol 

\ N I,C 
\ 0 
\ 
\ 
\ 
\ 
\ -
\ 

0 

z \ \ 8 \ z \ 
0 \ 0 \ 0:::: 
Q_ \ Q_ \ U") 

0 

U") N 
0 - -.:r -X 0 c) 0 0 ......... 

(',I 

u. 



I 
. I -.0 -

-·ru -

N - . 
0'1 

i.i: 

•' 

'. 



6. 

5. 

r-o "Vcj2 = 5 GeV 
N 

> ' 
~ 

<:)4 . . 
.0 
::1. 
~ 

-111 
c 
o3. ·'fcj2 =4GeV 
a. 
"'0 
~ 
Cl .... 
ro 
-52. -

bll>< 
'fc12=3 GeV uu 

N 
CT 

1. 

0.1 0.3 0.5 0.7 0.9 

X :2P 
fc12 

Fig.3 



I 

+'i 7. 
b'i -. VI 6; 
c 
e u 5. 
ro 

.t::. 

t ". 
I 
+t:.) 3. 
t:.) -
b 2. 
II 

n:: 
1. 

o CEA 
o FRASCA Tl y Y. 
A FRASCATI J.L rt 
n SLAC (preliminary) 
U Physics Today Mar.' 7t. 

1. 2. 

I 

Fig.t. 

3. ". 5. 
Vcj2 [GeV]