SLAC-PUB-662
October 1969
STATISTICAL MODEL FOR ELECTRON-POSITRON tTH) and tEXP)
ANNRIILATION INTO HADRONS*
James D. Bjorken and Stanley J. Brodslcy
B
Stanford Linear Accelerator Center
Stanford University, Stanford) California
94305
4
ABSTRACT
A statistical model for the production of multibody hadronic states
by e+e- annihilation is discussed, We associate the secondary hadron
momentum distributions for colliding beam processes with the exponen-
tially falling transverse-momentum distributions in hadron-hadron col-
lisions,
The consequence of this picture is that at high energies hadron
multiplicity rises linearly with center-of-mass energy unlike the log
s
behavior for the multiplicity of secondaries in hadron-hadron collisions.
If the total annihilat:ion cross section is assumed to have a power fall-
off rvs-m,
the n-pion cross sections follow a Poisson distribution with
the most probable multiplicity n,
= &/<E,> + 3-m and <ET>-375 MeV. An
alternate statistical model based on jets is also briefly discussed.
The
storage rings now .being constructed or envisaged should easily distin-
guish between the various possibilities.
J
%
* Work supported by the U. S. Atomic Energy Commission.
I. INTRODUCTION
In the next few years, electron-positron storage rings will be developed capable
of producing hadron systems of total mass js up to -6 CeV or higher. Aside from
predictions for the energy dependence of tho total annihilation cross section into
hadrons,
192
there has been little discussion concerning the composition, multi-
plicity, and other properties expected for the multibody hadron final states. 3
It is not so clear what to expect, even qualitatively.
The process e++ e- * hadrons
at high energy differs from almost all other hadron processes inasmuch as (within
the one-photon-exchange approximation) the hadrons are produced via the decay
of an arbitrarily heavy virtual photon.
One picture of such a decay would be that
the virtual photon decays into an intermediate state consisting of a virtual pair
of “bare” constituent partons (such as a bare quark-antiquark pair) which subse-
quently decay in some way into hddrons-mainly pions. If this were the case, one
could anticipate anisotropy and the existence of an axis in the distribution of had-
ron products; in other words the hadrons “remember” the direction along which the
bare constituents were emitted. Under these circumstances, the transverse mo-
menta pL of the secondaries relative to the axis for a given event would be no
more than a few hundred MeV, while the longitudinal momenta could be much
larger. In other words the momentum distributions and the (slow) increase of
multiplicity with energy would be much l.ike the situation in ordinary hadron-
hadron collisions.
The observation of such “jets” in colliding-beam processes would be most
spectacular.
It is notour intention here to study such a possibility further. In-
stead, we consider the case in which there are no high hadron momenta in the
-
final state.
Because all directions are equivalent in the center-of-mass frame,
we associate the secondary hadron momentum distributions for colliding beam
-2-
processes with the transverse-momentum distributions in hadron-hadron colli-
sions . These fall off exponentially and are characterized by a mean momentum
of a few hundred MeV. The most immediate consequence of this picture is that
the hadron multiplicity rises linearly iyith center-of-mass energy, quite unlike
the case in hadron-hadron collisons.
For example, given this picture we predict
that at an energy of N
1.5
GeV/lepton ( 6 = 3 GeV), states containing 8 to 10
pions will be most prevalent; at
-3 GeV/le@on typical hadron states are expected
to contain on the order of 15-20 pions; at 6 GeV/lepton the number is 30-40 pions.
II, THE STATISTICAL MODEL
A striking phenomenological feature of high energy hadron collisions is the
fact that the distribution of transverse momenta of the secondaries is quite well
represented by an exponential law5:
PI
--
N(P$ dpI CC Pi e
b
dq
(1)
with b = I/2 <pl>
x 150-206 MeV/c depending upon the mass of the secondary0
For pions, <p,>
w 300 MeV/c.
The relation (1) has been checked over a range
of transverse momentum from
-10 MeV/c to 1.5 GeV/c, The approximate con-
stancy of <p,> with energy has been checked for incident nucleon energies ex-
tending from a few GeV to cosmic-ray energies of w104-lo5 GeV.
The longitu-
dinal momentum distribution is much broader; as a consequence of the slow in-
crease of multiplicity with energy, the energy per
secondary increases
as the
center-of-mass energy increases.
In going over to electron-positron collisions we take the same exponential
form (I), with pl replaced by 1 p 1 and l/2 dp:
nm
= pldpl replaced by the invariant
-3-
phase- space d3p/E
’
-4
’ d3
N(g) d3p oc e- b -#
(2)
If we choose b1 such that <p,> remains unchanged from its value in hadron-
hadron processes, we find, for pions
2b = (Pl)
N-
; b’
.
P
oh x3 (x2 + m2b12)-1’2e-x
2
S
m& x2 (x2 + m2/bt2)-1’2e-x
0
(3)
This factor in brackets, for reasonable b’, is nearly unity and gives
br e 1.2b zz 175 MeV.
An immediate result is that if, as we expect to be the
case, pions dominate the secondaries, the mean energy of a pion is
<E?,> = 2b’
2f 375 MeV
-4-
(4)
By equipartition this leads to a crude estimltte of multiplicity
In the following, we try to improve and refine this estimate,
III. ANNIHILATION INTO n PIONS
The differential cross section for annihilation into n pions may be written as
1
dan = m
with the hadron matrix element dv given by
ti” = ~<O~jEL(O)~n><n~j”(O)~O> (2r)484 (Pn- q) n”
d3pi
i=l 2Ei(2~)3
(6)
(7)
The statistical assumption that we adopt is that in the center-of-mass frame
&pv =
a,t&? - 8’ q2J e
-Fa Igil
n d3pi
t2d~4tPn-s) n
i=l 2Ei(2n)3
(8)
with an a slowly varying func tion of q2 and a=2/<B>
We then find for don, the expression
2 2
don = Sy
an
(9)
We discuss in the appendix the relationship of this form with the single-pion
momentum distribution (2).
Clearly the essentially uncorrelated distribution
(9) should only be expected to have validity, if at all, for large n.
We shall only
apply the results consequent from (9) for n 14,
All correlation effects, isospin
requirements, Bose-statistics, and effects associated with higher mass particles
(K, N, i?, Y, y) have been neglected.
In a statistical model such as this, we
-5-
