SLAC-I’UB-779
jimc 1970
(TH) and (EXP)
SCALING, DUALITY, AND THE BEHAVIOR OF .RESONANCES IN
INELASTIC ELECTRON-PROTON SCATTERING?
E. D. Bloom and F. J. Gilman
Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305
Al3STRACT
We propose that a substantial part of the observed behavior of inelastic electron-proton
scattering is due to a non-diffractive component of virtual photon-proton scattering.
The behavior of resonance electroproduction is shown to be related in a striking way
to that of deep inelastic electron-proton scattering.
Relations between the elastic
and inelastic form factors and the threshold behavior of the inelastic structure
functions in the scaling limit arc found.
(Submitted to Physical Review Letters)
High energy inelastic electron-nucleon scattering provides a unique way to
probe the instantaneous charge distribution of the nucleon and to search for possible
substructure.’
If one observes only the scattered electrons’ energy and angle, then
the results of such scatterings are summarized in the structure functions Wl and
W2, which depend on the virtual photon’s laboratory energy, v, and invariant mass
squared, q2. Considered as a collision between the exchanged virtual photon and
the proton, one is studying the total cross section for the process I1 y” + p - !iadrons,
where the hadrons have an invariant mass W which is related to v and q2 by
w2=
s=2Mv +
M? -q2.
t Work supported by the U. S. Atomic Energy Commission.
-2 -
Experiments have revealed a very large cross section for inelastic ep
scattering -
a cross section which when integrated over ZJ at fixed q2 is the
same order of magnitude as the Mott cross section for scattering from a point
protonl.
This has led to descriptions of the scattering in terms of point-like
constituents of the proton (partons), and to the proposal of scaling’:
as !, and
q2- 03,
W,(v, q2) and v W2(v ,q2) are to become functions of the single variable
0 = 2Mv/q2.
If we restrict ourselves to the region W 2 2.0 GeV (above the pro-
minent resonances) and q
2
2 0.5 GeV2, then the resulting subset of data is con-
sistent with scaling, i. e. ,
with a single smooth curve for v W2 (and Wl) as a
function of w . This curve (for v W2) starts at zero at o = 1, the position of the
elastic peak, rises to a maximum at LJ = 5, and then appears to fall off at large
w. I,3
Since v W2 is proportional to the virtual photon-proton total cross section,
such a fall off of v W2 at large w implies the presence of a non-diffractive (non-
Pomeranchukon exchange) component of virtual photon-proton scattering.
In
hadronic reactions, at least, such a non-diffractive component at high energy is
correlated wi.th the presence and behavior of resonances at low energy.
For
example, the K’p total cross section, which shows no prominent resonance bumps
at low energy, is constant at high energy, while the K-p total cross section, with
many Y* resonances at low energy, falls at high energy.
This correlation bet-
ween resonances at low energy and non-Pomeranchukon exchanges (falling total
cross sections) at high energies is part of the more general concept of duality,
and takes quantitative form in terms of finite energy sum rules.
This directs our attention to the behavior of the resonances in electro-
production and the comparison of their behavior to that of v W2 in the scaling
limit, v and q2-+ ~0. In particular we want to investigate whether the resonances
disappear at large q2 relative to a “background” which has the scaling behavior,
or whether the resonances and any “background” have the same behavior, which
-3-
might then be related to scaling and the apparent fall off in v W2 at large w. When
71
W2
is considered as a function of (J, the resonances occur at values of w > 1,
with the position of any given resonance moving towards Q = 1 as q2
increases.
On the other hand, the zeroth resonance or nucleon pole, corresponding to elastic
scattering, always occurs at a fixed value of LJ = 1.
There have been recent
attempts4 within the framework of parton models to derive a connection between
the q2 dependence of the elastic form factors and the behavior of v W2 in the
scaling limit near c3 = 1.
But when v W2 is considered as a function of o the
elastic peak is always at w = 1 where v W2 vanishes in the scaling limit.
With
the nucleon pole, always at w = 1, and the resonances, at varying values of
w > 1, on a different footing,
the connection of either elastic scattering or
resonance electroproduction to the scaling behavior of t, W2 is difficult to see.
To easily see the behavior of the resonances and of elastic scattering in
comparison to I’ W2 in the scaling limit, one should plot the data for I/ W2 versus
the variable 0 l = (2Mv +
M2)/q2 = 1 + s/q2 = G, + M2/q2 (or more generally,
w f = CIJ + ,m2/q2 with m2
= 1 GeV2).
This variable originally arose in the analysis5
of the large angle inelastic ep data near o = 1.
In the scaling limit where v and
s2- co,
the variables 6, 1 and CLI are clearly the same.
For finite values of q2
there is a difference; in particular,
the elastic peak is no longer at o’ = 1, but
appears at W’ = 1 + m2/q2 > 1, and moves to smaller values of of as q2 increases,
just as the other resonances do6,
The striking results of making such a plot versus o’ = 1 + s/q2= 0 -t
M?/q2
are shown in Figure 1. The dashed line, which is the same in all cases, is a
smooth curve through the high energy 0 = 10’ data7 in the region beyond the pro-
minent resonances (W > 2.0 GeV) and with large q2 (3 < q2 < 7 Ge
? ).
This is a
region where the scaling behavior has occurred experimentally, and we call this
the “scaling limit curve”, v W2(01).
The solid lines are smooth curves through
-4-
6’ data at incident electron energies of 7, 10, 13.5, and 16 GeV, and typical values
of q2 of 0.4, 1.0, 1.7, and 2.4 GeV2,
respectively. As q2 increases the resonances
move toward w’ = 1, each clearly following in magnitude the smooth scaling limit
curve. As similar graphs of the 10’ data in the resonance region also show, the
prominent resonances do not disappear at large q2 relative to a
1’background11
under them, but instead fall at roughly the same rate as any “backgrounds and
closely, resonance by resonance, follow the scaling limit curve. We emphasize
that this behavior of the resonances, which is of central importance in our argu-
ments, can be seen by careful examination of the data when they are plotted with
respect to other variables; with respect to o1 it just becomes obvious at a glance.
Thus the resonances have a behavior which is closely related to that of
v W2 in the scaling limit. For large values of o1 , the data for v W2 with
q2 > 0.5 Ge
3
are consistently on a single curve which falls with increasing o’,
just as when plotted versus w. We therefore propose that the resonances are not
a separate entity, but are an intrinsic part of the scaling behavior of v W2 and that
a substantial part of the observed scaling behavior of inelastic electron-proton
scattering is non-diffractive in nature. Appropriately averaged, the nucleon and
the resonances at low energy build, in the strong interaction duality sense, the
relevant non-Pomeranchukon exchanges at high energy, which result in a falling
v W
2
curve.
What is unique to electroproduction is the experimentally observed scaling
behavior which allows us to consider points at the same o’ arising from different
values of q2 and s =
2
, both within and outside the low energy resonance region.
If we choose v
m
and q2 tn the region where v W2
scales, i. e. , beyond the region
of prominent resonances and where v W,(v, q2) = v W2(w1) = a smooth function of v
(see Figure I), then a finite energy sum rule for v W2 at fixed q2
tells us that
-5-
V
2M m
s
+m2)/q2
70
dv v W2(v,q2) =
do’ v W2( w’),
(1)
2 2
since the integrands are the same for v > vm or G;’ > (2Mvm+ m )/q (by the
assumption
that v
m and q2 are in the region where v W2 scales).
Eq. (1) states
that for v < vm,
v W2( w’) acts as a smooth average function for v W2(v,q2) in
the sense of finite energy sum rules.
Thus, because we can vary the external
photon mass in electroproduction and have scaling, we can directly measure a
smooth curve which averages the resonances in the finite energy sum rule and
duality sense.
High energy electroproduction thus becomes a beautiful example
of the duality between resonances and non-Pomeranchukon exchanges at high
energy.
Looked at the other way, by appropriate averages over the resonances we
would build up the curve for v W2 in the scaling limit.
But how can resonances,
which have form factors which fall rapidly with q2,
be consistent with a scaling
li-mit curve which is supposed to characterize a very slow q2 variation?
Let us
fix s =
Iv?
R, the mass squared of a given resonance (possibly this could be the
zeroth resonance, the nucleon) and vary q2.
Then if G(q2) is the excitation form
factor of the resonance,
VW2 =
2Mv [G(q2)126(s-M2,) = (dR-M2+ q2) [G(q2q26(s- gR)
(2)
is its contribution to v W2 in the narrow resonance approximation.
For large q2,
the form factor falls off as some power, say
G(s2 1
- (l/q2)n’2 ’
(3)
As q2 increases, the resonance is pushed down toward
W’ =
1, where v W2( w’)
can be parametrized by some power behavior,