INFN - Laboratori Nazionali di Frascati, May 1995
From: The Second DAFNE Physics Handbook
Chapter 9: One Photon Initiated Processes
Eds. L. Maiani, G. Pancheri, N. Paver
Supported by the INFN, by the EC under the HCM contract number CHRX-CT920026 and by the authors home institutions.
at DA
NE
Dipartimento di Fisica
Università di Roma, La Sapienza
P.le Aldo Moro, Roma, Italy
The muon anomalous magnetic moment is in principle sensitive to
electroweak effects and could be a probe for new phenomena at very large
energy scales. Computations of the hadronic contributions to the muon
anomaly are beyond present theoretical understanding. This can however
be circumvented by good measurements of the 
annihilation cross
section into hadrons. The purpose of this note is to discuss the
feasibility of such measurements.
1. THE MUON GYROMAGNETIC RATIO.The g-2 value of the muon is about 10 times more sensitive
than the electron g-2 value to large mass states[1].
We can therefore hope to
experimentally verify the renormalizability of the electroweak
interaction. Defining the anomaly as 
, the weak
contribution to the muon anomaly is
=(19.5
0.1)
[2].
The pure QED anomaly[3],
to 
, is 
.
Hadronic contributions to the anomaly are not computed from prime
principles. The hadronic contributions to the photon propagator are
obtained from 
annihilations into hadrons. Including next to
leading QED corrections and light by light scattering, the authors of
reference 3 estimate 
introducing an error in the theory larger than the weak
contributions.
The complete value for the anomaly is given below compared to the
experimental result, an average of the 
and 
data:
An experiment whose goal is to measure the muon anomaly approximately 20
times better than previous efforts, is under construction at
Brookhaven[1,4]. Improved estimates of the hadronic
contributions to 
are necessary to test the validity of the weak
interaction correction as well as probe new phenomena at very high
energy scales.
2. 
AND 
HADRONS.The lowest order contribution to the muon anomaly 
due to
hadronic corrections to the photon propagator is given
by[3]
where
with
where 
is the c.m. energy.
The authors of reference 3 have used all the available experimental
data to obtain the value 
a_(hadr, lo)
, where
and 
are respectively statistical
and systematic errors.
Several years ago it was suggested[5] that better
measurements of 
(-
-
) should be possible at -
colliders.
Unfortunately such colliders did not appear to be available. With the
advent of DA
NE it will become very easy to perform such measurements.
3. THE REACTION -
-
.The cross section for -
hadrons is dominated by 
production
below 1.08 GeV. For simplicity, we
consider in the following only the contribution to 
from
-
-
,
which represents 
70% of the lowest order estimate of 
.
For the 
production cross section we use the expression[6]:
which peaks at about 765 MeV with a value of 1205 nb. Performing the
integral up to W=1.08 GeV
gives
a_(hadr, lo)
=0.019 GeV
nb=488
to be
compared with a_(hadr, lo)
=506(2)(15)
given in reference 3
for -
-
.
The 
contribution should be measured to an accuracy of at least
one half of the uncertainty in the pure QED contribution to 
, i.e. \
.5,-10,. This correspond to a measurement of the integral above to
an accuracy of .3%.
Measuring 
(-
-
) at 100 equally spaced values of W, for
1.08 GeV, each resulting in an accuracy of
.5,-10,
=.5,-11, in 
, gives the integral of eq. 1
to an
accuracy of .5,-10,. This in turn requires measuring the -
cross
section with a fractional accuracy of .9% per point around the 
peak and considerably lower at other energies.
An integrated luminosity of nb
per point is required for
measurements around W=765 MeV. The total integrated luminosity needed
for the complete measurement is nb
.
4. THE MACHINE ENERGY SCALE.The absolute energy of the colliding electrons, for a contribution to
the final error of the magnitude discussed above, must be known to a
fractional accuracy of
(10
). This can be easily achieved by the g-2
depolarizing resonance method[7].
Hopefully
measurements at a few energies, combined with accurate magnetic measurements
will be enough.
5. IDENTIFYING THE -
FINAL STATE.The -
channel must be distinguished from the competing two-body processes
e^+e^-
and Bhabha scattering. The latter can easily be
recognized by calorimetry, the former by momentum measurements, time of
flight and calorimetry in KLOE.
The helium drift chamber in KLOE gives excellent momentum resolution.
Additional 
pair rejection is provided by the angular distribution
of pions,
and muons, 
.
The KLOE shower detector is very effective in
positively identifying Bhabha scattering (and
converted 
events) as necessary
for measuring correctly 
(-
-
)
and to obtain good luminosity measurements as discussed
later.
The shower detector will provide detection of
final states with 
's.
6. A QUESTION OF NORMALIZATION. Absolute cross sections measurements are necessary and
precise measurements of the luminosity are therefore required. For the
error
in luminosity to be negligible the luminosity should be measured to at
least
twice the accuracy of the pion yield measurements. Thus at the 
peak
the luminosity must be measured to an accuracy of at least 0.4%.
The differential cross section for -
, to
lowest order in QED, is given by[8]:
and the total cross section by
As shown in the table, the dimuon yield cannot be used to measure the
luminosity with the required accuracy in most of the energy range of
interest, without requiring considerably more running time. At the 
peak,
for instance, the dipion cross section is about eight times the dimuon
cross
section. Together with the need of observing four times as many 
events, the
required luminosity is increased by a factor of 32 and severe systematic
errors might be incurred.
7. BHABHA SCATTERING AS A MEASURE OF THE LUMINOSITY.Bhabha scattering, even at medium large angles, say >20êlse
, is much
larger than muon pair production and can be used to measure the
luminosity. The Bhabha scattering cross section,
integrated over azimuth is given by:
Thus, for instance, in the center of mass angular range
20
160êlse
, which is well covered by the KLOE detector, the
Bhabha cross section is given by 
(10 nb)/(s
GeV), more than one hundred times the muon pair production cross
section. Radiative correction are ignored here, they become important in
order to reach accuracies of 
(10
) and they are a job
for theorist.
8. LUMINOSITY REQUIREMENTS.While there are no statistical limitations in performing excellent
luminosity measurements, the very strong angular dependence of the
Bhabha
scattering and radiative Bhabha events, -
-
, requires
great care in obtaining correct values. In practice
this demands very good knowledge of beam and tracking system positions,
of
efficiency vs angle, of all materials responsible for multiple
scattering
and so on. Folding in all of these effects,
and the effect of radiative corrections as well, requires extensive
Monte
Carlo calculations, quite beyond the scope of this note.
500 nb
corresponds to 14 hours of collisions at a luminosity
=10
cm
s
, well below the expected
maximum DA
NE luminosity of 10
. In practice it will take
considerably longer time, in order to perform repeated measurements, and
the always
necessary auxiliary measurements, in particular the determination of the
absolute machine energy scale.
DA
NE appears to be an ideal machine to achieve the goal of a
definitive
measurements of the contributions of hadronics states to the spectral
function[9] of the photon propagator, 
.[10]
We have not discussed measurements of 
. This channel
contributes about 
to the muon anomaly. Any luminosity
adequate for the two pions case, will amply do for the three pions one.
The next largest
contribution is due to the 
meson. The two pions contribution
becomes
negligible beyond w=1.08 GeV.
9. CONCLUSION.We have shown that a very modest luminosity is necessary to perform
measurements of 
e^+e^-
, to the accuracy required to
evaluate the muon anomaly to the accuracy necessary to measure
electroweak effects and/or signals from new physics. It appears quite
feasible to identify
the reactions -
-
and -
or 4 pions
in a large background of dimuon and Bhabha events by well
established techniques. Measurements of the luminosity will require most
care and possibly some help from expert theorists, but there is ample
yield of Bhabha scattering.
ACKNOWLEDGEMENTSI wish to acknowledge discussions with T. Kinoshita, M. Greco, J. Lee- Franzini, P. J. Franzini and M. Baillargeon.