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.
Dip. di Fisica, Univ. di Pisa and INFN, Sez. di Pisa
and
Ettore Remiddi
Dip. di Fisica, Univ. di Bologna and INFN, Sez. di Bologna
cm
The obvious target in the domain of the precision measurements of the muon g-2 is the detection of the electroweak corrections due to vector and Higgs boson exchange. The present experimental value of the muon anomaly is [1]
with the electroweak contribution predicted to be:
To identify the effect (2), we need to reduce the error in the
theoretical prediction of the other contributions to 
, besides
reducing the experimental error in (1) by more than one order of
magnitude.
The main source of these errors are the hadronic vacuum polarization
corrections, Fig. 1(a), given in terms of an appropriate integral over
the hadronic (one-photon) 
cross section. With present data,
one has [2,3]
(the first is the statistical, the second the systematic error).
The error in (3) is mainly due to the imperfect knowledge of 
at low energy, as shown in Tab. 1
[2].
Table 1: Analysis [2] of the present statistical and systematic error of
the hadronic contribution to the muon anomaly, in units of 
. Totals
are obtained in quadrature.
DA
NE can improve considerably on the statistical and systematic errors
in the 
, 
and 
regions, reducing the overall error
to something like 
, sufficient to identify the bulk of the
electroweak correction.
This is not the full story, however.
Hadronic corrections, to the next order, enter via the light-by-light
scattering diagram, Fig 1(b), and we have to make sure that the error on
this contribution is also sufficiently small. This question was
considered in ref.[2]
, where various estimates are given, see Tab. 2,
replacing the hadron bubble in Fig. 1(b) with a pointlike quark loop and
adding the contribution arising from the 
coupling, see Fig. 1(d).
For the charged pion loop, the pointlike
case has been considered, as well as the case where 
-dominated
form factors are associated with the 
vertices (form
factors softening is always necessary for the neutral pion contribution,
which would be UV divergent otherwise).
Table 2: Contributions to the muon anomaly in units of 
, from
different estimates of light-by-light scattering [2], see fig. 1. Numbers
in parenthesis are the contributions from two different, gauge invariant
set of diagrams, see ref. [2]. In the quark loop, (
=0.3,
=0.5,
=1.5 GeV)
The quark loop result, with the masses indicated in the Table, agrees remarkably well with the sum of the contributions of the neutral and charged pions (with form factors) leading the authors to quote, as best value:
The error is their estimate of the model dependence of the result.
There are reasons to think, however, that the error has been quite
underestimated. For one, the pion result arises from the cancellation of
different (gauge invariant) contributions, each of which is larger than
the electroweak correction itself, see Tab. 2 and ref [2],
so that the
uncertainty of the result is unlikely to be as small as quoted in (4).
Moreover, the quark model result is very sensitive to the values of the
assumed light quark masses (it goes approximately like 
) and the
agreement of the quark and meson results looks suspiciously close to
being accidental. A quark loop estimate of 
, with
the quark masses given in Tab. 2 yields
One could think to fix the light quark mass by requiring the quark loop
to reproduce the value (3). This requires a light quark mass of about 0.
16 GeV (a not unreasonable value) and would give 
, a much less favorable result.
The problem appears still open. More thinking is needed at least to
estimate the size of the contribution from 
scattering.
In the meanwhile, a good measurement of the hadron cross-section at low
energy, with DA
NE, cannot but be welcome.
After the first writing of this brief report [4], some new contributions on the issue under discussion have appeared in the literature. They are:
Their content can be summarised as follows.
From Ref. [5], the hadronic vacuum polarisation corrections are given by
to be compared with equation (3).
As to the light-light hadronic contribution , both
Refs. [6,7]
agree on the statement that more work needs to be done in order to reduce
the error below the expected sensitivity of the foreseen Brookhaven
experiment [8] on the muon anomaly.
This might be possible by resorting to
theoretical models that are thought to represent a fairly good
approximation of low energy QCD.