Chiral Dynamics: Theory and Experiment: Proceedings of the Workshop Held in Mainz, Germany, 1–5 September 1997

381 Pages·1998·18.269 MB·English

by  A. M. Bernstein (auth.), Aron M. Bernstein, Dieter Drechsel, Thomas Walcher (eds.)

Part I
Chiral Dynamics and QCD
Introduction to Chiral Dynamics:
Theory and Experiment
A.M. Bernstein
Physics Department and Laboratory for Nuclear Science
M.I.T., Cambridge Mass.,USA
At very high energies Quantum Chromodynamics (QCD) predictions are
made by perturbation theory based on the weak interactions of the quarks
and gluons (Donoghue et al. (1992), Sterman et al. (1995)). At low energies
the interaction between quarks and gluons is extremely strong and leads to
confinement. Nevertheless, approximate QCD solutions can be obtained by an
effective field theory known as chiral perturbation theory (ChPT) or Chiral
Dynamics (Donoghue et al. (1992), Chiral Perturbation Theory (1), Chiral
Perturbation Theory (2), Chiral Perturbation Theory (3)). This is based on
the chiral symmetry present in the QCD Lagrangian in the limit of massless
light quarks (up, down, and possibly strange), but which is broken in the
ground state of matter. In such a situation, Goldstone's theorem tells us
that there are massless, pseudoscalar Bosons whose interactions with other
hadrons vanish at zero momentum (Donoghue et al. (1992), Goldstone (1961),
Leutwyler (1994)). In the case of two assumed massless quarks there are three
Goldstone Bosons which are identified as the pion triplet. Ift he strange quark
is also included as massless then there are eight Goldstone Bosons which are
identified as ,r7 7, K (more precisely the heaviest Goldstone Boson is a mixture
of the 1 r and 7' mesons). The relatively weak interactions of Goldstone Bosons
at low energies invites a perturbation scheme based on chiral symmetry and
hadronic degrees of freedom.
However, in the real world the light quark masses are small, but not
zero (Weinberg (1977), Gasser and Leutwyler (1996)). Therefore, for strong
interaction theory to have predictive power, calculations must be performed
taking the deviations from the pure Goldstone theorem into account. As an
example, the s wave scattering length, a, vanishes for a Goldstone Boson
scattering from any hadron in the tow energy limit. However for a physical
meson with finite mass ,rT( ,~r K) one would intuitively expect a -~ I/As where
As is the chiral symmetry breaking scale -~ 47rf~ - i GeV and where f~ -~
92 MeV is the pion decay constant. This intuitive expectation is validated by
the original calculation of Weinberg (Weinberg (1966)) in which the scattering
lengths of pions from any hadrons were first obtained by current algebra
techniques (a precursor of ChPT). The order of magnitude of the s wave
scattering lengths is (Weinberg (1966)):
rTar m~ 1.5 (1)
ao = 47rf~ -- -Axf~ - ~ ~----A
4 Aron M. Bernstein
One observes that ao ~ 0 when m~ +-- 0 (the chiral limit) and also that
ao "" 1/Ax.
Similarly, on an intuitive basis, I would expect the production and decay
amplitudes of Goldstone Bosons to vanish in the chiral limit. Some examples,
which can be obtained from ChPT calculations (Chiral Perturbation Theory
(1), Chiral Perturbation Theory (2)), are the threshold electric dipole am-
plitude, E0+(vN +-- lr°N) for s wave photo-pion production, the Z term of
7rN scattering , the isospin breaking ~ ~-- r73 decay, and the form factors for
Kl4 decays .1 In a similar vein, there are some observables that diverge in
the chiral limit, such as the charge radii and polarizabilities of nucleons and
pions (Chiral Perturbation Theory (1), Chiral Perturbation Theory (2)). In
this case, the physical interpretation is that the meson cloud extends beyond
the hadron and in the chiral limit extends to infinity.
Pion-hadron scattering (Eq.1) and the amplitudes given above are exam-
ples of quantities that either vanish or blow up in the chiral limit. In the real
world, where the light quark masses are non-zero, chiral symmetry is explic-
itly broken and these quantities are finite and non-zero. Their precise (finite)
values are measures of explicit chiral symmetry breaking. As such, it is a the-
oreticM challenge to calculate them. Quantities which either vanish or diverge
in the chiral limit point to an experimental opportunity to perform precise
experiments, not only to check ChPT calculations, but also as fundamental
quantities which must be predicted by any theory of the strong interaction.
Consequently, I would define Chiral Dynamics as the study of the properties,
production and decay amplitudes, and low energy interactions of the almost
Goldstone Bosons (zr, ,77 K) with themselves and with other hadrons.
A partial list of the processes which I would include in Chiral Dynamics
includes the RMS charge radii, transition radii, and electric and magnetic
polarizabilities. The production amplitudes include nucleonic processes such
as 7N ~ ~rN and vN ~ AK, as well as purely Goldstone Boson reactions
such as e+e - ~-- VV *-- °r7 and r7,3 ~ rTr7 which proceed primarily via the
chiral anomaly. The completion of this program requires measurements of
the interaction of the almost Goldstone Bosons with each other and other
hadrons; e.g. ~rlr, 7rK, and 7rN scattering. The experimental program outlined
above is of immediate interest because it tests the predictions of the chiral
anomaly and ChPT (which is also based on the same physics as Goldstone's
theorem). These provide a direct link to confinement scale QCD. Accurate
data obtained in this experimental program will test any theory of the strong
interactions at low energies (e.g. lattice gauge theory).
It has been recognized for a long time that the theoretical aspects of this
program represent a frontier of QCD research since full predictions of the
1 There are known exceptions to this rule. One is a transition due to the chiral
anomaly which is finite in the chirM limit. Another is due to charge couplings
such as the threshold PV -~ ~r+n reaction which goes to the Kroll Ruderman term
in the chiral limit.
Introductiotno Chiral Dynamics: Theory and Experiment 5
theory are not possible without accurate predictions for both the "hard" and
the "soft physics". As an example of this we quote from the Handbook of
Perturbative QCD "Every experiment in strong interactions tests QCD from
some fixed 'short' distance to its very longest distance scales, over which the
value of the strong coupling may change radically" (Sterman et al. (1995)).
What I believe is less recognized is that the program of chiral dynamics
outlined above is also at the cutting edge of experimental physics. It can
utilize all of the new accelerators and detection techniques that have been
recently completed, or are presently under development, including high inten-
sity, polarized, CW electron beams, high luminosity e+e - colliders, as well as
the use of hadron beams of high energies. Also required is the development of
new techniques to accurately measure some otfh ese processes, most of which
involve unstable particles.
From an experimental point of view one of the interesting things about
Chiral Dynamics is the variety of physical processes that are studied and,
correspondingly, of accelerators and techniques that are used. Some exam-
ples discussed at this workshop include the physics that is accessed with
relatively low energy electron accelerators (Bates, Bonn, Jefferson Lab (CE-
BAF), Saskatoon, and Mainz), e.g. to photo- and electro- produce Goldstone
Bosons. Meson factories (LAMPF, PSI, TRIUMF) are used to measure Nr7
and rTr~ interactions although there are interesting opportunities for electro-
magnetic accelerators (as will be discussed below) to study Nr7 interaction as
well. High energy pion beams can be used to study the r~r interaction from
low t, Nr7 scattering with extrapolation to the pion pole, or by producing
pionium. One of the most promising techniques to study the r~r7 interac-
tion is through the Ke4 decay in which the pions interact in the final state
(Brookhaven AGS, Frascati). Measurements of the pion polarizabilityc an be
performed at low energy electron accelerators by the yp *-- 7r+n~ reaction
extrapolated to the pion pole, by the use of the Primakoff effect with high
energy hadron beams (Fermilab, CERN), and by e+e - ~ rTr~ (Frascati).
While not wanting to present a summary of the workshop I do want to
briefly mention a few topics of personM interest (with apologies to those
whose favorite topic is not included here), which I hope will illustrate some
of the recent interest and activity in the field and some of the open questions.
The first of these is one of my favorite topics, threshold electromagnetic pion
production, where considerable progress has been made in the past few years.
The unpolarized cross section for the 7P *-- P°r7 reaction has been measured
at Mainz (Bernstein et al. (1996)) and Saskatoon (Bergstrom et al. (1997))
and the ChPT calculation to one loop O(p )4 has been worked out in the
heavyb aryon formulation (Bernard et al. (1992-96)). There is generallgyo od
agreement between the ChPT calculations and experiment. This agreement
includes the newly derived low energy theorem for the p wave amplitudes. In
addition the predicted unitary cusp in the s wave electric dipole amplitude
E0+ has been observed experimentally (Bernstein et al. (1996), Bergstrom et
6 Aron M. Bernstein
al. (1997)). The results for ReEo+ is presented in Fig. 1. The unitary cusp
is of particular interest since it represents the contribution of the two step
reaction 7P ~-- r+n *-- 7r°P, which is the same order of magnitude as the
single step 7P ~-- 7r°P amplitude. The magnitude of the unitary cusp depends
on/~ = Eo+(yp ~ ~+n). a~:(Tc+n ~ 7r°p), where a~:~(1r+n *--- 7r°p) is the
s wave scattering charge exchange scattering length. Thus an accurate mea-
surement of the energy dependence of the near threshold 7P *-- 7r°P reaction
will enable us to obtain an experimentM result for a~(Tr+n *--- 7r°p). This is
of particular importance since Weinberg has predicted that the s wave 7rN
scattering lengths will exhibit substantial isospin violation due to the up,
down quark mass difference (Weinberg (1977)). The use of photo-pion reac-
tions is a promising new technique that can perform accurate measurements
in different charge states than are usually accessible with conventional pion
beams.
0.0
, I , I , I , I , I , I , I ,
- - T : TT
~-0.5
7
o
2"
-1.0 -
5.1-
441 ' 641 I 841 r i 051 I ' 251I ' I 451 651' 851I 061 ' I
Lab Photon Energy [MeV]
Fig. 1. ReEo+ (in units of 10-3/m~) for the 7P *-- P°r7 reaction versus photon
energy k(MeV). The solid curve is the ChPT fit (Bernard et al. (1992-96)) and the
dashed curve is the unitary fit (Bernstein et al. (1996)). The circles (filled diamonds)
are the Mainz (Saskatoon) points (Bernstein et al. (1996), Bergstrom et al. (1997)).
The errors are statistical only.
A new area that has opened up since the last workshop is the study of
the coherent near threshold photo- and electro-pion production 7*D *-- 7r°D
reaction. There have been ChPT calculations (Beane) and experiments at
Mainz (Merkel) and Saskatoon (Bergstrom) with interesting results. In a
Introduction to Chiral Dynamics: Theory and Experiment 7
sense the study of the chiral dynamics of Deuterium has commenced in the
past few years. This activity was motivated in part by the desire to study
the 7*n *-- 7r°n amplitude. However, as illustrated in Fig. 2, which shows the
lowest order Feynman diagrams for this process, there are two nucleon and
meson exchange current contributions to the amplitude which are of interest
in their own right. The final state interactions in which the photon is absorbed
on one nucleon followed by a pion exchange with the other nucleon can be
considered as interesting as the situation for pion production on the nucleon
in which there is pion-nucleon rescattering in the final state (pion loop). This
is an important bridge between chiral dynamics and nuclear physics [van
Kolk].
7N . ~z°N
7D • ~° D
Fig. 2. Feynman diagrams for the p/~ *-- p°c7 and D~" *-- 7r°D processes.
There are several areas where better theoretical calculations are needed to
make experimental progress. In order to study the properties of the pion, or
pion-pion interactions one is led to the use of virtual pions as a target. This
requires an extrapolation to the pion pole. The convergence of this procedure
is rarely explored theoretically. For example one method to measure the pion
polarizability is to study the 7P -~ 7v+n reaction for small values of t (the
four momentum transfer squared) and extrapolate to t = m~, which is outside
of the physical region (t < 0). One problem with this is that there is not a
complete theoretical calculation of the background processes. In order to
work at small magnitudes of t one needs to perform the experiment at high
8 Aron M. Bernstein
photon energies so that ChPT is not applicable and also there are significant
contributions from the A or higher resonances. It might in fact be more
productive to perform the measurement of the cross section at low energies
for which ChPT calculations could be performed and then use the amplitude
in the physical region to extract the polarizability from the data. At the
previous workshop (Chiral Perturbation Theory (3)) this question was raised
for the extraction of the rTr~ scattering length from the rN -~ ~r~rN reaction
in the low energy regime. As a consequence the ChPT for this process was
worked out (Bernard et al (1997)). It would be of interest if this could also
be done for the PV -+ 7~r+n reaction.
A topic which appeared to be reasonably well settled at the time of the
last workshop was the value of the Z term in rN scattering as discussed by
Sainio in Chiral Perturbation Theory (3). Despite this conclusion, there are
some underlying questions about the role of isospin violation in rN scatter-
ing which have been ignored in all previous studies. In addition, there were
worries that errors in the data base could propagate into the value of the E
term. It is also a source of some interest that the dispersion analysis of the
t dependence of the Z term indicates a nucleon scalar radius of -~ 1.6 fm,
about twice the size of the electromagnetic radius of the proton. It would be
very nice to have an independent verification of this value. However, to date,
no technique has been suggested which can do this. Despite these pedagogi-
cal concerns the situation seemed to be settled until a recent work presented
at this workshop challenged the empirical underpinning s of the previous so-
lution [Pavan]. The ingredients of this analysis are the new data and lower
value of the 7rN coupling constant. The new analysis indicates a larger value
then was previously accepted [Pavan]. It is clear that this will be the subject
of further work, and we will probably hear more about this basic topic at the
next workshop. It is also possible that we might get an empirical value for
the ~ term in lrTr scattering from the new Ke4 data at Brookhaven [Lowe].
A basic characteristic of QCD at low energies is the appearance of the
chiral anomaly which, for example, provides the dominant contribution to
the r~ ° ~ 77 decay rate (Donoghue et al. (1992)). An anomaly occurs when a
symmetry of the classical action is not a true symmetry of the full quantum
theory. The relevance of the anomaly to Goldstone Bosons production and
decay amplitudes was presented at the workshop by Bijnens. Here I want
to discuss the experimental opportunities to test the predictions in the case
of the lr ° +-- V/~ decay rate (Bernstein et al (1997)). There are three known
methods to measure the r7 ° lifetime (Barnett et al. 1996): direct observation
of the r7 ° flight path, the Primakoff effect in the V~/(Z) ~ r7 ° ~ ~/y reac-
tion(where v(Z) represents the Coulomb field of a heavy nucleus), and the
e+e - -~ e+e-v*V * ~-- e+e-lr ° +-- e+e-vv reaction with colliding beams. The
results for the width (lifetime) of the r~ ° meson are shown in Fig. 3 with the ex-
perimental results from the latest experiment of each type, the particle data
book (Barnett et al. 1996) average, and the predictions of the chiral anomaly.
Introduction to Chiral Dynamics: Theory and Experiment 9
An estimate of the theoretical uncertainty of order (rn~/1GeV) 2 = ±2% is
given for chiral loop corrections to the lifetime (Donoghue et al. (1985), Bij-
nens et al. (1988)). In addition the model dependent loop corrections (Bijnens
et al. (1988)) are also shown in Fig. 3.
8.4"
8.0"
r-
"O
.m
m
7.6"
O
7.2"
1) Direct 2) Primakoff 3)e+e- 4)Av(PDB)
6.8
Fig. 3. r~ ° *-- ~/decay width in eV. The horizontal solid lines are the prediction of
the chiral anomaly (Donoghue et al. (1992)) with an estimated 2% error (see text
and Bernstein et al (1997) for discussion). The horizontal dashed lines are model
dependent chiral loop corrections (Bijnens et al. (1988)). The experimental results
with errors are the latest results for:l) direct method; 2) Primakoff method; e+e-;
and the particle data book average (Barnett et al. 1996).
The experiments for the ~0 lifetime are in good agreement with the pre-
diction at the present estimated level of errors. This fundamental prediction
10 Aron M. Bernstein
should be tested with more accurate, modern experiments. In particular the
used of tagged photon beams could be used to perform a more accurate Pri-
makoff experiment. Another CEBAF possibility is the use of virtual photons
in the ev(Z ) *-- e'er ° reaction where the target photon -y(Z) comes from the
Coulomb field of a heavy nucleus (Primakoff effect). The DAPHNE e+e -
collider ring constructed at Frascati and the KLOE detector could be used
to perform a more accurate measurement of the r~ ° *-- VV decay width, per-
haps to the few % level. These are very important possibilities. The model
dependent theoretical corrections to the chiral anomaly predictions are also
needed.
I hope that these few examples show the richness of Chiral Dynamics and
the wide range of experimental and theoretical challenges in this field.
Acknowledgments: I would like to thank my co-organizers Dieter Drech-
sel and Thomas Walcher for their collaboration in the organization of the
workshop. I would like to thank Barry Holstein and Marcello Pavan for their
helpful comments on this manuscript. This work was supported in part at
MIT through a grant from the Dept. of Energy and in Mainz by a Research
Award from the Alexander von Humboldt Foundation.
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