INFN - Laboratori Nazionali di Frascati, May 1995
From: The Second DAFNE Physics Handbook
Chapter 5: The $pipi$ Interaction
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.
Amplitude in
M. Knecht B. Moussallam and J. Stern
Division de Physique Théorique
, Institut de Physique Nucléaire
F-91406 Orsay Cedex, France
interaction is studied at order one loop in the framework of
Generalized Chiral Perturbation Theory.
The general requirements of analyticity, unitarity and crossing symmetry,
together with the Goldstone nature of the pion and
isospin invariance, determine a low energy representation of the 
scattering amplitude up to corrections of order 
[1], where 
denotes the scale at which particles other
than pions can appear as intermediate states. This
general representation depends on six independent low energy subtraction
constants, which can be determined from experiment, given sufficiently precise
data. Furthermore, only four of these constants (denoted as 
,
, 
and 
in the sequel) appear if two loop
contributions are neglected [1]:
with
and where 
stands for the usual loop function,
The corresponding scattering lengths 
and slope parameters 
are
given as follows:
and
The above parametrization of the low energy behaviour of the 
amplitude
and of the threshold parameters holds independently of more detailed
considerations like, for instance, the value of the quark-antiquark condensate
in the chiral limit. On the other hand, the expansions of 
,
, 
, 
in powers of quark masses up to a given order
will differ
according to whether one assumes 
1 GeV, or 
100 MeV.
In the sequel, we shall consider the amplitude 
up to and
including order 
contributions only, within the framework of
SU(3)
SU(3) Generalized Chiral Perturbation Theory (G
PT)
[1,2].
This means that both in Eq.
(1) for the 
scattering amplitude and in the expressions (4)
and (5) for the threshold parameters, only the leading order expressions
are to be retained for the parts that are quadratic in 
and 
. In
particular, 
of Eq. (1) should be replaced by
at that order, since the difference
only affects contributions at orders 
and 
. The leading order
expressions 
and 
and the higher order corrections
are obtained from the effective lagrangian 
given
in [2]. Expanding the contributions
from 
and 
intermediate states in powers of 
, 
(
) and retaining only the dominant terms at order 
,
one obtains the following result for 
and for 
(
is
an arbitrary subtraction scale):
In these expressions, one recognizes the contributions coming from the tree
diagrams, given in terms of the low energy constants 
, 
and 
,
and the chiral logarithms coming from the loops. While these different
contributions separately depend on the subtraction scale 
,
and 
are
-independent. The same holds for 
and 
, for which
one obtains a similar decomposition:
The tree level contributions read:
with
whereas
Notice that 
varies from 1 (the standard case) up to
= 4 for r = 
, 
.
In the above expressions, 
and
represent contributions from 
, 
and 
. Their scale
dependences are compensated by the 
-dependences of the loop
contributions, which read:
and
with 
denoting the combination
of pseudoscalar masses, and 
, 
.
In the standard case, 
and 
are
relegated to higher orders, while the quark mass ratio 
takes the value 
, with [3]
To the order we consider here, the standard expressions for 
and
are
while the splitting of the decay constants is expressed as [3]
with
(
, 
and 
are 
-independent).
The expressions (6) for
and for 
remain unchanged, while one finds that 
and 
become
These formulae could also be obtained directly from the standard expansion of
the SU(3)
SU(3) effective lagrangian. The above exercise is a rather
non trivial illustration of how the standard case arises as a special case
of the
generalized 
PT. Notice also that restricting further the formulae
(6), (19) and (20) to the SU(2)
SU(2) chiral limit
reproduces the 
scattering amplitude of Ref. [3].
Let us now come back to the expressions of Eqs. (8) and (11)
for 
and Eqs. (10), (12) for 
. Apart from the observable
pseudoscalar
masses and decay constants, they involve various other quantities which we
discuss in turn:
i) As already mentioned earlier, 
and
collect tree level contributions from
, 
and
. The various
low energy constants involved in these pieces of the effective lagrangian,
and which would appear only starting from order 
in the standard case,
are not under quantitative control at present. As an estimate of the
uncertainties in 
and in 
coming from our lack of knowledge of
and of 
, we shall take the
changes in 
and in 
as the subtraction
scale 
is varied between 
= 547.5 MeV and 
=
770 MeV.
ii) The two parameters 
(
) and 
are
also not known. Their values are expected to remain small as compared to unity,
due to the Zweig rule. At leading order, the same vacuum stability argument
that requires 
demands that 
remains bounded as 
varies between 
and 
,
In what follows, we shall assume that these bounds still provide, at order
, a good estimate of the uncertainty on 
, which we further
restrict not to become larger than 0.5, i.e.,
As for the remaining Zweig rule violating quantity 
, we allow
it to vary between --0.2 and +0.2,
Notice that 
appears in the expression for the
cross section already at order one loop in G
PT
[4], and
also contributes to the 
form factors,
especially to F. Its value may in principle be obtained from very
accurate 
data, together with 
, 
and 
.
Before discussing the
values of these latter low energy constants, let us consider
the ranges of values
accessible to 
and 
according to the preceeding
discussion . We
show on Fig. 1 the bands of allowed values as functions of the quark mass ratio
r. The bands delimited by the dotted curves correspond to the variations of
and of 
in the ranges (22) and (23),
respectively, and to the estimates, via
the 
-dependences of 
and of 
, of
the values and of the uncertainties associated to 
and
to 
, respectively. The solid curves show the range of
variation of 
and of 
when only these last uncertainties are
taken into account, with 
and 
both taken to vanish. In
the standard case, with the values of 
, 
, 
and 
as given in
Ref. [3], we obtain, from Eqs. (19) and (20)
in agreement with the values read from Fig. 1 for 
25.
iii) The low energy constants 
, i=1, 2, 3, can be extracted from
the data on 
decays [5,6]. The expressions of the
corresponding axial and vector current matrix elements are known at the
one loop level
both in the standard case [5,7] and in generalized
PT [8]. In the latter case, the form factors depend on r in
addition. This dependence on the quark mass ratio may affect the values of the
's one extracts from the data. In the standard case,
the most recent
analysis [7] of the data of Rosselet et al. [9] leads to
the following values,
which give
A preliminary study of the generalized case shows that these values of 
,
and 
tend to
decrease in absolute value with r [8].
The constant 
is the most sensitive
to variations in the quark mass ratio r and its central value becomes
, for 
, with an uncertainty
comparable to the one shown in Eq. (25).
The variations in the values of 
and
of 
, however, are smaller and affected by larger error bars. Within these,
they remain compatible with the standard values (25). For 
and for 
, 
as before, 
remains unchanged, but
decreases (in absolute value) by 
30% and its central value
becomes 
.
In Figs. 2 to 4 we have plotted the behaviours of the scattering lengths
and 
, and of the slope parameter 
(all in units of
) as the quark mass ratio
varies between 8 and 30. The two error bands (solid lines and
dotted lines) come from the corresponding errors on 
and on 
shown in Fig. 1. The plots correspond to the central values of the 
's,
i=1,2,3,
as given by Eq. (25) above. In Fig. 5 we show the difference
, which can be obtained directly from the lifetime measurement of
atoms, an experiment planed at CERN [10].
The 
phase shift combination 
can be extracted down to very small energies by analyzing
decay data.
In order to calculate the phase shifts, one first constructs
the amplitude 
for a given partial wave l and
isospin I from 
(see e.g. the appendix of Ref [1]).
At order 
the phase shifts are given as usual by
This expression for 
should only be used close to the threshold,
since the perturbative 
amplitude 
violates unitarity for
energies 
above 430-440 MeV.
The predictions of the
G
PT for 
is shown
in Fig. 6, for r=10, and compared to the data of
Rosselet et al. [9]. The predictions of the standard
PT are also plotted. In the latter case, we use the values (25) of the low
energy constants 
, and take for 
and 
the
central values in Eq. (24), whereas the leading order values of 
and 
are both equal to 1. At present, both results are compatible
with the data, but it is clear that a reduction of the experimental error bars
by a factor of two could be enough to make the
distinction between the alternatives 
and 
.
Notice also that the uncertainties materialized by the dotted lines in Fig. 6
include the variations of the values of 
between 
and
. The solid lines, which represent the variations induced by
changing the subtraction scale 
in 
and in 
between 
and 
, correspond to 
. For
, the solid lines would lie at the center of the dotted
band.
Finally, we have gathered, in Table 1, the central values of the threshold
parameters of Eqs. (4) and (5), for r=10. The value r=10 taken in the
table below and in Fig. 6 showing the phase shifts, is suggested both by the
analysis of the deviation from the Goldberger-Treiman relation [11], and
by a recent analysis of the process 
in G
PT
[4].
The main differences between the standard predictions and the generalized case
for r=10 arise in 
(30%), in 
(50%) and in 
(for the
lowest value of 
).
In all the figures presented here, the error bands
delimited by the dotted lines are in principle reducible, if our knowledge of
the two Zweig rule violating parameters 
and 
were to
improve. A simultaneous determination of r and of 
is in principle
possible from very accurate data on both 
and 
scattering. As
for the value of 
, it might be obtained from more precise data
on 
decays and on 
.
The
smaller error bands, delimited by the solid curves, reflect our lack of
information concerning the low energy constants of 
, 
and 
, and a significant improvement seems
unlikely so far. Living with only these last uncertainties still offers a good
possibility to desentangle the low 
and large 
alternatives, provided
the experimental data become accurate enough. On the theoretical side, it is in
principle possible to study the effect of higher orders on the values
of the parameters
, 
and 
, i=1, 2, by comparing a fit to
the data obtained from
the parametrization of the amplitude given in Eq. [1], with a similar
fit done by using the general parametrization of 
up to and
including the order two loop given in Ref. [1].
true cm
Table 1: Results for the threshold parameters for r=10 and for two
values of 
, as compared to the standard predictions. The values shown
correspond to the central values 
, 
, 
(for 
) or 
(for 
), and 
.
Figure 1: The bands of allowed values of 
(a) and of 
(b)
as functions of
the quark mass ratio 
. The dotted curves correspond to
variations of 
and of 
in the ranges set by Eqs.
(22) and (23), respectively, and with 
and
estimated by the respective variations of
and of 
for 
between 547.5 MeV
and 770 MeV. The solid curves show the allowed values of 
and of
when only the latter uncertainties are taken into account, having put
and 
to zero.
Figure 2: The I=0 S-wave scattering lenth 
as 
varies between 8 and
30. The band delimited by the solid lines and the band delimited by the dotted
lines arise from the corresponding uncertainties in 
and in 
shown in Fig. 1.
Figure 3: The I=0 S-wave slope parameter 
as 
varies between 8 and
30. The band delimited by the solid lines and the band delimited by the dotted
lines arise from the corresponding uncertainties in 
and in 
shown in Fig. 1.
Figure 4: The I=1 P-wave scattering lenth 10
as 
varies between 8 and
30. The band delimited by the solid lines and the band delimited by the dotted
lines arise from the corresponding uncertainties in 
and in 
shown in Fig. 1.
Figure 5: The difference 
of S-wave scattering lengths as a
function of 
.
Figure 6: Plot of the phase shifts 
as a
function of energy. The dashed line corresponds to the predictions of the
standard case. The solid and dotted lines give the allowed values for r=10 in
the generalized case (see text). The data points are from Rosselet et al.
[8].
