arXiv:0704.1201v3  [hep-ph]  1 Jun 2007
FTUAM 06-07
April 9, 2007
arXiv: 0704.1201 [hep-ph]
Evaluation of the Axial Vector Commutator Sum Rule
for Pion-Pion Scattering
Stephen L. Adler
Institute for Advanced Study
Einstein Drive
Princeton, NJ 08540, USA
F. J. Ynduráin
Departamento de F́ısica Teórica, C-XI
Universidad Autónoma de Madrid,
Canto Blanco,
E-28049, Madrid, Spain
Abstract
We consider the sum rule proposed by one of us (SLA), obtained by
taking the expectation value of an axial vector commutator in a state with one
pion. The sum rule relates the pion decay constant to integrals of pion-pion cross
sections, with one pion off the mass shell. We remark that recent data on pion-
pion scattering allow a precise evaluation of the sum rule. We also discuss the
related Adler–Weisberger sum rule (obtained by taking the expectation value of
the same commutator in a state with one nucleon), especially in connection with
the problem of extrapolation of the pion momentum off its mass shell. We find,
with current data, that both the pion-pion and pion-nucleon sum rules are satisfied
to better than six percent, and we give detailed estimates of the experimental and
extrapolation errors in the closure discrepancies.
http://arxiv.org/abs/0704.1201v3
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-evaluation of the axial vector commutator sum rule for pion-pion scattering-
1. Introduction.
In refs. 1, 2 a sum rule was obtained by taking matrix elements of the commutator,
[χ+(t), χ−(t)] = 2I3, (1.1)
between one-nucleon states. Here χ± are the chiral charges,
χ±(x0) =
d3xA0
I3 is the third component of isospin and the A
are axial currents. In terms of quarks,
+(x) = ū(x)γ
µγ5d(x), A
(x) = d̄(x)γµγ5u(x).
One may note that, in QCD, the commutation relation (1.1) is an exact theorem, that follows from
the global symmetries of the QCD Lagrangian; in fact, (1.1) may be obtained by integrating the local axial
vector commutation relation
δ(x0 − y0)[A0+(x), Aν−(y)] = 2δ4(x− y)V ν3 (x)
with V ν3 (x) =
(ūγνu − d̄γνd) the third component of the vector (isospin) current. This commutation
relation is exact in QCD, irrespective of the value of the u, d quark masses.
The sum rule obtained in refs. 1, 2 relates the axial weak charge, gA, to integrals over pion-nucleon
cross sections, with the pion off-shell (and of zero mass),
g2rKNNπ(0)
(MN+Mπ)2
s−M2N
σ+0p(s)− σ
0p(s)
. (1.2)
Here, σ±0p is the total cross section for scattering of a zero mass π
± on a proton, this latter on its mass shell
(with averages over the spin of the nucleon implicit).
Later, in ref. 3, it was noted that one can also take matrix elements of (1.1) between one pion states.
In this way one obtains the relation
g2rKNNπ(0)
s−M2π
σ−0π(s)− σ
0π(s)
, (1.3)
where σ±0π are the respective total cross sections for scattering of a zero mass π
± on an on-shell π−. Both in
(1.2) and (1.3) the exact Goldberger–Treiman relation[4] (see Sect. 2 below for the meaning of this)
grKNNπ(0)Fπ
was used to relate the χ± to the pion field. In these formulas, Mπ = 139.57 MeV is the (charged) pion mass,
MN the nucleon mass, gA is the weak axial coupling, gr the on-shell strong NNπ coupling, and KNNπ(0)
the nucleon-nucleon-pion vertex, for a zero mass pion, normalized to KNNπ(M
π) = 1. Fπ is the pion decay
constant.
One can rewrite (1.2), (1.3) in a form that is more convenient by expressing the constant prefactor
in terms of Fπ, corresponding to writing the PCAC definition of the pion field in terms of Fπ,
∂ ·A± =
πφ± (1.4)
(φ± are the fields for π
±) and also expressing the sum rule in terms of scattering amplitudes, rather than
cross sections. In this case we find that (1.2), (1.3) are replaced by the relations
g2A = 1 + 8πF
(MN+Mπ)2
(s−M2N)2
Im Fπ+,p(k
2 = 0; s, t = 0)− Im Fπ−,p(k2 = 0; s, t = 0)
(1.5)
1 = 4πF 2π
(s−M2π)2
Im Fπ+,π−(k
2 = 0; s, t = 0)− Im Fπ−,π−(k2 = 0; s, t = 0)
. (1.6)
– 1 –
-s. l. adler and f. j. ynduráin-
Here Fπ±,p(k
2, s, t = 0) stands for the forward pion-proton scattering amplitude,
π±(k) +N(p) → π±(k) +N(p)
with the nucleon N(p) on its mass shell, p2 = M2N , while the momentum of the pion is off-shell, k
2 = 0 in
(1.5). Likewise, Fπ±,π−(k
2 = 0; s, t = 0) is the forward scattering amplitude for a π±(k) with momentum
k2 = 0 on a π−(p) on its mass shell, p2 = M2π :
π±(k) + π−(p) → π±(k) + π−(p).
In the form (1.6), the pionic sum rule is seen as a remarkable relation that allows one to calculate
the pion decay constant in terms of (off-shell) pion-pion scattering amplitudes. In ref. 3, this sum rule for
the pion case could not be fully evaluated. In 1965 data for ππ scattering in the S0 wave were so scanty
that it was only possible to note that fulfillment of the sum rule requires a strong ππ interaction in this S0
wave. The situation has improved dramatically at present; precise and reliable experimental data on the P
and S0 waves for ππ scattering, and also on high energy scattering, have appeared in the last years, which
will allow us to present an evaluation of (1.3), (1.6) to a few percent accuracy.
The nucleon sum rule, known as the Adler–Weisberger sum rule, (1.2), (1.5) was already evaluated
in refs. 1, 2, 3; however, we will present here a new calculation: the precision of the pionic sum rule is
such that the question of the extrapolation to the off-shell pion in (1.3), (1.6) becomes important, and the
corresponding discussion is illuminated by considering also what happens in the nucleon case.
In the present paper we will only consider the sum rules (1.5), (1.6); but, by taking expectation
values of (1.1) between kaon states one can get a sum rule identical to (1.5), replacing the g2A in the left
hand side by zero, the nucleon mass by the kaon mass, and pion-nucleon by pion-kaon amplitudes: e.g., for
a K+ state,
1 = 8πF 2π
(MK+Mπ)2
(s−M2K)2
Im Fπ−,K+(k
π = 0; s, t = 0)− Im Fπ+,K+(k2π = 0; s, t = 0)
. (1.7)
It is also possible to find a complementary relation to (1.7), but extrapolating in the mass of the
kaon. Consider the currents A
= ūγµγ5s, A
= s̄γµγ5u; A
= d̄γµγ5s, A
= s̄γµγ5d, and the
corresponding chiral charges, χK+ , χK− ; χK0 , χK̄0 . We have
[χK+ , χK− ]− [χK0 , χK̄0 ] = 2I3.
Taking expectation values on a π− state we now find the sum rule
1 = 4πF 2K
(MK+Mπ)2
(s−M2π)2
Im FK+,π−(k
K = 0; s, t = 0) + FK̄0,π−(k
K = 0; s, t = 0)
− Im FK−,π−(k2K = 0; s, t = 0)− Im FK0,π−(k2K = 0; s, t = 0)
=8πF 2K
(MK+Mπ)2
(s−M2π)2
Im FK+,π−(k
K = 0; s, t = 0)− Im FK+,π+(k2K = 0; s, t = 0)
(1.8)
the last relation using isospin and charge conjugation invariance. We here leave the pion on its mass shell, and
extrapolate in the kaon mass. Although there are some studies on pion-kaon scattering, variously using chiral
perturbation theory, dispersive techniques and phenomenological information (see ref. 5 at low energies, and
at high energies, the Regge calculations of ref. 10), the experimental information on πK scattering amplitudes
is much less precise than that for ππ scattering. However, having two different masses to extrapolate (Mπ
and MK) would perhaps give some insight on the matter of extrapolation: the only differences between (1.7)
and (1.8) are FK = 113.00 ± 1.06 MeV instead of Fπ, the different masses that are sent to zero, and the
masses that appear in the denominators in the integrals.
Similarly, as discussed by Weisberger, ref. 5, by taking appropriate chiral charges in SU(3), and
taking expectation values on nucleons one obtains relations involving kaon-nucleon scattering. The pion-
pion, pion-kaon or pion-nucleon sum rules require only an extrapolation in the pion mass to yield an on-shell
relation, while the kaon-nucleon [or kaon-pion, Eq. (1.8)] sum rule requires extrapolation in the much larger
kaon mass to give a physical relation, which is therefore expected to be less precise than the ones studied here,
but could also give information on the question of extrapolation on the meson masses. Nevertheless, and
as already stated we will, in the present paper, concern ourselves only with the pion-pion and pion-nucleon
– 2 –
-evaluation of the axial vector commutator sum rule for pion-pion scattering-
sum rules, for which good experimental data exist, and which require extrapolation only to M2π ; leaving the
rest for future work.
2. The Goldberger–Treiman relation
Although the Goldberger–Treiman relation[4] is not the object of this paper, we say a few words about it
as we will use some information connected with it. We first have what may be called the exact or off-shell
Goldberger–Treiman relation,
KNNπ(0)grFπ
= 1. (2.1)
This involves the pion-nucleon coupling, with the pion off its mass shell (with momentum k2 = 0). One can
define the quantity (the G.T. discrepancy)
∆G.T. ≡ 1−
, (2.2)
which measures the validity of the approximationKNNπ(0) ≃ KNNπ(M2π) = 1 that is used to get the on-shell
Goldberger–Treiman relation Fπ = MNgA/gr.
With the present values gA = 1.2695 ± 0.0029, Fπ = 92.42 ± 0.26 (both from the Particle Data
Tables[6]) and with MN = 938.9 (average n-p mass) and taking the on-shell pion nucleon coupling constant
gr = 13.2± 0.2, the relation (2.2) equals
∆G.T. = 0.023± 0.016, (2.3)
so the effect of approximating KNNπ(0) by unity appears to be small. This corrects most of the mismatch
studied by Pagels and Zepeda,[8] which turns out to be largely due to an underestimated gA (a possibility
that they actually considered); the remainder in (2.3) can easily be attributed, as was done in ref. 8, to the
contribution of the π(1300) resonance to KNNπ(0), expected to be of O[M
π(1300)
] ∼ 1%.
One can look at the off-shell Goldberger–Treiman relation in a different way, as providing the value
of the quantity KNNπ(0): (2.1) tells us that
KNNπ(0) =
= 0.977± 0.015. (2.4)
In fact, from the careful analysis of Pagels and Zepeda, it follows that most of the deviation of KNNπ(0)
from unity is due to the fact that, since the pion is off-shell, and the corresponding Green’s function is not
amputated, the NNπ vertex must contain a factor M2πΠ(0), where Π(k
2) is the pion propagator normalized
to 1 = (M2π − k2)Π(k2)|k2=M2π . Thus, one expects
KNNπ(0) ≃ M2πΠ(0), (2.5)
which will play a role in the extrapolation discussion for the sum rules below. In fact, the same factor
M2πΠ(0) will appear in the sum rules because, both in the pion-pion and pion-nucleon cases, the propagator
corresponding to the off-shell pion line is not amputated. Use of ∆G.T. to estimate off-shell extrapolation
corrections has also been discussed by Dominguez.[8]
3. Calculation of the sum rule on pions
3.1. The sum rules
We present here a sketch of the derivation of the sum rules, for ease of reference; more details may be found
in refs. 1, 2, 3. We will treat in detail the pionic case derived in ref. 3, and indicate the modifications
necessary for the nucleon case.
– 3 –
-s. l. adler and f. j. ynduráin-
We first take the expectation value of the commutator (1.1) between physical π− states, and intro-
duce a sum over a complete set of states: we find
〈π−(p′)|2I3|π−(p)〉 = − 2× 2p0δ(p− p′)
d(q2)
〈π−(p′)|χ+(t)|q; INT〉〈q; INT|χ−(t)|π−(p)〉 − (+ ↔ −),
(3.1)
where by |q; INT〉 we denote a physical intermediate state with total momentum q and internal degrees of
freedom “INT”. Next, using the PCAC definition (1.4), one can relate
〈q; INT|χ±(x0)|π−(p)〉 =
p0 − q0
π〈q; INT|
d3xφ±(x)|π−(p)〉
with φ±(x) the field operator for π
±. Working in the infinite momentum frame (p → ∞) gives the sum rule
1 = 2πF 2π
(s−M2π)2
|F (π+(k2 = 0), π−(p) → q; INT)|2 − |F (π−(k2 = 0), π−(p) → q; INT)|2
(3.2a)
here s = q2, and
F (π±(k2 = 0), π−(p) → q; INT) = (2π)5/2〈q; INT|M2πφ±(0)|π−(p)〉 (3.2b)
is the amplitude for a pseudoscalar current, with virtual four momentum k, k2 = 0, to scatter off a physical
π−, p2 = M2π , into the physical intermediate state |q; INT〉. Using extended unitarity, this may be written
as the sum rule (1.6), which we repeat here in the form of a discrepancy, ∆π = 0, with
∆π ≡ 4πF 2π
(s−M2π)2
Im Fπ+,π−(k
2 = 0; s, t = 0)− Im Fπ−,π−(k2 = 0; s, t = 0)
− 1. (3.3)
In these formulas, the Fπ±,π−(k
2 = 0; s, t = 0) are the forward scattering amplitudes for an off-shell
pion with zero mass. For an on-shell pion, the corresponding scattering amplitude is obtained by replacing,
in (3.2b),
F (π±(k2 = 0), π−(p) → q; INT) = (2π)5/2〈q; INT|M2πφ±(0)|π−(p)〉
= (2π)5/2〈q; INT|(M2π + )φ±(0)|π−(p)〉
→ (2π)5/2〈q; INT|(M2π + )φ±(0)|π−(p)〉
k2=M2
(3.4)
and using unitarity to perform the sum over intermediate states.
In QCD one expects the mass scale for the internal dynamics to be given by a parameter µ0 of the
order of the parameter Λ ∼ 0.4 GeV or the rho resonance mass, Mρ; thus, to an error M2π/µ20 ∼ 10% or
smaller, we can relate (3.3) to physical quantities by approximating the off-shell scattering amplitudes by
the physical scattering amplitudes as in (3.4).
The sum rule (1.2), (1.5) on nucleons is derived in a similar manner. The only differences are the
replacement of the pion mass by the nucleon mass in the denominator corresponding to (3.3), the different
isospin of the proton (that results in a factor −1/2 with respect to that for the π−) and that, due to the
existence of the neutron intermediate state, we find the extra term proportional to g2A in (1.5) because the
proton-neutron matrix element of the divergence 〈p|(∂ · A|n〉 is proportional to gA. The pion-kaon and
kaon-pion sum rules are also similar to the pion-pion one.
3.2. The pion sum rule in the on-shell approximation
In the form (3.3), the sum rule is an exact theorem, following from the commutation relation (1.1) and
the definition (1.4); but, of course, direct comparison with experiment is precluded by the fact that the
amplitudes that appear in (3.3) involve a pion off its mass shell. As stated in the previous subsection, a first
approximation is obtained by neglecting the fact that (3.3) is defined for off-shell pseudoscalar currents, i.e.,
working in the approximation (3.4). The sum rule is then,
– 4 –
-evaluation of the axial vector commutator sum rule for pion-pion scattering-
1 ≃ 4πF 2π
(s−M2π)2
{ImFπ−(s)− Im Fπ+(s)} (3.5)
with F±(s) physical, forward π
±π− scattering amplitudes. One can write these scattering amplitudes in
terms of amplitudes with well-defined isospin in the t or s channels,
Fπ−(s)− Fπ+(s) = F (It=1) = 13F
(Is=0) + 1
F (Is=1) − 5
F (Is=0); (3.6)
the F are normalized so that, for pions on their mass shell, and in the elastic region, one has
F (Is) = 2
2s1/2
(2l + 1) sin δ
l . (3.7)
The factor 2 in front of the right hand side is due to identity of the particles and k = 1
s− 4M2π is the
center of mass momentum, for physical pions.
In this case, we can use the precise determinations of the pion-pion scattering amplitudes, obtained
fitting experimental data, that have been found recently[9,10] thanks to the availability of very precise data:
on the low energy S0 wave from kaon decays and, for the P wave, from determinations of the pion form
factor.[11] One finds, for the contributions of the various waves to the right hand side of (3.5) for energy
below 1420 MeV,[9]
S0; s1/2 ≤ 932 MeV : 0.408± 0.013
S0; 932 ≤ s1/2 ≤ 1420 MeV : 0.043
D0 : 0.097± 0.003
P : 0.403± 0.003
F : 0.0016
S2 : −0.090± 0.005
D2 : −0.0023;
(3.8a)
errors are only given for the more significant pieces. The results are similar to those already obtained in
ref. 3 (although, of course, now much more precise) for the contributions of P, D0, S2 waves. What is new
is the contribution of the S0 wave, which turns out to be the most important of all, thus confirming the
prediction in ref. 3 that an important S0 wave contribution is needed to saturate the sum rule.
For the (Regge) contribution above 1420 MeV we use the Regge formula
ImF (1)(s) = (1.22± 0.14)(s/1 GeV2)0.42
and then find[10]
s1/2 > 1420 MeV : 0.167± 0.017. (3.8b)
Altogether, the right hand side of (3.5) now reads
1.027± 0.022, i.e., ∆π = 0.027± 0.022. (3.9)
The error is due to the experimental errors in the pion-pion scattering amplitudes in (3.8). Therefore, we
only have a discrepancy of (2.7± 2.2)% in the fulfillment of the sum rule in this approximation.
A few words may be said on the smallness of the error in (3.9). This is due to the fact that, as stated,
recent experimental results have allowed us to get very precise fits to data at low energy. Moreover, the larger
contributions to the final result come from independent sources, so one can add their errors quadratically.
And, finally, all the large contributions (S0, P, D0 waves and Regge region) are positive: only the relatively
small S2 and D2 contributions produce cancellations. As we will see, the situation is less favourable for the
sum rule on nucleons, where large cancellations take place.
– 5 –
-s. l. adler and f. j. ynduráin-
3.3. Extrapolation
To improve the evaluation using (3.5) one can calculate by taking the recipe of ref. 3 for extrapolation, which
takes into account threshold kinematic effects by replacing (3.7), in the elastic region,1 by
4s1/2
(2l + 1)
sin δ
l , (3.10)
where k(0) = (s −M2π)/2s1/2 is the center of mass momentum for a pion of zero mass incident on a target
pion that is on mass shell. This does not take into account the effects of the extrapolation at high energies,
s1/2 > 1.42 GeV, likely below the 1% level, which we neglect. We have verified that this recipe works in a
model calculation in which the interactions are generated by effective Lagrangians
gρ(~φ×
∂ µ~φ)ρµ, gf ~φ( gµν − ∂µ∂ν)~φfµν2 , . . .
coupling pions to various resonances [ρ, f2(1275), . . . ]. In these models, the off shell correction is valid not
only at threshold but throughout the resonance region, which justifies our using it here in the elastic region.
The recipe in (3.10) then gives a discrepancy
∆π = 0.069± 0.023. (3.11)
This deteriorates the sum rule, which seems to imply that the dynamical effects of the extrapolation are not
negligible. In fact, this could be expected; the replacement (3.10) does not affect the S waves, in particular
the S0 wave, which is the one that contributes most to the sum rule.
A possible way to estimate at least part of the dynamical correction is to assume it to be universal,
and take it from the Goldberger–Treiman relation: thus multiplying the right hand side of (3.11) by the
factor K2NNπ(0) = 0.955±0.03, as was done in ref. 3; see also ref. 8. A motivation for this was already given
when we discussed the Goldberger–Treiman relation (end of Sect. 2); the motivation for this in ref. 3 was
the observation that in a field theory of pions and nucleons, a zero mass pion must couple to a physical pion
through a virtual nucleon loop, and so the factor KNNπ(0) should be present (of course, both motivations
are not exclusive, but complementary). In terms of QCD, an analogous observation is that an off-shell pion
couples to both nucleons and pions through a coupling to a single quark line, suggesting that a universal
off-shell factor may be present. It would be of interest to pursue this idea further within a QCD framework;
see also the discussion in Sect. 5 below.
Including this factor K2NNπ(0) improves the fulfillment of the sum rule to
∆π = 0.021± 0.023, (3.12)
an agreement as good as one can wish. However, as we will see in the case of the nucleon sum rule,
following the same procedure of including a K2NNπ(0) factor depreciates, rather than improves, agreement
with experiment. Hence a more conservative procedure is to take the difference between (3.11) and (3.12) as
a measure of the uncertainty in the extrapolation procedure and thus write for the discrepancy, Eq. (3.3b),
∆π = 0.021± 0.023 (Exp.)± 0.048 (Extr.), (3.13)
showing explicitly the error arising from experimental errors in the pion-pion amplitudes and the estimated
error of the extrapolation procedure. This will be our final result for the sum rule on pions.
3.4. Connection with chiral perturbation theory
Consider the forward dispersion relation for the (physical) amplitude F (It=1)(s),
F (It=1)(s) =
2s− 4M2π
ImF (It=1)(s′)
(s′ − s)(s′ + s− 4M2π)
1 Actually, we make the corresponding replacement up to the Regge region, s1/2 ≃ 1.42 GeV.
– 6 –
-evaluation of the axial vector commutator sum rule for pion-pion scattering-
Evaluating it at threshold, and writing F (It=1)(4M2π) in terms of scattering lengths, we find the so-called
(first) Olsson sum rule[12] [which is thus an exact consequence of the dispersion relation, independent of the
current algebra commutator (1.1)]:
0 − 5a
0 = 3Mπ
ImF (It=1)(s)
s(s− 4M2π)
. (3.14)
One can evaluate the scattering lengths to lowest order (l.o.) in chiral perturbation theory, i.e., to lowest
order in M2π (actually, this is strictly equivalent to the old Weinberg
[13] evaluation) finding
0 − 5a
4πF 2π
and, substituting in (3.14), the l.o. relation
1 = 4πF 2π
ImF (It=1)(s)
s(s− 4M2π)
. (3.15)
To l.o. (which implies M2π ≈ 0) Eq. (3.15) is equivalent to (1.6) since
ImF (It=1)(s)
s(s− 4M2π)
ImF (It=1)(s)
(s−M2π)2
ImF (It=1)(k2 = 0; s, t = 0)
(s−M2π)2
The same conclusion is reached if we use the Froissart–Gribov representation of the P wave scattering length.
Of course, (3.15) differs in status from (1.6) as the latter is valid exactly, whereas the l.o. expression
0 − 5a
= 3Mπ/4πF
π is known to have large loop corrections.
Another connection with results based on chiral perturbation theory is obtained by remarking that,
using chiral methods to two loops, plus analyticity (in the form of Roy equations) and extra experimental
information, has led Colangelo, Gasser and Leutwyler[15] to propose parametrizations of the S0, S2 and P
waves at low energy consistent with these requirements. Substituting them into Eq. (3.5) gives the result
∆π = 0.046, to be compared with what we found using experimental data in Eq. (3.9), 0.027±0.022. Likewise,
implementing off-shell corrections as in (3.10) gives ∆π = 0.086 [to be now compared with 0.069 ± 0.023
from Eq. (3.11)] and, finally, applying the correction deduced from the off-shell Goldberger–Treiman relation
under the universality assumption gives ∆π = 0.033, a number which is also similar to what was found using
experimental ππ data in Eq. (3.12), viz., 0.021± 0.023.2
We emphasize that it is not our intention in this paper to give a detailed review of either chiral
perturbation theory or the Olsson sum rule, which are substantial topics in their own right. We are only
interested in elucidating the connection between the Olsson relation and the pion-pion sum rule (1.6), which,
as far as we know, can only be established to leading order in chiral perturbation theory. Tests of the Olsson
relation by itself have been made extensively in the literature; see, for example, refs. 14, 15 for evaluations
using two-loop chiral perturbation theory, and the papers of Kamiński, Peláez and Ynduráin and of Peláez
and Ynduráin in ref. 9 for calculations using experimental data prior to 2006. The Olsson relation has also
been verified including more recent data (those of the NA48/2 collaboration, ref. 11) as in our evaluation
of the pion-pion sum rule here, by Kamiński, Peláez, and Ynduráin (unpublished work in progress).
4. The Adler–Weisberger sum rule
4.1. An approximate calculation
The Adler–Weisberger sum rule (1.2), (1.5), has been evaluated in a number of papers (besides the original
ones); a recent article is ref. 17. If we approximate the amplitudes in (1.5) by the scattering amplitudes for
2 The numbers 0.046, 0.086 and 0.033 were kindly communicated to us by H. Leutwyler[16] as an independent check
on the evaluation of Sect. 3.2, using different methods.
– 7 –
-s. l. adler and f. j. ynduráin-
a physical pion, k2 = M2π , then we get the sum rule
g2A ≃ 1 + 8πf2π
(MN+Mπ)2
(s−M2N)2
ImFπ+(s)− Im Fπ−(s)
(4.1)
with Fπ±(s) the physical, forward π
±p scattering amplitudes, normalized so that the pion-proton cross
sections are
σπ±p(s) =
λ1/2(s,M2π ,M
ImFπ±(s),
with the function λ(a, b, c) as defined below. In terms of s-channel isospin amplitudes, we have
Fπ+(s)− Fπ−(s) = 23
F (Is=3/2) − F (Is=1/2)
where the partial wave expansion in the elastic region is
F (I)(s) =
2s1/2
(l + 1) sin δ
l+ + l sin δ
; (4.2)
the phase shifts δ
l± correspond to isospin I, orbital angular momentum l, and total angular momentum
j = l± 1
. The center of mass momentum is
k = 1
λ(s,M2π ,M
, λ(a, b, c) = [a2 − (b + c)2][a2 − (b− c)2].
The high energy part, s1/2 > 2374 MeV of the integral in (4.1) is easily evaluated with the fit to
πN cross sections in ref. 10: Regge formulas are good approximations for kinetic energies above 1 GeV. We
will use the Regge formula
ImFπ+(s)− Im Fπ−(s) = (0.42± 0.04)(s/1 GeV2)0.42,
and integrate this from s1/2 = 2374 MeV to infinity. This would give a result of −0.091. Alternatively, we
can use numerical values of the cross sections given in ref. 18 from s1/2 = 2374 MeV to s1/2 = 3004 MeV,
and use the Regge formula above this latter energy. This would give −0.087. We consider this last number
to be the more reliable one and then write, taking errors into account,
8πf2π
(2374 MeV)2
(s−M2N )2
ImFπ+(s)− Im Fπ−(s)
= −0.087± 0.006. (4.3a)
On the other hand, numerical evaluation of the low energy piece, using the numerical cross sections
collected in ref. 18 gives
8πf2π
∫ (2374 MeV)2
(MN+Mπ)2
(s−M2N )2
ImFπ+(s)− Im Fπ−(s)
= 0.460± 0.024. (4.3b)
This large error is due to the fact that the number in the right hand side is the difference between two large
numbers: specifically,
0.46 = 1.71 [from π+]− 1.25 [from π−]. (4.4)
Substituting the value of gA the sum rule (4.1) reads
1.612± 0.006 = 1 + 0.373± 0.025. (4.5)
We may define a discrepancy ∆A.W. as the difference between g
A and the right hand side of Eq. (1.5), and
express the result in (4.5) as a largish mismatch,
∆A.W. = 0.239± 0.025.
In the present case, there are various substantial cancellations: as already stated, there are cancel-
lations between the π+p and π−p cross sections at low energy, but there is also a cancellation between the
– 8 –
-evaluation of the axial vector commutator sum rule for pion-pion scattering-
low energy region and higher energy (s1/2 > 1.390 GeV) contributions. For example, if we only integrated
up to and including the ∆(3, 3) resonance region, we would have obtained
8πf2π
∫ (1390 MeV)2
(MN+Mπ)2
(s−M2N )2
ImFπ+(s)− Im Fπ−(s)
= 0.70.
These cancellations amplify the errors in the sum rule, and indicate that the effects of the extrapolation to
a zero mass pion, which affect mostly low energies, are now very important, as already remarked in refs. 1
and 3.
4.2. Extrapolation
To perform the extrapolation, we repeat the method used for the case of the sum rule on pions, and replace
the expression (4.2) for the scattering amplitude by
0 (s) =
2s1/2
(l + 1) sin δ
l+ + l sin δ
(4.7)
with k(0) the momentum for an incident pion of zero mass,
k(0) = 1
λ(s, 0,M2N)
We integrate this in the elastic region, s1/2 <∼ 1.5 GeV, using the parametrizations of ref. 19. These
parametrizations have been obtained by fitting up to energies of, respectively, 1.3 GeV and 1.38 GeV;
however, we have verified that they continue to fit the experimental cross sections up to ∼ 150 MeV above
their nominal maximum. Between these energies and 1.9 GeV, one can use a resonance saturation model,
with the resonance parameters taken from the PDT;[6] plus a background, estimated as the tail of the lower
energy resonances. Above 1.9 GeV, there are not well determined values for the resonance parameters and,
moreover, a resonance model will cease to be valid as one is entering the Regge regime. Fortunately, one can
likely neglect the effects of the extrapolation above such energy, and so this will be done here.
The results one gets for the extrapolation correction are
0.225 : up to 1460 MeV; −0.065 : resonances, from 1460 MeV; 0.005 : background, from 1460 MeV.
(4.8)
Adding this and considering as an error estimate the variation of the result if we vary the matching point of
the parametrizations and the resonance model from 1460 to 1420 MeV or 1520 MeV, we get a correction of
0.165± 0.009, and hence the sum rule becomes
1.612± 0.006 = 1 + (0.373± 0.025) + (0.165± 0.009) = 1.538± 0.034. (4.9)
We have added the errors linearly, as they are clearly correlated. The results show reasonable fulfillment of
the Adler–Weisberger sum rule, ∆A.W. = 0.074± 0.034.
If we include a global correction, as we did in the pionic case, multiplying the r.h.s. of (4.9) by the
factor K2NNπ(0) = 0.955± 0.03, the sum rule now deteriorates to
1.612± 0.006 = 1 + [(0.373± 0.025) + (0.165± 0.009)]× (0.955± 0.030) = 1.514± 0.038. (4.10)
We can now write our final result, as we did for the pionic case, as (4.10), adding as an extra error the
difference between (4.10) and (4.9):
∆A.W. = 0.098± 0.006 [g2A]± 0.031 [Exp.]± 0.035 [Extr.] (4.11)
To compare with the results obtained in refs. 1 and 3, we note that these papers did not multiply
through by a factor of g2A = 1.612, and included the K
NNπ(0) extrapolation factor. Thus, the relevant
number to use is [cf. (4.10)] (0.514± 0.038)/1.612 = 0.319± 0.024. The comparable number in refs. 1 and
3 is (4M2N/g
r)(R1 + R2 + R3) = 0.254 + 0.155− 0.061 = 0.348, with a roughly estimated error of ±0.025.
Hence our current evaluation, and the 1965 evaluation of refs. 1 and 3, are in satisfactory agreement. This
should come as no surprise, since good pion-nucleon cross sections were already available in 1965. What
– 9 –
-s. l. adler and f. j. ynduráin-
has changed dramatically since then, and is the motivation for the present paper, is the status of data on
pion-pion scattering.
4.3. Connection with lowest order chiral perturbation theory
As we did for the pion case, the Adler–Weisberger relation can be connected with (lowest order) chiral
perturbation theory, by comparing e.g., (4.1) with the forward dispersion relation for the πp scattering
amplitude for exchange of isospin unity at threshold,3 evaluating the amplitude at threshold in terms of
the scattering lengths combination a3 − a1, and calculating the latter in terms of Fπ, as in the Tomozawa–
Weinberg articles.[21] Details of this may be found in ref. 17.
5. Comments
We have shown that with current data, the pion-pion sum rule, as well as the pion-nucleon one, is satisfied to
better than six percent. To improve on this precision, it will be necessary to have an improved understanding
of dynamical extrapolation corrections that account for the appearance of a zero mass, off-shell pion in the
sum rules. We make two remarks in this regard. The first is that extrapolation of the incident pion to
zero mass is not the same as taking the chiral limit of QCD, since the target pion or nucleon, and all
intermediate state particles, remain on mass shell. The second is that while for a generic pion interpolating
field the results of this extrapolation are not well-defined, the sum rules involve a very specific choice of pion
interpolating field: the divergence of the axial vector current, which is a well defined entity in QCD, as are the
on shell pion-pion and pion-nucleon scattering amplitudes. Thus, the question of estimating extrapolation
corrections, that are needed for a very accurate comparison of the sum rules with experiment, is a well-posed
one in QCD. Modern lattice methods may permit improvement on the estimates that have been used here
and in refs. 1 and 3.
In this respect, it is amusing to remark that, unlike the situation in 1965, the precision of the pionic
sum rule is now greater than that of the pion-nucleon one. This is very likely due to the fact that the latter
involves small differences of large numbers, so any small alteration is amplified here.
3 This is generally known as the Goldberger–Miyazawa–Oehme sum rule.[20]
– 10 –
-evaluation of the axial vector commutator sum rule for pion-pion scattering-
Acknowledgements
We wish to thank H. Leutwyler for supplying numbers that give an independent check of the pion-pion sum
rule evaluation, and also for correspondence that assisted one of us (SLA) in reconstructing the reasoning
behind the extrapolation method used for the pion-pion case in ref. 3. The work of SLA was supported
in part by the U. S. Department of Energy under Grant No. DE-FG02-90ER40542. FJY is grateful to
J. R. Peláez for interesting discussions in the preliminary stages of this paper, and to the Spanish DGI of
the MEC under contract FPA2003-04597 for financial support.
References
1 Adler, S. L., Phys. Rev. Letters 14, 1051 (1965).
2 Weisberger, W. I., Phys. Rev. Letters 14, 1047 (1965).
3 Adler, S. L., Phys. Rev. 140, B736 (1965) and (E) Phys. Rev. 149, 1294 (1966) and Phys. Rev. 175,
2224 (1968).
4 Goldberger, M. L., and Treiman, S. B., Phys. Rev. 109, 193 (1958).
5 References on pion-kaon scattering: M. Jamin, J., Oller, J. A., and Pich, A., Nucl. Phys. B587, 331
(2000); Gómez-Nicola, A., and Peláez, J. R., Phys. Rev. D65, 054009 (2002); Descotes-Genon, S., and
Moussallam, B., Eur. Phys. J. C33, 409 (2004); Zhou, Z. Y., and Zheng, H. Q., Nucl. Phys. A775,
212 (2006). Theoretical proposal of kaon-nucleon sum rules: Weisberger, W. I., Phys. Rev. 143, 1302
(1966).
6 Particle Data Tables: Yao, W.-M., et al., Journal of Physics G33, 1 (2006).
7 Saino, M. E., hep-ph/9912337 and, especially, Schindler, M. R., et al., nucl-th/0611083. The second
article reviews previous evaluations.
8 Pagels, H., and Zepeda, A., Phys. Rev. D5, 3262 (1972). For a more recent discussion of correc-
tions to the Goldberger–Treiman relation, and their relation to off-shell extrapolation corrections, see
Dominguez, C. A., Phys. Rev. D15, 1350 (1977) and Phys. Rev. D16, 2320 (1977).
9 S0 wave, low energy (s1/2 ≤ 932 MeV): Ynduráin, F. J., Garćıa Mart́ın, R., and Peláez, J. R., hep-
ph/0701025. P wave, low energy (s1/2 ≤ 932 MeV): de Trocóniz, J. F., and Ynduráin, F. J., Phys. Rev.,
D65, 093001, (2002), and Phys. Rev. D71, 073008 (2005). S2 wave, D2 and F wave, s1/2 ≤ 1420 MeV:
Peláez, J. R., and Ynduráin, F. J., Phys. Rev. D71, 074016 (2005). S0 and P waves, medium energy
(up to s1/2 ≤ 1420 MeV); D0 wave: Kamiński, R., Peláez, J. R., and Ynduráin, F. J., Phys.Rev. D74,
014001 (2006) and (E) D74, 079903 (2006).
10 High energy, s1/2 > 1420 MeV: Peláez, J. R., and Ynduráin, F. J., Phys. Rev. D69, 114001 (2004).
Actually, in the present calculation one uses the slightly improved Regge parametrization in the paper
of Kamiński, Peláez and Ynduráin in ref. 9; see also Cudell, J. R., et al., Phys. Letters B587, 78 (2004)
and Peláez, J. R., in Proc. Blois. Conf. on Elastic and Diffractive Scattering (hep-ph/0510005) for the
pion-nucleon case.
11 Kl4 decays: Rosselet, L., et al., Phys. Rev. D15, 574 (1977); Pislak, S., et al., Phys. Rev. Lett.,
87, 221801 (2001); NA48/2 ( CERN/SPS experiment); Bloch-Devaux, B., presented at QCD06 in
Montpellier (France), 3-7 July 2006 and Masetti, L., presented at ICHEP06 in Moscow (Russia), 26 July
to 2 August 2006. K → 2π decays: Aloisio, A., et al., Phys. Letters, B538, 21 (2002). Pion form factor
data: Novosibirsk, ρ region: L. M. Barkov et al., Nucl. Phys. B256, 365 (1985); R. R. Akhmetshin et
al., Phys. Letters B527, 161 (2002). τ decays: ALEPH: R. Barate et al., Z. Phys. C76, 15 (1997);
OPAL: K. Ackerstaff et al., Eur. Phys. J. C7, 571 (1999); CLEO: S. Anderson et al., Phys. Rev., D61,
112002 (2000).
12 Olsson, M. G., Phys. Rev.162, 1338 (1967).
13 Weinberg, S., Phys. Rev. Letters 17, 616 (1966).
14 Gasser, J., and Leutwyler, H., Ann. Phys. (N.Y.) 158, 14 (1984).
15 Colangelo, G., Gasser, J., and Leutwyler, H., Nucl. Phys. B603, 125, (2001). See also Ananthanarayan,
B., et al., Phys. Rep., 353, 207, (2001). The Roy equations were derived in Roy, S. M., Phys. Letters,
36B, 353, (1971).
– 11 –
http://arxiv.org/abs/hep-ph/9912337
http://arxiv.org/abs/nucl-th/0611083
http://arxiv.org/abs/hep-ph/0701025
http://arxiv.org/abs/hep-ph/0701025
http://arxiv.org/abs/hep-ph/0510005
-s. l. adler and f. j. ynduráin-
16 Leutwyler, H. (private communication).
17 Kondratyuk, S., et al., Nucl. Phys. A736, 339 (2004).
18 The COMPAS Group compilations can be traced from Hagiwara, K., et al., Phys. Rev. D66, 010001
(2002).
19 Rowe, G., Salomon, M., and Landau, R. H., Phys. Rev. C18, 584 (1978); Ebrahim, A. A., and
Peterson, R. J., Phys. Rev. C54, 2499 (1996).
20 Goldberger, M. L, Miyazawa, H., and Oehme, R., Phys. Rev. 99, 986 (1955). For a very recent
evaluation and further references, see Abaev, V. V., Metsä, P., and Sainio, M. E., arXiv:0704.3167
[hep-ph].
21 Tomozawa, Y., Nuovo Cimento 46A, 707 (1966); Weinberg, S., Phys. Rev. Lett. 19, 616 (1966).
– 12 –
http://arxiv.org/abs/0704.3167

  We consider the sum rule proposed by one of us (SLA), obtained by taking the
expectation value of an axial vector commutator in a state with one pion. The
sum rule relates the pion decay constant to integrals of pion-pion cross
sections, with one pion off the mass shell. We remark that recent data on
pion-pion scattering allow a precise evaluation of the sum rule. We also
discuss the related Adler--Weisberger sum rule (obtained by taking the
expectation value of the same commutator in a state with one nucleon),
especially in connection with the problem of extrapolation of the pion momentum
off its mass shell. We find, with current data, that both the pion-pion and
pion-nucleon sum rules are satisfied to better than six percent, and we give
detailed estimates of the experimental and extrapolation errors in the closure
discrepancies.

Introduction.
In refs. 1, 2 a sum rule was obtained by taking matrix elements of the commutator,
[χ+(t), χ−(t)] = 2I3, (1.1)
between one-nucleon states. Here χ± are the chiral charges,
χ±(x0) =
d3xA0
I3 is the third component of isospin and the A
are axial currents. In terms of quarks,
+(x) = ū(x)γ
µγ5d(x), A
(x) = d̄(x)γµγ5u(x).
One may note that, in QCD, the commutation relation (1.1) is an exact theorem, that follows from
the global symmetries of the QCD Lagrangian; in fact, (1.1) may be obtained by integrating the local axial
vector commutation relation
δ(x0 − y0)[A0+(x), Aν−(y)] = 2δ4(x− y)V ν3 (x)
with V ν3 (x) =
(ūγνu − d̄γνd) the third component of the vector (isospin) current. This commutation
relation is exact in QCD, irrespective of the value of the u, d quark masses.
The sum rule obtained in refs. 1, 2 relates the axial weak charge, gA, to integrals over pion-nucleon
cross sections, with the pion off-shell (and of zero mass),
g2rKNNπ(0)
(MN+Mπ)2
s−M2N
σ+0p(s)− σ
0p(s)
. (1.2)
Here, σ±0p is the total cross section for scattering of a zero mass π
± on a proton, this latter on its mass shell
(with averages over the spin of the nucleon implicit).
Later, in ref. 3, it was noted that one can also take matrix elements of (1.1) between one pion states.
In this way one obtains the relation
g2rKNNπ(0)
s−M2π
σ−0π(s)− σ
0π(s)
, (1.3)
where σ±0π are the respective total cross sections for scattering of a zero mass π
± on an on-shell π−. Both in
(1.2) and (1.3) the exact Goldberger–Treiman relation[4] (see Sect. 2 below for the meaning of this)
grKNNπ(0)Fπ
was used to relate the χ± to the pion field. In these formulas, Mπ = 139.57 MeV is the (charged) pion mass,
MN the nucleon mass, gA is the weak axial coupling, gr the on-shell strong NNπ coupling, and KNNπ(0)
the nucleon-nucleon-pion vertex, for a zero mass pion, normalized to KNNπ(M
π) = 1. Fπ is the pion decay
constant.
One can rewrite (1.2), (1.3) in a form that is more convenient by expressing the constant prefactor
in terms of Fπ, corresponding to writing the PCAC definition of the pion field in terms of Fπ,
∂ ·A± =
πφ± (1.4)
(φ± are the fields for π
±) and also expressing the sum rule in terms of scattering amplitudes, rather than
cross sections. In this case we find that (1.2), (1.3) are replaced by the relations
g2A = 1 + 8πF
(MN+Mπ)2
(s−M2N)2
Im Fπ+,p(k
2 = 0; s, t = 0)− Im Fπ−,p(k2 = 0; s, t = 0)
(1.5)
1 = 4πF 2π
(s−M2π)2
Im Fπ+,π−(k
2 = 0; s, t = 0)− Im Fπ−,π−(k2 = 0; s, t = 0)
. (1.6)
– 1 –
-s. l. adler and f. j. ynduráin-
Here Fπ±,p(k
2, s, t = 0) stands for the forward pion-proton scattering amplitude,
π±(k) +N(p) → π±(k) +N(p)
with the nucleon N(p) on its mass shell, p2 = M2N , while the momentum of the pion is off-shell, k
2 = 0 in
(1.5). Likewise, Fπ±,π−(k
2 = 0; s, t = 0) is the forward scattering amplitude for a π±(k) with momentum
k2 = 0 on a π−(p) on its mass shell, p2 = M2π :
π±(k) + π−(p) → π±(k) + π−(p).
In the form (1.6), the pionic sum rule is seen as a remarkable relation that allows one to calculate
the pion decay constant in terms of (off-shell) pion-pion scattering amplitudes. In ref. 3, this sum rule for
the pion case could not be fully evaluated. In 1965 data for ππ scattering in the S0 wave were so scanty
that it was only possible to note that fulfillment of the sum rule requires a strong ππ interaction in this S0
wave. The situation has improved dramatically at present; precise and reliable experimental data on the P
and S0 waves for ππ scattering, and also on high energy scattering, have appeared in the last years, which
will allow us to present an evaluation of (1.3), (1.6) to a few percent accuracy.
The nucleon sum rule, known as the Adler–Weisberger sum rule, (1.2), (1.5) was already evaluated
in refs. 1, 2, 3; however, we will present here a new calculation: the precision of the pionic sum rule is
such that the question of the extrapolation to the off-shell pion in (1.3), (1.6) becomes important, and the
corresponding discussion is illuminated by considering also what happens in the nucleon case.
In the present paper we will only consider the sum rules (1.5), (1.6); but, by taking expectation
values of (1.1) between kaon states one can get a sum rule identical to (1.5), replacing the g2A in the left
hand side by zero, the nucleon mass by the kaon mass, and pion-nucleon by pion-kaon amplitudes: e.g., for
a K+ state,
1 = 8πF 2π
(MK+Mπ)2
(s−M2K)2
Im Fπ−,K+(k
π = 0; s, t = 0)− Im Fπ+,K+(k2π = 0; s, t = 0)
. (1.7)
It is also possible to find a complementary relation to (1.7), but extrapolating in the mass of the
kaon. Consider the currents A
= ūγµγ5s, A
= s̄γµγ5u; A
= d̄γµγ5s, A
= s̄γµγ5d, and the
corresponding chiral charges, χK+ , χK− ; χK0 , χK̄0 . We have
[χK+ , χK− ]− [χK0 , χK̄0 ] = 2I3.
Taking expectation values on a π− state we now find the sum rule
1 = 4πF 2K
(MK+Mπ)2
(s−M2π)2
Im FK+,π−(k
K = 0; s, t = 0) + FK̄0,π−(k
K = 0; s, t = 0)
− Im FK−,π−(k2K = 0; s, t = 0)− Im FK0,π−(k2K = 0; s, t = 0)
=8πF 2K
(MK+Mπ)2
(s−M2π)2
Im FK+,π−(k
K = 0; s, t = 0)− Im FK+,π+(k2K = 0; s, t = 0)
(1.8)
the last relation using isospin and charge conjugation invariance. We here leave the pion on its mass shell, and
extrapolate in the kaon mass. Although there are some studies on pion-kaon scattering, variously using chiral
perturbation theory, dispersive techniques and phenomenological information (see ref. 5 at low energies, and
at high energies, the Regge calculations of ref. 10), the experimental information on πK scattering amplitudes
is much less precise than that for ππ scattering. However, having two different masses to extrapolate (Mπ
and MK) would perhaps give some insight on the matter of extrapolation: the only differences between (1.7)
and (1.8) are FK = 113.00 ± 1.06 MeV instead of Fπ, the different masses that are sent to zero, and the
masses that appear in the denominators in the integrals.
Similarly, as discussed by Weisberger, ref. 5, by taking appropriate chiral charges in SU(3), and
taking expectation values on nucleons one obtains relations involving kaon-nucleon scattering. The pion-
pion, pion-kaon or pion-nucleon sum rules require only an extrapolation in the pion mass to yield an on-shell
relation, while the kaon-nucleon [or kaon-pion, Eq. (1.8)] sum rule requires extrapolation in the much larger
kaon mass to give a physical relation, which is therefore expected to be less precise than the ones studied here,
but could also give information on the question of extrapolation on the meson masses. Nevertheless, and
as already stated we will, in the present paper, concern ourselves only with the pion-pion and pion-nucleon
– 2 –
-evaluation of the axial vector commutator sum rule for pion-pion scattering-
sum rules, for which good experimental data exist, and which require extrapolation only to M2π ; leaving the
rest for future work.
2. The Goldberger–Treiman relation
Although the Goldberger–Treiman relation[4] is not the object of this paper, we say a few words about it
as we will use some information connected with it. We first have what may be called the exact or off-shell
Goldberger–Treiman relation,
KNNπ(0)grFπ
= 1. (2.1)
This involves the pion-nucleon coupling, with the pion off its mass shell (with momentum k2 = 0). One can
define the quantity (the G.T. discrepancy)
∆G.T. ≡ 1−
, (2.2)
which measures the validity of the approximationKNNπ(0) ≃ KNNπ(M2π) = 1 that is used to get the on-shell
Goldberger–Treiman relation Fπ = MNgA/gr.
With the present values gA = 1.2695 ± 0.0029, Fπ = 92.42 ± 0.26 (both from the Particle Data
Tables[6]) and with MN = 938.9 (average n-p mass) and taking the on-shell pion nucleon coupling constant
gr = 13.2± 0.2, the relation (2.2) equals
∆G.T. = 0.023± 0.016, (2.3)
so the effect of approximating KNNπ(0) by unity appears to be small. This corrects most of the mismatch
studied by Pagels and Zepeda,[8] which turns out to be largely due to an underestimated gA (a possibility
that they actually considered); the remainder in (2.3) can easily be attributed, as was done in ref. 8, to the
contribution of the π(1300) resonance to KNNπ(0), expected to be of O[M
π(1300)
] ∼ 1%.
One can look at the off-shell Goldberger–Treiman relation in a different way, as providing the value
of the quantity KNNπ(0): (2.1) tells us that
KNNπ(0) =
= 0.977± 0.015. (2.4)
In fact, from the careful analysis of Pagels and Zepeda, it follows that most of the deviation of KNNπ(0)
from unity is due to the fact that, since the pion is off-shell, and the corresponding Green’s function is not
amputated, the NNπ vertex must contain a factor M2πΠ(0), where Π(k
2) is the pion propagator normalized
to 1 = (M2π − k2)Π(k2)|k2=M2π . Thus, one expects
KNNπ(0) ≃ M2πΠ(0), (2.5)
which will play a role in the extrapolation discussion for the sum rules below. In fact, the same factor
M2πΠ(0) will appear in the sum rules because, both in the pion-pion and pion-nucleon cases, the propagator
corresponding to the off-shell pion line is not amputated. Use of ∆G.T. to estimate off-shell extrapolation
corrections has also been discussed by Dominguez.[8]
3. Calculation of the sum rule on pions
3.1. The sum rules
We present here a sketch of the derivation of the sum rules, for ease of reference; more details may be found
in refs. 1, 2, 3. We will treat in detail the pionic case derived in ref. 3, and indicate the modifications
necessary for the nucleon case.
– 3 –
-s. l. adler and f. j. ynduráin-
We first take the expectation value of the commutator (1.1) between physical π− states, and intro-
duce a sum over a complete set of states: we find
〈π−(p′)|2I3|π−(p)〉 = − 2× 2p0δ(p− p′)
d(q2)
〈π−(p′)|χ+(t)|q; INT〉〈q; INT|χ−(t)|π−(p)〉 − (+ ↔ −),
(3.1)
where by |q; INT〉 we denote a physical intermediate state with total momentum q and internal degrees of
freedom “INT”. Next, using the PCAC definition (1.4), one can relate
〈q; INT|χ±(x0)|π−(p)〉 =
p0 − q0
π〈q; INT|
d3xφ±(x)|π−(p)〉
with φ±(x) the field operator for π
±. Working in the infinite momentum frame (p → ∞) gives the sum rule
1 = 2πF 2π
(s−M2π)2
|F (π+(k2 = 0), π−(p) → q; INT)|2 − |F (π−(k2 = 0), π−(p) → q; INT)|2
(3.2a)
here s = q2, and
F (π±(k2 = 0), π−(p) → q; INT) = (2π)5/2〈q; INT|M2πφ±(0)|π−(p)〉 (3.2b)
is the amplitude for a pseudoscalar current, with virtual four momentum k, k2 = 0, to scatter off a physical
π−, p2 = M2π , into the physical intermediate state |q; INT〉. Using extended unitarity, this may be written
as the sum rule (1.6), which we repeat here in the form of a discrepancy, ∆π = 0, with
∆π ≡ 4πF 2π
(s−M2π)2
Im Fπ+,π−(k
2 = 0; s, t = 0)− Im Fπ−,π−(k2 = 0; s, t = 0)
− 1. (3.3)
In these formulas, the Fπ±,π−(k
2 = 0; s, t = 0) are the forward scattering amplitudes for an off-shell
pion with zero mass. For an on-shell pion, the corresponding scattering amplitude is obtained by replacing,
in (3.2b),
F (π±(k2 = 0), π−(p) → q; INT) = (2π)5/2〈q; INT|M2πφ±(0)|π−(p)〉
= (2π)5/2〈q; INT|(M2π + )φ±(0)|π−(p)〉
→ (2π)5/2〈q; INT|(M2π + )φ±(0)|π−(p)〉
k2=M2
(3.4)
and using unitarity to perform the sum over intermediate states.
In QCD one expects the mass scale for the internal dynamics to be given by a parameter µ0 of the
order of the parameter Λ ∼ 0.4 GeV or the rho resonance mass, Mρ; thus, to an error M2π/µ20 ∼ 10% or
smaller, we can relate (3.3) to physical quantities by approximating the off-shell scattering amplitudes by
the physical scattering amplitudes as in (3.4).
The sum rule (1.2), (1.5) on nucleons is derived in a similar manner. The only differences are the
replacement of the pion mass by the nucleon mass in the denominator corresponding to (3.3), the different
isospin of the proton (that results in a factor −1/2 with respect to that for the π−) and that, due to the
existence of the neutron intermediate state, we find the extra term proportional to g2A in (1.5) because the
proton-neutron matrix element of the divergence 〈p|(∂ · A|n〉 is proportional to gA. The pion-kaon and
kaon-pion sum rules are also similar to the pion-pion one.
3.2. The pion sum rule in the on-shell approximation
In the form (3.3), the sum rule is an exact theorem, following from the commutation relation (1.1) and
the definition (1.4); but, of course, direct comparison with experiment is precluded by the fact that the
amplitudes that appear in (3.3) involve a pion off its mass shell. As stated in the previous subsection, a first
approximation is obtained by neglecting the fact that (3.3) is defined for off-shell pseudoscalar currents, i.e.,
working in the approximation (3.4). The sum rule is then,
– 4 –
-evaluation of the axial vector commutator sum rule for pion-pion scattering-
1 ≃ 4πF 2π
(s−M2π)2
{ImFπ−(s)− Im Fπ+(s)} (3.5)
with F±(s) physical, forward π
±π− scattering amplitudes. One can write these scattering amplitudes in
terms of amplitudes with well-defined isospin in the t or s channels,
Fπ−(s)− Fπ+(s) = F (It=1) = 13F
(Is=0) + 1
F (Is=1) − 5
F (Is=0); (3.6)
the F are normalized so that, for pions on their mass shell, and in the elastic region, one has
F (Is) = 2
2s1/2
(2l + 1) sin δ
l . (3.7)
The factor 2 in front of the right hand side is due to identity of the particles and k = 1
s− 4M2π is the
center of mass momentum, for physical pions.
In this case, we can use the precise determinations of the pion-pion scattering amplitudes, obtained
fitting experimental data, that have been found recently[9,10] thanks to the availability of very precise data:
on the low energy S0 wave from kaon decays and, for the P wave, from determinations of the pion form
factor.[11] One finds, for the contributions of the various waves to the right hand side of (3.5) for energy
below 1420 MeV,[9]
S0; s1/2 ≤ 932 MeV : 0.408± 0.013
S0; 932 ≤ s1/2 ≤ 1420 MeV : 0.043
D0 : 0.097± 0.003
P : 0.403± 0.003
F : 0.0016
S2 : −0.090± 0.005
D2 : −0.0023;
(3.8a)
errors are only given for the more significant pieces. The results are similar to those already obtained in
ref. 3 (although, of course, now much more precise) for the contributions of P, D0, S2 waves. What is new
is the contribution of the S0 wave, which turns out to be the most important of all, thus confirming the
prediction in ref. 3 that an important S0 wave contribution is needed to saturate the sum rule.
For the (Regge) contribution above 1420 MeV we use the Regge formula
ImF (1)(s) = (1.22± 0.14)(s/1 GeV2)0.42
and then find[10]
s1/2 > 1420 MeV : 0.167± 0.017. (3.8b)
Altogether, the right hand side of (3.5) now reads
1.027± 0.022, i.e., ∆π = 0.027± 0.022. (3.9)
The error is due to the experimental errors in the pion-pion scattering amplitudes in (3.8). Therefore, we
only have a discrepancy of (2.7± 2.2)% in the fulfillment of the sum rule in this approximation.
A few words may be said on the smallness of the error in (3.9). This is due to the fact that, as stated,
recent experimental results have allowed us to get very precise fits to data at low energy. Moreover, the larger
contributions to the final result come from independent sources, so one can add their errors quadratically.
And, finally, all the large contributions (S0, P, D0 waves and Regge region) are positive: only the relatively
small S2 and D2 contributions produce cancellations. As we will see, the situation is less favourable for the
sum rule on nucleons, where large cancellations take place.
– 5 –
-s. l. adler and f. j. ynduráin-
3.3. Extrapolation
To improve the evaluation using (3.5) one can calculate by taking the recipe of ref. 3 for extrapolation, which
takes into account threshold kinematic effects by replacing (3.7), in the elastic region,1 by
4s1/2
(2l + 1)
sin δ
l , (3.10)
where k(0) = (s −M2π)/2s1/2 is the center of mass momentum for a pion of zero mass incident on a target
pion that is on mass shell. This does not take into account the effects of the extrapolation at high energies,
s1/2 > 1.42 GeV, likely below the 1% level, which we neglect. We have verified that this recipe works in a
model calculation in which the interactions are generated by effective Lagrangians
gρ(~φ×
∂ µ~φ)ρµ, gf ~φ( gµν − ∂µ∂ν)~φfµν2 , . . .
coupling pions to various resonances [ρ, f2(1275), . . . ]. In these models, the off shell correction is valid not
only at threshold but throughout the resonance region, which justifies our using it here in the elastic region.
The recipe in (3.10) then gives a discrepancy
∆π = 0.069± 0.023. (3.11)
This deteriorates the sum rule, which seems to imply that the dynamical effects of the extrapolation are not
negligible. In fact, this could be expected; the replacement (3.10) does not affect the S waves, in particular
the S0 wave, which is the one that contributes most to the sum rule.
A possible way to estimate at least part of the dynamical correction is to assume it to be universal,
and take it from the Goldberger–Treiman relation: thus multiplying the right hand side of (3.11) by the
factor K2NNπ(0) = 0.955±0.03, as was done in ref. 3; see also ref. 8. A motivation for this was already given
when we discussed the Goldberger–Treiman relation (end of Sect. 2); the motivation for this in ref. 3 was
the observation that in a field theory of pions and nucleons, a zero mass pion must couple to a physical pion
through a virtual nucleon loop, and so the factor KNNπ(0) should be present (of course, both motivations
are not exclusive, but complementary). In terms of QCD, an analogous observation is that an off-shell pion
couples to both nucleons and pions through a coupling to a single quark line, suggesting that a universal
off-shell factor may be present. It would be of interest to pursue this idea further within a QCD framework;
see also the discussion in Sect. 5 below.
Including this factor K2NNπ(0) improves the fulfillment of the sum rule to
∆π = 0.021± 0.023, (3.12)
an agreement as good as one can wish. However, as we will see in the case of the nucleon sum rule,
following the same procedure of including a K2NNπ(0) factor depreciates, rather than improves, agreement
with experiment. Hence a more conservative procedure is to take the difference between (3.11) and (3.12) as
a measure of the uncertainty in the extrapolation procedure and thus write for the discrepancy, Eq. (3.3b),
∆π = 0.021± 0.023 (Exp.)± 0.048 (Extr.), (3.13)
showing explicitly the error arising from experimental errors in the pion-pion amplitudes and the estimated
error of the extrapolation procedure. This will be our final result for the sum rule on pions.
3.4. Connection with chiral perturbation theory
Consider the forward dispersion relation for the (physical) amplitude F (It=1)(s),
F (It=1)(s) =
2s− 4M2π
ImF (It=1)(s′)
(s′ − s)(s′ + s− 4M2π)
1 Actually, we make the corresponding replacement up to the Regge region, s1/2 ≃ 1.42 GeV.
– 6 –
-evaluation of the axial vector commutator sum rule for pion-pion scattering-
Evaluating it at threshold, and writing F (It=1)(4M2π) in terms of scattering lengths, we find the so-called
(first) Olsson sum rule[12] [which is thus an exact consequence of the dispersion relation, independent of the
current algebra commutator (1.1)]:
0 − 5a
0 = 3Mπ
ImF (It=1)(s)
s(s− 4M2π)
. (3.14)
One can evaluate the scattering lengths to lowest order (l.o.) in chiral perturbation theory, i.e., to lowest
order in M2π (actually, this is strictly equivalent to the old Weinberg
[13] evaluation) finding
0 − 5a
4πF 2π
and, substituting in (3.14), the l.o. relation
1 = 4πF 2π
ImF (It=1)(s)
s(s− 4M2π)
. (3.15)
To l.o. (which implies M2π ≈ 0) Eq. (3.15) is equivalent to (1.6) since
ImF (It=1)(s)
s(s− 4M2π)
ImF (It=1)(s)
(s−M2π)2
ImF (It=1)(k2 = 0; s, t = 0)
(s−M2π)2
The same conclusion is reached if we use the Froissart–Gribov representation of the P wave scattering length.
Of course, (3.15) differs in status from (1.6) as the latter is valid exactly, whereas the l.o. expression
0 − 5a
= 3Mπ/4πF
π is known to have large loop corrections.
Another connection with results based on chiral perturbation theory is obtained by remarking that,
using chiral methods to two loops, plus analyticity (in the form of Roy equations) and extra experimental
information, has led Colangelo, Gasser and Leutwyler[15] to propose parametrizations of the S0, S2 and P
waves at low energy consistent with these requirements. Substituting them into Eq. (3.5) gives the result
∆π = 0.046, to be compared with what we found using experimental data in Eq. (3.9), 0.027±0.022. Likewise,
implementing off-shell corrections as in (3.10) gives ∆π = 0.086 [to be now compared with 0.069 ± 0.023
from Eq. (3.11)] and, finally, applying the correction deduced from the off-shell Goldberger–Treiman relation
under the universality assumption gives ∆π = 0.033, a number which is also similar to what was found using
experimental ππ data in Eq. (3.12), viz., 0.021± 0.023.2
We emphasize that it is not our intention in this paper to give a detailed review of either chiral
perturbation theory or the Olsson sum rule, which are substantial topics in their own right. We are only
interested in elucidating the connection between the Olsson relation and the pion-pion sum rule (1.6), which,
as far as we know, can only be established to leading order in chiral perturbation theory. Tests of the Olsson
relation by itself have been made extensively in the literature; see, for example, refs. 14, 15 for evaluations
using two-loop chiral perturbation theory, and the papers of Kamiński, Peláez and Ynduráin and of Peláez
and Ynduráin in ref. 9 for calculations using experimental data prior to 2006. The Olsson relation has also
been verified including more recent data (those of the NA48/2 collaboration, ref. 11) as in our evaluation
of the pion-pion sum rule here, by Kamiński, Peláez, and Ynduráin (unpublished work in progress).
4. The Adler–Weisberger sum rule
4.1. An approximate calculation
The Adler–Weisberger sum rule (1.2), (1.5), has been evaluated in a number of papers (besides the original
ones); a recent article is ref. 17. If we approximate the amplitudes in (1.5) by the scattering amplitudes for
2 The numbers 0.046, 0.086 and 0.033 were kindly communicated to us by H. Leutwyler[16] as an independent check
on the evaluation of Sect. 3.2, using different methods.
– 7 –
-s. l. adler and f. j. ynduráin-
a physical pion, k2 = M2π , then we get the sum rule
g2A ≃ 1 + 8πf2π
(MN+Mπ)2
(s−M2N)2
ImFπ+(s)− Im Fπ−(s)
(4.1)
with Fπ±(s) the physical, forward π
±p scattering amplitudes, normalized so that the pion-proton cross
sections are
σπ±p(s) =
λ1/2(s,M2π ,M
ImFπ±(s),
with the function λ(a, b, c) as defined below. In terms of s-channel isospin amplitudes, we have
Fπ+(s)− Fπ−(s) = 23
F (Is=3/2) − F (Is=1/2)
where the partial wave expansion in the elastic region is
F (I)(s) =
2s1/2
(l + 1) sin δ
l+ + l sin δ
; (4.2)
the phase shifts δ
l± correspond to isospin I, orbital angular momentum l, and total angular momentum
j = l± 1
. The center of mass momentum is
k = 1
λ(s,M2π ,M
, λ(a, b, c) = [a2 − (b + c)2][a2 − (b− c)2].
The high energy part, s1/2 > 2374 MeV of the integral in (4.1) is easily evaluated with the fit to
πN cross sections in ref. 10: Regge formulas are good approximations for kinetic energies above 1 GeV. We
will use the Regge formula
ImFπ+(s)− Im Fπ−(s) = (0.42± 0.04)(s/1 GeV2)0.42,
and integrate this from s1/2 = 2374 MeV to infinity. This would give a result of −0.091. Alternatively, we
can use numerical values of the cross sections given in ref. 18 from s1/2 = 2374 MeV to s1/2 = 3004 MeV,
and use the Regge formula above this latter energy. This would give −0.087. We consider this last number
to be the more reliable one and then write, taking errors into account,
8πf2π
(2374 MeV)2
(s−M2N )2
ImFπ+(s)− Im Fπ−(s)
= −0.087± 0.006. (4.3a)
On the other hand, numerical evaluation of the low energy piece, using the numerical cross sections
collected in ref. 18 gives
8πf2π
∫ (2374 MeV)2
(MN+Mπ)2
(s−M2N )2
ImFπ+(s)− Im Fπ−(s)
= 0.460± 0.024. (4.3b)
This large error is due to the fact that the number in the right hand side is the difference between two large
numbers: specifically,
0.46 = 1.71 [from π+]− 1.25 [from π−]. (4.4)
Substituting the value of gA the sum rule (4.1) reads
1.612± 0.006 = 1 + 0.373± 0.025. (4.5)
We may define a discrepancy ∆A.W. as the difference between g
A and the right hand side of Eq. (1.5), and
express the result in (4.5) as a largish mismatch,
∆A.W. = 0.239± 0.025.
In the present case, there are various substantial cancellations: as already stated, there are cancel-
lations between the π+p and π−p cross sections at low energy, but there is also a cancellation between the
– 8 –
-evaluation of the axial vector commutator sum rule for pion-pion scattering-
low energy region and higher energy (s1/2 > 1.390 GeV) contributions. For example, if we only integrated
up to and including the ∆(3, 3) resonance region, we would have obtained
8πf2π
∫ (1390 MeV)2
(MN+Mπ)2
(s−M2N )2
ImFπ+(s)− Im Fπ−(s)
= 0.70.
These cancellations amplify the errors in the sum rule, and indicate that the effects of the extrapolation to
a zero mass pion, which affect mostly low energies, are now very important, as already remarked in refs. 1
and 3.
4.2. Extrapolation
To perform the extrapolation, we repeat the method used for the case of the sum rule on pions, and replace
the expression (4.2) for the scattering amplitude by
0 (s) =
2s1/2
(l + 1) sin δ
l+ + l sin δ
(4.7)
with k(0) the momentum for an incident pion of zero mass,
k(0) = 1
λ(s, 0,M2N)
We integrate this in the elastic region, s1/2 <∼ 1.5 GeV, using the parametrizations of ref. 19. These
parametrizations have been obtained by fitting up to energies of, respectively, 1.3 GeV and 1.38 GeV;
however, we have verified that they continue to fit the experimental cross sections up to ∼ 150 MeV above
their nominal maximum. Between these energies and 1.9 GeV, one can use a resonance saturation model,
with the resonance parameters taken from the PDT;[6] plus a background, estimated as the tail of the lower
energy resonances. Above 1.9 GeV, there are not well determined values for the resonance parameters and,
moreover, a resonance model will cease to be valid as one is entering the Regge regime. Fortunately, one can
likely neglect the effects of the extrapolation above such energy, and so this will be done here.
The results one gets for the extrapolation correction are
0.225 : up to 1460 MeV; −0.065 : resonances, from 1460 MeV; 0.005 : background, from 1460 MeV.
(4.8)
Adding this and considering as an error estimate the variation of the result if we vary the matching point of
the parametrizations and the resonance model from 1460 to 1420 MeV or 1520 MeV, we get a correction of
0.165± 0.009, and hence the sum rule becomes
1.612± 0.006 = 1 + (0.373± 0.025) + (0.165± 0.009) = 1.538± 0.034. (4.9)
We have added the errors linearly, as they are clearly correlated. The results show reasonable fulfillment of
the Adler–Weisberger sum rule, ∆A.W. = 0.074± 0.034.
If we include a global correction, as we did in the pionic case, multiplying the r.h.s. of (4.9) by the
factor K2NNπ(0) = 0.955± 0.03, the sum rule now deteriorates to
1.612± 0.006 = 1 + [(0.373± 0.025) + (0.165± 0.009)]× (0.955± 0.030) = 1.514± 0.038. (4.10)
We can now write our final result, as we did for the pionic case, as (4.10), adding as an extra error the
difference between (4.10) and (4.9):
∆A.W. = 0.098± 0.006 [g2A]± 0.031 [Exp.]± 0.035 [Extr.] (4.11)
To compare with the results obtained in refs. 1 and 3, we note that these papers did not multiply
through by a factor of g2A = 1.612, and included the K
NNπ(0) extrapolation factor. Thus, the relevant
number to use is [cf. (4.10)] (0.514± 0.038)/1.612 = 0.319± 0.024. The comparable number in refs. 1 and
3 is (4M2N/g
r)(R1 + R2 + R3) = 0.254 + 0.155− 0.061 = 0.348, with a roughly estimated error of ±0.025.
Hence our current evaluation, and the 1965 evaluation of refs. 1 and 3, are in satisfactory agreement. This
should come as no surprise, since good pion-nucleon cross sections were already available in 1965. What
– 9 –
-s. l. adler and f. j. ynduráin-
has changed dramatically since then, and is the motivation for the present paper, is the status of data on
pion-pion scattering.
4.3. Connection with lowest order chiral perturbation theory
As we did for the pion case, the Adler–Weisberger relation can be connected with (lowest order) chiral
perturbation theory, by comparing e.g., (4.1) with the forward dispersion relation for the πp scattering
amplitude for exchange of isospin unity at threshold,3 evaluating the amplitude at threshold in terms of
the scattering lengths combination a3 − a1, and calculating the latter in terms of Fπ, as in the Tomozawa–
Weinberg articles.[21] Details of this may be found in ref. 17.
5. Comments
We have shown that with current data, the pion-pion sum rule, as well as the pion-nucleon one, is satisfied to
better than six percent. To improve on this precision, it will be necessary to have an improved understanding
of dynamical extrapolation corrections that account for the appearance of a zero mass, off-shell pion in the
sum rules. We make two remarks in this regard. The first is that extrapolation of the incident pion to
zero mass is not the same as taking the chiral limit of QCD, since the target pion or nucleon, and all
intermediate state particles, remain on mass shell. The second is that while for a generic pion interpolating
field the results of this extrapolation are not well-defined, the sum rules involve a very specific choice of pion
interpolating field: the divergence of the axial vector current, which is a well defined entity in QCD, as are the
on shell pion-pion and pion-nucleon scattering amplitudes. Thus, the question of estimating extrapolation
corrections, that are needed for a very accurate comparison of the sum rules with experiment, is a well-posed
one in QCD. Modern lattice methods may permit improvement on the estimates that have been used here
and in refs. 1 and 3.
In this respect, it is amusing to remark that, unlike the situation in 1965, the precision of the pionic
sum rule is now greater than that of the pion-nucleon one. This is very likely due to the fact that the latter
involves small differences of large numbers, so any small alteration is amplified here.
3 This is generally known as the Goldberger–Miyazawa–Oehme sum rule.[20]
– 10 –
-evaluation of the axial vector commutator sum rule for pion-pion scattering-
Acknowledgements
We wish to thank H. Leutwyler for supplying numbers that give an independent check of the pion-pion sum
rule evaluation, and also for correspondence that assisted one of us (SLA) in reconstructing the reasoning
behind the extrapolation method used for the pion-pion case in ref. 3. The work of SLA was supported
in part by the U. S. Department of Energy under Grant No. DE-FG02-90ER40542. FJY is grateful to
J. R. Peláez for interesting discussions in the preliminary stages of this paper, and to the Spanish DGI of
the MEC under contract FPA2003-04597 for financial support.
References
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2 Weisberger, W. I., Phys. Rev. Letters 14, 1047 (1965).
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2224 (1968).
4 Goldberger, M. L., and Treiman, S. B., Phys. Rev. 109, 193 (1958).
5 References on pion-kaon scattering: M. Jamin, J., Oller, J. A., and Pich, A., Nucl. Phys. B587, 331
(2000); Gómez-Nicola, A., and Peláez, J. R., Phys. Rev. D65, 054009 (2002); Descotes-Genon, S., and
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212 (2006). Theoretical proposal of kaon-nucleon sum rules: Weisberger, W. I., Phys. Rev. 143, 1302
(1966).
6 Particle Data Tables: Yao, W.-M., et al., Journal of Physics G33, 1 (2006).
7 Saino, M. E., hep-ph/9912337 and, especially, Schindler, M. R., et al., nucl-th/0611083. The second
article reviews previous evaluations.
8 Pagels, H., and Zepeda, A., Phys. Rev. D5, 3262 (1972). For a more recent discussion of correc-
tions to the Goldberger–Treiman relation, and their relation to off-shell extrapolation corrections, see
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9 S0 wave, low energy (s1/2 ≤ 932 MeV): Ynduráin, F. J., Garćıa Mart́ın, R., and Peláez, J. R., hep-
ph/0701025. P wave, low energy (s1/2 ≤ 932 MeV): de Trocóniz, J. F., and Ynduráin, F. J., Phys. Rev.,
D65, 093001, (2002), and Phys. Rev. D71, 073008 (2005). S2 wave, D2 and F wave, s1/2 ≤ 1420 MeV:
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(up to s1/2 ≤ 1420 MeV); D0 wave: Kamiński, R., Peláez, J. R., and Ynduráin, F. J., Phys.Rev. D74,
014001 (2006) and (E) D74, 079903 (2006).
10 High energy, s1/2 > 1420 MeV: Peláez, J. R., and Ynduráin, F. J., Phys. Rev. D69, 114001 (2004).
Actually, in the present calculation one uses the slightly improved Regge parametrization in the paper
of Kamiński, Peláez and Ynduráin in ref. 9; see also Cudell, J. R., et al., Phys. Letters B587, 78 (2004)
and Peláez, J. R., in Proc. Blois. Conf. on Elastic and Diffractive Scattering (hep-ph/0510005) for the
pion-nucleon case.
11 Kl4 decays: Rosselet, L., et al., Phys. Rev. D15, 574 (1977); Pislak, S., et al., Phys. Rev. Lett.,
87, 221801 (2001); NA48/2 ( CERN/SPS experiment); Bloch-Devaux, B., presented at QCD06 in
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to 2 August 2006. K → 2π decays: Aloisio, A., et al., Phys. Letters, B538, 21 (2002). Pion form factor
data: Novosibirsk, ρ region: L. M. Barkov et al., Nucl. Phys. B256, 365 (1985); R. R. Akhmetshin et
al., Phys. Letters B527, 161 (2002). τ decays: ALEPH: R. Barate et al., Z. Phys. C76, 15 (1997);
OPAL: K. Ackerstaff et al., Eur. Phys. J. C7, 571 (1999); CLEO: S. Anderson et al., Phys. Rev., D61,
112002 (2000).
12 Olsson, M. G., Phys. Rev.162, 1338 (1967).
13 Weinberg, S., Phys. Rev. Letters 17, 616 (1966).
14 Gasser, J., and Leutwyler, H., Ann. Phys. (N.Y.) 158, 14 (1984).
15 Colangelo, G., Gasser, J., and Leutwyler, H., Nucl. Phys. B603, 125, (2001). See also Ananthanarayan,
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– 11 –
http://arxiv.org/abs/hep-ph/9912337
http://arxiv.org/abs/nucl-th/0611083
http://arxiv.org/abs/hep-ph/0701025
http://arxiv.org/abs/hep-ph/0701025
http://arxiv.org/abs/hep-ph/0510005
-s. l. adler and f. j. ynduráin-
16 Leutwyler, H. (private communication).
17 Kondratyuk, S., et al., Nucl. Phys. A736, 339 (2004).
18 The COMPAS Group compilations can be traced from Hagiwara, K., et al., Phys. Rev. D66, 010001
(2002).
19 Rowe, G., Salomon, M., and Landau, R. H., Phys. Rev. C18, 584 (1978); Ebrahim, A. A., and
Peterson, R. J., Phys. Rev. C54, 2499 (1996).
20 Goldberger, M. L, Miyazawa, H., and Oehme, R., Phys. Rev. 99, 986 (1955). For a very recent
evaluation and further references, see Abaev, V. V., Metsä, P., and Sainio, M. E., arXiv:0704.3167
[hep-ph].
21 Tomozawa, Y., Nuovo Cimento 46A, 707 (1966); Weinberg, S., Phys. Rev. Lett. 19, 616 (1966).
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http://arxiv.org/abs/0704.3167
